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1508.00305
| 7 |
The ï¬nal dataset contains 22,033 examples on 2,108 tables. We set aside 20% of the tables and their associated questions as the test set and de- velop on the remaining examples. Simple pre- processing was done on the tables: We omit all non-textual contents of the tables, and if there is a merged cell spanning many rows or columns, we unmerge it and duplicate its content into each un- merged cell. Section 7.2 analyzes various aspects of the dataset and compares it to other datasets.
# 3 Approach
We now describe our semantic parsing framework for answering a given question and for training the model with question-answer pairs.
Prediction. Given a table ¢ and a question x, we predict an answer y using the framework il- lustrated in Figure We first convert the table t into a knowledge graph w (âworldâ) which en- codes different relations in the table (Section 4). Next, we generate a set of candidate logical forms Z,, by parsing the question x using the informa- tion from w (Section . Each generated logical form z ⬠Zz, is a graph query that can be exe- cuted on the knowledge graph w to get a denota- tion [z]]w. We extract a feature vector $(2, w, z) for each z ⬠2, (Section [6.2) and define a log- linear distribution over the candidates:
|
1508.00305#7
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 7,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "The ï¬nal dataset contains 22,033 examples on 2,108 tables. We set aside 20% of the tables and their associated questions as the test set and de- velop on the remaining examples. Simple pre- processing was done on the tables: We omit all non-textual contents of the tables, and if there is a merged cell spanning many rows or columns, we unmerge it and duplicate its content into each un- merged cell. Section 7.2 analyzes various aspects of the dataset and compares it to other datasets.\n# 3 Approach\nWe now describe our semantic parsing framework for answering a given question and for training the model with question-answer pairs.\nPrediction. Given a table ¢ and a question x, we predict an answer y using the framework il- lustrated in Figure We first convert the table t into a knowledge graph w (âworldâ) which en- codes different relations in the table (Section 4). Next, we generate a set of candidate logical forms Z,, by parsing the question x using the informa- tion from w (Section . Each generated logical form z ⬠Zz, is a graph query that can be exe- cuted on the knowledge graph w to get a denota- tion [z]]w. We extract a feature vector $(2, w, z) for each z ⬠2, (Section [6.2) and define a log- linear distribution over the candidates:",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 8 |
po(z | x,w) x exp{O'd(a,w,z)}, (1)
where θ is the parameter vector. Finally, we choose the logical form z with the highest model probability and execute it on w to get the answer denotation y =
[z]wTraining. Given training examples D = {(xi, ti, yi)}N i=1, we seek a parameter vector θ that maximizes the regularized log-likelihood of the correct denotation yi marginalized over logi- cal forms z. Formally, we maximize the objective function
N 1 J(9) = S> log po(yi | as, wi) â [|], , 2) i=l
where wi is deterministically generated from ti, and
pθ(y | x, w) = pθ(z | x, w). zâZx;y= (3)
2â¬Z7;y=[2]v We optimize @ using AdaGrad (Duchi et al.,| )), running 3 passes over the data. We use Ly regularization with \ = 3 x 107° obtained from cross-validation.
The following sections explain individual sys- tem components in more detail.
# 4 Knowledge graph
|
1508.00305#8
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 8,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "po(z | x,w) x exp{O'd(a,w,z)}, (1)\nwhere θ is the parameter vector. Finally, we choose the logical form z with the highest model probability and execute it on w to get the answer denotation y =\n[z]wTraining. Given training examples D = {(xi, ti, yi)}N i=1, we seek a parameter vector θ that maximizes the regularized log-likelihood of the correct denotation yi marginalized over logi- cal forms z. Formally, we maximize the objective function\nN 1 J(9) = S> log po(yi | as, wi) â [|], , 2) i=l\nwhere wi is deterministically generated from ti, and\npθ(y | x, w) = pθ(z | x, w). zâZx;y= (3)\n2â¬Z7;y=[2]v We optimize @ using AdaGrad (Duchi et al.,| )), running 3 passes over the data. We use Ly regularization with \\ = 3 x 107° obtained from cross-validation.\nThe following sections explain individual sys- tem components in more detail.\n# 4 Knowledge graph",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 9 |
The following sections explain individual sys- tem components in more detail.
# 4 Knowledge graph
Inspired by the graph representation of knowledge bases, we preprocess the table t by deterministi- cally converting it into a knowledge graph w as illustrated in Figure 3. In the most basic form, ta- ble rows become row nodes, strings in table cells become entity nodes,1 and table columns become directed edges from the row nodes to the entity
# 1Two occurrences of the same string constitute one node.
Index ~,7 \ Year \city \ Country y~ bs 0 1896 Athens Greece Index _â. / yo | Next 1 1900 \y \cit: \Count Year city Country Paris France : Number} : Â¥ 1900.0 1900-XX-XX
Part of the knowledge graph corre- Figure 3: sponding to the table in Figure 1. Circular nodes are row nodes. We augment the graph with dif- ferent entity normalization nodes such as Number and Date (red) and additional row node relations Next and Index (blue).
|
1508.00305#9
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 9,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "The following sections explain individual sys- tem components in more detail.\n# 4 Knowledge graph\nInspired by the graph representation of knowledge bases, we preprocess the table t by deterministi- cally converting it into a knowledge graph w as illustrated in Figure 3. In the most basic form, ta- ble rows become row nodes, strings in table cells become entity nodes,1 and table columns become directed edges from the row nodes to the entity\n# 1Two occurrences of the same string constitute one node.\nIndex ~,7 \\ Year \\city \\ Country y~ bs 0 1896 Athens Greece Index _â. / yo | Next 1 1900 \\y \\cit: \\Count Year city Country Paris France : Number} : Â¥ 1900.0 1900-XX-XX\nPart of the knowledge graph corre- Figure 3: sponding to the table in Figure 1. Circular nodes are row nodes. We augment the graph with dif- ferent entity normalization nodes such as Number and Date (red) and additional row node relations Next and Index (blue).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 10 |
nodes of that column. The column headers are used as edge labels for these row-entity relations. The knowledge graph representation is conve- nient for three reasons. First, we can encode dif- ferent forms of entity normalization in the graph. Some entity strings (e.g., â1900â) can be inter- preted as a number, a date, or a proper name de- pending on the context, while some other strings (e.g., â200 kmâ) have multiple parts. Instead of committing to one normalization scheme, we in- troduce edges corresponding to different normal- ization methods from the entity nodes. For exam- ple, the node 1900 will have an edge called Date to another node 1900-XX-XX of type date. Apart these normalization nodes from type checking, also aid learning by providing signals on the ap- propriate answer type. For instance, we can deï¬ne a feature that associates the phrase âhow manyâ with a logical form that says âtraverse a row-entity edge, then a Number edgeâ instead of just âtraverse a row-entity edge.â
|
1508.00305#10
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 10,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "nodes of that column. The column headers are used as edge labels for these row-entity relations. The knowledge graph representation is conve- nient for three reasons. First, we can encode dif- ferent forms of entity normalization in the graph. Some entity strings (e.g., â1900â) can be inter- preted as a number, a date, or a proper name de- pending on the context, while some other strings (e.g., â200 kmâ) have multiple parts. Instead of committing to one normalization scheme, we in- troduce edges corresponding to different normal- ization methods from the entity nodes. For exam- ple, the node 1900 will have an edge called Date to another node 1900-XX-XX of type date. Apart these normalization nodes from type checking, also aid learning by providing signals on the ap- propriate answer type. For instance, we can deï¬ne a feature that associates the phrase âhow manyâ with a logical form that says âtraverse a row-entity edge, then a Number edgeâ instead of just âtraverse a row-entity edge.â",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 11 |
The second beneï¬t of the graph representation is its ability to handle various logical phenomena via graph augmentation. For example, to answer questions of the form âWhat is the next . . . ?â or âWho came before . . . ?â, we augment each row node with an edge labeled Next pointing to the next row node, after which the questions can be answered by traversing the Next edge. In this work, we choose to add two special edges on each row node: the Next edge mentioned above and an Index edge pointing to the row index number (0, 1, 2, . . . ).
Finally, with a graph representation, we can query it directly using a logical formalism for knowledge graphs, which we turn to next.
Name Example Join City.Athens (row nodes with a City edge to Athens) Union City.(Athens L| Beijing) Intersection City.Athens M Year.Number.<./990 Reverse RlYear].City.Athens (entities where a row in City.Athens has a Year edge to) Aggregation count(City.Athens) (the number of rows with city Athens) Superlative | argmax(City.Athens, Index) (the last row with city Athens) Arithmetic sub(204,201) (= 204 â 201) Lambda Aa[Year.Date.2] (a binary: composition of two relations)
|
1508.00305#11
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 11,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "The second beneï¬t of the graph representation is its ability to handle various logical phenomena via graph augmentation. For example, to answer questions of the form âWhat is the next . . . ?â or âWho came before . . . ?â, we augment each row node with an edge labeled Next pointing to the next row node, after which the questions can be answered by traversing the Next edge. In this work, we choose to add two special edges on each row node: the Next edge mentioned above and an Index edge pointing to the row index number (0, 1, 2, . . . ).\nFinally, with a graph representation, we can query it directly using a logical formalism for knowledge graphs, which we turn to next.\nName Example Join City.Athens (row nodes with a City edge to Athens) Union City.(Athens L| Beijing) Intersection City.Athens M Year.Number.<./990 Reverse RlYear].City.Athens (entities where a row in City.Athens has a Year edge to) Aggregation count(City.Athens) (the number of rows with city Athens) Superlative | argmax(City.Athens, Index) (the last row with city Athens) Arithmetic sub(204,201) (= 204 â 201) Lambda Aa[Year.Date.2] (a binary: composition of two relations)",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 12 |
Table 1: The lambda DCS operations we use.
# 5 Logical forms
forms, we use As our lambda dependency-based compositional seman- tics (Liang, 2013), or lambda DCS, which we brieï¬y describe here. Each lambda DCS logical form is either a unary (denoting a list of values) or a binary (denoting a list of pairs). The most basic unaries are singletons (e.g., China represents an entity node, and 30 represents a single number), while the most basic binaries are relations (e.g., City maps rows to city entities, Next maps rows to rows, and >= maps numbers to numbers). Log- ical forms can be combined into larger ones via various operations listed in Table 1. Each opera- tion produces a unary except lambda abstraction: λx[f (x)] is a binary mapping x to f (x).
# 6 Parsing and ranking
Given the knowledge graph w, we now describe how to parse the utterance x into a set of candidate logical forms Zx
# 6.1 Parsing algorithm
We propose a new ï¬oating parser which is more ï¬exible than a standard chart parser. Both parsers recursively build up derivations and corresponding logical forms by repeatedly applying deduction rules, but the ï¬oating parser allows logical form predicates to be generated independently from the utterance.
|
1508.00305#12
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 12,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Table 1: The lambda DCS operations we use.\n# 5 Logical forms\nforms, we use As our lambda dependency-based compositional seman- tics (Liang, 2013), or lambda DCS, which we brieï¬y describe here. Each lambda DCS logical form is either a unary (denoting a list of values) or a binary (denoting a list of pairs). The most basic unaries are singletons (e.g., China represents an entity node, and 30 represents a single number), while the most basic binaries are relations (e.g., City maps rows to city entities, Next maps rows to rows, and >= maps numbers to numbers). Log- ical forms can be combined into larger ones via various operations listed in Table 1. Each opera- tion produces a unary except lambda abstraction: λx[f (x)] is a binary mapping x to f (x).\n# 6 Parsing and ranking\nGiven the knowledge graph w, we now describe how to parse the utterance x into a set of candidate logical forms Zx\n# 6.1 Parsing algorithm\nWe propose a new ï¬oating parser which is more ï¬exible than a standard chart parser. Both parsers recursively build up derivations and corresponding logical forms by repeatedly applying deduction rules, but the ï¬oating parser allows logical form predicates to be generated independently from the utterance.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 13 |
Chart parser. We brieï¬y review the CKY al- gorithm for chart parsing to introduce notation. Given an utterance with tokens x1, . . . , xn, the CKY algorithm applies deduction rules of the folSemantics Anchored to the utterance match(z1) (match(s) = entity with name s) Rule Example TokenSpan â Entity Greece anchored to âGreeceâ TokenSpan â Atomic val(z1) (val(s) = interpreted value) 2012-07-XX anchored to âJuly 2012â Unanchored (ï¬oating) â
â Relation r Country (r = row-entity relation) â
â Relation λx[r.p.x] λx[Year.Date.x] (p = normalization relation) â
â Records â
â RecordFn Type.Row Index (list of all rows) (row â row index)
Table 2: Base deduction rules. Entities and atomic values (e.g., numbers, dates) are anchored to token spans, while other predicates are kept ï¬oating. (a â b represents a binary mapping b to a.)
lowing two kinds:
â &)
|
1508.00305#13
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 13,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Chart parser. We brieï¬y review the CKY al- gorithm for chart parsing to introduce notation. Given an utterance with tokens x1, . . . , xn, the CKY algorithm applies deduction rules of the folSemantics Anchored to the utterance match(z1) (match(s) = entity with name s) Rule Example TokenSpan â Entity Greece anchored to âGreeceâ TokenSpan â Atomic val(z1) (val(s) = interpreted value) 2012-07-XX anchored to âJuly 2012â Unanchored (ï¬oating) â
â Relation r Country (r = row-entity relation) â
â Relation λx[r.p.x] λx[Year.Date.x] (p = normalization relation) â
â Records â
â RecordFn Type.Row Index (list of all rows) (row â row index)\nTable 2: Base deduction rules. Entities and atomic values (e.g., numbers, dates) are anchored to token spans, while other predicates are kept ï¬oating. (a â b represents a binary mapping b to a.)\nlowing two kinds:\nâ &)",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 14 |
lowing two kinds:
â &)
(TokenSpan, i, j)[s] â (c, i, j)[f (s)], (c1, i, k)[z1] + (c2, k + 1, j)[z2]
(5) â (c, i, j)[f (z1, z2)].
The ï¬rst rule is a lexical rule that matches an utter- ance token span xi · · · xj (e.g., s = âNew Yorkâ) form (e.g., f (s) = and produces a logical NewYorkCity) with category c (e.g., Entity). The second rule takes two adjacent spans giv- ing rise to logical forms z1 and z2 and builds a new logical form f (z1, z2). Algorithmically, CKY stores derivations of category c covering the span xi · · · xj in a cell (c, i, j). CKY ï¬lls in the cells of increasing span lengths, and the logical forms in the top cell (ROOT, 1, n) are returned.
|
1508.00305#14
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 14,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "lowing two kinds:\nâ &)\n(TokenSpan, i, j)[s] â (c, i, j)[f (s)], (c1, i, k)[z1] + (c2, k + 1, j)[z2]\n(5) â (c, i, j)[f (z1, z2)].\nThe ï¬rst rule is a lexical rule that matches an utter- ance token span xi · · · xj (e.g., s = âNew Yorkâ) form (e.g., f (s) = and produces a logical NewYorkCity) with category c (e.g., Entity). The second rule takes two adjacent spans giv- ing rise to logical forms z1 and z2 and builds a new logical form f (z1, z2). Algorithmically, CKY stores derivations of category c covering the span xi · · · xj in a cell (c, i, j). CKY ï¬lls in the cells of increasing span lengths, and the logical forms in the top cell (ROOT, 1, n) are returned.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 15 |
Floating parser. Chart parsing uses lexical rules (4) to generate relevant logical predicates, but in our setting of semantic parsing on tables, we do not have the luxury of starting with or inducing a full-ï¬edged lexicon. Moreover, there is a mismatch between words in the utterance and predicates in the logical form. For in- stance, consider the question âGreece held its last Summer Olympics in which year?â on the table in Figure 1 and the correct logical form R[λx[Year.Date.x]].argmax(Country.Greece, Index). While the entity Greece can be anchored to the token âGreeceâ, some logical predicates (e.g., Country) cannot be clearly anchored to a token span. We could potentially learn to anchor the logical form Country.Greece to âGreeceâ, but if the relation Country is not seen during training, such a mapping is impossible to learn from the
|
1508.00305#15
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 15,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Floating parser. Chart parsing uses lexical rules (4) to generate relevant logical predicates, but in our setting of semantic parsing on tables, we do not have the luxury of starting with or inducing a full-ï¬edged lexicon. Moreover, there is a mismatch between words in the utterance and predicates in the logical form. For in- stance, consider the question âGreece held its last Summer Olympics in which year?â on the table in Figure 1 and the correct logical form R[λx[Year.Date.x]].argmax(Country.Greece, Index). While the entity Greece can be anchored to the token âGreeceâ, some logical predicates (e.g., Country) cannot be clearly anchored to a token span. We could potentially learn to anchor the logical form Country.Greece to âGreeceâ, but if the relation Country is not seen during training, such a mapping is impossible to learn from the",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 16 |
Rule Semantics Example Join + Aggregate Entity or Atomic + Values ZA China Atomic â Values C21 >=.30 (at least 30) (c ⬠{<, >, <=, >=}) Relation + Values â> Records 21.22 Relation + Records + Values Riz1]-22 Records â Records Next.2z1 Records â Records R[Next].z1 Values â Atomic a(z1) (a ⬠{count, max, min, sum, avg}) Values + ROOT ZA Country.China R[Year].Country.China Next.Country.China R[Next].Country.China count (Country.China) (events (rows) where the country is China) (years of events in China) (... before China) (... after China) (How often did China ...) Superlative Relation + RecordFn 2 Records + RecordFn â Records 8(21, 22) (s ⬠{argmax, argmin}) Relation + ValueFn Relation + Relation + ValueFn R{Az[a(z1.2)]] Ax[R{z1].z2.2] Values + ValueFn â Values 8(21, 22) Az[Nations.Number.z] argmax(Type.Row, \x[Nations.Number.z])
|
1508.00305#16
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 16,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Rule Semantics Example Join + Aggregate Entity or Atomic + Values ZA China Atomic â Values C21 >=.30 (at least 30) (c ⬠{<, >, <=, >=}) Relation + Values â> Records 21.22 Relation + Records + Values Riz1]-22 Records â Records Next.2z1 Records â Records R[Next].z1 Values â Atomic a(z1) (a ⬠{count, max, min, sum, avg}) Values + ROOT ZA Country.China R[Year].Country.China Next.Country.China R[Next].Country.China count (Country.China) (events (rows) where the country is China) (years of events in China) (... before China) (... after China) (How often did China ...) Superlative Relation + RecordFn 2 Records + RecordFn â Records 8(21, 22) (s ⬠{argmax, argmin}) Relation + ValueFn Relation + Relation + ValueFn R{Az[a(z1.2)]] Ax[R{z1].z2.2] Values + ValueFn â Values 8(21, 22) Az[Nations.Number.z] argmax(Type.Row, \\x[Nations.Number.z])",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 17 |
Values + ValueFn â Values 8(21, 22) Az[Nations.Number.z] argmax(Type.Row, \x[Nations.Number.z]) argmin(City.Athens, Index) R[Az[count(City.z)]] Az[R[City].Nations.Number.z] argmax(..., R[Ax[count(City.x)]]) (row < value in Nations column) (events with the most participating nations) (first event in Athens) (city + num. of rows with that city) (city + value in Nations column) (most frequent city) Other operations ValueFn + Values + Values â> Values o(R[z1].22, R[z1].23) (o ⬠{add, sub, mul, div}) sub(R[Number].R[Nations].City.London, ...) (How many more participants were in London than ...) Entity + Entity > Values 21 U ze Chinal/France (China or France) Records + Records + Records z1 M1 22 City.BeijingCountry.China (...in Beijing, China)
|
1508.00305#17
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 17,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Values + ValueFn â Values 8(21, 22) Az[Nations.Number.z] argmax(Type.Row, \\x[Nations.Number.z]) argmin(City.Athens, Index) R[Az[count(City.z)]] Az[R[City].Nations.Number.z] argmax(..., R[Ax[count(City.x)]]) (row < value in Nations column) (events with the most participating nations) (first event in Athens) (city + num. of rows with that city) (city + value in Nations column) (most frequent city) Other operations ValueFn + Values + Values â> Values o(R[z1].22, R[z1].23) (o ⬠{add, sub, mul, div}) sub(R[Number].R[Nations].City.London, ...) (How many more participants were in London than ...) Entity + Entity > Values 21 U ze Chinal/France (China or France) Records + Records + Records z1 M1 22 City.BeijingCountry.China (...in Beijing, China)",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 18 |
Table 3: Compositional deduction rules. Each rule c1, . . . , ck â c takes logical forms z1, . . . , zk constructed over categories c1, . . . , ck, respectively, and produces a logical form based on the semantics.
training data. Similarly, some prominent tokens (e.g., âOlympicsâ) are irrelevant and have no predicates anchored to them.
Therefore, instead of anchoring each predicate in the logical form to tokens in the utterance via lexical rules, we propose parsing more freely. We replace the anchored cells (c, i, j) with ï¬oating cells (c, s) of category c and logical form size s. Then we apply rules of the following three kinds:
(TokenSpan, i, j)[s] â (c, 1)[f (s)], (6)
â
â (c, 1)[f ()], (7)
(c1, s1)[z1] + (c2, s2)[z2]
(8) â (c, s1 + s2 + 1)[f (z1, z2)].
(Values, 8)
|
1508.00305#18
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 18,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Table 3: Compositional deduction rules. Each rule c1, . . . , ck â c takes logical forms z1, . . . , zk constructed over categories c1, . . . , ck, respectively, and produces a logical form based on the semantics.\ntraining data. Similarly, some prominent tokens (e.g., âOlympicsâ) are irrelevant and have no predicates anchored to them.\nTherefore, instead of anchoring each predicate in the logical form to tokens in the utterance via lexical rules, we propose parsing more freely. We replace the anchored cells (c, i, j) with ï¬oating cells (c, s) of category c and logical form size s. Then we apply rules of the following three kinds:\n(TokenSpan, i, j)[s] â (c, 1)[f (s)], (6)\nâ
â (c, 1)[f ()], (7)\n(c1, s1)[z1] + (c2, s2)[z2]\n(8) â (c, s1 + s2 + 1)[f (z1, z2)].\n(Values, 8)",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 20 |
Note that rules (6) are similar to (4) in chart parsing except that the ï¬oating cell (c, 1) only keeps track of the category and its size 1, not the span (i, j). Rules (7) allow us to construct predicates out of thin air. For example, we can construct a logical form representing a table rela- tion Country in cell (Relation, 1) using the rule â
â Relation [Country] independent of the ut- terance. Rules (8) perform composition, where the induction is on the size s of the logical form rather than the span length. The algorithm stops when the speciï¬ed maximum size is reached, after which the logical forms in cells (ROOT, s) for any
Figure 4: A derivation for the utterance âGreece held its last Summer Olympics in which year?â Only Greece is anchored to a phrase âGreeceâ; Year and other predicates are ï¬oating.
s are included in Zx. Figure 4 shows an example derivation generated by our ï¬oating parser.
The ï¬oating parser is very ï¬exible: it can skip tokens and combine logical forms in any order. This ï¬exibility might seem too unconstrained, but we can use strong typing constraints to prevent nonsensical derivations from being constructed.
|
1508.00305#20
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 20,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Note that rules (6) are similar to (4) in chart parsing except that the ï¬oating cell (c, 1) only keeps track of the category and its size 1, not the span (i, j). Rules (7) allow us to construct predicates out of thin air. For example, we can construct a logical form representing a table rela- tion Country in cell (Relation, 1) using the rule â
â Relation [Country] independent of the ut- terance. Rules (8) perform composition, where the induction is on the size s of the logical form rather than the span length. The algorithm stops when the speciï¬ed maximum size is reached, after which the logical forms in cells (ROOT, s) for any\nFigure 4: A derivation for the utterance âGreece held its last Summer Olympics in which year?â Only Greece is anchored to a phrase âGreeceâ; Year and other predicates are ï¬oating.\ns are included in Zx. Figure 4 shows an example derivation generated by our ï¬oating parser.\nThe ï¬oating parser is very ï¬exible: it can skip tokens and combine logical forms in any order. This ï¬exibility might seem too unconstrained, but we can use strong typing constraints to prevent nonsensical derivations from being constructed.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 21 |
âGreece held its last Summer Olympics in which year?â z = R[λx[Year.Number.x]].argmax(Type.Row, Index) y = {2012} (type: NUM, column: YEAR)
lex unlex (âµ âyearâ = YEAR) lex lex unlex (âµ âyearâ = YEAR)
Table 4: Example features that ï¬re for the (incor- rect) logical form z. All features are binary. (lex = lexicalized)
rules we use. We assume that all named entities will explicitly appear in the question x, so we an- chor all entity predicates (e.g., Greece) to token spans (e.g., âGreeceâ). We also anchor all numer- ical values (numbers, dates, percentages, etc.) de- tected by an NER system. In contrast, relations (e.g., Country) and operations (e.g., argmax) are kept ï¬oating since we want to learn how they are expressed in language. Connections between phrases in x and the generated relations and op- erations in z are established in the ranking model through features.
# 6.2 Features
|
1508.00305#21
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 21,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "âGreece held its last Summer Olympics in which year?â z = R[λx[Year.Number.x]].argmax(Type.Row, Index) y = {2012} (type: NUM, column: YEAR)\nlex unlex (âµ âyearâ = YEAR) lex lex unlex (âµ âyearâ = YEAR)\nTable 4: Example features that ï¬re for the (incor- rect) logical form z. All features are binary. (lex = lexicalized)\nrules we use. We assume that all named entities will explicitly appear in the question x, so we an- chor all entity predicates (e.g., Greece) to token spans (e.g., âGreeceâ). We also anchor all numer- ical values (numbers, dates, percentages, etc.) de- tected by an NER system. In contrast, relations (e.g., Country) and operations (e.g., argmax) are kept ï¬oating since we want to learn how they are expressed in language. Connections between phrases in x and the generated relations and op- erations in z are established in the ranking model through features.\n# 6.2 Features",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 22 |
# 6.2 Features
We define features 6(x,w,z) for our log-linear model to capture the relationship between the question x and the candidate z. Table [4] shows some example features from each feature type. Most features are of the form (f(x), g(z)) or (f(x), h(y)) where y = [[z]w is the denotation, and f, g, and h extract some information (e.g., identity, POS tags) from x, z, or y, respectively.
phrase-predicate: Conjunctions between n- grams f (x) from x and predicates g(z) from z. We use both lexicalized features, where all possi- ble pairs (f (x), g(z)) form distinct features, and binary unlexicalized features indicating whether f (x) and g(z) have a string match.
missing-predicate: Indicators on whether there are entities or relations mentioned in x but not in z. These features are unlexicalized.
denotation: Size and type of the denotation y = [2]. The type can be either a primitive type (e.g., NUM, DATE, ENTITY) or the name of the column containing the entity in y (e.g., CITY).
|
1508.00305#22
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 22,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "# 6.2 Features\nWe define features 6(x,w,z) for our log-linear model to capture the relationship between the question x and the candidate z. Table [4] shows some example features from each feature type. Most features are of the form (f(x), g(z)) or (f(x), h(y)) where y = [[z]w is the denotation, and f, g, and h extract some information (e.g., identity, POS tags) from x, z, or y, respectively.\nphrase-predicate: Conjunctions between n- grams f (x) from x and predicates g(z) from z. We use both lexicalized features, where all possi- ble pairs (f (x), g(z)) form distinct features, and binary unlexicalized features indicating whether f (x) and g(z) have a string match.\nmissing-predicate: Indicators on whether there are entities or relations mentioned in x but not in z. These features are unlexicalized.\ndenotation: Size and type of the denotation y = [2]. The type can be either a primitive type (e.g., NUM, DATE, ENTITY) or the name of the column containing the entity in y (e.g., CITY).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 23 |
phrase-denotation: Conjunctions between ngrams from x and the types of y. Similar to the phrase-predicate features, we use both lexicalized and unlexicalized features.
headword-denotation: Conjunctions between the question word Q (e.g., what, who, how many) or the headword H (the ï¬rst noun after the ques- tion word) with the types of y.
# 6.3 Generation and pruning
Due to their recursive nature, the rules allow us to generate highly compositional logical forms. However, the compositionality comes at the cost of generating exponentially many logical forms, most of which are redundant (e.g., logical forms with an argmax operation on a set of size 1). We employ several methods to deal with this combi- natorial explosion:
Beam search. We compute the model proba- bility of each partial logical form based on avail- able features (i.e., features that do not depend on the ï¬nal denotation) and keep only the K = 200 highest-scoring logical forms in each cell.
|
1508.00305#23
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 23,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "phrase-denotation: Conjunctions between ngrams from x and the types of y. Similar to the phrase-predicate features, we use both lexicalized and unlexicalized features.\nheadword-denotation: Conjunctions between the question word Q (e.g., what, who, how many) or the headword H (the ï¬rst noun after the ques- tion word) with the types of y.\n# 6.3 Generation and pruning\nDue to their recursive nature, the rules allow us to generate highly compositional logical forms. However, the compositionality comes at the cost of generating exponentially many logical forms, most of which are redundant (e.g., logical forms with an argmax operation on a set of size 1). We employ several methods to deal with this combi- natorial explosion:\nBeam search. We compute the model proba- bility of each partial logical form based on avail- able features (i.e., features that do not depend on the ï¬nal denotation) and keep only the K = 200 highest-scoring logical forms in each cell.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 24 |
Pruning. We prune partial logical forms that lead to invalid or redundant final logical forms. For example, we eliminate any logical form that does not type check (e.g., Beijing L! Greece), executes to an empty list (e.g., Year.Number.24), includes an aggregate or superlative on a singleton set (e.g., argmax(Year.Number.20/2, Index)), or joins two relations that are the reverses of each other (e.g., R[City].City.Beijing).
# 7 Experiments
# 7.1 Main evaluation
We evaluate the system on the development sets (three random 80:20 splits of the training data) and the test data. In both settings, the tables we test on do not appear during training.
Evaluation metrics. Our main metric is accu- racy, which is the number of examples (x, t, y) on which the system outputs the correct answer y. We also report the oracle score, which counts the number of examples where at least one generated candidate z â Zx executes to y.
|
1508.00305#24
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 24,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Pruning. We prune partial logical forms that lead to invalid or redundant final logical forms. For example, we eliminate any logical form that does not type check (e.g., Beijing L! Greece), executes to an empty list (e.g., Year.Number.24), includes an aggregate or superlative on a singleton set (e.g., argmax(Year.Number.20/2, Index)), or joins two relations that are the reverses of each other (e.g., R[City].City.Beijing).\n# 7 Experiments\n# 7.1 Main evaluation\nWe evaluate the system on the development sets (three random 80:20 splits of the training data) and the test data. In both settings, the tables we test on do not appear during training.\nEvaluation metrics. Our main metric is accu- racy, which is the number of examples (x, t, y) on which the system outputs the correct answer y. We also report the oracle score, which counts the number of examples where at least one generated candidate z â Zx executes to y.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 25 |
Baselines. We compare the system to two base- lines. The ï¬rst baseline (IR), which simulates in- formation retrieval, selects an answer y among the entities in the table using a log-linear model over entities (table cells) rather than logical forms. The features are conjunctions between phrases in x and
dev test IR baseline WQ baseline Our system acc 13.4 23.6 37.0 ora 69.1 34.4 76.7 acc 12.7 24.3 37.1 ora 70.6 35.6 76.6
Table 5: Accuracy (acc) and oracle scores (ora) on the development sets (3 random splits of the training data) and the test data.
|
1508.00305#25
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 25,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Baselines. We compare the system to two base- lines. The ï¬rst baseline (IR), which simulates in- formation retrieval, selects an answer y among the entities in the table using a log-linear model over entities (table cells) rather than logical forms. The features are conjunctions between phrases in x and\ndev test IR baseline WQ baseline Our system acc 13.4 23.6 37.0 ora 69.1 34.4 76.7 acc 12.7 24.3 37.1 ora 70.6 35.6 76.6\nTable 5: Accuracy (acc) and oracle scores (ora) on the development sets (3 random splits of the training data) and the test data.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 26 |
Table 5: Accuracy (acc) and oracle scores (ora) on the development sets (3 random splits of the training data) and the test data.
acc ora Our system 37.0 76.7 (a) Rule Ablation join only 10.6 15.7 join + count (= WQ baseline) 23.6 344 join + count + superlative 30.7 68.6 all â {N, Li} 34.8 75.1 (b) Feature Ablation all â features involving predicate 11.8 74.5 all â phrase-predicate 16.9 74.5 all â lex phrase-predicate 17.6 75.9 all â unlex phrase-predicate 34.3, 76.7 all â missing-predicate 35.9 76.7 all â features involving denotation 33.5 76.8 all â denotation 34.3 76.6 all â phrase-denotation 35.7 76.8 all â headword-denotation 36.0 76.7 (c) Anchor operations to trigger words 37.1 59.4
Table 6: Average accuracy and oracle scores on development data in various system settings.
properties of the answers y, which cover all fea- tures in our main system that do not involve the logical form. As an upper bound of this baseline, 69.1% of the development examples have the an- swer appearing as an entity in the table.
|
1508.00305#26
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 26,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Table 5: Accuracy (acc) and oracle scores (ora) on the development sets (3 random splits of the training data) and the test data.\nacc ora Our system 37.0 76.7 (a) Rule Ablation join only 10.6 15.7 join + count (= WQ baseline) 23.6 344 join + count + superlative 30.7 68.6 all â {N, Li} 34.8 75.1 (b) Feature Ablation all â features involving predicate 11.8 74.5 all â phrase-predicate 16.9 74.5 all â lex phrase-predicate 17.6 75.9 all â unlex phrase-predicate 34.3, 76.7 all â missing-predicate 35.9 76.7 all â features involving denotation 33.5 76.8 all â denotation 34.3 76.6 all â phrase-denotation 35.7 76.8 all â headword-denotation 36.0 76.7 (c) Anchor operations to trigger words 37.1 59.4\nTable 6: Average accuracy and oracle scores on development data in various system settings.\nproperties of the answers y, which cover all fea- tures in our main system that do not involve the logical form. As an upper bound of this baseline, 69.1% of the development examples have the an- swer appearing as an entity in the table.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 27 |
In the second baseline (WQ), we only allow de- duction rules that produce join and count logical forms. This rule subset has the same logical cov- erage as Berant and Liang (2014), which is de- signed to handle the WEBQUESTIONS (Berant et al., 2013) and FREE917 (Cai and Yates, 2013) datasets.
Results. Table 5 shows the results compared to the baselines. Our system gets an accuracy of 37.1% on the test data, which is signiï¬cantly higher than both baselines, while the oracle is 76.6%. The next subsections analyze the system components in more detail.
# 7.2 Dataset statistics
In this section, we analyze the breadth and depth of the WIKITABLEQUESTIONS dataset, and how the system handles them.
Number of relations. With 3,929 unique col- umn headers (relations) among 13,396 columns,
Operation Amount join (table lookup) 13.5% + join with Next +5.5% + aggregate (count, sum,max,...) + 15.0% + superlative (argmax, argmin) + 24.5% + arithmetic, 7, U + 20.5% + other phenomena + 21.0%
Table 7: The logical operations required to answer the questions in 200 random examples.
|
1508.00305#27
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 27,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "In the second baseline (WQ), we only allow de- duction rules that produce join and count logical forms. This rule subset has the same logical cov- erage as Berant and Liang (2014), which is de- signed to handle the WEBQUESTIONS (Berant et al., 2013) and FREE917 (Cai and Yates, 2013) datasets.\nResults. Table 5 shows the results compared to the baselines. Our system gets an accuracy of 37.1% on the test data, which is signiï¬cantly higher than both baselines, while the oracle is 76.6%. The next subsections analyze the system components in more detail.\n# 7.2 Dataset statistics\nIn this section, we analyze the breadth and depth of the WIKITABLEQUESTIONS dataset, and how the system handles them.\nNumber of relations. With 3,929 unique col- umn headers (relations) among 13,396 columns,\nOperation Amount join (table lookup) 13.5% + join with Next +5.5% + aggregate (count, sum,max,...) + 15.0% + superlative (argmax, argmin) + 24.5% + arithmetic, 7, U + 20.5% + other phenomena + 21.0%\nTable 7: The logical operations required to answer the questions in 200 random examples.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 28 |
Table 7: The logical operations required to answer the questions in 200 random examples.
the tables in the WIKITABLEQUESTIONS dataset contain many more relations than closed-domain datasets such as Geoquery (Zelle and Mooney, 1996) and ATIS (Price, 1990). Additionally, the logical forms that execute to the correct denota- tions refer to a total of 2,056 unique column head- ers, which is greater than the number of relations in the FREE917 dataset (635 Freebase relations). Knowledge coverage. We sampled 50 exam- ples from the dataset and tried to answer them manually using Freebase. Even though Free- base contains some information extracted from Wikipedia, we can answer only 20% of the ques- indicating that WIKITABLEQUESTIONS tions, contains a broad set of facts beyond Freebase.
Logical operation coverage. The dataset cov- ers a wide range of question types and logical operations. Table 6(a) shows the drop in oracle scores when different subsets of rules are used to generate candidates logical forms. The join only subset corresponds to simple table lookup, while join + count is the WQ baseline for Freebase ques- tion answering on the WEBQUESTIONS dataset. Finally, join + count + superlative roughly corre- sponds to the coverage of the Geoquery dataset.
|
1508.00305#28
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 28,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Table 7: The logical operations required to answer the questions in 200 random examples.\nthe tables in the WIKITABLEQUESTIONS dataset contain many more relations than closed-domain datasets such as Geoquery (Zelle and Mooney, 1996) and ATIS (Price, 1990). Additionally, the logical forms that execute to the correct denota- tions refer to a total of 2,056 unique column head- ers, which is greater than the number of relations in the FREE917 dataset (635 Freebase relations). Knowledge coverage. We sampled 50 exam- ples from the dataset and tried to answer them manually using Freebase. Even though Free- base contains some information extracted from Wikipedia, we can answer only 20% of the ques- indicating that WIKITABLEQUESTIONS tions, contains a broad set of facts beyond Freebase.\nLogical operation coverage. The dataset cov- ers a wide range of question types and logical operations. Table 6(a) shows the drop in oracle scores when different subsets of rules are used to generate candidates logical forms. The join only subset corresponds to simple table lookup, while join + count is the WQ baseline for Freebase ques- tion answering on the WEBQUESTIONS dataset. Finally, join + count + superlative roughly corre- sponds to the coverage of the Geoquery dataset.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 29 |
To better understand the distribution of log- ical operations in the WIKITABLEQUESTIONS dataset, we manually classiï¬ed 200 examples based on the types of operations required to an- swer the question. The statistics in Table 7 shows that while a few questions only require simple operations such as table lookup, the majority of the questions demands more advanced operations. Additionally, 21% of the examples cannot be an- swered using any logical form generated from the current deduction rules; these examples are dis- cussed in Section 7.4.
Compositionality. From each example, we compute the logical form size (number of rules applied) of the highest-scoring candidate that exe- cutes to the correct denotation. The histogram in Figure 5 shows that a signiï¬cant number of logical
1500] 1000] 500 0 frequency 203 4 5 6 7 8 9 formula size 10 11
Figure 5: Sizes of the highest-scoring correct can- didate logical forms in development examples.
with pruning 80] » so ra © 60 5 40 2) 4o| h 20f B a 0 oF 0 2% 50 75 100 0 2% 50 75 100 beam size without pruning 80 beam size
Figure 6: Accuracy (solid red) and oracle (dashed blue) scores with different beam sizes.
forms are non-trivial.
|
1508.00305#29
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 29,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "To better understand the distribution of log- ical operations in the WIKITABLEQUESTIONS dataset, we manually classiï¬ed 200 examples based on the types of operations required to an- swer the question. The statistics in Table 7 shows that while a few questions only require simple operations such as table lookup, the majority of the questions demands more advanced operations. Additionally, 21% of the examples cannot be an- swered using any logical form generated from the current deduction rules; these examples are dis- cussed in Section 7.4.\nCompositionality. From each example, we compute the logical form size (number of rules applied) of the highest-scoring candidate that exe- cutes to the correct denotation. The histogram in Figure 5 shows that a signiï¬cant number of logical\n1500] 1000] 500 0 frequency 203 4 5 6 7 8 9 formula size 10 11\nFigure 5: Sizes of the highest-scoring correct can- didate logical forms in development examples.\nwith pruning 80] » so ra © 60 5 40 2) 4o| h 20f B a 0 oF 0 2% 50 75 100 0 2% 50 75 100 beam size without pruning 80 beam size\nFigure 6: Accuracy (solid red) and oracle (dashed blue) scores with different beam sizes.\nforms are non-trivial.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 30 |
Figure 6: Accuracy (solid red) and oracle (dashed blue) scores with different beam sizes.
forms are non-trivial.
Beam size and pruning. Figure 6 shows the results with and without pruning on various beam sizes. Apart from saving time, pruning also pre- vents bad logical forms from clogging up the beam which hurts both oracle and accuracy metrics.
# 7.3 Features
Effect of features. Table 6(b) shows the accu- racy when some feature types are ablated. The most inï¬uential features are lexicalized phrase- predicate features, which capture the relationship between phrases and logical operations (e.g., relat- ing âlastâ to argmax) as well as between phrases and relations (e.g., relating âbeforeâ to < or Next, and relating âwhoâ to the relation Name).
Anchoring with trigger words. In our parsing algorithm, relations and logical operations are not anchored to the utterance. We consider an alter- native approach where logical operations are an- chored to âtriggerâ phrases, which are hand-coded based on co-occurrence statistics (e.g., we trigger a count logical form with how, many, and total).
|
1508.00305#30
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 30,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Figure 6: Accuracy (solid red) and oracle (dashed blue) scores with different beam sizes.\nforms are non-trivial.\nBeam size and pruning. Figure 6 shows the results with and without pruning on various beam sizes. Apart from saving time, pruning also pre- vents bad logical forms from clogging up the beam which hurts both oracle and accuracy metrics.\n# 7.3 Features\nEffect of features. Table 6(b) shows the accu- racy when some feature types are ablated. The most inï¬uential features are lexicalized phrase- predicate features, which capture the relationship between phrases and logical operations (e.g., relat- ing âlastâ to argmax) as well as between phrases and relations (e.g., relating âbeforeâ to < or Next, and relating âwhoâ to the relation Name).\nAnchoring with trigger words. In our parsing algorithm, relations and logical operations are not anchored to the utterance. We consider an alter- native approach where logical operations are an- chored to âtriggerâ phrases, which are hand-coded based on co-occurrence statistics (e.g., we trigger a count logical form with how, many, and total).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 31 |
Table 6(c) shows that the trigger words do not signiï¬cantly impact the accuracy, suggesting that the original system is already able to learn the re- lationship between phrases and operations even without a manual lexicon. As an aside, the huge drop in oracle is because fewer âsemantically in- correctâ logical forms are generated; we discuss this phenomenon in the next subsection.
# 7.4 Semantically correct logical forms
In our setting, we face a new challenge that arises from learning with denotations: with deeper com- positionality, a larger number of nonsensical log- ical forms can execute to the correct denotation. For example, if the target answer is a small num- ber (say, 2), it is possible to count the number of rows with some random properties and arrive at the correct answer. However, as the system en- counters more examples, it can potentially learn to disfavor them by recognizing the characteristics of semantically correct logical forms.
|
1508.00305#31
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 31,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Table 6(c) shows that the trigger words do not signiï¬cantly impact the accuracy, suggesting that the original system is already able to learn the re- lationship between phrases and operations even without a manual lexicon. As an aside, the huge drop in oracle is because fewer âsemantically in- correctâ logical forms are generated; we discuss this phenomenon in the next subsection.\n# 7.4 Semantically correct logical forms\nIn our setting, we face a new challenge that arises from learning with denotations: with deeper com- positionality, a larger number of nonsensical log- ical forms can execute to the correct denotation. For example, if the target answer is a small num- ber (say, 2), it is possible to count the number of rows with some random properties and arrive at the correct answer. However, as the system en- counters more examples, it can potentially learn to disfavor them by recognizing the characteristics of semantically correct logical forms.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 32 |
logical semantically forms. The system can learn the features of semantically correct logical forms only if it can generate them in the ï¬rst place. To see how well the system can generate correct logical forms, looking at the oracle score is insufï¬cient since bad logical forms can execute to the correct denotations. Instead, we randomly chose 200 ex- amples and manually annotated them with logical forms to see if a trained system can produce the annotated logical form as a candidate.
|
1508.00305#32
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 32,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "logical semantically forms. The system can learn the features of semantically correct logical forms only if it can generate them in the ï¬rst place. To see how well the system can generate correct logical forms, looking at the oracle score is insufï¬cient since bad logical forms can execute to the correct denotations. Instead, we randomly chose 200 ex- amples and manually annotated them with logical forms to see if a trained system can produce the annotated logical form as a candidate.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 33 |
Out of 200 examples, we find that 79% can be manually annotated. The remaining ones in- clude artifacts such as unhandled question types (e.g., yes-no questions, or questions with phrases âsameâ or âconsecutiveâ), table cells that require advanced normalization methods (e.g., cells with comma-separated lists), and incorrect annotations. The system generates the annotated logical form among the candidates in 53.5% of the ex- amples. The missing examples are mostly caused by anchoring errors due to lexical mismatch (e.g., âTtalianâ â Italy, or âno zip codeâ â> an empty cell in the zip code column) or the need to generate complex logical forms from a single phrase (e.g., âMay 2010â â >=.2010-05-0111<=.2010-05-31).
# 7.5 Error analysis
|
1508.00305#33
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 33,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Out of 200 examples, we find that 79% can be manually annotated. The remaining ones in- clude artifacts such as unhandled question types (e.g., yes-no questions, or questions with phrases âsameâ or âconsecutiveâ), table cells that require advanced normalization methods (e.g., cells with comma-separated lists), and incorrect annotations. The system generates the annotated logical form among the candidates in 53.5% of the ex- amples. The missing examples are mostly caused by anchoring errors due to lexical mismatch (e.g., âTtalianâ â Italy, or âno zip codeâ â> an empty cell in the zip code column) or the need to generate complex logical forms from a single phrase (e.g., âMay 2010â â >=.2010-05-0111<=.2010-05-31).\n# 7.5 Error analysis",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 34 |
# 7.5 Error analysis
The errors on the development data can be divided into four groups. The ï¬rst two groups are unhan- dled question types (21%) and the failure to an- chor entities (25%) as described in Section 7.4. The third group is normalization and type errors (29%): although we handle some forms of en- tity normalization, we observe many unhandled string formats such as times (e.g., 3:45.79) and city-country pairs (e.g., Beijing, China), as well as complex calculation such as computing time peri- ods (e.g., 12pmâ1am â 1 hour). Finally, we have
ranking errors (25%) which mostly occur when the utterance phrase and the relation are obliquely re- lated (e.g., âairplaneâ and Model).
# 8 Discussion
Our work simultaneously increases the breadth of knowledge source and the depth of compositional- ity in semantic parsing. This section explores the connections in both aspects to related work.
|
1508.00305#34
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 34,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "# 7.5 Error analysis\nThe errors on the development data can be divided into four groups. The ï¬rst two groups are unhan- dled question types (21%) and the failure to an- chor entities (25%) as described in Section 7.4. The third group is normalization and type errors (29%): although we handle some forms of en- tity normalization, we observe many unhandled string formats such as times (e.g., 3:45.79) and city-country pairs (e.g., Beijing, China), as well as complex calculation such as computing time peri- ods (e.g., 12pmâ1am â 1 hour). Finally, we have\nranking errors (25%) which mostly occur when the utterance phrase and the relation are obliquely re- lated (e.g., âairplaneâ and Model).\n# 8 Discussion\nOur work simultaneously increases the breadth of knowledge source and the depth of compositional- ity in semantic parsing. This section explores the connections in both aspects to related work.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 35 |
# 8 Discussion
Our work simultaneously increases the breadth of knowledge source and the depth of compositional- ity in semantic parsing. This section explores the connections in both aspects to related work.
Logical coverage. Different semantic parsing systems are designed to handle different sets of logical operations and degrees of compositional- ity. For example, form-ï¬lling systems (Wang et al., 2011) usually cover a smaller scope of opera- tions and compositionality, while early statistical semantic parsers for question answering (Wong and Mooney, 2007; Zettlemoyer and Collins, 2007) and high-accuracy natural language inter- faces for databases (Androutsopoulos et al., 1995; Popescu et al., 2003) target more compositional utterances with a wide range of logical opera- tions. This work aims to increase the logical coverage even further. For example, compared to the Geoquery dataset, the WIKITABLEQUES- TIONS dataset includes a move diverse set of log- ical operations, and while it does not have ex- tremely compositional questions like in Geoquery (e.g., âWhat states border states that border states that border Florida?â), our dataset contains fairly compositional questions on average.
|
1508.00305#35
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 35,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "# 8 Discussion\nOur work simultaneously increases the breadth of knowledge source and the depth of compositional- ity in semantic parsing. This section explores the connections in both aspects to related work.\nLogical coverage. Different semantic parsing systems are designed to handle different sets of logical operations and degrees of compositional- ity. For example, form-ï¬lling systems (Wang et al., 2011) usually cover a smaller scope of opera- tions and compositionality, while early statistical semantic parsers for question answering (Wong and Mooney, 2007; Zettlemoyer and Collins, 2007) and high-accuracy natural language inter- faces for databases (Androutsopoulos et al., 1995; Popescu et al., 2003) target more compositional utterances with a wide range of logical opera- tions. This work aims to increase the logical coverage even further. For example, compared to the Geoquery dataset, the WIKITABLEQUES- TIONS dataset includes a move diverse set of log- ical operations, and while it does not have ex- tremely compositional questions like in Geoquery (e.g., âWhat states border states that border states that border Florida?â), our dataset contains fairly compositional questions on average.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 36 |
To parse a compositional utterance, many works rely on a lexicon that translates phrases to enti- ties, relations, and logical operations. A lexicon can be automatically generated (Unger and Cimi- ano, 2011; Unger et al., 2012), learned from data (Zettlemoyer and Collins, 2007; Kwiatkowski et al., 2011), or extracted from external sources (Cai and Yates, 2013; Berant et al., 2013), but requires some techniques to generalize to unseen data. Our work takes a different approach similar to the log- ical form growing algorithm in Berant and Liang (2014) by not anchoring relations and operations to the utterance.
Knowledge domain. Recent works on seman- tic parsing for question answering operate on more open and diverse data domains. In particular, large-scale knowledge bases have gained popular- ity in the semantic parsing community (Cai and Yates, 2013; Berant et al., 2013; Fader et al., 2014). The increasing number of relations and en- tities motivates new resources and techniques for
improving the accuracy, including the use of ontol- ogy matching models (Kwiatkowski et al., 2013), paraphrase models (Fader et al., 2013; Berant and Liang, 2014), and unlabeled sentences (Krishna- murthy and Kollar, 2013; Reddy et al., 2014).
|
1508.00305#36
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 36,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "To parse a compositional utterance, many works rely on a lexicon that translates phrases to enti- ties, relations, and logical operations. A lexicon can be automatically generated (Unger and Cimi- ano, 2011; Unger et al., 2012), learned from data (Zettlemoyer and Collins, 2007; Kwiatkowski et al., 2011), or extracted from external sources (Cai and Yates, 2013; Berant et al., 2013), but requires some techniques to generalize to unseen data. Our work takes a different approach similar to the log- ical form growing algorithm in Berant and Liang (2014) by not anchoring relations and operations to the utterance.\nKnowledge domain. Recent works on seman- tic parsing for question answering operate on more open and diverse data domains. In particular, large-scale knowledge bases have gained popular- ity in the semantic parsing community (Cai and Yates, 2013; Berant et al., 2013; Fader et al., 2014). The increasing number of relations and en- tities motivates new resources and techniques for\nimproving the accuracy, including the use of ontol- ogy matching models (Kwiatkowski et al., 2013), paraphrase models (Fader et al., 2013; Berant and Liang, 2014), and unlabeled sentences (Krishna- murthy and Kollar, 2013; Reddy et al., 2014).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 37 |
Our work leverages open-ended data from the Web through semi-structured tables. There have been several studies on analyzing or inferring the table schemas (Cafarella et al., 2008; Venetis et al., 2011; Syed et al., 2010; Limaye et al., 2010) and answering search queries by joining tables on sim- ilar columns (Cafarella et al., 2008; Gonzalez et al., 2010; Pimplikar and Sarawagi, 2012). While the latter is similar to question answering, the queries tend to be keyword lists instead of natural language sentences. In parallel, open information extraction (Wu and Weld, 2010; Masaum et al., 2012) and knowledge base population (Ji and Gr- ishman, 2011) extract information from web pages and compile them into structured data. The result- ing knowledge base is systematically organized, but as a trade-off, some knowledge is inevitably lost during extraction and the information is forced to conform to a speciï¬c schema. To avoid these is- sues, we choose to work on HTML tables directly. In future work, we wish to draw informa- tion from other semi-structured formats such as colon-delimited
|
1508.00305#37
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 37,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Our work leverages open-ended data from the Web through semi-structured tables. There have been several studies on analyzing or inferring the table schemas (Cafarella et al., 2008; Venetis et al., 2011; Syed et al., 2010; Limaye et al., 2010) and answering search queries by joining tables on sim- ilar columns (Cafarella et al., 2008; Gonzalez et al., 2010; Pimplikar and Sarawagi, 2012). While the latter is similar to question answering, the queries tend to be keyword lists instead of natural language sentences. In parallel, open information extraction (Wu and Weld, 2010; Masaum et al., 2012) and knowledge base population (Ji and Gr- ishman, 2011) extract information from web pages and compile them into structured data. The result- ing knowledge base is systematically organized, but as a trade-off, some knowledge is inevitably lost during extraction and the information is forced to conform to a speciï¬c schema. To avoid these is- sues, we choose to work on HTML tables directly. In future work, we wish to draw informa- tion from other semi-structured formats such as colon-delimited",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 38 |
to work on HTML tables directly. In future work, we wish to draw informa- tion from other semi-structured formats such as colon-delimited pairs (Wong et al., 2009), bulleted lists (Gupta and Sarawagi, 2009), and top-k lists (Zhang et al., 2013). Pasupat and Liang (2014) used a framework similar to ours to extract entities from web pages, where the âlogical formsâ were XPath expressions. A natural direction is to com- bine the logical compositionality of this work with the even broader knowledge source of general web pages.
|
1508.00305#38
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 38,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "to work on HTML tables directly. In future work, we wish to draw informa- tion from other semi-structured formats such as colon-delimited pairs (Wong et al., 2009), bulleted lists (Gupta and Sarawagi, 2009), and top-k lists (Zhang et al., 2013). Pasupat and Liang (2014) used a framework similar to ours to extract entities from web pages, where the âlogical formsâ were XPath expressions. A natural direction is to com- bine the logical compositionality of this work with the even broader knowledge source of general web pages.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 39 |
Acknowledgements. We gratefully acknowl- edge the support of the Google Natural Language Understanding Focused Program and the Defense Advanced Research Projects Agency (DARPA) Deep Exploration and Filtering of Text (DEFT) Program under Air Force Research Laboratory (AFRL) contract no. FA8750-13-2-0040.
Data and reproducibility. The WIKITABLE- QUESTIONS dataset can be downloaded at http: //nlp.stanford.edu/software/sempre/wikitable/. Additionally, code, data, and experiments for this paper are available on the CodaLab plat- form at https://www.codalab.org/worksheets/ 0xf26cd79d4d734287868923ad1067cf4c/.
# References
Androutsopoulos, G. D. Ritchie, and P. Thanisch. 1995. Natural language interfaces to databases â an introduction. Journal of Natural Language Engineering, 1:29â81.
[Berant and Liang2014] J. Berant and P. Liang. 2014. Semantic parsing via paraphrasing. In Association for Computational Linguistics (ACL).
|
1508.00305#39
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 39,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "Acknowledgements. We gratefully acknowl- edge the support of the Google Natural Language Understanding Focused Program and the Defense Advanced Research Projects Agency (DARPA) Deep Exploration and Filtering of Text (DEFT) Program under Air Force Research Laboratory (AFRL) contract no. FA8750-13-2-0040.\nData and reproducibility. The WIKITABLE- QUESTIONS dataset can be downloaded at http: //nlp.stanford.edu/software/sempre/wikitable/. Additionally, code, data, and experiments for this paper are available on the CodaLab plat- form at https://www.codalab.org/worksheets/ 0xf26cd79d4d734287868923ad1067cf4c/.\n# References\nAndroutsopoulos, G. D. Ritchie, and P. Thanisch. 1995. Natural language interfaces to databases â an introduction. Journal of Natural Language Engineering, 1:29â81.\n[Berant and Liang2014] J. Berant and P. Liang. 2014. Semantic parsing via paraphrasing. In Association for Computational Linguistics (ACL).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 40 |
[Berant and Liang2014] J. Berant and P. Liang. 2014. Semantic parsing via paraphrasing. In Association for Computational Linguistics (ACL).
[Berant et al.2013] J. Berant, A. Chou, R. Frostig, and P. Liang. 2013. Semantic parsing on Freebase from question-answer pairs. In Empirical Methods in Natural Language Processing (EMNLP).
[Cafarella et al.2008] M. J. Cafarella, A. Halevy, D. Z. Wang, E. Wu, and Y. Zhang. 2008. WebTables: exploring the power of tables on the web. In Very Large Data Bases (VLDB), pages 538â549.
[Cai and Yates2013] Q. Cai and A. Yates. 2013. Large- scale semantic parsing via schema matching and lex- In Association for Computational icon extension. Linguistics (ACL).
[Duchi et al.2010] J. Duchi, E. Hazan, and Y. Singer. 2010. Adaptive subgradient methods for online learning and stochastic optimization. In Conference on Learning Theory (COLT).
|
1508.00305#40
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 40,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Berant and Liang2014] J. Berant and P. Liang. 2014. Semantic parsing via paraphrasing. In Association for Computational Linguistics (ACL).\n[Berant et al.2013] J. Berant, A. Chou, R. Frostig, and P. Liang. 2013. Semantic parsing on Freebase from question-answer pairs. In Empirical Methods in Natural Language Processing (EMNLP).\n[Cafarella et al.2008] M. J. Cafarella, A. Halevy, D. Z. Wang, E. Wu, and Y. Zhang. 2008. WebTables: exploring the power of tables on the web. In Very Large Data Bases (VLDB), pages 538â549.\n[Cai and Yates2013] Q. Cai and A. Yates. 2013. Large- scale semantic parsing via schema matching and lex- In Association for Computational icon extension. Linguistics (ACL).\n[Duchi et al.2010] J. Duchi, E. Hazan, and Y. Singer. 2010. Adaptive subgradient methods for online learning and stochastic optimization. In Conference on Learning Theory (COLT).",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 41 |
[Fader et al.2013] A. Fader, L. Zettlemoyer, and O. Et- zioni. 2013. Paraphrase-driven learning for open In Association for Computa- question answering. tional Linguistics (ACL).
[Fader et al.2014] A. Fader, L. Zettlemoyer, and O. Et- zioni. 2014. Open question answering over curated In International and extracted knowledge bases. Conference on Knowledge Discovery and Data Min- ing (KDD), pages 1156â1165.
[Gonzalez et al.2010] H. Gonzalez, A. Y. Halevy, C. S. Jensen, A. Langen, J. Madhavan, R. Shapley, W. Shen, and J. Goldberg-Kidon. 2010. Google fu- sion tables: web-centered data management and col- In Proceedings of the 2010 ACM SIG- laboration. MOD International Conference on Management of data, pages 1061â1066.
[Gupta and Sarawagi2009] R. Gupta and S. Sarawagi. 2009. Answering table augmentation queries from In Very Large Data unstructured lists on the web. Bases (VLDB), number 1, pages 289â300.
|
1508.00305#41
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 41,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Fader et al.2013] A. Fader, L. Zettlemoyer, and O. Et- zioni. 2013. Paraphrase-driven learning for open In Association for Computa- question answering. tional Linguistics (ACL).\n[Fader et al.2014] A. Fader, L. Zettlemoyer, and O. Et- zioni. 2014. Open question answering over curated In International and extracted knowledge bases. Conference on Knowledge Discovery and Data Min- ing (KDD), pages 1156â1165.\n[Gonzalez et al.2010] H. Gonzalez, A. Y. Halevy, C. S. Jensen, A. Langen, J. Madhavan, R. Shapley, W. Shen, and J. Goldberg-Kidon. 2010. Google fu- sion tables: web-centered data management and col- In Proceedings of the 2010 ACM SIG- laboration. MOD International Conference on Management of data, pages 1061â1066.\n[Gupta and Sarawagi2009] R. Gupta and S. Sarawagi. 2009. Answering table augmentation queries from In Very Large Data unstructured lists on the web. Bases (VLDB), number 1, pages 289â300.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
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] |
1508.00305
| 42 |
[Ji and Grishman2011] H. Ji and R. Grishman. 2011. Knowledge base population: Successful approaches In Association for Computational and challenges. Linguistics (ACL), pages 1148â1158.
[Krishnamurthy and Kollar2013] J. Krishnamurthy and T. Kollar. 2013. Jointly learning to parse and per- ceive: Connecting natural language to the physical world. Transactions of the Association for Compu- tational Linguistics (TACL), 1:193â206.
[Kwiatkowski et al.2011] T. Kwiatkowski, L. Zettle- moyer, S. Goldwater, and M. Steedman. 2011. Lex- ical generalization in CCG grammar induction for semantic parsing. In Empirical Methods in Natural Language Processing (EMNLP), pages 1512â1523.
[Kwiatkowski et al.2013] T. Kwiatkowski, E. Choi, Y. Artzi, and L. Zettlemoyer. 2013. Scaling seman- tic parsers with on-the-ï¬y ontology matching. In Empirical Methods in Natural Language Processing (EMNLP).
2013. Lambda dependency- based compositional semantics. Technical report, arXiv.
|
1508.00305#42
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 42,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Ji and Grishman2011] H. Ji and R. Grishman. 2011. Knowledge base population: Successful approaches In Association for Computational and challenges. Linguistics (ACL), pages 1148â1158.\n[Krishnamurthy and Kollar2013] J. Krishnamurthy and T. Kollar. 2013. Jointly learning to parse and per- ceive: Connecting natural language to the physical world. Transactions of the Association for Compu- tational Linguistics (TACL), 1:193â206.\n[Kwiatkowski et al.2011] T. Kwiatkowski, L. Zettle- moyer, S. Goldwater, and M. Steedman. 2011. Lex- ical generalization in CCG grammar induction for semantic parsing. In Empirical Methods in Natural Language Processing (EMNLP), pages 1512â1523.\n[Kwiatkowski et al.2013] T. Kwiatkowski, E. Choi, Y. Artzi, and L. Zettlemoyer. 2013. Scaling seman- tic parsers with on-the-ï¬y ontology matching. In Empirical Methods in Natural Language Processing (EMNLP).\n2013. Lambda dependency- based compositional semantics. Technical report, arXiv.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 43 |
2013. Lambda dependency- based compositional semantics. Technical report, arXiv.
and S. Chakrabarti. 2010. Annotating and searching web tables using entities, types and relationships. In Very Large Data Bases (VLDB), volume 3, pages 1338â1347.
[Masaum et al.2012] Masaum, M. Schmitz, R. Bart, Open S. Soderland, and O. Etzioni. language learning for information extraction. In Empirical Methods in Natural Language Process- ing and Computational Natural Language Learning (EMNLP/CoNLL), pages 523â534.
[Pasupat and Liang2014] P. Pasupat and P. Liang. 2014. Zero-shot entity extraction from web pages. In Association for Computational Linguistics (ACL).
and S. Sarawagi. 2012. Answering table queries on the In Very Large Data web using column keywords. Bases (VLDB), volume 5, pages 908â919.
and H. Kautz. 2003. Towards a theory of natural lan- guage interfaces to databases. In International Con- ference on Intelligent User Interfaces (IUI), pages 149â157.
|
1508.00305#43
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 43,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "2013. Lambda dependency- based compositional semantics. Technical report, arXiv.\nand S. Chakrabarti. 2010. Annotating and searching web tables using entities, types and relationships. In Very Large Data Bases (VLDB), volume 3, pages 1338â1347.\n[Masaum et al.2012] Masaum, M. Schmitz, R. Bart, Open S. Soderland, and O. Etzioni. language learning for information extraction. In Empirical Methods in Natural Language Process- ing and Computational Natural Language Learning (EMNLP/CoNLL), pages 523â534.\n[Pasupat and Liang2014] P. Pasupat and P. Liang. 2014. Zero-shot entity extraction from web pages. In Association for Computational Linguistics (ACL).\nand S. Sarawagi. 2012. Answering table queries on the In Very Large Data web using column keywords. Bases (VLDB), volume 5, pages 908â919.\nand H. Kautz. 2003. Towards a theory of natural lan- guage interfaces to databases. In International Con- ference on Intelligent User Interfaces (IUI), pages 149â157.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 44 |
[Price1990] P. Price. 1990. Evaluation of spoken lan- guage systems: The ATIS domain. In Proceedings of the Third DARPA Speech and Natural Language Workshop, pages 91â95.
[Reddy et al.2014] S. Reddy, M. Lapata, and M. Steed- man. 2014. Large-scale semantic parsing with- the out question-answer pairs. Association for Computational Linguistics (TACL), 2(10):377â392.
[Syed et al.2010] Z. Syed, T. Finin, V. Mulwad, and A. Joshi. 2010. Exploiting a web of semantic data for interpreting tables. In Proceedings of the Second Web Science Conference.
[Unger and Cimiano2011] C. Unger and P. Cimiano. 2011. Pythia: compositional meaning construction for ontology-based question answering on the se- In Proceedings of the 16th interna- mantic web. tional conference on Natural language processing and information systems, pages 153â160.
[Unger et al.2012] C. Unger, L. B¨uhmann, J. Lehmann, 2012. A. Ngonga, D. Gerber, and P. Cimiano. Template-based question answering over RDF data. In World Wide Web (WWW), pages 639â648.
|
1508.00305#44
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 44,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Price1990] P. Price. 1990. Evaluation of spoken lan- guage systems: The ATIS domain. In Proceedings of the Third DARPA Speech and Natural Language Workshop, pages 91â95.\n[Reddy et al.2014] S. Reddy, M. Lapata, and M. Steed- man. 2014. Large-scale semantic parsing with- the out question-answer pairs. Association for Computational Linguistics (TACL), 2(10):377â392.\n[Syed et al.2010] Z. Syed, T. Finin, V. Mulwad, and A. Joshi. 2010. Exploiting a web of semantic data for interpreting tables. In Proceedings of the Second Web Science Conference.\n[Unger and Cimiano2011] C. Unger and P. Cimiano. 2011. Pythia: compositional meaning construction for ontology-based question answering on the se- In Proceedings of the 16th interna- mantic web. tional conference on Natural language processing and information systems, pages 153â160.\n[Unger et al.2012] C. Unger, L. B¨uhmann, J. Lehmann, 2012. A. Ngonga, D. Gerber, and P. Cimiano. Template-based question answering over RDF data. In World Wide Web (WWW), pages 639â648.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 45 |
[Venetis et al.2011] P. Venetis, A. Halevy, J. Madhavan, M. Pas¸ca, W. Shen, F. Wu, G. Miao, and C. Wu. 2011. Recovering semantics of tables on the web. In Very Large Data Bases (VLDB), volume 4, pages 528â538.
[Wang et al.2011] Y. Wang, L. Deng, and A. Acero. 2011. Semantic frame-based spoken language un- Spoken Language Understanding: derstanding. Systems for Extracting Semantic Information from Speech, pages 41â91.
J. Mooney. 2007. Learning synchronous grammars for semantic parsing with lambda calculus. In Asso- ciation for Computational Linguistics (ACL), pages 960â967.
D. Widdows, [Wong et al.2009] Y. W. Wong, Scalable T. Lokovic, and K. Nigam. attribute-value extraction from semi-structured text. In IEEE International Conference on Data Mining Workshops, pages 302â307.
[Wu and Weld2010] F. Wu and D. S. Weld. 2010. Open In Asso- information extraction using Wikipedia. ciation for Computational Linguistics (ACL), pages 118â127.
|
1508.00305#45
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 45,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Venetis et al.2011] P. Venetis, A. Halevy, J. Madhavan, M. Pas¸ca, W. Shen, F. Wu, G. Miao, and C. Wu. 2011. Recovering semantics of tables on the web. In Very Large Data Bases (VLDB), volume 4, pages 528â538.\n[Wang et al.2011] Y. Wang, L. Deng, and A. Acero. 2011. Semantic frame-based spoken language un- Spoken Language Understanding: derstanding. Systems for Extracting Semantic Information from Speech, pages 41â91.\nJ. Mooney. 2007. Learning synchronous grammars for semantic parsing with lambda calculus. In Asso- ciation for Computational Linguistics (ACL), pages 960â967.\nD. Widdows, [Wong et al.2009] Y. W. Wong, Scalable T. Lokovic, and K. Nigam. attribute-value extraction from semi-structured text. In IEEE International Conference on Data Mining Workshops, pages 302â307.\n[Wu and Weld2010] F. Wu and D. S. Weld. 2010. Open In Asso- information extraction using Wikipedia. ciation for Computational Linguistics (ACL), pages 118â127.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1508.00305
| 46 |
[Zelle and Mooney1996] M. Zelle and R. J. Mooney. 1996. Learning to parse database queries using in- ductive logic programming. In Association for the Advancement of Artiï¬cial Intelligence (AAAI), pages 1050â1055.
[Zettlemoyer and Collins2007] L. S. Zettlemoyer and 2007. Online learning of relaxed M. Collins. CCG grammars for parsing to logical form. In Empirical Methods in Natural Language Process- ing and Computational Natural Language Learning (EMNLP/CoNLL), pages 678â687.
[Zhang et al.2013] Z. Zhang, K. Q. Zhu, H. Wang, and H. Li. 2013. Automatic extraction of top-k lists from the web. In International Conference on Data Engineering.
|
1508.00305#46
|
Compositional Semantic Parsing on Semi-Structured Tables
|
Two important aspects of semantic parsing for question answering are the
breadth of the knowledge source and the depth of logical compositionality.
While existing work trades off one aspect for another, this paper
simultaneously makes progress on both fronts through a new task: answering
complex questions on semi-structured tables using question-answer pairs as
supervision. The central challenge arises from two compounding factors: the
broader domain results in an open-ended set of relations, and the deeper
compositionality results in a combinatorial explosion in the space of logical
forms. We propose a logical-form driven parsing algorithm guided by strong
typing constraints and show that it obtains significant improvements over
natural baselines. For evaluation, we created a new dataset of 22,033 complex
questions on Wikipedia tables, which is made publicly available.
|
http://arxiv.org/pdf/1508.00305
|
Panupong Pasupat, Percy Liang
|
cs.CL
| null | null |
cs.CL
|
20150803
|
20150803
|
[] |
{
"authors": "Panupong Pasupat, Percy Liang",
"chunk_id": 46,
"doc_id": "1508.00305",
"primary_category": "cs.CL",
"published": 20150803,
"source": "http://arxiv.org/pdf/1508.00305",
"summary": "Two important aspects of semantic parsing for question answering are the\nbreadth of the knowledge source and the depth of logical compositionality.\nWhile existing work trades off one aspect for another, this paper\nsimultaneously makes progress on both fronts through a new task: answering\ncomplex questions on semi-structured tables using question-answer pairs as\nsupervision. The central challenge arises from two compounding factors: the\nbroader domain results in an open-ended set of relations, and the deeper\ncompositionality results in a combinatorial explosion in the space of logical\nforms. We propose a logical-form driven parsing algorithm guided by strong\ntyping constraints and show that it obtains significant improvements over\nnatural baselines. For evaluation, we created a new dataset of 22,033 complex\nquestions on Wikipedia tables, which is made publicly available.",
"text": "[Zelle and Mooney1996] M. Zelle and R. J. Mooney. 1996. Learning to parse database queries using in- ductive logic programming. In Association for the Advancement of Artiï¬cial Intelligence (AAAI), pages 1050â1055.\n[Zettlemoyer and Collins2007] L. S. Zettlemoyer and 2007. Online learning of relaxed M. Collins. CCG grammars for parsing to logical form. In Empirical Methods in Natural Language Process- ing and Computational Natural Language Learning (EMNLP/CoNLL), pages 678â687.\n[Zhang et al.2013] Z. Zhang, K. Q. Zhu, H. Wang, and H. Li. 2013. Automatic extraction of top-k lists from the web. In International Conference on Data Engineering.",
"title": "Compositional Semantic Parsing on Semi-Structured Tables",
"year": 2015
}
|
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] |
1506.02626
| 1 |
# John Tran NVIDIA [email protected]
William J. Dally Stanford University NVIDIA [email protected]
# Abstract
Neural networks are both computationally intensive and memory intensive, making them difï¬cult to deploy on embedded systems. Also, conventional networks ï¬x the architecture before training starts; as a result, training cannot improve the architecture. To address these limitations, we describe a method to reduce the storage and computation required by neural networks by an order of magnitude without affecting their accuracy by learning only the important connections. Our method prunes redundant connections using a three-step method. First, we train the network to learn which connections are important. Next, we prune the unim- portant connections. Finally, we retrain the network to ï¬ne tune the weights of the remaining connections. On the ImageNet dataset, our method reduced the number of parameters of AlexNet by a factor of 9Ã, from 61 million to 6.7 million, without incurring accuracy loss. Similar experiments with VGG-16 found that the total number of parameters can be reduced by 13Ã, from 138 million to 10.3 million, again with no loss of accuracy.
# Introduction
|
1506.02626#1
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 1,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# John Tran NVIDIA [email protected]\nWilliam J. Dally Stanford University NVIDIA [email protected]\n# Abstract\nNeural networks are both computationally intensive and memory intensive, making them difï¬cult to deploy on embedded systems. Also, conventional networks ï¬x the architecture before training starts; as a result, training cannot improve the architecture. To address these limitations, we describe a method to reduce the storage and computation required by neural networks by an order of magnitude without affecting their accuracy by learning only the important connections. Our method prunes redundant connections using a three-step method. First, we train the network to learn which connections are important. Next, we prune the unim- portant connections. Finally, we retrain the network to ï¬ne tune the weights of the remaining connections. On the ImageNet dataset, our method reduced the number of parameters of AlexNet by a factor of 9Ã, from 61 million to 6.7 million, without incurring accuracy loss. Similar experiments with VGG-16 found that the total number of parameters can be reduced by 13Ã, from 138 million to 10.3 million, again with no loss of accuracy.\n# Introduction",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 2 |
# Introduction
Neural networks have become ubiquitous in applications ranging from computer vision [1] to speech recognition [2] and natural language processing [3]. We consider convolutional neural networks used for computer vision tasks which have grown over time. In 1998 Lecun et al. designed a CNN model LeNet-5 with less than 1M parameters to classify handwritten digits [4], while in 2012, Krizhevsky et al. [1] won the ImageNet competition with 60M parameters. Deepface classiï¬ed human faces with 120M parameters [5], and Coates et al. [6] scaled up a network to 10B parameters.
|
1506.02626#2
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 2,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# Introduction\nNeural networks have become ubiquitous in applications ranging from computer vision [1] to speech recognition [2] and natural language processing [3]. We consider convolutional neural networks used for computer vision tasks which have grown over time. In 1998 Lecun et al. designed a CNN model LeNet-5 with less than 1M parameters to classify handwritten digits [4], while in 2012, Krizhevsky et al. [1] won the ImageNet competition with 60M parameters. Deepface classiï¬ed human faces with 120M parameters [5], and Coates et al. [6] scaled up a network to 10B parameters.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 3 |
While these large neural networks are very powerful, their size consumes considerable storage, memory bandwidth, and computational resources. For embedded mobile applications, these resource demands become prohibitive. Figure 1 shows the energy cost of basic arithmetic and memory operations in a 45nm CMOS process. From this data we see the energy per connection is dominated by memory access and ranges from 5pJ for 32 bit coefï¬cients in on-chip SRAM to 640pJ for 32bit coefï¬cients in off-chip DRAM [7]. Large networks do not ï¬t in on-chip storage and hence require the more costly DRAM accesses. Running a 1 billion connection neural network, for example, at 20Hz would require (20Hz)(1G)(640pJ) = 12.8W just for DRAM access - well beyond the power envelope of a typical mobile device. Our goal in pruning networks is to reduce the energy required to run such large networks so they can run in real time on mobile devices. The model size reduction from pruning also facilitates storage and transmission of mobile applications incorporating DNNs.
1
Relative Energy Cost
|
1506.02626#3
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 3,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "While these large neural networks are very powerful, their size consumes considerable storage, memory bandwidth, and computational resources. For embedded mobile applications, these resource demands become prohibitive. Figure 1 shows the energy cost of basic arithmetic and memory operations in a 45nm CMOS process. From this data we see the energy per connection is dominated by memory access and ranges from 5pJ for 32 bit coefï¬cients in on-chip SRAM to 640pJ for 32bit coefï¬cients in off-chip DRAM [7]. Large networks do not ï¬t in on-chip storage and hence require the more costly DRAM accesses. Running a 1 billion connection neural network, for example, at 20Hz would require (20Hz)(1G)(640pJ) = 12.8W just for DRAM access - well beyond the power envelope of a typical mobile device. Our goal in pruning networks is to reduce the energy required to run such large networks so they can run in real time on mobile devices. The model size reduction from pruning also facilitates storage and transmission of mobile applications incorporating DNNs.\n1\nRelative Energy Cost",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
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] |
1506.02626
| 4 |
1
Relative Energy Cost
Operation Energy [pJ] Relative Cost 32 bit int ADD 32 bit ï¬oat ADD 32 bit Register File 32 bit int MULT 32 bit ï¬oat MULT 32 bit SRAM Cache 32 bit DRAM Memory 0.1 0.9 1 3.1 3.7 5 640 1 9 10 31 37 50 6400
1 10 100 1000
= 10000
Figure 1: Energy table for 45nm CMOS process [7]. Memory access is 3 orders of magnitude more energy expensive than simple arithmetic.
To achieve this goal, we present a method to prune network connections in a manner that preserves the original accuracy. After an initial training phase, we remove all connections whose weight is lower than a threshold. This pruning converts a dense, fully-connected layer to a sparse layer. This ï¬rst phase learns the topology of the networks â learning which connections are important and removing the unimportant connections. We then retrain the sparse network so the remaining connections can compensate for the connections that have been removed. The phases of pruning and retraining may be repeated iteratively to further reduce network complexity. In effect, this training process learns the network connectivity in addition to the weights - much as in the mammalian brain [8][9], where synapses are created in the ï¬rst few months of a childâs development, followed by gradual pruning of little-used connections, falling to typical adult values.
|
1506.02626#4
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 4,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "1\nRelative Energy Cost\nOperation Energy [pJ] Relative Cost 32 bit int ADD 32 bit ï¬oat ADD 32 bit Register File 32 bit int MULT 32 bit ï¬oat MULT 32 bit SRAM Cache 32 bit DRAM Memory 0.1 0.9 1 3.1 3.7 5 640 1 9 10 31 37 50 6400\n1 10 100 1000\n= 10000\nFigure 1: Energy table for 45nm CMOS process [7]. Memory access is 3 orders of magnitude more energy expensive than simple arithmetic.\nTo achieve this goal, we present a method to prune network connections in a manner that preserves the original accuracy. After an initial training phase, we remove all connections whose weight is lower than a threshold. This pruning converts a dense, fully-connected layer to a sparse layer. This ï¬rst phase learns the topology of the networks â learning which connections are important and removing the unimportant connections. We then retrain the sparse network so the remaining connections can compensate for the connections that have been removed. The phases of pruning and retraining may be repeated iteratively to further reduce network complexity. In effect, this training process learns the network connectivity in addition to the weights - much as in the mammalian brain [8][9], where synapses are created in the ï¬rst few months of a childâs development, followed by gradual pruning of little-used connections, falling to typical adult values.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 5 |
# 2 Related Work
Neural networks are typically over-parameterized, and there is signiï¬cant redundancy for deep learn- ing models [10]. This results in a waste of both computation and memory. There have been various proposals to remove the redundancy: Vanhoucke et al. [11] explored a ï¬xed-point implementation with 8-bit integer (vs 32-bit ï¬oating point) activations. Denton et al. [12] exploited the linear structure of the neural network by ï¬nding an appropriate low-rank approximation of the parameters and keeping the accuracy within 1% of the original model. With similar accuracy loss, Gong et al. [13] compressed deep convnets using vector quantization. These approximation and quantization techniques are orthogonal to network pruning, and they can be used together to obtain further gains [14].
|
1506.02626#5
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 5,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 2 Related Work\nNeural networks are typically over-parameterized, and there is signiï¬cant redundancy for deep learn- ing models [10]. This results in a waste of both computation and memory. There have been various proposals to remove the redundancy: Vanhoucke et al. [11] explored a ï¬xed-point implementation with 8-bit integer (vs 32-bit ï¬oating point) activations. Denton et al. [12] exploited the linear structure of the neural network by ï¬nding an appropriate low-rank approximation of the parameters and keeping the accuracy within 1% of the original model. With similar accuracy loss, Gong et al. [13] compressed deep convnets using vector quantization. These approximation and quantization techniques are orthogonal to network pruning, and they can be used together to obtain further gains [14].",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 6 |
There have been other attempts to reduce the number of parameters of neural networks by replacing the fully connected layer with global average pooling. The Network in Network architecture [15] and GoogLenet [16] achieves state-of-the-art results on several benchmarks by adopting this idea. However, transfer learning, i.e. reusing features learned on the ImageNet dataset and applying them to new tasks by only ï¬ne-tuning the fully connected layers, is more difï¬cult with this approach. This problem is noted by Szegedy et al. [16] and motivates them to add a linear layer on the top of their networks to enable transfer learning.
Network pruning has been used both to reduce network complexity and to reduce over-ï¬tting. An early approach to pruning was biased weight decay [17]. Optimal Brain Damage [18] and Optimal Brain Surgeon [19] prune networks to reduce the number of connections based on the Hessian of the loss function and suggest that such pruning is more accurate than magnitude-based pruning such as weight decay. However, second order derivative needs additional computation.
|
1506.02626#6
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 6,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "There have been other attempts to reduce the number of parameters of neural networks by replacing the fully connected layer with global average pooling. The Network in Network architecture [15] and GoogLenet [16] achieves state-of-the-art results on several benchmarks by adopting this idea. However, transfer learning, i.e. reusing features learned on the ImageNet dataset and applying them to new tasks by only ï¬ne-tuning the fully connected layers, is more difï¬cult with this approach. This problem is noted by Szegedy et al. [16] and motivates them to add a linear layer on the top of their networks to enable transfer learning.\nNetwork pruning has been used both to reduce network complexity and to reduce over-ï¬tting. An early approach to pruning was biased weight decay [17]. Optimal Brain Damage [18] and Optimal Brain Surgeon [19] prune networks to reduce the number of connections based on the Hessian of the loss function and suggest that such pruning is more accurate than magnitude-based pruning such as weight decay. However, second order derivative needs additional computation.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 7 |
HashedNets [20] is a recent technique to reduce model sizes by using a hash function to randomly group connection weights into hash buckets, so that all connections within the same hash bucket share a single parameter value. This technique may beneï¬t from pruning. As pointed out in Shi et al. [21] and Weinberger et al. [22], sparsity will minimize hash collision making feature hashing even more effective. HashedNets may be used together with pruning to give even better parameter savings.
2
Train Connectivity wu Prune Connections we Train Weights
before pruning after pruning pruning synapses --> pruning neurons
Figure 2: Three-Step Training Pipeline.
Figure 3: Synapses and neurons before and after pruning.
# 3 Learning Connections in Addition to Weights
Our pruning method employs a three-step process, as illustrated in Figure 2, which begins by learning the connectivity via normal network training. Unlike conventional training, however, we are not learning the ï¬nal values of the weights, but rather we are learning which connections are important.
|
1506.02626#7
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 7,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "HashedNets [20] is a recent technique to reduce model sizes by using a hash function to randomly group connection weights into hash buckets, so that all connections within the same hash bucket share a single parameter value. This technique may beneï¬t from pruning. As pointed out in Shi et al. [21] and Weinberger et al. [22], sparsity will minimize hash collision making feature hashing even more effective. HashedNets may be used together with pruning to give even better parameter savings.\n2\nTrain Connectivity wu Prune Connections we Train Weights\nbefore pruning after pruning pruning synapses --> pruning neurons\nFigure 2: Three-Step Training Pipeline.\nFigure 3: Synapses and neurons before and after pruning.\n# 3 Learning Connections in Addition to Weights\nOur pruning method employs a three-step process, as illustrated in Figure 2, which begins by learning the connectivity via normal network training. Unlike conventional training, however, we are not learning the ï¬nal values of the weights, but rather we are learning which connections are important.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 8 |
The second step is to prune the low-weight connections. All connections with weights below a threshold are removed from the network â converting a dense network into a sparse network, as shown in Figure 3. The ï¬nal step retrains the network to learn the ï¬nal weights for the remaining sparse connections. This step is critical. If the pruned network is used without retraining, accuracy is signiï¬cantly impacted.
# 3.1 Regularization
Choosing the correct regularization impacts the performance of pruning and retraining. L1 regulariza- tion penalizes non-zero parameters resulting in more parameters near zero. This gives better accuracy after pruning, but before retraining. However, the remaining connections are not as good as with L2 regularization, resulting in lower accuracy after retraining. Overall, L2 regularization gives the best pruning results. This is further discussed in experiment section.
# 3.2 Dropout Ratio Adjustment
|
1506.02626#8
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 8,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "The second step is to prune the low-weight connections. All connections with weights below a threshold are removed from the network â converting a dense network into a sparse network, as shown in Figure 3. The ï¬nal step retrains the network to learn the ï¬nal weights for the remaining sparse connections. This step is critical. If the pruned network is used without retraining, accuracy is signiï¬cantly impacted.\n# 3.1 Regularization\nChoosing the correct regularization impacts the performance of pruning and retraining. L1 regulariza- tion penalizes non-zero parameters resulting in more parameters near zero. This gives better accuracy after pruning, but before retraining. However, the remaining connections are not as good as with L2 regularization, resulting in lower accuracy after retraining. Overall, L2 regularization gives the best pruning results. This is further discussed in experiment section.\n# 3.2 Dropout Ratio Adjustment",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 9 |
# 3.2 Dropout Ratio Adjustment
Dropout [23] is widely used to prevent over-ï¬tting, and this also applies to retraining. During retraining, however, the dropout ratio must be adjusted to account for the change in model capacity. In dropout, each parameter is probabilistically dropped during training, but will come back during inference. In pruning, parameters are dropped forever after pruning and have no chance to come back during both training and inference. As the parameters get sparse, the classiï¬er will select the most informative predictors and thus have much less prediction variance, which reduces over-ï¬tting. As pruning already reduced model capacity, the retraining dropout ratio should be smaller.
Quantitatively, let Ci be the number of connections in layer i, Cio for the original network, Cir for the network after retraining, Ni be the number of neurons in layer i. Since dropout works on neurons, and Ci varies quadratically with Ni, according to Equation 1 thus the dropout ratio after pruning the parameters should follow Equation 2, where Do represent the original dropout rate, Dr represent the dropout rate during retraining.
Ci = NiNiâ1 (1) Dr = Do (2)
# 3.3 Local Pruning and Parameter Co-adaptation
|
1506.02626#9
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 9,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 3.2 Dropout Ratio Adjustment\nDropout [23] is widely used to prevent over-ï¬tting, and this also applies to retraining. During retraining, however, the dropout ratio must be adjusted to account for the change in model capacity. In dropout, each parameter is probabilistically dropped during training, but will come back during inference. In pruning, parameters are dropped forever after pruning and have no chance to come back during both training and inference. As the parameters get sparse, the classiï¬er will select the most informative predictors and thus have much less prediction variance, which reduces over-ï¬tting. As pruning already reduced model capacity, the retraining dropout ratio should be smaller.\nQuantitatively, let Ci be the number of connections in layer i, Cio for the original network, Cir for the network after retraining, Ni be the number of neurons in layer i. Since dropout works on neurons, and Ci varies quadratically with Ni, according to Equation 1 thus the dropout ratio after pruning the parameters should follow Equation 2, where Do represent the original dropout rate, Dr represent the dropout rate during retraining.\nCi = NiNiâ1 (1) Dr = Do (2)\n# 3.3 Local Pruning and Parameter Co-adaptation",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 10 |
Ci = NiNiâ1 (1) Dr = Do (2)
# 3.3 Local Pruning and Parameter Co-adaptation
During retraining, it is better to retain the weights from the initial training phase for the connections that survived pruning than it is to re-initialize the pruned layers. CNNs contain fragile co-adapted features [24]: gradient descent is able to ï¬nd a good solution when the network is initially trained, but not after re-initializing some layers and retraining them. So when we retrain the pruned layers, we should keep the surviving parameters instead of re-initializing them.
3
Table 1: Network pruning can save 9Ã to 13Ã parameters with no drop in predictive performance.
Network Top-1 Error Top-5 Error Parameters Compression Rate LeNet-300-100 Ref LeNet-300-100 Pruned LeNet-5 Ref LeNet-5 Pruned AlexNet Ref AlexNet Pruned VGG-16 Ref VGG-16 Pruned 1.64% 1.59% 0.80% 0.77% 42.78% 42.77% 31.50% 31.34% - - - - 19.73% 19.67% 11.32% 10.88% 267K 22K 431K 36K 61M 6.7M 138M 10.3M 12Ã 12Ã 9Ã 13Ã
|
1506.02626#10
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 10,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Ci = NiNiâ1 (1) Dr = Do (2)\n# 3.3 Local Pruning and Parameter Co-adaptation\nDuring retraining, it is better to retain the weights from the initial training phase for the connections that survived pruning than it is to re-initialize the pruned layers. CNNs contain fragile co-adapted features [24]: gradient descent is able to ï¬nd a good solution when the network is initially trained, but not after re-initializing some layers and retraining them. So when we retrain the pruned layers, we should keep the surviving parameters instead of re-initializing them.\n3\nTable 1: Network pruning can save 9à to 13à parameters with no drop in predictive performance.\nNetwork Top-1 Error Top-5 Error Parameters Compression Rate LeNet-300-100 Ref LeNet-300-100 Pruned LeNet-5 Ref LeNet-5 Pruned AlexNet Ref AlexNet Pruned VGG-16 Ref VGG-16 Pruned 1.64% 1.59% 0.80% 0.77% 42.78% 42.77% 31.50% 31.34% - - - - 19.73% 19.67% 11.32% 10.88% 267K 22K 431K 36K 61M 6.7M 138M 10.3M 12à 12à 9à 13Ã",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 11 |
Retraining the pruned layers starting with retained weights requires less computation because we donât have to back propagate through the entire network. Also, neural networks are prone to suffer the vanishing gradient problem [25] as the networks get deeper, which makes pruning errors harder to recover for deep networks. To prevent this, we ï¬x the parameters for CONV layers and only retrain the FC layers after pruning the FC layers, and vice versa.
# Iterative Pruning
Learning the right connections is an iterative process. Pruning followed by a retraining is one iteration, after many such iterations the minimum number connections could be found. Without loss of accuracy, this method can boost pruning rate from 5à to 9à on AlexNet compared with single-step aggressive pruning. Each iteration is a greedy search in that we ï¬nd the best connections. We also experimented with probabilistically pruning parameters based on their absolute value, but this gave worse results.
# 3.5 Pruning Neurons
|
1506.02626#11
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 11,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Retraining the pruned layers starting with retained weights requires less computation because we donât have to back propagate through the entire network. Also, neural networks are prone to suffer the vanishing gradient problem [25] as the networks get deeper, which makes pruning errors harder to recover for deep networks. To prevent this, we ï¬x the parameters for CONV layers and only retrain the FC layers after pruning the FC layers, and vice versa.\n# Iterative Pruning\nLearning the right connections is an iterative process. Pruning followed by a retraining is one iteration, after many such iterations the minimum number connections could be found. Without loss of accuracy, this method can boost pruning rate from 5à to 9à on AlexNet compared with single-step aggressive pruning. Each iteration is a greedy search in that we ï¬nd the best connections. We also experimented with probabilistically pruning parameters based on their absolute value, but this gave worse results.\n# 3.5 Pruning Neurons",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 12 |
# 3.5 Pruning Neurons
After pruning connections, neurons with zero input connections or zero output connections may be safely pruned. This pruning is furthered by removing all connections to or from a pruned neuron. The retraining phase automatically arrives at the result where dead neurons will have both zero input connections and zero output connections. This occurs due to gradient descent and regularization. A neuron that has zero input connections (or zero output connections) will have no contribution to the ï¬nal loss, leading the gradient to be zero for its output connection (or input connection), respectively. Only the regularization term will push the weights to zero. Thus, the dead neurons will be automatically removed during retraining.
# 4 Experiments
We implemented network pruning in Caffe [26]. Caffe was modiï¬ed to add a mask which disregards pruned parameters during network operation for each weight tensor. The pruning threshold is chosen as a quality parameter multiplied by the standard deviation of a layerâs weights. We carried out the experiments on Nvidia TitanX and GTX980 GPUs.
We pruned four representative networks: Lenet-300-100 and Lenet-5 on MNIST, together with AlexNet and VGG-16 on ImageNet. The network parameters and accuracy 1 before and after pruning are shown in Table 1.
# 4.1 LeNet on MNIST
|
1506.02626#12
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 12,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 3.5 Pruning Neurons\nAfter pruning connections, neurons with zero input connections or zero output connections may be safely pruned. This pruning is furthered by removing all connections to or from a pruned neuron. The retraining phase automatically arrives at the result where dead neurons will have both zero input connections and zero output connections. This occurs due to gradient descent and regularization. A neuron that has zero input connections (or zero output connections) will have no contribution to the ï¬nal loss, leading the gradient to be zero for its output connection (or input connection), respectively. Only the regularization term will push the weights to zero. Thus, the dead neurons will be automatically removed during retraining.\n# 4 Experiments\nWe implemented network pruning in Caffe [26]. Caffe was modiï¬ed to add a mask which disregards pruned parameters during network operation for each weight tensor. The pruning threshold is chosen as a quality parameter multiplied by the standard deviation of a layerâs weights. We carried out the experiments on Nvidia TitanX and GTX980 GPUs.\nWe pruned four representative networks: Lenet-300-100 and Lenet-5 on MNIST, together with AlexNet and VGG-16 on ImageNet. The network parameters and accuracy 1 before and after pruning are shown in Table 1.\n# 4.1 LeNet on MNIST",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 13 |
# 4.1 LeNet on MNIST
We ï¬rst experimented on MNIST dataset with the LeNet-300-100 and LeNet-5 networks [4]. LeNet- 300-100 is a fully connected network with two hidden layers, with 300 and 100 neurons each, which achieves 1.6% error rate on MNIST. LeNet-5 is a convolutional network that has two convolutional layers and two fully connected layers, which achieves 0.8% error rate on MNIST. After pruning, the network is retrained with 1/10 of the original networkâs original learning rate. Table 1 shows
1Reference model is from Caffe model zoo, accuracy is measured without data augmentation
4
Table 2: For Lenet-300-100, pruning reduces the number of weights by 12Ã and computation by 12Ã.
Layer Weights fc1 fc2 fc3 Total 235K 30K 1K 266K FLOP Act% Weights% FLOP% 8% 470K 38% 65% 60K 9% 100% 26% 2K 8% 532K 46% 8% 4% 17% 8%
Table 3: For Lenet-5, pruning reduces the number of weights by 12Ã and computation by 6Ã.
|
1506.02626#13
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 13,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 4.1 LeNet on MNIST\nWe ï¬rst experimented on MNIST dataset with the LeNet-300-100 and LeNet-5 networks [4]. LeNet- 300-100 is a fully connected network with two hidden layers, with 300 and 100 neurons each, which achieves 1.6% error rate on MNIST. LeNet-5 is a convolutional network that has two convolutional layers and two fully connected layers, which achieves 0.8% error rate on MNIST. After pruning, the network is retrained with 1/10 of the original networkâs original learning rate. Table 1 shows\n1Reference model is from Caffe model zoo, accuracy is measured without data augmentation\n4\nTable 2: For Lenet-300-100, pruning reduces the number of weights by 12à and computation by 12Ã.\nLayer Weights fc1 fc2 fc3 Total 235K 30K 1K 266K FLOP Act% Weights% FLOP% 8% 470K 38% 65% 60K 9% 100% 26% 2K 8% 532K 46% 8% 4% 17% 8%\nTable 3: For Lenet-5, pruning reduces the number of weights by 12à and computation by 6Ã.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 15 |
Figure 4: Visualization of the ï¬rst FC layerâs sparsity pattern of Lenet-300-100. It has a banded structure repeated 28 times, which correspond to the un-pruned parameters in the center of the images, since the digits are written in the center.
pruning saves 12à parameters on these networks. For each layer of the network the table shows (left to right) the original number of weights, the number of ï¬oating point operations to compute that layerâs activations, the average percentage of activations that are non-zero, the percentage of non-zero weights after pruning, and the percentage of actually required ï¬oating point operations.
|
1506.02626#15
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 15,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Figure 4: Visualization of the ï¬rst FC layerâs sparsity pattern of Lenet-300-100. It has a banded structure repeated 28 times, which correspond to the un-pruned parameters in the center of the images, since the digits are written in the center.\npruning saves 12à parameters on these networks. For each layer of the network the table shows (left to right) the original number of weights, the number of ï¬oating point operations to compute that layerâs activations, the average percentage of activations that are non-zero, the percentage of non-zero weights after pruning, and the percentage of actually required ï¬oating point operations.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 16 |
An interesting byproduct is that network pruning detects visual attention regions. Figure 4 shows the sparsity pattern of the ï¬rst fully connected layer of LeNet-300-100, the matrix size is 784 â 300. It has 28 bands, each bandâs width 28, corresponding to the 28 à 28 input pixels. The colored regions of the ï¬gure, indicating non-zero parameters, correspond to the center of the image. Because digits are written in the center of the image, these are the important parameters. The graph is sparse on the left and right, corresponding to the less important regions on the top and bottom of the image. After pruning, the neural network ï¬nds the center of the image more important, and the connections to the peripheral regions are more heavily pruned.
# 4.2 AlexNet on ImageNet
|
1506.02626#16
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 16,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "An interesting byproduct is that network pruning detects visual attention regions. Figure 4 shows the sparsity pattern of the ï¬rst fully connected layer of LeNet-300-100, the matrix size is 784 â 300. It has 28 bands, each bandâs width 28, corresponding to the 28 à 28 input pixels. The colored regions of the ï¬gure, indicating non-zero parameters, correspond to the center of the image. Because digits are written in the center of the image, these are the important parameters. The graph is sparse on the left and right, corresponding to the less important regions on the top and bottom of the image. After pruning, the neural network ï¬nds the center of the image more important, and the connections to the peripheral regions are more heavily pruned.\n# 4.2 AlexNet on ImageNet",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 17 |
# 4.2 AlexNet on ImageNet
We further examine the performance of pruning on the ImageNet ILSVRC-2012 dataset, which has 1.2M training examples and 50k validation examples. We use the AlexNet Caffe model as the reference model, which has 61 million parameters across 5 convolutional layers and 3 fully connected layers. The AlexNet Caffe model achieved a top-1 accuracy of 57.2% and a top-5 accuracy of 80.3%. The original AlexNet took 75 hours to train on NVIDIA Titan X GPU. After pruning, the whole network is retrained with 1/100 of the original networkâs initial learning rate. It took 173 hours to retrain the pruned AlexNet. Pruning is not used when iteratively prototyping the model, but rather used for model reduction when the model is ready for deployment. Thus, the retraining time is less a concern. Table 1 shows that AlexNet can be pruned to 1/9 of its original size without impacting accuracy, and the amount of computation can be reduced by 3Ã.
5
Table 4: For AlexNet, pruning reduces the number of weights by 9Ã and computation by 3Ã.
|
1506.02626#17
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 17,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 4.2 AlexNet on ImageNet\nWe further examine the performance of pruning on the ImageNet ILSVRC-2012 dataset, which has 1.2M training examples and 50k validation examples. We use the AlexNet Caffe model as the reference model, which has 61 million parameters across 5 convolutional layers and 3 fully connected layers. The AlexNet Caffe model achieved a top-1 accuracy of 57.2% and a top-5 accuracy of 80.3%. The original AlexNet took 75 hours to train on NVIDIA Titan X GPU. After pruning, the whole network is retrained with 1/100 of the original networkâs initial learning rate. It took 173 hours to retrain the pruned AlexNet. Pruning is not used when iteratively prototyping the model, but rather used for model reduction when the model is ready for deployment. Thus, the retraining time is less a concern. Table 1 shows that AlexNet can be pruned to 1/9 of its original size without impacting accuracy, and the amount of computation can be reduced by 3Ã.\n5\nTable 4: For AlexNet, pruning reduces the number of weights by 9Ã and computation by 3Ã.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 18 |
5
Table 4: For AlexNet, pruning reduces the number of weights by 9Ã and computation by 3Ã.
Layer Weights conv1 conv2 conv3 conv4 conv5 fc1 fc2 fc3 Total 35K 307K 885K 663K 442K 38M 17M 4M 61M FLOP Act% Weights% FLOP% 84% 211M 88% 38% 448M 52% 35% 299M 37% 37% 224M 40% 37% 150M 34% 9% 75M 36% 9% 34M 40% 25% 100% 8M 11% 54% 1.5B 84% 33% 18% 14% 14% 3% 3% 10% 30%
Table 5: For VGG-16, pruning reduces the number of weights by 12Ã and computation by 5Ã.
58% 12% 30% 29% 43% 16% 29% 21% 14% 15% 12% 9% 11% 1% 2% 9% 21% 100% 23% 7.5%
# 4.3 VGG-16 on ImageNet
|
1506.02626#18
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 18,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "5\nTable 4: For AlexNet, pruning reduces the number of weights by 9Ã and computation by 3Ã.\nLayer Weights conv1 conv2 conv3 conv4 conv5 fc1 fc2 fc3 Total 35K 307K 885K 663K 442K 38M 17M 4M 61M FLOP Act% Weights% FLOP% 84% 211M 88% 38% 448M 52% 35% 299M 37% 37% 224M 40% 37% 150M 34% 9% 75M 36% 9% 34M 40% 25% 100% 8M 11% 54% 1.5B 84% 33% 18% 14% 14% 3% 3% 10% 30%\nTable 5: For VGG-16, pruning reduces the number of weights by 12Ã and computation by 5Ã.\n58% 12% 30% 29% 43% 16% 29% 21% 14% 15% 12% 9% 11% 1% 2% 9% 21% 100% 23% 7.5%\n# 4.3 VGG-16 on ImageNet",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 19 |
# 4.3 VGG-16 on ImageNet
With promising results on AlexNet, we also looked at a larger, more recent network, VGG-16 [27], on the same ILSVRC-2012 dataset. VGG-16 has far more convolutional layers but still only three fully-connected layers. Following a similar methodology, we aggressively pruned both convolutional and fully-connected layers to realize a signiï¬cant reduction in the number of weights, shown in Table 5. We used ï¬ve iterations of pruning an retraining.
The VGG-16 results are, like those for AlexNet, very promising. The network as a whole has been reduced to 7.5% of its original size (13Ã smaller). In particular, note that the two largest fully-connected layers can each be pruned to less than 4% of their original size. This reduction is critical for real time image processing, where there is little reuse of fully connected layers across images (unlike batch processing during training).
# 5 Discussion
|
1506.02626#19
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 19,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 4.3 VGG-16 on ImageNet\nWith promising results on AlexNet, we also looked at a larger, more recent network, VGG-16 [27], on the same ILSVRC-2012 dataset. VGG-16 has far more convolutional layers but still only three fully-connected layers. Following a similar methodology, we aggressively pruned both convolutional and fully-connected layers to realize a signiï¬cant reduction in the number of weights, shown in Table 5. We used ï¬ve iterations of pruning an retraining.\nThe VGG-16 results are, like those for AlexNet, very promising. The network as a whole has been reduced to 7.5% of its original size (13à smaller). In particular, note that the two largest fully-connected layers can each be pruned to less than 4% of their original size. This reduction is critical for real time image processing, where there is little reuse of fully connected layers across images (unlike batch processing during training).\n# 5 Discussion",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 20 |
# 5 Discussion
The trade-off curve between accuracy and number of parameters is shown in Figure 5. The more parameters pruned away, the less the accuracy. We experimented with L1 and L2 regularization, with and without retraining, together with iterative pruning to give ï¬ve trade off lines. Comparing solid and dashed lines, the importance of retraining is clear: without retraining, accuracy begins dropping much sooner â with 1/3 of the original connections, rather than with 1/10 of the original connections. Itâs interesting to see that we have the âfree lunchâ of reducing 2à the connections without losing accuracy even without retraining; while with retraining we are ably to reduce connections by 9Ã.
6
-O-L2 regularization w/o retrain -A-L1 regularization w/o retrain -&L1 regularization w/ retrain ~OL2 regularization w/ retrain -®L2 regularization w/ iterative prune and retrain 0.5% 0.0% -0.5% 1.0% â1.5% -2.0% -2.5% -3.0% -3.5% -4.0% -4.5% 40% 50% 60% 70% 80% 90% 100% Parametes Pruned Away Accuracy Loss
|
1506.02626#20
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 20,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 5 Discussion\nThe trade-off curve between accuracy and number of parameters is shown in Figure 5. The more parameters pruned away, the less the accuracy. We experimented with L1 and L2 regularization, with and without retraining, together with iterative pruning to give ï¬ve trade off lines. Comparing solid and dashed lines, the importance of retraining is clear: without retraining, accuracy begins dropping much sooner â with 1/3 of the original connections, rather than with 1/10 of the original connections. Itâs interesting to see that we have the âfree lunchâ of reducing 2à the connections without losing accuracy even without retraining; while with retraining we are ably to reduce connections by 9Ã.\n6\n-O-L2 regularization w/o retrain -A-L1 regularization w/o retrain -&L1 regularization w/ retrain ~OL2 regularization w/ retrain -®L2 regularization w/ iterative prune and retrain 0.5% 0.0% -0.5% 1.0% â1.5% -2.0% -2.5% -3.0% -3.5% -4.0% -4.5% 40% 50% 60% 70% 80% 90% 100% Parametes Pruned Away Accuracy Loss",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 21 |
Figure 5: Trade-off curve for parameter reduction and loss in top-5 accuracy. L1 regularization performs better than L2 at learning the connections without retraining, while L2 regularization performs better than L1 at retraining. Iterative pruning gives the best result.
âconv! ~conv2 fconvs cond -*-convS fet 0% XE SX Bax x18: 0% 0040-0 6 5% § B-10% 3-10% g â15% â18% -20% 25% 50% 75% 400% 0% 25% 50% 75% 100% #Parameters #Parameters
âconv! ~conv2 fconvs cond -*-convS 0% XE SX Bax x18: 6 E | B-10% Es â15% 25% 50% 75% 400% #Parameters
fet 0% 0040-0 5% § 3-10% g â18% -20% 0% 25% 50% 75% 100% #Parameters
Figure 6: Pruning sensitivity for CONV layer (left) and FC layer (right) of AlexNet.
|
1506.02626#21
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 21,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Figure 5: Trade-off curve for parameter reduction and loss in top-5 accuracy. L1 regularization performs better than L2 at learning the connections without retraining, while L2 regularization performs better than L1 at retraining. Iterative pruning gives the best result.\nâconv! ~conv2 fconvs cond -*-convS fet 0% XE SX Bax x18: 0% 0040-0 6 5% § B-10% 3-10% g â15% â18% -20% 25% 50% 75% 400% 0% 25% 50% 75% 100% #Parameters #Parameters\nâconv! ~conv2 fconvs cond -*-convS 0% XE SX Bax x18: 6 E | B-10% Es â15% 25% 50% 75% 400% #Parameters\nfet 0% 0040-0 5% § 3-10% g â18% -20% 0% 25% 50% 75% 100% #Parameters\nFigure 6: Pruning sensitivity for CONV layer (left) and FC layer (right) of AlexNet.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 22 |
Figure 6: Pruning sensitivity for CONV layer (left) and FC layer (right) of AlexNet.
L1 regularization gives better accuracy than L2 directly after pruning (dotted blue and purple lines) since it pushes more parameters closer to zero. However, comparing the yellow and green lines shows that L2 outperforms L1 after retraining, since there is no beneï¬t to further pushing values towards zero. One extension is to use L1 regularization for pruning and then L2 for retraining, but this did not beat simply using L2 for both phases. Parameters from one mode do not adapt well to the other.
The biggest gain comes from iterative pruning (solid red line with solid circles). Here we take the pruned and retrained network (solid green line with circles) and prune and retrain it again. The leftmost dot on this curve corresponds to the point on the green line at 80% (5Ã pruning) pruned to 8Ã. Thereâs no accuracy loss at 9Ã. Not until 10Ã does the accuracy begin to drop sharply.
Two green points achieve slightly better accuracy than the original model. We believe this accuracy improvement is due to pruning ï¬nding the right capacity of the network and hence reducing overï¬tting.
|
1506.02626#22
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 22,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Figure 6: Pruning sensitivity for CONV layer (left) and FC layer (right) of AlexNet.\nL1 regularization gives better accuracy than L2 directly after pruning (dotted blue and purple lines) since it pushes more parameters closer to zero. However, comparing the yellow and green lines shows that L2 outperforms L1 after retraining, since there is no beneï¬t to further pushing values towards zero. One extension is to use L1 regularization for pruning and then L2 for retraining, but this did not beat simply using L2 for both phases. Parameters from one mode do not adapt well to the other.\nThe biggest gain comes from iterative pruning (solid red line with solid circles). Here we take the pruned and retrained network (solid green line with circles) and prune and retrain it again. The leftmost dot on this curve corresponds to the point on the green line at 80% (5à pruning) pruned to 8Ã. Thereâs no accuracy loss at 9Ã. Not until 10à does the accuracy begin to drop sharply.\nTwo green points achieve slightly better accuracy than the original model. We believe this accuracy improvement is due to pruning ï¬nding the right capacity of the network and hence reducing overï¬tting.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 23 |
Both CONV and FC layers can be pruned, but with different sensitivity. Figure 6 shows the sensitivity of each layer to network pruning. The ï¬gure shows how accuracy drops as parameters are pruned on a layer-by-layer basis. The CONV layers (on the left) are more sensitive to pruning than the fully connected layers (on the right). The ï¬rst convolutional layer, which interacts with the input image directly, is most sensitive to pruning. We suspect this sensitivity is due to the input layer having only 3 channels and thus less redundancy than the other convolutional layers. We used the sensitivity results to ï¬nd each layerâs threshold: for example, the smallest threshold was applied to the most sensitive layer, which is the ï¬rst convolutional layer.
Storing the pruned layers as sparse matrices has a storage overhead of only 15.6%. Storing relative rather than absolute indices reduces the space taken by the FC layer indices to 5 bits. Similarly, CONV layer indices can be represented with only 8 bits.
7
|
1506.02626#23
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 23,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Both CONV and FC layers can be pruned, but with different sensitivity. Figure 6 shows the sensitivity of each layer to network pruning. The ï¬gure shows how accuracy drops as parameters are pruned on a layer-by-layer basis. The CONV layers (on the left) are more sensitive to pruning than the fully connected layers (on the right). The ï¬rst convolutional layer, which interacts with the input image directly, is most sensitive to pruning. We suspect this sensitivity is due to the input layer having only 3 channels and thus less redundancy than the other convolutional layers. We used the sensitivity results to ï¬nd each layerâs threshold: for example, the smallest threshold was applied to the most sensitive layer, which is the ï¬rst convolutional layer.\nStoring the pruned layers as sparse matrices has a storage overhead of only 15.6%. Storing relative rather than absolute indices reduces the space taken by the FC layer indices to 5 bits. Similarly, CONV layer indices can be represented with only 8 bits.\n7",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 24 |
7
Table 6: Comparison with other model reduction methods on AlexNet. Data-free pruning [28] saved only 1.5Ã parameters with much loss of accuracy. Deep Fried Convnets [29] worked on fully connected layers only and reduced the parameters by less than 4Ã. [30] reduced the parameters by 4Ã with inferior accuracy. Naively cutting the layer size saves parameters but suffers from 4% loss of accuracy. [12] exploited the linear structure of convnets and compressed each layer individually, where model compression on a single layer incurred 0.9% accuracy penalty with biclustering + SVD.
|
1506.02626#24
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 24,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "7\nTable 6: Comparison with other model reduction methods on AlexNet. Data-free pruning [28] saved only 1.5Ã parameters with much loss of accuracy. Deep Fried Convnets [29] worked on fully connected layers only and reduced the parameters by less than 4Ã. [30] reduced the parameters by 4Ã with inferior accuracy. Naively cutting the layer size saves parameters but suffers from 4% loss of accuracy. [12] exploited the linear structure of convnets and compressed each layer individually, where model compression on a single layer incurred 0.9% accuracy penalty with biclustering + SVD.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 25 |
Network Baseline Caffemodel [26] Data-free pruning [28] Fastfood-32-AD [29] Fastfood-16-AD [29] Collins & Kohli [30] Naive Cut SVD [12] Network Pruning Top-1 Error Top-5 Error 42.78% 44.40% 41.93% 42.90% 44.40% 47.18% 44.02% 42.77% 19.73% - - - - 23.23% 20.56% 19.67% Parameters 61.0M 39.6M 32.8M 16.4M 15.2M 13.8M 11.9M 6.7M Compression Rate 1Ã 1.5Ã 2Ã 3.7Ã 4Ã 4.4Ã 5Ã 9Ã
# Count
Figure 7: Weight distribution before and after parameter pruning. The right ï¬gure has 10à smaller scale.
|
1506.02626#25
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 25,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "Network Baseline Caffemodel [26] Data-free pruning [28] Fastfood-32-AD [29] Fastfood-16-AD [29] Collins & Kohli [30] Naive Cut SVD [12] Network Pruning Top-1 Error Top-5 Error 42.78% 44.40% 41.93% 42.90% 44.40% 47.18% 44.02% 42.77% 19.73% - - - - 23.23% 20.56% 19.67% Parameters 61.0M 39.6M 32.8M 16.4M 15.2M 13.8M 11.9M 6.7M Compression Rate 1à 1.5à 2à 3.7à 4à 4.4à 5à 9à \n# Count\nFigure 7: Weight distribution before and after parameter pruning. The right ï¬gure has 10à smaller scale.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 26 |
# Count
Figure 7: Weight distribution before and after parameter pruning. The right ï¬gure has 10à smaller scale.
After pruning, the storage requirements of AlexNet and VGGNet are are small enough that all weights can be stored on chip, instead of off-chip DRAM which takes orders of magnitude more energy to access (Table 1). We are targeting our pruning method for ï¬xed-function hardware specialized for sparse DNN, given the limitation of general purpose hardware on sparse computation. Figure 7 shows histograms of weight distribution before (left) and after (right) pruning. The weight is from the ï¬rst fully connected layer of AlexNet. The two panels have different y-axis scales. The original distribution of weights is centered on zero with tails dropping off quickly. Almost all parameters are between [â0.015, 0.015]. After pruning the large center region is removed. The network parameters adjust themselves during the retraining phase. The result is that the parameters form a bimodal distribution and become more spread across the x-axis, between [â0.025, 0.025].
# 6 Conclusion
|
1506.02626#26
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 26,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# Count\nFigure 7: Weight distribution before and after parameter pruning. The right ï¬gure has 10à smaller scale.\nAfter pruning, the storage requirements of AlexNet and VGGNet are are small enough that all weights can be stored on chip, instead of off-chip DRAM which takes orders of magnitude more energy to access (Table 1). We are targeting our pruning method for ï¬xed-function hardware specialized for sparse DNN, given the limitation of general purpose hardware on sparse computation. Figure 7 shows histograms of weight distribution before (left) and after (right) pruning. The weight is from the ï¬rst fully connected layer of AlexNet. The two panels have different y-axis scales. The original distribution of weights is centered on zero with tails dropping off quickly. Almost all parameters are between [â0.015, 0.015]. After pruning the large center region is removed. The network parameters adjust themselves during the retraining phase. The result is that the parameters form a bimodal distribution and become more spread across the x-axis, between [â0.025, 0.025].\n# 6 Conclusion",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 27 |
# 6 Conclusion
We have presented a method to improve the energy efï¬ciency and storage of neural networks without affecting accuracy by ï¬nding the right connections. Our method, motivated in part by how learning works in the mammalian brain, operates by learning which connections are important, pruning the unimportant connections, and then retraining the remaining sparse network. We highlight our experiments on AlexNet and VGGNet on ImageNet, showing that both fully connected layer and convolutional layer can be pruned, reducing the number of connections by 9à to 13à without loss of accuracy. This leads to smaller memory capacity and bandwidth requirements for real-time image processing, making it easier to be deployed on mobile systems.
# References
[1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classiï¬cation with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097â1105, 2012.
8
[2] Alex Graves and J¨urgen Schmidhuber. Framewise phoneme classiï¬cation with bidirectional lstm and other neural network architectures. Neural Networks, 18(5):602â610, 2005.
|
1506.02626#27
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 27,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "# 6 Conclusion\nWe have presented a method to improve the energy efï¬ciency and storage of neural networks without affecting accuracy by ï¬nding the right connections. Our method, motivated in part by how learning works in the mammalian brain, operates by learning which connections are important, pruning the unimportant connections, and then retraining the remaining sparse network. We highlight our experiments on AlexNet and VGGNet on ImageNet, showing that both fully connected layer and convolutional layer can be pruned, reducing the number of connections by 9à to 13à without loss of accuracy. This leads to smaller memory capacity and bandwidth requirements for real-time image processing, making it easier to be deployed on mobile systems.\n# References\n[1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classiï¬cation with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097â1105, 2012.\n8\n[2] Alex Graves and J¨urgen Schmidhuber. Framewise phoneme classiï¬cation with bidirectional lstm and other neural network architectures. Neural Networks, 18(5):602â610, 2005.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 28 |
[3] Ronan Collobert, Jason Weston, L´eon Bottou, Michael Karlen, Koray Kavukcuoglu, and Pavel Kuksa. Natural language processing (almost) from scratch. JMLR, 12:2493â2537, 2011.
[4] Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278â2324, 1998.
[5] Yaniv Taigman, Ming Yang, MarcâAurelio Ranzato, and Lior Wolf. Deepface: Closing the gap to human-level performance in face veriï¬cation. In CVPR, pages 1701â1708. IEEE, 2014.
[6] Adam Coates, Brody Huval, Tao Wang, David Wu, Bryan Catanzaro, and Ng Andrew. Deep learning with cots hpc systems. In 30th ICML, pages 1337â1345, 2013.
[7] Mark Horowitz. Energy table for 45nm process, Stanford VLSI wiki. [8] JP Rauschecker. Neuronal mechanisms of developmental plasticity in the catâs visual system. Human
neurobiology, 3(2):109â114, 1983.
|
1506.02626#28
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 28,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "[3] Ronan Collobert, Jason Weston, L´eon Bottou, Michael Karlen, Koray Kavukcuoglu, and Pavel Kuksa. Natural language processing (almost) from scratch. JMLR, 12:2493â2537, 2011.\n[4] Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278â2324, 1998.\n[5] Yaniv Taigman, Ming Yang, MarcâAurelio Ranzato, and Lior Wolf. Deepface: Closing the gap to human-level performance in face veriï¬cation. In CVPR, pages 1701â1708. IEEE, 2014.\n[6] Adam Coates, Brody Huval, Tao Wang, David Wu, Bryan Catanzaro, and Ng Andrew. Deep learning with cots hpc systems. In 30th ICML, pages 1337â1345, 2013.\n[7] Mark Horowitz. Energy table for 45nm process, Stanford VLSI wiki. [8] JP Rauschecker. Neuronal mechanisms of developmental plasticity in the catâs visual system. Human\nneurobiology, 3(2):109â114, 1983.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 29 |
neurobiology, 3(2):109â114, 1983.
[9] Christopher A Walsh. Peter huttenlocher (1931-2013). Nature, 502(7470):172â172, 2013. [10] Misha Denil, Babak Shakibi, Laurent Dinh, Nando de Freitas, et al. Predicting parameters in deep learning.
In Advances in Neural Information Processing Systems, pages 2148â2156, 2013.
[11] Vincent Vanhoucke, Andrew Senior, and Mark Z Mao. Improving the speed of neural networks on cpus. In Proc. Deep Learning and Unsupervised Feature Learning NIPS Workshop, 2011.
[12] Emily L Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efï¬cient evaluation. In NIPS, pages 1269â1277, 2014.
[13] Yunchao Gong, Liu Liu, Ming Yang, and Lubomir Bourdev. Compressing deep convolutional networks using vector quantization. arXiv preprint arXiv:1412.6115, 2014.
|
1506.02626#29
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 29,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "neurobiology, 3(2):109â114, 1983.\n[9] Christopher A Walsh. Peter huttenlocher (1931-2013). Nature, 502(7470):172â172, 2013. [10] Misha Denil, Babak Shakibi, Laurent Dinh, Nando de Freitas, et al. Predicting parameters in deep learning.\nIn Advances in Neural Information Processing Systems, pages 2148â2156, 2013.\n[11] Vincent Vanhoucke, Andrew Senior, and Mark Z Mao. Improving the speed of neural networks on cpus. In Proc. Deep Learning and Unsupervised Feature Learning NIPS Workshop, 2011.\n[12] Emily L Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efï¬cient evaluation. In NIPS, pages 1269â1277, 2014.\n[13] Yunchao Gong, Liu Liu, Ming Yang, and Lubomir Bourdev. Compressing deep convolutional networks using vector quantization. arXiv preprint arXiv:1412.6115, 2014.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 30 |
[14] Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural network with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.
[15] Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. arXiv preprint arXiv:1312.4400, 2013. [16] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. arXiv preprint arXiv:1409.4842, 2014.
[17] Stephen Jos´e Hanson and Lorien Y Pratt. Comparing biases for minimal network construction with back-propagation. In Advances in neural information processing systems, pages 177â185, 1989.
[18] Yann Le Cun, John S. Denker, and Sara A. Solla. Optimal brain damage. In Advances in Neural Information Processing Systems, pages 598â605. Morgan Kaufmann, 1990.
|
1506.02626#30
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 30,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "[14] Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural network with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149, 2015.\n[15] Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. arXiv preprint arXiv:1312.4400, 2013. [16] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. arXiv preprint arXiv:1409.4842, 2014.\n[17] Stephen Jos´e Hanson and Lorien Y Pratt. Comparing biases for minimal network construction with back-propagation. In Advances in neural information processing systems, pages 177â185, 1989.\n[18] Yann Le Cun, John S. Denker, and Sara A. Solla. Optimal brain damage. In Advances in Neural Information Processing Systems, pages 598â605. Morgan Kaufmann, 1990.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 31 |
[19] Babak Hassibi, David G Stork, et al. Second order derivatives for network pruning: Optimal brain surgeon. Advances in neural information processing systems, pages 164â164, 1993.
[20] Wenlin Chen, James T. Wilson, Stephen Tyree, Kilian Q. Weinberger, and Yixin Chen. Compressing neural networks with the hashing trick. arXiv preprint arXiv:1504.04788, 2015.
[21] Qinfeng Shi, James Petterson, Gideon Dror, John Langford, Alex Smola, and SVN Vishwanathan. Hash kernels for structured data. The Journal of Machine Learning Research, 10:2615â2637, 2009.
[22] Kilian Weinberger, Anirban Dasgupta, John Langford, Alex Smola, and Josh Attenberg. Feature hashing for large scale multitask learning. In ICML, pages 1113â1120. ACM, 2009.
[23] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overï¬tting. JMLR, 15:1929â1958, 2014.
|
1506.02626#31
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 31,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "[19] Babak Hassibi, David G Stork, et al. Second order derivatives for network pruning: Optimal brain surgeon. Advances in neural information processing systems, pages 164â164, 1993.\n[20] Wenlin Chen, James T. Wilson, Stephen Tyree, Kilian Q. Weinberger, and Yixin Chen. Compressing neural networks with the hashing trick. arXiv preprint arXiv:1504.04788, 2015.\n[21] Qinfeng Shi, James Petterson, Gideon Dror, John Langford, Alex Smola, and SVN Vishwanathan. Hash kernels for structured data. The Journal of Machine Learning Research, 10:2615â2637, 2009.\n[22] Kilian Weinberger, Anirban Dasgupta, John Langford, Alex Smola, and Josh Attenberg. Feature hashing for large scale multitask learning. In ICML, pages 1113â1120. ACM, 2009.\n[23] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overï¬tting. JMLR, 15:1929â1958, 2014.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.02626
| 32 |
[24] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in Neural Information Processing Systems, pages 3320â3328, 2014.
[25] Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difï¬cult. Neural Networks, IEEE Transactions on, 5(2):157â166, 1994.
[26] Yangqing Jia, et al. Caffe: Convolutional architecture for fast feature embedding. arXiv preprint arXiv:1408.5093, 2014.
[27] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recogni- tion. CoRR, abs/1409.1556, 2014.
[28] Suraj Srinivas and R Venkatesh Babu. Data-free parameter pruning for deep neural networks. arXiv preprint arXiv:1507.06149, 2015.
[29] Zichao Yang, Marcin Moczulski, Misha Denil, Nando de Freitas, Alex Smola, Le Song, and Ziyu Wang. Deep fried convnets. arXiv preprint arXiv:1412.7149, 2014.
|
1506.02626#32
|
Learning both Weights and Connections for Efficient Neural Networks
|
Neural networks are both computationally intensive and memory intensive,
making them difficult to deploy on embedded systems. Also, conventional
networks fix the architecture before training starts; as a result, training
cannot improve the architecture. To address these limitations, we describe a
method to reduce the storage and computation required by neural networks by an
order of magnitude without affecting their accuracy by learning only the
important connections. Our method prunes redundant connections using a
three-step method. First, we train the network to learn which connections are
important. Next, we prune the unimportant connections. Finally, we retrain the
network to fine tune the weights of the remaining connections. On the ImageNet
dataset, our method reduced the number of parameters of AlexNet by a factor of
9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar
experiments with VGG-16 found that the number of parameters can be reduced by
13x, from 138 million to 10.3 million, again with no loss of accuracy.
|
http://arxiv.org/pdf/1506.02626
|
Song Han, Jeff Pool, John Tran, William J. Dally
|
cs.NE, cs.CV, cs.LG
|
Published as a conference paper at NIPS 2015
| null |
cs.NE
|
20150608
|
20151030
|
[
{
"id": "1507.06149"
},
{
"id": "1504.04788"
},
{
"id": "1510.00149"
}
] |
{
"authors": "Song Han, Jeff Pool, John Tran, William J. Dally",
"chunk_id": 32,
"doc_id": "1506.02626",
"primary_category": "cs.NE",
"published": 20150608,
"source": "http://arxiv.org/pdf/1506.02626",
"summary": "Neural networks are both computationally intensive and memory intensive,\nmaking them difficult to deploy on embedded systems. Also, conventional\nnetworks fix the architecture before training starts; as a result, training\ncannot improve the architecture. To address these limitations, we describe a\nmethod to reduce the storage and computation required by neural networks by an\norder of magnitude without affecting their accuracy by learning only the\nimportant connections. Our method prunes redundant connections using a\nthree-step method. First, we train the network to learn which connections are\nimportant. Next, we prune the unimportant connections. Finally, we retrain the\nnetwork to fine tune the weights of the remaining connections. On the ImageNet\ndataset, our method reduced the number of parameters of AlexNet by a factor of\n9x, from 61 million to 6.7 million, without incurring accuracy loss. Similar\nexperiments with VGG-16 found that the number of parameters can be reduced by\n13x, from 138 million to 10.3 million, again with no loss of accuracy.",
"text": "[24] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in Neural Information Processing Systems, pages 3320â3328, 2014.\n[25] Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difï¬cult. Neural Networks, IEEE Transactions on, 5(2):157â166, 1994.\n[26] Yangqing Jia, et al. Caffe: Convolutional architecture for fast feature embedding. arXiv preprint arXiv:1408.5093, 2014.\n[27] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recogni- tion. CoRR, abs/1409.1556, 2014.\n[28] Suraj Srinivas and R Venkatesh Babu. Data-free parameter pruning for deep neural networks. arXiv preprint arXiv:1507.06149, 2015.\n[29] Zichao Yang, Marcin Moczulski, Misha Denil, Nando de Freitas, Alex Smola, Le Song, and Ziyu Wang. Deep fried convnets. arXiv preprint arXiv:1412.7149, 2014.",
"title": "Learning both Weights and Connections for Efficient Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 1 |
# Abstract
It is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically elim- inates the need to experimentally ï¬nd the best values and schedule for the global learning rates. Instead of mono- tonically decreasing the learning rate, this method lets the learning rate cyclically vary between reasonable bound- ary values. Training with cyclical learning rates instead of ï¬xed values achieves improved classiï¬cation accuracy without a need to tune and often in fewer iterations. This paper also describes a simple way to estimate âreasonable boundsâ â linearly increasing the learning rate of the net- work for a few epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10 and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets, and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These are practical tools for everyone who trains neural networks.
âXponential âCLR (our approach) 0 1 2 3 4 5 6 7 Iteration x 10°
Figure 1. Classiï¬cation accuracy while training CIFAR-10. The red curve shows the result of training with one of the new learning rate policies.
|
1506.01186#1
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 1,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "# Abstract\nIt is known that the learning rate is the most important hyper-parameter to tune for training deep neural networks. This paper describes a new method for setting the learning rate, named cyclical learning rates, which practically elim- inates the need to experimentally ï¬nd the best values and schedule for the global learning rates. Instead of mono- tonically decreasing the learning rate, this method lets the learning rate cyclically vary between reasonable bound- ary values. Training with cyclical learning rates instead of ï¬xed values achieves improved classiï¬cation accuracy without a need to tune and often in fewer iterations. This paper also describes a simple way to estimate âreasonable boundsâ â linearly increasing the learning rate of the net- work for a few epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10 and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets, and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These are practical tools for everyone who trains neural networks.\nâXponential âCLR (our approach) 0 1 2 3 4 5 6 7 Iteration x 10°\nFigure 1. Classiï¬cation accuracy while training CIFAR-10. The red curve shows the result of training with one of the new learning rate policies.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 2 |
Figure 1. Classiï¬cation accuracy while training CIFAR-10. The red curve shows the result of training with one of the new learning rate policies.
ing training. This paper demonstrates the surprising phe- nomenon that a varying learning rate during training is ben- eï¬cial overall and thus proposes to let the global learning rate vary cyclically within a band of values instead of set- ting it to a ï¬xed value. In addition, this cyclical learning rate (CLR) method practically eliminates the need to tune the learning rate yet achieve near optimal classiï¬cation accu- racy. Furthermore, unlike adaptive learning rates, the CLR methods require essentially no additional computation.
# 1. Introduction
Deep neural networks are the basis of state-of-the-art re- sults for image recognition [17, 23, 25], object detection [7], face recognition [26], speech recognition [8], machine translation [24], image caption generation [28], and driver- less car technology [14]. However, training a deep neural network is a difï¬cult global optimization problem.
|
1506.01186#2
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 2,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 1. Classiï¬cation accuracy while training CIFAR-10. The red curve shows the result of training with one of the new learning rate policies.\ning training. This paper demonstrates the surprising phe- nomenon that a varying learning rate during training is ben- eï¬cial overall and thus proposes to let the global learning rate vary cyclically within a band of values instead of set- ting it to a ï¬xed value. In addition, this cyclical learning rate (CLR) method practically eliminates the need to tune the learning rate yet achieve near optimal classiï¬cation accu- racy. Furthermore, unlike adaptive learning rates, the CLR methods require essentially no additional computation.\n# 1. Introduction\nDeep neural networks are the basis of state-of-the-art re- sults for image recognition [17, 23, 25], object detection [7], face recognition [26], speech recognition [8], machine translation [24], image caption generation [28], and driver- less car technology [14]. However, training a deep neural network is a difï¬cult global optimization problem.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 3 |
The potential beneï¬ts of CLR can be seen in Figure 1, which shows the test data classiï¬cation accuracy of the CIFAR-10 dataset during training1. The baseline (blue curve) reaches a ï¬nal accuracy of 81.4% after 70, 000 it- erations. In contrast, it is possible to fully train the network using the CLR method instead of tuning (red curve) within 25,000 iterations and attain the same accuracy.
The contributions of this paper are:
A deep neural network is typically updated by stochastic gradient descent and the parameters 0 (weights) are updated by 0 = ott â 5, where L is a loss function and â¬; is the learning rate. It is well known that too small a learning rate will make a training algorithm converge slowly while too large a learning rate will make the training algorithm diverge [2]. Hence, one must experiment with a variety of learning rates and schedules.
1. A methodology for setting the global learning rates for training neural networks that eliminates the need to perform numerous experiments to ï¬nd the best values and schedule with essentially no additional computa- tion.
2. A surprising phenomenon is demonstrated - allowing
the learning rate should be a single value that monotonically decreases dur1Hyper-parameters and architecture were obtained in April 2015 from caffe.berkeleyvision.org/gathered/examples/cifar10.html
|
1506.01186#3
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 3,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "The potential beneï¬ts of CLR can be seen in Figure 1, which shows the test data classiï¬cation accuracy of the CIFAR-10 dataset during training1. The baseline (blue curve) reaches a ï¬nal accuracy of 81.4% after 70, 000 it- erations. In contrast, it is possible to fully train the network using the CLR method instead of tuning (red curve) within 25,000 iterations and attain the same accuracy.\nThe contributions of this paper are:\nA deep neural network is typically updated by stochastic gradient descent and the parameters 0 (weights) are updated by 0 = ott â 5, where L is a loss function and â¬; is the learning rate. It is well known that too small a learning rate will make a training algorithm converge slowly while too large a learning rate will make the training algorithm diverge [2]. Hence, one must experiment with a variety of learning rates and schedules.\n1. A methodology for setting the global learning rates for training neural networks that eliminates the need to perform numerous experiments to ï¬nd the best values and schedule with essentially no additional computa- tion.\n2. A surprising phenomenon is demonstrated - allowing\nthe learning rate should be a single value that monotonically decreases dur1Hyper-parameters and architecture were obtained in April 2015 from caffe.berkeleyvision.org/gathered/examples/cifar10.html",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 4 |
the learning rate to rise and fall is beneï¬cial overall even though it might temporarily harm the networkâs performance.
3. Cyclical learning rates are demonstrated with ResNets, Stochastic Depth networks, and DenseNets on the CIFAR-10 and CIFAR-100 datasets, and on ImageNet with two well-known architectures: AlexNet [17] and GoogleNet [25].
# 2. Related work
The book âNeural Networks: Tricks of the Tradeâ is a terriï¬c source of practical advice. In particular, Yoshua Bengio [2] discusses reasonable ranges for learning rates and stresses the importance of tuning the learning rate. A technical report by Breuel [3] provides guidance on a vari- ety of hyper-parameters. There are also a numerous web- sites giving practical suggestions for setting the learning rates.
Adaptive learning rates: Adaptive learning rates can be considered a competitor to cyclical learning rates because one can rely on local adaptive learning rates in place of global learning rate experimentation but there is a signiï¬- cant computational cost in doing so. CLR does not possess this computational costs so it can be used freely.
|
1506.01186#4
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 4,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "the learning rate to rise and fall is beneï¬cial overall even though it might temporarily harm the networkâs performance.\n3. Cyclical learning rates are demonstrated with ResNets, Stochastic Depth networks, and DenseNets on the CIFAR-10 and CIFAR-100 datasets, and on ImageNet with two well-known architectures: AlexNet [17] and GoogleNet [25].\n# 2. Related work\nThe book âNeural Networks: Tricks of the Tradeâ is a terriï¬c source of practical advice. In particular, Yoshua Bengio [2] discusses reasonable ranges for learning rates and stresses the importance of tuning the learning rate. A technical report by Breuel [3] provides guidance on a vari- ety of hyper-parameters. There are also a numerous web- sites giving practical suggestions for setting the learning rates.\nAdaptive learning rates: Adaptive learning rates can be considered a competitor to cyclical learning rates because one can rely on local adaptive learning rates in place of global learning rate experimentation but there is a signiï¬- cant computational cost in doing so. CLR does not possess this computational costs so it can be used freely.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 5 |
A review of the early work on adaptive learning rates can be found in George and Powell [6]. Duchi, et al. [5] pro- posed AdaGrad, which is one of the early adaptive methods that estimates the learning rates from the gradients.
RMSProp is discussed in the slides by Geoffrey Hinton2 [27]. RMSProp is described there as âDivide the learning rate for a weight by a running average of the magnitudes of recent gradients for that weight.â RMSProp is a funda- mental adaptive learning rate method that others have built on.
Schaul et al. [22] discuss an adaptive learning rate based on a diagonal estimation of the Hessian of the gradients. One of the features of their method is that they allow their automatic method to decrease or increase the learning rate. However, their paper seems to limit the idea of increasing learning rate to non-stationary problems. On the other hand, this paper demonstrates that a schedule of increasing the learning rate is more universally valuable.
Zeiler [29] describes his AdaDelta method, which im- proves on AdaGrad based on two ideas: limiting the sum of squared gradients over all time to a limited window, and making the parameter update rule consistent with a units evaluation on the relationship between the update and the Hessian.
|
1506.01186#5
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 5,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "A review of the early work on adaptive learning rates can be found in George and Powell [6]. Duchi, et al. [5] pro- posed AdaGrad, which is one of the early adaptive methods that estimates the learning rates from the gradients.\nRMSProp is discussed in the slides by Geoffrey Hinton2 [27]. RMSProp is described there as âDivide the learning rate for a weight by a running average of the magnitudes of recent gradients for that weight.â RMSProp is a funda- mental adaptive learning rate method that others have built on.\nSchaul et al. [22] discuss an adaptive learning rate based on a diagonal estimation of the Hessian of the gradients. One of the features of their method is that they allow their automatic method to decrease or increase the learning rate. However, their paper seems to limit the idea of increasing learning rate to non-stationary problems. On the other hand, this paper demonstrates that a schedule of increasing the learning rate is more universally valuable.\nZeiler [29] describes his AdaDelta method, which im- proves on AdaGrad based on two ideas: limiting the sum of squared gradients over all time to a limited window, and making the parameter update rule consistent with a units evaluation on the relationship between the update and the Hessian.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 6 |
More recently, several papers have appeared on adaptive learning rates. Gulcehre and Bengio [9] propose an adaptive learning rate algorithm, called AdaSecant, that utilizes the
2www.cs.toronto.edu/ tijmen/csc321/slides/lecture slides lec6.pdf
root mean square statistics and variance of the gradients. Dauphin et al. [4] show that RMSProp provides a biased estimate and go on to describe another estimator, named ESGD, that is unbiased. Kingma and Lei-Ba [16] introduce Adam that is designed to combine the advantages from Ada- Grad and RMSProp. Bache, et al. [1] propose exploiting solutions to a multi-armed bandit problem for learning rate selection. A summary and tutorial of adaptive learning rates can be found in a recent paper by Ruder [20].
Adaptive learning rates are fundamentally different from CLR policies, and CLR can be combined with adaptive learning rates, as shown in Section 4.1. In addition, CLR policies are computationally simpler than adaptive learning rates. CLR is likely most similar to the SGDR method [18] that appeared recently.
# 3. Optimal Learning Rates
# 3.1. Cyclical Learning Rates
|
1506.01186#6
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 6,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "More recently, several papers have appeared on adaptive learning rates. Gulcehre and Bengio [9] propose an adaptive learning rate algorithm, called AdaSecant, that utilizes the\n2www.cs.toronto.edu/ tijmen/csc321/slides/lecture slides lec6.pdf\nroot mean square statistics and variance of the gradients. Dauphin et al. [4] show that RMSProp provides a biased estimate and go on to describe another estimator, named ESGD, that is unbiased. Kingma and Lei-Ba [16] introduce Adam that is designed to combine the advantages from Ada- Grad and RMSProp. Bache, et al. [1] propose exploiting solutions to a multi-armed bandit problem for learning rate selection. A summary and tutorial of adaptive learning rates can be found in a recent paper by Ruder [20].\nAdaptive learning rates are fundamentally different from CLR policies, and CLR can be combined with adaptive learning rates, as shown in Section 4.1. In addition, CLR policies are computationally simpler than adaptive learning rates. CLR is likely most similar to the SGDR method [18] that appeared recently.\n# 3. Optimal Learning Rates\n# 3.1. Cyclical Learning Rates",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 7 |
# 3. Optimal Learning Rates
# 3.1. Cyclical Learning Rates
The essence of this learning rate policy comes from the observation that increasing the learning rate might have a short term negative effect and yet achieve a longer term ben- eï¬cial effect. This observation leads to the idea of letting the learning rate vary within a range of values rather than adopt- ing a stepwise ï¬xed or exponentially decreasing value. That is, one sets minimum and maximum boundaries and the learning rate cyclically varies between these bounds. Ex- periments with numerous functional forms, such as a trian- gular window (linear), a Welch window (parabolic) and a Hann window (sinusoidal) all produced equivalent results This led to adopting a triangular window (linearly increas- ing then linearly decreasing), which is illustrated in Figure 2, because it is the simplest function that incorporates this idea. The rest of this paper refers to this as the triangular learning rate policy.
Maximum bound (max_Ir) Minimum bound - (base_Ir) stepsize
Figure 2. Triangular learning rate policy. The blue lines represent learning rate values changing between bounds. The input parame- ter stepsize is the number of iterations in half a cycle.
|
1506.01186#7
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 7,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "# 3. Optimal Learning Rates\n# 3.1. Cyclical Learning Rates\nThe essence of this learning rate policy comes from the observation that increasing the learning rate might have a short term negative effect and yet achieve a longer term ben- eï¬cial effect. This observation leads to the idea of letting the learning rate vary within a range of values rather than adopt- ing a stepwise ï¬xed or exponentially decreasing value. That is, one sets minimum and maximum boundaries and the learning rate cyclically varies between these bounds. Ex- periments with numerous functional forms, such as a trian- gular window (linear), a Welch window (parabolic) and a Hann window (sinusoidal) all produced equivalent results This led to adopting a triangular window (linearly increas- ing then linearly decreasing), which is illustrated in Figure 2, because it is the simplest function that incorporates this idea. The rest of this paper refers to this as the triangular learning rate policy.\nMaximum bound (max_Ir) Minimum bound - (base_Ir) stepsize\nFigure 2. Triangular learning rate policy. The blue lines represent learning rate values changing between bounds. The input parame- ter stepsize is the number of iterations in half a cycle.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 8 |
Figure 2. Triangular learning rate policy. The blue lines represent learning rate values changing between bounds. The input parame- ter stepsize is the number of iterations in half a cycle.
An intuitive understanding of why CLR methods work comes from considering the loss function topology. Dauphin et al. [4] argue that the difï¬culty in minimizing the loss arises from saddle points rather than poor local minima.
Saddle points have small gradients that slow the learning process. However, increasing the learning rate allows more rapid traversal of saddle point plateaus. A more practical reason as to why CLR works is that, by following the meth- ods in Section 3.3, it is likely the optimum learning rate will be between the bounds and near optimal learning rates will be used throughout training.
The red curve in Figure 1 shows the result of the triangular policy on CIFAR-10. The settings used to cre- ate the red curve were a minimum learning rate of 0.001 (as in the original parameter ï¬le) and a maximum of 0.006. Also, the cycle length (i.e., the number of iterations until the learning rate returns to the initial value) is set to 4, 000 iterations (i.e., stepsize = 2000) and Figure 1 shows that the accuracy peaks at the end of each cycle.
|
1506.01186#8
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 8,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 2. Triangular learning rate policy. The blue lines represent learning rate values changing between bounds. The input parame- ter stepsize is the number of iterations in half a cycle.\nAn intuitive understanding of why CLR methods work comes from considering the loss function topology. Dauphin et al. [4] argue that the difï¬culty in minimizing the loss arises from saddle points rather than poor local minima.\nSaddle points have small gradients that slow the learning process. However, increasing the learning rate allows more rapid traversal of saddle point plateaus. A more practical reason as to why CLR works is that, by following the meth- ods in Section 3.3, it is likely the optimum learning rate will be between the bounds and near optimal learning rates will be used throughout training.\nThe red curve in Figure 1 shows the result of the triangular policy on CIFAR-10. The settings used to cre- ate the red curve were a minimum learning rate of 0.001 (as in the original parameter ï¬le) and a maximum of 0.006. Also, the cycle length (i.e., the number of iterations until the learning rate returns to the initial value) is set to 4, 000 iterations (i.e., stepsize = 2000) and Figure 1 shows that the accuracy peaks at the end of each cycle.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 9 |
Implementation of the code for a new learning rate policy is straightforward. An example of the code added to Torch 7 in the experiments shown in Section 4.1.2 is the following few lines:
l o c a l c y c l e = math . f l o o r ( 1 + e p o c h C o u n t e r / ( 2 â s t e p s i z e ) ) l o c a l x = math . a b s ( e p o c h C o u n t e r / s t e p s i z e â 2â c y c l e + 1 ) l o c a l l r = o p t . LR + ( maxLR â o p t . LR ) (1âx ) ) â math . max ( 0 ,
where opt.LR is the speciï¬ed lower (i.e., base) learning rate, epochCounter is the number of epochs of training, and lr is the computed learning rate. This policy is named triangular and is as described above, with two new in- put parameters deï¬ned: stepsize (half the period or cycle length) and max lr (the maximum learning rate boundary). This code varies the learning rate linearly between the min- imum (base lr) and the maximum (max lr).
|
1506.01186#9
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 9,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Implementation of the code for a new learning rate policy is straightforward. An example of the code added to Torch 7 in the experiments shown in Section 4.1.2 is the following few lines:\nl o c a l c y c l e = math . f l o o r ( 1 + e p o c h C o u n t e r / ( 2 â s t e p s i z e ) ) l o c a l x = math . a b s ( e p o c h C o u n t e r / s t e p s i z e â 2â c y c l e + 1 ) l o c a l l r = o p t . LR + ( maxLR â o p t . LR ) (1âx ) ) â math . max ( 0 ,\nwhere opt.LR is the speciï¬ed lower (i.e., base) learning rate, epochCounter is the number of epochs of training, and lr is the computed learning rate. This policy is named triangular and is as described above, with two new in- put parameters deï¬ned: stepsize (half the period or cycle length) and max lr (the maximum learning rate boundary). This code varies the learning rate linearly between the min- imum (base lr) and the maximum (max lr).",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 10 |
In addition to the triangular policy, the following CLR policies are discussed in this paper:
1. triangular2; the same as the triangular policy ex- cept the learning rate difference is cut in half at the end of each cycle. This means the learning rate difference drops after each cycle.
2. exp range; the learning rate varies between the min- imum and maximum boundaries and each bound- factor of ary value declines by an exponential gammaiteration.
# 3.2. How can one estimate a good value for the cycle length?
The length of a cycle and the input parameter stepsize can be easily computed from the number of iterations in an epoch. An epoch is calculated by dividing the number of training images by the batchsize used. For example, CIFAR-10 has 50, 000 training images and the batchsize is 100 so an epoch = 50, 000/100 = 500 iterations. The ï¬nal
accuracy results are actually quite robust to cycle length but experiments show that it often is good to set stepsize equal to 2 â 10 times the number of iterations in an epoch. For example, setting stepsize = 8 â epoch with the CIFAR-10 training run (as shown in Figure 1) only gives slightly better results than setting stepsize = 2 â epoch.
|
1506.01186#10
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
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},
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},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 10,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "In addition to the triangular policy, the following CLR policies are discussed in this paper:\n1. triangular2; the same as the triangular policy ex- cept the learning rate difference is cut in half at the end of each cycle. This means the learning rate difference drops after each cycle.\n2. exp range; the learning rate varies between the min- imum and maximum boundaries and each bound- factor of ary value declines by an exponential gammaiteration.\n# 3.2. How can one estimate a good value for the cycle length?\nThe length of a cycle and the input parameter stepsize can be easily computed from the number of iterations in an epoch. An epoch is calculated by dividing the number of training images by the batchsize used. For example, CIFAR-10 has 50, 000 training images and the batchsize is 100 so an epoch = 50, 000/100 = 500 iterations. The ï¬nal\naccuracy results are actually quite robust to cycle length but experiments show that it often is good to set stepsize equal to 2 â 10 times the number of iterations in an epoch. For example, setting stepsize = 8 â epoch with the CIFAR-10 training run (as shown in Figure 1) only gives slightly better results than setting stepsize = 2 â epoch.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 11 |
Furthermore, there is a certain elegance to the rhythm of these cycles and it simpliï¬es the decision of when to drop learning rates and when to stop the current training run. Experiments show that replacing each step of a con- stant learning rate with at least 3 cycles trains the network weights most of the way and running for 4 or more cycles will achieve even better performance. Also, it is best to stop training at the end of a cycle, which is when the learning rate is at the minimum value and the accuracy peaks.
# 3.3. How can one estimate reasonable minimum and maximum boundary values?
There is a simple way to estimate reasonable minimum and maximum boundary values with one training run of the network for a few epochs. It is a âLR range testâ; run your model for several epochs while letting the learning rate in- crease linearly between low and high LR values. This test is enormously valuable whenever you are facing a new ar- chitecture or dataset.
CIFAR-10 0.6 Accuracy 0.1 0 0.005 0.01 Learning rate 0.015 0.02
Figure 3. Classiï¬cation accuracy as a function of increasing learn- ing rate for 8 epochs (LR range test).
|
1506.01186#11
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 11,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Furthermore, there is a certain elegance to the rhythm of these cycles and it simpliï¬es the decision of when to drop learning rates and when to stop the current training run. Experiments show that replacing each step of a con- stant learning rate with at least 3 cycles trains the network weights most of the way and running for 4 or more cycles will achieve even better performance. Also, it is best to stop training at the end of a cycle, which is when the learning rate is at the minimum value and the accuracy peaks.\n# 3.3. How can one estimate reasonable minimum and maximum boundary values?\nThere is a simple way to estimate reasonable minimum and maximum boundary values with one training run of the network for a few epochs. It is a âLR range testâ; run your model for several epochs while letting the learning rate in- crease linearly between low and high LR values. This test is enormously valuable whenever you are facing a new ar- chitecture or dataset.\nCIFAR-10 0.6 Accuracy 0.1 0 0.005 0.01 Learning rate 0.015 0.02\nFigure 3. Classiï¬cation accuracy as a function of increasing learn- ing rate for 8 epochs (LR range test).",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 12 |
Figure 3. Classiï¬cation accuracy as a function of increasing learn- ing rate for 8 epochs (LR range test).
The triangular learning rate policy provides a simple mechanism to do this. For example, in Caffe, set base lr to the minimum value and set max lr to the maximum value. Set both the stepsize and max iter to the same number of iterations. In this case, the learning rate will increase lin- early from the minimum value to the maximum value dur- ing this short run. Next, plot the accuracy versus learning rate. Note the learning rate value when the accuracy starts to increase and when the accuracy slows, becomes ragged, or
|
1506.01186#12
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 12,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 3. Classiï¬cation accuracy as a function of increasing learn- ing rate for 8 epochs (LR range test).\nThe triangular learning rate policy provides a simple mechanism to do this. For example, in Caffe, set base lr to the minimum value and set max lr to the maximum value. Set both the stepsize and max iter to the same number of iterations. In this case, the learning rate will increase lin- early from the minimum value to the maximum value dur- ing this short run. Next, plot the accuracy versus learning rate. Note the learning rate value when the accuracy starts to increase and when the accuracy slows, becomes ragged, or",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 13 |
Dataset CIFAR-10 CIFAR-10 CIFAR-10 CIFAR-10 CIFAR-10 AlexNet AlexNet AlexNet AlexNet AlexNet GoogLeNet GoogLeNet GoogLeNet GoogLeNet LR policy f ixed triangular2 decay exp exp range f ixed triangular2 exp exp exp range f ixed triangular2 exp exp range Iterations Accuracy (%) 70,000 25, 000 25,000 70,000 42,000 400,000 400,000 300,000 460,000 300,000 420,000 420,000 240,000 240,000 81.4 81.4 78.5 79.1 82.2 58.0 58.4 56.0 56.5 56.5 63.0 64.4 58.2 60.2
Table 1. Comparison of accuracy results on test/validation data at the end of the training.
starts to fall. These two learning rates are good choices for bounds; that is, set base lr to the ï¬rst value and set max lr to the latter value. Alternatively, one can use the rule of thumb that the optimum learning rate is usually within a factor of two of the largest one that converges [2] and set base lr to 1
|
1506.01186#13
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 13,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Dataset CIFAR-10 CIFAR-10 CIFAR-10 CIFAR-10 CIFAR-10 AlexNet AlexNet AlexNet AlexNet AlexNet GoogLeNet GoogLeNet GoogLeNet GoogLeNet LR policy f ixed triangular2 decay exp exp range f ixed triangular2 exp exp exp range f ixed triangular2 exp exp range Iterations Accuracy (%) 70,000 25, 000 25,000 70,000 42,000 400,000 400,000 300,000 460,000 300,000 420,000 420,000 240,000 240,000 81.4 81.4 78.5 79.1 82.2 58.0 58.4 56.0 56.5 56.5 63.0 64.4 58.2 60.2\nTable 1. Comparison of accuracy results on test/validation data at the end of the training.\nstarts to fall. These two learning rates are good choices for bounds; that is, set base lr to the ï¬rst value and set max lr to the latter value. Alternatively, one can use the rule of thumb that the optimum learning rate is usually within a factor of two of the largest one that converges [2] and set base lr to 1",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 14 |
Figure 3 shows an example of making this type of run with the CIFAR-10 dataset, using the architecture and hyper-parameters provided by Caffe. One can see from Fig- ure 3 that the model starts converging right away, so it is rea- sonable to set base lr = 0.001. Furthermore, above a learn- ing rate of 0.006 the accuracy rise gets rough and eventually begins to drop so it is reasonable to set max lr = 0.006.
Whenever one is starting with a new architecture or dataset, a single LR range test provides both a good LR value and a good range. Then one should compare runs with a ï¬xed LR versus CLR with this range. Whichever wins can be used with conï¬dence for the rest of oneâs experiments.
# 4. Experiments
The purpose of this section is to demonstrate the effec- tiveness of the CLR methods on some standard datasets and with a range of architectures. In the subsections below, CLR policies are used for training with the CIFAR-10, CIFAR- 100, and ImageNet datasets. These three datasets and a va- riety of architectures demonstrate the versatility of CLR.
# 4.1. CIFAR-10 and CIFAR-100
# 4.1.1 Caffeâs CIFAR-10 architecture
|
1506.01186#14
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 14,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 3 shows an example of making this type of run with the CIFAR-10 dataset, using the architecture and hyper-parameters provided by Caffe. One can see from Fig- ure 3 that the model starts converging right away, so it is rea- sonable to set base lr = 0.001. Furthermore, above a learn- ing rate of 0.006 the accuracy rise gets rough and eventually begins to drop so it is reasonable to set max lr = 0.006.\nWhenever one is starting with a new architecture or dataset, a single LR range test provides both a good LR value and a good range. Then one should compare runs with a ï¬xed LR versus CLR with this range. Whichever wins can be used with conï¬dence for the rest of oneâs experiments.\n# 4. Experiments\nThe purpose of this section is to demonstrate the effec- tiveness of the CLR methods on some standard datasets and with a range of architectures. In the subsections below, CLR policies are used for training with the CIFAR-10, CIFAR- 100, and ImageNet datasets. These three datasets and a va- riety of architectures demonstrate the versatility of CLR.\n# 4.1. CIFAR-10 and CIFAR-100\n# 4.1.1 Caffeâs CIFAR-10 architecture",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 15 |
# 4.1. CIFAR-10 and CIFAR-100
# 4.1.1 Caffeâs CIFAR-10 architecture
The CIFAR-10 architecture and hyper-parameter settings on the Caffe website are fairly standard and were used here as a baseline. As discussed in Section 3.2, an epoch is equal
CIFAR-10 ---Exp policy ââExp Range 0.2 1 2 3 4 5 6 7 Iteration * 10°
Figure 4. Classiï¬cation accuracy as a function of iteration for 70, 000 iterations.
CIFAR10; Combining adaptive LR and CLR âNesterov + CLR â Adam > 03 âAdam + CLR Iteration % isâ
Figure 5. Classiï¬cation accuracy as a function of iteration for the CIFAR-10 dataset using adaptive learning methods. See text for explanation.
to 500 iterations and a good setting for stepsize is 2, 000. Section 3.3 discussed how to estimate reasonable minimum and maximum boundary values for the learning rate from Figure 3. All that is needed to optimally train the network is to set base lr = 0.001 and max lr = 0.006. This is all that is needed to optimally train the network. For the triangular2 policy run shown in Figure 1, the stepsize and learning rate bounds are shown in Table 2.
|
1506.01186#15
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 15,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "# 4.1. CIFAR-10 and CIFAR-100\n# 4.1.1 Caffeâs CIFAR-10 architecture\nThe CIFAR-10 architecture and hyper-parameter settings on the Caffe website are fairly standard and were used here as a baseline. As discussed in Section 3.2, an epoch is equal\nCIFAR-10 ---Exp policy ââExp Range 0.2 1 2 3 4 5 6 7 Iteration * 10°\nFigure 4. Classiï¬cation accuracy as a function of iteration for 70, 000 iterations.\nCIFAR10; Combining adaptive LR and CLR âNesterov + CLR â Adam > 03 âAdam + CLR Iteration % isâ\nFigure 5. Classiï¬cation accuracy as a function of iteration for the CIFAR-10 dataset using adaptive learning methods. See text for explanation.\nto 500 iterations and a good setting for stepsize is 2, 000. Section 3.3 discussed how to estimate reasonable minimum and maximum boundary values for the learning rate from Figure 3. All that is needed to optimally train the network is to set base lr = 0.001 and max lr = 0.006. This is all that is needed to optimally train the network. For the triangular2 policy run shown in Figure 1, the stepsize and learning rate bounds are shown in Table 2.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 16 |
base lr 0.001 0.0001 0.00001 max lr 0.005 0.0005 0.00005 stepsize 2,000 1,000 500 start 0 16,000 22,000 max iter 16,000 22,000 25,000
Table 2. Hyper-parameter settings for CIFAR-10 example in Fig- ure 1.
running with the the result of triangular2 policy with the parameter setting in Table 2. As shown in Table 1, one obtains the same test classiï¬ca- tion accuracy of 81.4% after only 25, 000 iterations with the triangular2 policy as obtained by running the standard hyper-parameter settings for 70, 000 iterations.
8 CIFAR10; Sigmoid + Batch Normalization â . 3.0.6 a g4 to2 %% 1 3 3 4 5 6 Iteration x104
Figure 6. Batch Normalization CIFAR-10 example (provided with the Caffe download).
|
1506.01186#16
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 16,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "base lr 0.001 0.0001 0.00001 max lr 0.005 0.0005 0.00005 stepsize 2,000 1,000 500 start 0 16,000 22,000 max iter 16,000 22,000 25,000\nTable 2. Hyper-parameter settings for CIFAR-10 example in Fig- ure 1.\nrunning with the the result of triangular2 policy with the parameter setting in Table 2. As shown in Table 1, one obtains the same test classiï¬ca- tion accuracy of 81.4% after only 25, 000 iterations with the triangular2 policy as obtained by running the standard hyper-parameter settings for 70, 000 iterations.\n8 CIFAR10; Sigmoid + Batch Normalization â . 3.0.6 a g4 to2 %% 1 3 3 4 5 6 Iteration x104\nFigure 6. Batch Normalization CIFAR-10 example (provided with the Caffe download).",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 17 |
Figure 6. Batch Normalization CIFAR-10 example (provided with the Caffe download).
from the triangular policy derive from reducing the learning rate because this is when the accuracy climbs the most. As a test, a decay policy was implemented where the learn- ing rate starts at the max lr value and then is linearly re- duced to the base lr value for stepsize number of itera- tions. After that, the learning rate is ï¬xed to base lr. For the decay policy, max lr = 0.007, base lr = 0.001, and stepsize = 4000. Table 1 shows that the ï¬nal accuracy is only 78.5%, providing evidence that both increasing and decreasing the learning rate are essential for the beneï¬ts of the CLR method.
Figure 4 compares the exp learning rate policy in Caffe with the new exp range policy using gamma = 0.99994 for both policies. is that when using the exp range policy one can stop training at iteration 42, 000 with a test accuracy of 82.2% (going to iteration 70, 000 does not improve on this result). This is substantially better than the best test accuracy of 79.1% one obtains from using the exp learning rate policy.
|
1506.01186#17
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 17,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 6. Batch Normalization CIFAR-10 example (provided with the Caffe download).\nfrom the triangular policy derive from reducing the learning rate because this is when the accuracy climbs the most. As a test, a decay policy was implemented where the learn- ing rate starts at the max lr value and then is linearly re- duced to the base lr value for stepsize number of itera- tions. After that, the learning rate is ï¬xed to base lr. For the decay policy, max lr = 0.007, base lr = 0.001, and stepsize = 4000. Table 1 shows that the ï¬nal accuracy is only 78.5%, providing evidence that both increasing and decreasing the learning rate are essential for the beneï¬ts of the CLR method.\nFigure 4 compares the exp learning rate policy in Caffe with the new exp range policy using gamma = 0.99994 for both policies. is that when using the exp range policy one can stop training at iteration 42, 000 with a test accuracy of 82.2% (going to iteration 70, 000 does not improve on this result). This is substantially better than the best test accuracy of 79.1% one obtains from using the exp learning rate policy.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 18 |
The current Caffe download contains additional archi- tectures and hyper-parameters for CIFAR-10 and in partic- ular there is one with sigmoid non-linearities and batch nor- malization. Figure 6 compares the training accuracy using the downloaded hyper-parameters with a ï¬xed learning rate (blue curve) to using a cyclical learning rate (red curve). As can be seen in this Figure, the ï¬nal accuracy for the ï¬xed learning rate (60.8%) is substantially lower than the cyclical learning rate ï¬nal accuracy (72.2%). There is clear perfor- mance improvement when using CLR with this architecture containing sigmoids and batch normalization.
Experiments were carried out with architectures featur- ing both adaptive learning rate methods and CLR. Table 3 lists the ï¬nal accuracy values from various adaptive learning rate methods, run with and without CLR. All of the adap- tive methods in Table 3 were run by invoking the respective option in Caffe. The learning rate boundaries are given in Table 3 (just below the methodâs name), which were deter- mined by using the technique described in Section 3.3. Just the lower bound was used for base lr for the f ixed policy.
|
1506.01186#18
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 18,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "The current Caffe download contains additional archi- tectures and hyper-parameters for CIFAR-10 and in partic- ular there is one with sigmoid non-linearities and batch nor- malization. Figure 6 compares the training accuracy using the downloaded hyper-parameters with a ï¬xed learning rate (blue curve) to using a cyclical learning rate (red curve). As can be seen in this Figure, the ï¬nal accuracy for the ï¬xed learning rate (60.8%) is substantially lower than the cyclical learning rate ï¬nal accuracy (72.2%). There is clear perfor- mance improvement when using CLR with this architecture containing sigmoids and batch normalization.\nExperiments were carried out with architectures featur- ing both adaptive learning rate methods and CLR. Table 3 lists the ï¬nal accuracy values from various adaptive learning rate methods, run with and without CLR. All of the adap- tive methods in Table 3 were run by invoking the respective option in Caffe. The learning rate boundaries are given in Table 3 (just below the methodâs name), which were deter- mined by using the technique described in Section 3.3. Just the lower bound was used for base lr for the f ixed policy.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 19 |
LR type/bounds Nesterov [19] 0.001 - 0.006 ADAM [16] 0.0005 - 0.002 RMSprop [27] 0.0001 - 0.0003 AdaGrad [5] 0.003 - 0.035 AdaDelta [29] 0.01 - 0.1 LR policy f ixed triangular f ixed triangular triangular f ixed triangular triangular f ixed triangular f ixed triangular Iterations Accuracy (%) 70,000 25,000 70,000 25,000 70,000 70,000 25,000 70,000 70,000 25,000 70,000 25,000 82.1 81.3 81.4 79.8 81.1 75.2 72.8 75.1 74.6 76.0 67.3 67.3
Table 3. Comparison of CLR with adaptive learning rate methods. The table shows accuracy results for the CIFAR-10 dataset on test data at the end of the training.
|
1506.01186#19
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 19,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "LR type/bounds Nesterov [19] 0.001 - 0.006 ADAM [16] 0.0005 - 0.002 RMSprop [27] 0.0001 - 0.0003 AdaGrad [5] 0.003 - 0.035 AdaDelta [29] 0.01 - 0.1 LR policy f ixed triangular f ixed triangular triangular f ixed triangular triangular f ixed triangular f ixed triangular Iterations Accuracy (%) 70,000 25,000 70,000 25,000 70,000 70,000 25,000 70,000 70,000 25,000 70,000 25,000 82.1 81.3 81.4 79.8 81.1 75.2 72.8 75.1 74.6 76.0 67.3 67.3\nTable 3. Comparison of CLR with adaptive learning rate methods. The table shows accuracy results for the CIFAR-10 dataset on test data at the end of the training.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 20 |
Table 3. Comparison of CLR with adaptive learning rate methods. The table shows accuracy results for the CIFAR-10 dataset on test data at the end of the training.
Table 3 shows that for some adaptive learning rate meth- ods combined with CLR, the ï¬nal accuracy after only 25,000 iterations is equivalent to the accuracy obtained without CLR after 70,000 iterations. For others, it was nec- essary (even with CLR) to run until 70,000 iterations to ob- tain similar results. Figure 5 shows the curves from running the Nesterov method with CLR (reached 81.3% accuracy in only 25,000 iterations) and the Adam method both with and without CLR (both needed 70,000 iterations). When using adaptive learning rate methods, the beneï¬ts from CLR are sometimes reduced, but CLR can still valuable as it some- times provides beneï¬t at essentially no cost.
# 4.1.2 ResNets, Stochastic Depth, and DenseNets
|
1506.01186#20
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
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{
"id": "1603.09382"
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"id": "1608.06993"
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] |
{
"authors": "Leslie N. Smith",
"chunk_id": 20,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Table 3. Comparison of CLR with adaptive learning rate methods. The table shows accuracy results for the CIFAR-10 dataset on test data at the end of the training.\nTable 3 shows that for some adaptive learning rate meth- ods combined with CLR, the ï¬nal accuracy after only 25,000 iterations is equivalent to the accuracy obtained without CLR after 70,000 iterations. For others, it was nec- essary (even with CLR) to run until 70,000 iterations to ob- tain similar results. Figure 5 shows the curves from running the Nesterov method with CLR (reached 81.3% accuracy in only 25,000 iterations) and the Adam method both with and without CLR (both needed 70,000 iterations). When using adaptive learning rate methods, the beneï¬ts from CLR are sometimes reduced, but CLR can still valuable as it some- times provides beneï¬t at essentially no cost.\n# 4.1.2 ResNets, Stochastic Depth, and DenseNets",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 21 |
# 4.1.2 ResNets, Stochastic Depth, and DenseNets
Residual networks [10, 11], and the family of variations that have subsequently emerged, achieve state-of-the-art re- sults on a variety of tasks. Here we provide comparison experiments between the original implementations and ver- sions with CLR for three members of this residual net- work family: the original ResNet [10], Stochastic Depth networks [13], and the recent DenseNets [12]. Our ex- periments can be readily replicated because the authors of these papers make their Torch code available3. Since all three implementation are available using the Torch 7 frame- work, the experiments in this section were performed using Torch. In addition to the experiment in the previous Sec- tion, these networks also incorporate batch normalization [15] and demonstrate the value of CLR for architectures with batch normalization.
Both CIFAR-10 and the CIFAR-100 datasets were used
# 3https://github.com/facebook/fb.resnet.torch, https://github.com/yueatsprograms/Stochastic Depth, https://github.com/liuzhuang13/DenseNet
in these experiments. The CIFAR-100 dataset is similar to the CIFAR-10 data but it has 100 classes instead of 10 and each class has 600 labeled examples.
|
1506.01186#21
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
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},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 21,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "# 4.1.2 ResNets, Stochastic Depth, and DenseNets\nResidual networks [10, 11], and the family of variations that have subsequently emerged, achieve state-of-the-art re- sults on a variety of tasks. Here we provide comparison experiments between the original implementations and ver- sions with CLR for three members of this residual net- work family: the original ResNet [10], Stochastic Depth networks [13], and the recent DenseNets [12]. Our ex- periments can be readily replicated because the authors of these papers make their Torch code available3. Since all three implementation are available using the Torch 7 frame- work, the experiments in this section were performed using Torch. In addition to the experiment in the previous Sec- tion, these networks also incorporate batch normalization [15] and demonstrate the value of CLR for architectures with batch normalization.\nBoth CIFAR-10 and the CIFAR-100 datasets were used\n# 3https://github.com/facebook/fb.resnet.torch, https://github.com/yueatsprograms/Stochastic Depth, https://github.com/liuzhuang13/DenseNet\nin these experiments. The CIFAR-100 dataset is similar to the CIFAR-10 data but it has 100 classes instead of 10 and each class has 600 labeled examples.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 22 |
Architecture ResNet ResNet ResNet ResNet+CLR SD SD SD SD+CLR DenseNet DenseNet DenseNet CIFAR-10 (LR) CIFAR-100 (LR) 92.8(0.1) 93.3(0.2) 91.8(0.3) 93.6(0.1 â 0.3) 94.6(0.1) 94.5(0.2) 94.2(0.3) 94.5(0.1 â 0.3) 94.5(0.1) 94.5(0.2) 94.2(0.3) 71.2(0.1) 71.6(0.2) 71.9(0.3) 72.5(0.1 â 0.3) 75.2(0.1) 75.2(0.2) 74.6(0.3) 75.4(0.1 â 0.3) 75.2(0.1) 75.3(0.2) 74.5(0.3) 75.9(0.1 â 0.2) DenseNet+CLR 94.9(0.1 â 0.2)
|
1506.01186#22
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 22,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Architecture ResNet ResNet ResNet ResNet+CLR SD SD SD SD+CLR DenseNet DenseNet DenseNet CIFAR-10 (LR) CIFAR-100 (LR) 92.8(0.1) 93.3(0.2) 91.8(0.3) 93.6(0.1 â 0.3) 94.6(0.1) 94.5(0.2) 94.2(0.3) 94.5(0.1 â 0.3) 94.5(0.1) 94.5(0.2) 94.2(0.3) 71.2(0.1) 71.6(0.2) 71.9(0.3) 72.5(0.1 â 0.3) 75.2(0.1) 75.2(0.2) 74.6(0.3) 75.4(0.1 â 0.3) 75.2(0.1) 75.3(0.2) 74.5(0.3) 75.9(0.1 â 0.2) DenseNet+CLR 94.9(0.1 â 0.2)",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 23 |
Table 4. Comparison of CLR with ResNets [10, 11], Stochastic Depth (SD) [13], and DenseNets [12]. The table shows the average accuracy of 5 runs for the CIFAR-10 and CIFAR-100 datasets on test data at the end of the training.
The results for these two datasets on these three archi- tectures are summarized in Table 4. The left column give the architecture and whether CLR was used in the experi- ments. The other two columns gives the average ï¬nal ac- curacy from ï¬ve runs and the initial learning rate or range used in parenthesis, which are reduced (for both the ï¬xed learning rate and the range) during the training according to the same schedule used in the original implementation. For all three architectures, the original implementation uses an initial LR of 0.1 which we use as a baseline.
|
1506.01186#23
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 23,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Table 4. Comparison of CLR with ResNets [10, 11], Stochastic Depth (SD) [13], and DenseNets [12]. The table shows the average accuracy of 5 runs for the CIFAR-10 and CIFAR-100 datasets on test data at the end of the training.\nThe results for these two datasets on these three archi- tectures are summarized in Table 4. The left column give the architecture and whether CLR was used in the experi- ments. The other two columns gives the average ï¬nal ac- curacy from ï¬ve runs and the initial learning rate or range used in parenthesis, which are reduced (for both the ï¬xed learning rate and the range) during the training according to the same schedule used in the original implementation. For all three architectures, the original implementation uses an initial LR of 0.1 which we use as a baseline.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 24 |
The accuracy results in Table 4 in the right two columns are the average ï¬nal test accuracies of ï¬ve runs. The Stochastic Depth implementation was slightly different than the ResNet and DenseNet implementation in that the au- thors split the 50,000 training images into 45,000 training images and 5,000 validation images. However, the reported results in Table 4 for the SD architecture is only test accura- cies for the ï¬ve runs. The learning rate range used by CLR was determined by the LR range test method and the cycle length was choosen as a tenth of the maximum number of epochs that was speciï¬ed in the original implementation.
In addition to the accuracy results shown in Table 4, similar results were obtained in Caffe for DenseNets [12] on CIFAR-10 using the prototxt ï¬les provided by the au- thors. The average accuracy of ï¬ve runs with learning rates of 0.1, 0.2, 0.3 was 91.67%, 92.17%, 92.46%, respectively, but running with CLR within the range of 0.1 to 0.3, the average accuracy was 93.33%.
The results from all of these experiments show similar or better accuracy performance when using CLR versus using a ï¬xed learning rate, even though the performance drops at
|
1506.01186#24
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 24,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "The accuracy results in Table 4 in the right two columns are the average ï¬nal test accuracies of ï¬ve runs. The Stochastic Depth implementation was slightly different than the ResNet and DenseNet implementation in that the au- thors split the 50,000 training images into 45,000 training images and 5,000 validation images. However, the reported results in Table 4 for the SD architecture is only test accura- cies for the ï¬ve runs. The learning rate range used by CLR was determined by the LR range test method and the cycle length was choosen as a tenth of the maximum number of epochs that was speciï¬ed in the original implementation.\nIn addition to the accuracy results shown in Table 4, similar results were obtained in Caffe for DenseNets [12] on CIFAR-10 using the prototxt ï¬les provided by the au- thors. The average accuracy of ï¬ve runs with learning rates of 0.1, 0.2, 0.3 was 91.67%, 92.17%, 92.46%, respectively, but running with CLR within the range of 0.1 to 0.3, the average accuracy was 93.33%.\nThe results from all of these experiments show similar or better accuracy performance when using CLR versus using a ï¬xed learning rate, even though the performance drops at",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 25 |
The results from all of these experiments show similar or better accuracy performance when using CLR versus using a ï¬xed learning rate, even though the performance drops at
ImageNet on AlexNet 0.2 Accuracy ° a we 0.05) ° 6.005 0.01 0.015 0.02 0.025 6.03 6.035 0.04 6.045 Learning rate
Figure 7. AlexNet LR range test; validation classiï¬cation accuracy as a function of increasing learning rate.
ImageNet/AlexNet architecture S a se 2s Row es) Validation Accuracy ses âob âTriangular2 os. 1 15.3225 °3°«35 Iteration x 10°
Figure 8. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.
some of the learning rate values within this range. These experiments conï¬rm that it is beneï¬cial to use CLR for a variety of residual architectures and for both CIFAR-10 and CIFAR-100.
# 4.2. ImageNet
The ImageNet dataset [21] is often used in deep learning literature as a standard for comparison. The ImageNet clas- siï¬cation challenge provides about 1, 000 training images for each of the 1, 000 classes, giving a total of 1, 281, 167 labeled training images.
# 4.2.1 AlexNet
|
1506.01186#25
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 25,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "The results from all of these experiments show similar or better accuracy performance when using CLR versus using a ï¬xed learning rate, even though the performance drops at\nImageNet on AlexNet 0.2 Accuracy ° a we 0.05) ° 6.005 0.01 0.015 0.02 0.025 6.03 6.035 0.04 6.045 Learning rate\nFigure 7. AlexNet LR range test; validation classiï¬cation accuracy as a function of increasing learning rate.\nImageNet/AlexNet architecture S a se 2s Row es) Validation Accuracy ses âob âTriangular2 os. 1 15.3225 °3°«35 Iteration x 10°\nFigure 8. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.\nsome of the learning rate values within this range. These experiments conï¬rm that it is beneï¬cial to use CLR for a variety of residual architectures and for both CIFAR-10 and CIFAR-100.\n# 4.2. ImageNet\nThe ImageNet dataset [21] is often used in deep learning literature as a standard for comparison. The ImageNet clas- siï¬cation challenge provides about 1, 000 training images for each of the 1, 000 classes, giving a total of 1, 281, 167 labeled training images.\n# 4.2.1 AlexNet",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 26 |
# 4.2.1 AlexNet
The Caffe website provides the architecture and hyper- parameter ï¬les for a slightly modiï¬ed AlexNet [17]. These were downloaded from the website and used as a baseline. In the training results reported in this section, all weights
ImageNet/AlexNet architecture = a S a S nS me (ees) Validation Accuracy âTriangular2 0 â F ; q 4 1 005 1 15. 2. 25. 3. 335 Iteration <i
Figure 9. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.
were initialized the same so as to avoid differences due to different random initializations.
Since the batchsize in the architecture ï¬le is 256, an epoch is equal to 1, 281, 167/256 = 5, 005 iterations. Hence, a reasonable setting for stepsize is 6 epochs or 30, 000 iterations.
|
1506.01186#26
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 26,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "# 4.2.1 AlexNet\nThe Caffe website provides the architecture and hyper- parameter ï¬les for a slightly modiï¬ed AlexNet [17]. These were downloaded from the website and used as a baseline. In the training results reported in this section, all weights\nImageNet/AlexNet architecture = a S a S nS me (ees) Validation Accuracy âTriangular2 0 â F ; q 4 1 005 1 15. 2. 25. 3. 335 Iteration <i\nFigure 9. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.\nwere initialized the same so as to avoid differences due to different random initializations.\nSince the batchsize in the architecture ï¬le is 256, an epoch is equal to 1, 281, 167/256 = 5, 005 iterations. Hence, a reasonable setting for stepsize is 6 epochs or 30, 000 iterations.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 27 |
Next, one can estimate reasonable minimum and maxi- mum boundaries for the learning rate from Figure 7. It can be seen from this ï¬gure that the training doesnât start con- verging until at least 0.006 so setting base lr = 0.006 is reasonable. However, for a fair comparison to the baseline where base lr = 0.01, it is necessary to set the base lr to 0.01 for the triangular and triangular2 policies or else the majority of the apparent improvement in the accuracy will be from the smaller learning rate. As for the maxi- mum boundary value, the training peaks and drops above a learning rate of 0.015 so max lr = 0.015 is reasonable. For comparing the exp range policy to the exp policy, set- ting base lr = 0.006 and max lr = 0.014 is reasonable and in this case one expects that the average accuracy of the exp range policy to be equal to the accuracy from the exp policy.
|
1506.01186#27
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 27,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Next, one can estimate reasonable minimum and maxi- mum boundaries for the learning rate from Figure 7. It can be seen from this ï¬gure that the training doesnât start con- verging until at least 0.006 so setting base lr = 0.006 is reasonable. However, for a fair comparison to the baseline where base lr = 0.01, it is necessary to set the base lr to 0.01 for the triangular and triangular2 policies or else the majority of the apparent improvement in the accuracy will be from the smaller learning rate. As for the maxi- mum boundary value, the training peaks and drops above a learning rate of 0.015 so max lr = 0.015 is reasonable. For comparing the exp range policy to the exp policy, set- ting base lr = 0.006 and max lr = 0.014 is reasonable and in this case one expects that the average accuracy of the exp range policy to be equal to the accuracy from the exp policy.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 28 |
Figure 9 compares the results of running with the f ixed versus the triangular2 policy for the AlexNet architecture. Here, the peaks at iterations that are multiples of 60,000 should produce a classiï¬cation accuracy that corresponds to the f ixed policy. Indeed, the accuracy peaks at the end of a cycle for the triangular2 policy are similar to the ac- curacies from the standard f ixed policy, which implies that the baseline learning rates are set quite well (this is also im- plied by Figure 7). As shown in Table 1, the ï¬nal accuracies from the CLR training run are only 0.4% better than the ac- curacies from the f ixed policy.
Figure 10 compares the results of running with the exp versus the exp range policy for the AlexNet architecture with gamma = 0.999995 for both policies. As expected,
ImageNet/AlexNet architecture 0.5 S & Validation Accuracy Ss bs 0.2 0.1 â Exp Range 0 I 2 3 4 Iteration x10°
Figure 10. Validation data classiï¬cation accuracy as a function of iteration for exp versus exp range.
ImageNet/GoogleNet architecture 0.08 id S HB Validation Accuracy 2 2° o Peg Ne eS 0 0.01 002 003 004 0.05 0.06 0.07 Learning rate
|
1506.01186#28
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 28,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 9 compares the results of running with the f ixed versus the triangular2 policy for the AlexNet architecture. Here, the peaks at iterations that are multiples of 60,000 should produce a classiï¬cation accuracy that corresponds to the f ixed policy. Indeed, the accuracy peaks at the end of a cycle for the triangular2 policy are similar to the ac- curacies from the standard f ixed policy, which implies that the baseline learning rates are set quite well (this is also im- plied by Figure 7). As shown in Table 1, the ï¬nal accuracies from the CLR training run are only 0.4% better than the ac- curacies from the f ixed policy.\nFigure 10 compares the results of running with the exp versus the exp range policy for the AlexNet architecture with gamma = 0.999995 for both policies. As expected,\nImageNet/AlexNet architecture 0.5 S & Validation Accuracy Ss bs 0.2 0.1 â Exp Range 0 I 2 3 4 Iteration x10°\nFigure 10. Validation data classiï¬cation accuracy as a function of iteration for exp versus exp range.\nImageNet/GoogleNet architecture 0.08 id S HB Validation Accuracy 2 2° o Peg Ne eS 0 0.01 002 003 004 0.05 0.06 0.07 Learning rate",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 29 |
Figure 11. GoogleNet LR range test; validation classiï¬cation ac- curacy as a function of increasing learning rate.
Figure 10 shows that the accuracies from the exp range policy do oscillate around the exp policy accuracies. The advantage of the exp range policy is that the accuracy of 56.5% is already obtained at iteration 300, 000 whereas the exp policy takes until iteration 460, 000 to reach 56.5%.
Finally, a comparison between the f ixed and exp poli- cies in Table 1 shows the f ixed and triangular2 policies produce accuracies that are almost 2% better than their ex- ponentially decreasing counterparts, but this difference is probably due to not having tuned gamma.
# 4.2.2 GoogLeNet/Inception Architecture
The GoogLeNet architecture was a winning entry to the ImageNet 2014 image classiï¬cation competition. Szegedy et al. [25] describe the architecture in detail but did not provide the architecture ï¬le. The architecture ï¬le publicly available from Princeton4 was used in the following exper- iments. The GoogLeNet paper does not state the learning rate values and the hyper-parameter solver ï¬le is not avail4vision.princeton.edu/pvt/GoogLeNet/
|
1506.01186#29
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 29,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Figure 11. GoogleNet LR range test; validation classiï¬cation ac- curacy as a function of increasing learning rate.\nFigure 10 shows that the accuracies from the exp range policy do oscillate around the exp policy accuracies. The advantage of the exp range policy is that the accuracy of 56.5% is already obtained at iteration 300, 000 whereas the exp policy takes until iteration 460, 000 to reach 56.5%.\nFinally, a comparison between the f ixed and exp poli- cies in Table 1 shows the f ixed and triangular2 policies produce accuracies that are almost 2% better than their ex- ponentially decreasing counterparts, but this difference is probably due to not having tuned gamma.\n# 4.2.2 GoogLeNet/Inception Architecture\nThe GoogLeNet architecture was a winning entry to the ImageNet 2014 image classiï¬cation competition. Szegedy et al. [25] describe the architecture in detail but did not provide the architecture ï¬le. The architecture ï¬le publicly available from Princeton4 was used in the following exper- iments. The GoogLeNet paper does not state the learning rate values and the hyper-parameter solver ï¬le is not avail4vision.princeton.edu/pvt/GoogLeNet/",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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] |
1506.01186
| 30 |
Imagenet with GoogLeNet architecture 2 g 2 a 2 in 2 5 2 io Validation Accuracy ° is e 0 05 1 15 2 25) 3 3.5 4 45 Iteration x10
Figure 12. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.
able for a baseline but not having these hyper-parameters is a typical situation when one is developing a new architec- ture or applying a network to a new dataset. This is a situa- tion that CLR readily handles. Instead of running numerous experiments to ï¬nd optimal learning rates, the base lr was set to a best guess value of 0.01.
|
1506.01186#30
|
Cyclical Learning Rates for Training Neural Networks
|
It is known that the learning rate is the most important hyper-parameter to
tune for training deep neural networks. This paper describes a new method for
setting the learning rate, named cyclical learning rates, which practically
eliminates the need to experimentally find the best values and schedule for the
global learning rates. Instead of monotonically decreasing the learning rate,
this method lets the learning rate cyclically vary between reasonable boundary
values. Training with cyclical learning rates instead of fixed values achieves
improved classification accuracy without a need to tune and often in fewer
iterations. This paper also describes a simple way to estimate "reasonable
bounds" -- linearly increasing the learning rate of the network for a few
epochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10
and CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,
and the ImageNet dataset with the AlexNet and GoogLeNet architectures. These
are practical tools for everyone who trains neural networks.
|
http://arxiv.org/pdf/1506.01186
|
Leslie N. Smith
|
cs.CV, cs.LG, cs.NE
|
Presented at WACV 2017; see https://github.com/bckenstler/CLR for
instructions to implement CLR in Keras
| null |
cs.CV
|
20150603
|
20170404
|
[
{
"id": "1504.01716"
},
{
"id": "1502.03167"
},
{
"id": "1600.04747"
},
{
"id": "1603.05027"
},
{
"id": "1508.02788"
},
{
"id": "1502.04390"
},
{
"id": "1608.03983"
},
{
"id": "1603.09382"
},
{
"id": "1608.06993"
}
] |
{
"authors": "Leslie N. Smith",
"chunk_id": 30,
"doc_id": "1506.01186",
"primary_category": "cs.CV",
"published": 20150603,
"source": "http://arxiv.org/pdf/1506.01186",
"summary": "It is known that the learning rate is the most important hyper-parameter to\ntune for training deep neural networks. This paper describes a new method for\nsetting the learning rate, named cyclical learning rates, which practically\neliminates the need to experimentally find the best values and schedule for the\nglobal learning rates. Instead of monotonically decreasing the learning rate,\nthis method lets the learning rate cyclically vary between reasonable boundary\nvalues. Training with cyclical learning rates instead of fixed values achieves\nimproved classification accuracy without a need to tune and often in fewer\niterations. This paper also describes a simple way to estimate \"reasonable\nbounds\" -- linearly increasing the learning rate of the network for a few\nepochs. In addition, cyclical learning rates are demonstrated on the CIFAR-10\nand CIFAR-100 datasets with ResNets, Stochastic Depth networks, and DenseNets,\nand the ImageNet dataset with the AlexNet and GoogLeNet architectures. These\nare practical tools for everyone who trains neural networks.",
"text": "Imagenet with GoogLeNet architecture 2 g 2 a 2 in 2 5 2 io Validation Accuracy ° is e 0 05 1 15 2 25) 3 3.5 4 45 Iteration x10\nFigure 12. Validation data classiï¬cation accuracy as a function of iteration for f ixed versus triangular.\nable for a baseline but not having these hyper-parameters is a typical situation when one is developing a new architec- ture or applying a network to a new dataset. This is a situa- tion that CLR readily handles. Instead of running numerous experiments to ï¬nd optimal learning rates, the base lr was set to a best guess value of 0.01.",
"title": "Cyclical Learning Rates for Training Neural Networks",
"year": 2015
}
|
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