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+"output of independently trained networks can improve accuracy, calibration, and robustness. conversely, frankle et al. (2020) showed that if we average together the weights to make one model, the network fails. fort et al. (2019) compared ensembling solutions that resulted from different initializations with ensembling solutions that were generated from the same original model. for example, in the latter case, they consider exploring around the solution in a limited subspace to find other appendixb.3.6 goodnearbypoints. theyfoundthatbothtechniquesprovidecomplementarybenefitsbutthat subspaces genuine ensembling from different random starting points provides a bigger improvement. anefficientwayofensemblingistocombinemodelsfromintermediatestagesoftraining. tothis end, izmailov et al. (2018) introduce stochastic weight averaging, in which the model weights are sampled at different time steps and averaged together. as the name suggests, snapshot ensembles (huang et al., 2017a) also store the models from different time steps and average their predictions. the diversity of these models can be improved by cyclically increasing and decreasing the learning rate. garipov et al. (2018) observed that different minima of the loss functionareoftenconnectedbyalow-energypath(i.e.,apathwithalowlosseverywherealong it). motivated by this observation, they developed a method that explores low-energy regions around an initial solution to provide diverse models without retraining. this is known as fast geometric ensembling. a review of ensembling methods can be found in ganaie et al. (2022). dropout: dropoutwasfirstintroducedbyhintonetal.(2012b)andsrivastavaetal.(2014). dropout is applied at the level of hidden units. dropping a hidden unit has the same effect as temporarily setting all the incoming and outgoing weights and the bias to zero. wan et al. (2013)generalizeddropoutbyrandomlysettingindividualweightstozero. gal&ghahramani (2016)andkendall&gal(2017)proposedmontecarlodropout,inwhichinferenceiscomputed withseveraldropoutpatterns,andtheresultsareaveragedtogether. gal&ghahramani(2016) argued that this could be interpreted as approximating bayesian inference. dropout is equivalent to applying multiplicative bernoulli noise to the hidden units. similar benefits derive from using other distributions, including the normal (srivastava et al., 2014; shen et al., 2017), uniform (shen et al., 2017), and beta distributions (liu et al., 2019b). adding noise: bishop (1995) and an (1996) added gaussian noise to the network inputs to improveperformance. bishop(1995)showedthatthisisequivalenttoweightdecay. an(1996) also investigated adding noise to the weights. devries & taylor (2017a) added gaussian noise tothehiddenunits. therandomized relu(xuetal.,2015)appliesnoiseinadifferentwayby making the activation functions stochastic. label smoothing: labelsmoothingwasintroducedbyszegedyetal.(2016)forimageclassi- ficationbuthassincebeenshowntobehelpfulinspeechrecognition(chorowski&jaitly,2017), machine translation (vaswani et al., 2017), and language modeling (pereyra et al., 2017). the precise mechanism by which label smoothing improves test performance isn’t well understood, although müller et al. (2019a) show that it improves the calibration of the predicted output probabilities. a closely related technique is disturblabel (xie et al., 2016), in which a certain percentage of the labels in each batch are randomly switched at each training iteration. finding wider minima: itisthoughtthatwiderminimageneralizebetter(seefigure20.11). here, the exact values of the weights are less important, so performance should be robust to errorsintheirestimates. oneofthereasonsthatapplyingnoisetopartsofthenetworkduring training is effective is that it encourages the network to be indifferent to their exact values. chaudhari et al. (2019) developed a variant of sgd that biases the optimization toward flat minima,whichtheycallentropy sgd.theideaistoincorporatelocalentropyasaterminthe loss function"