Robust training with ensemble consensus

Since deep neural networks are over-parameterized , they can memorize noisy examples . We address such a memorization issue in the presence of label noise . From the fact that deep neural networks can not generalize to neighborhoods of memorized features , we hypothesize that noisy examples do not consistently incur small losses on the network under a certain perturbation . Based on this , we propose a novel training method called Learning with Ensemble Consensus ( LEC ) that prevents overfitting to noisy examples by removing them based on the consensus of an ensemble of perturbed networks . One of the proposed LECs , LTEC outperforms the current state-of-the-art methods on noisy MNIST , CIFAR-10 , and CIFAR-100 in an efficient manner . 1 INTRODUCTION . Deep neural networks ( DNNs ) have shown excellent performance ( Krizhevsky et al. , 2012 ; He et al. , 2016 ) on visual recognition datasets ( Deng et al. , 2009 ) . However , it is difficult to obtain highquality labeled datasets in practice ( Wang et al. , 2018a ) . Even worse , DNNs might not learn patterns from the training data in the presence of noisy examples ( Zhang et al. , 2016 ) . Therefore , there is an increasing demand for robust training methods . In general , DNNs optimized with SGD first learn patterns relevant to clean examples under label noise ( Arpit et al. , 2017 ) . Based on this , recent studies regard examples that incur small losses on the network that does not overfit noisy examples as clean ( Han et al. , 2018 ; Shen & Sanghavi , 2019 ) . However , such small-loss examples could be noisy , especially under a high level of noise . Therefore , sampling trainable examples from a noisy dataset by relying on small-loss criteria might be impractical . To address this , we find the method to identify noisy examples among small-loss ones based on wellknown observations : ( i ) noisy examples are learned via memorization rather than via pattern learning and ( ii ) under a certain perturbation , network predictions for memorized features easily fluctuate , while those for generalized features do not . Based on these two observations , we hypothesize that out of small-loss examples , training losses of noisy examples would increase by injecting certain perturbation to network parameters , while those of clean examples would not . This suggests that examples that consistently incur small losses under multiple perturbations can be regarded as clean . This idea comes from an artifact of SGD optimization , thereby being applicable to any architecture optimized with SGD . In this work , we introduce a method to perturb parameters to distinguish noisy examples from smallloss examples . We then propose a method to robustly train neural networks under label noise , which is termed learning with ensemble consensus ( LEC ) . In LEC , the network is initially trained on the entire training set for a while and then trained on the intersection of small-loss examples of the ensemble of perturbed networks . We present three LECs with different perturbations and evaluate their effectiveness on three benchmark datasets with random label noise ( Goldberger & Ben-Reuven , 2016 ; Ma et al. , 2018 ) , open-set noise ( Wang et al. , 2018b ) , and semantic noise . Our proposed LEC outperforms existing robust training methods by efficiently removing noisy examples from training batches . 2 RELATED WORK . Generalization of DNNs . Although DNNs are over-parameterized , they have impressive generalization ability ( Krizhevsky et al. , 2012 ; He et al. , 2016 ) . Some studies argue that gradient-based optimization plays an important role in regularizing DNNs ( Neyshabur et al. , 2014 ; Zhang et al. , 2016 ) . Arpit et al . ( 2017 ) show that DNNs optimized with gradient-based methods learn patterns relevant to clean examples in the early stage of training . Since mislabeling reduces the correlation with other training examples , it is likely that noisy examples are learned via memorization . Therefore , we analyze the difference between generalized and memorized features to discriminate clean and noisy examples . Training DNNs with Noisy datasets . Label noise issues can be addressed by reducing negative impact of noisy examples . One direction is to train with a modified loss function based on the noise distribution . Most studies of this direction estimate the noise distribution prior to training as it is not accessible in general ( Sukhbaatar et al. , 2014 ; Goldberger & Ben-Reuven , 2016 ; Patrini et al. , 2017 ; Hendrycks et al. , 2018 ) . Another direction is to train with modified labels using the current model prediction ( Reed et al. , 2014 ; Ma et al. , 2018 ) . Aside from these directions , recent work suggests a method of exploiting small-loss examples ( Jiang et al. , 2017 ; Han et al. , 2018 ; Yu et al. , 2019 ; Shen & Sanghavi , 2019 ) based on the generalization ability of DNNs . However , it is still hard to find clean examples by relying on training losses . This study presents a simple method to overcome such a problem of small-loss criteria . 3 ROBUST TRAINING WITH ENSEMBLE CONSENSUS . 3.1 PROBLEM STATEMENT . Suppose that % of examples in a dataset D : = Dclean ∪Dnoisy are noisy . Let S , D , θ denote the set of ( 100- ) % small-loss examples of the network f parameterized by θ out of examples in D. Since it is generally hard to learn only all clean examples especially on the highly corrupted training set , it is problematic to regard all examples in S , D , θ as being clean . To mitigate this , we suggest a simple idea : to find noisy examples among examples in S , D , θ . 3.2 LEARNING WITH ENSEMBLE CONSENSUS ( LEC ) . Since noisy examples are little correlated with other training examples , they are likely to be learned via memorization . However , DNNs can not generalize to neighborhoods of the memorized features . This means that even if training losses of noisy examples are small , they can be easily increased under a certain perturbation δ , i.e. , for ( x , y ) ∈ Dnoisy , ( x , y ) ∈ S , D , θ ⇒ ( x , y ) /∈ S , D , θ+δ . Unlike noisy examples , the network f trained on the entire set D can learn patterns from some clean examples in the early stage of training . Thus , their training losses are consistently small in the presence of the perturbation δ , i.e. , for ( x , y ) ∈ Dclean , ( x , y ) ∈ S , D , θ ⇒ ( x , y ) ∈ S , D , θ+δ . This suggests that noisy examples can be identified from the inconsistency of losses under certain perturbation δ . Based on this , we regard examples in the intersection of ( 100- ) % small-loss examples of an ensemble of M networks generated by adding perturbations δ1 , δ2 , ... , δM to θ , i.e. , ∩Mm=1S , D , θ+δm as clean . We call it ensemble consensus filtering because examples are selected via ensemble consensus . With this filtering , we develop a training method termed learning with ensemble consensus ( LEC ) described in Algorithms 1 and 2 . Both algorithms consist of warming-up and filtering processes . The difference between these two lies in the filtering process . During the filtering process of Algorithm 1 , the network is trained on the intersection of ( 100- ) % small-loss examples of M networks within a mini batch B . Therefore , the number of examples updated at once is changing . We can encourage more stable training with a fixed number of examples to be updated at once as described in Algorithm 2 . During the filtering process of Algorithm 2 , we first obtain the intersection of small-loss examples of M networks within a full batch D at each epoch . We then sample a subset of batchsize from the intersection and train them at each update like a normal SGD . Algorithm 1 LEC Require : noisy datasetD with noise ratio % , duration of warming- up Tw , # of networks used for filteringM , perturbation δ 1 : Initialize θ randomly 2 : for epoch t = 1 : Tw do I Warming-up process 3 : for mini-batch index b = 1 : |D|batchsize do 4 : Sample a subset of batchsize Bb from a full batchD 5 : θ ← θ − α∇θ 1|Bb| ∑ ( x , y ) ∈Bb CE ( fθ ( x ) , y ) 6 : end for 7 : end for 8 : for epoch t = Tw + 1 : Tend do I Filtering process 9 : for mini-batch index b = 1 : |D|batchsize do 10 : Sample a subset of batchsize Bb from a full batchD 11 : form = 1 : M do 12 : θm = θ + δm , b , t . Adding perturbation 13 : S , Bb , θm : = ( 100− ) % small-loss examples of fθm within a mini batch Bb 14 : end for 15 : Bb′ = ∩Mm=1S , Bb , θm . Ensemble consensus filtering 16 : θ ← θ − α∇θ 1|Bb′| ∑ ( x , y ) ∈Bb′ CE ( fθ ( x ) , y ) 17 : end for 18 : end for Algorithm 2 LEC-full Require : noisy datasetD with noise ratio % , duration of warming- up Tw , # of networks used for filteringM , perturbation δ 1 : Initialize θ randomly 2 : for epoch t = 1 : Tw do I Warming-up process 3 : for mini-batch index b = 1 : |D|batchsize do 4 : Sample a subset of batchsize Bb from a full batchD 5 : θ ← θ − α∇θ 1|Bb| ∑ ( x , y ) ∈Bb CE ( fθ ( x ) , y ) 6 : end for 7 : end for 8 : for epoch t = Tw + 1 : Tend do I Filtering process 9 : form = 1 : M do 10 : θm = θ + δm , t . Adding perturbation 11 : S , D , θm : = ( 100− ) % small-loss examples of fθm within a full batchD 12 : end for 13 : D′t = ∩Mm=1S , D , θm . Ensemble consensus filtering 14 : for mini-batch index b = 1 : |D ′ t| batchsize do 15 : Sample a subset of batchsize B′b fromD′t 16 : θ ← θ − α∇θ 1|B′ b | ∑ ( x , y ) ∈B′ b CE ( fθ ( x ) , y ) 17 : end for 18 : end for 3.3 PERTURBATION TO IDENTIFY NOISY EXAMPLES . Now we aim to find a perturbation δ to be injected to discriminate memorized features from generalized ones . We present three LECs with different perturbations in the following . The pseudocodes can be found in Section A.1.3 . • Network-Ensemble Consensus ( LNEC ) : Inspired by the observation that an ensemble of networks with the same architecture is correlated during generalization and is decorrelated during memorization ( Morcos et al. , 2018 ) , the perturbation δ comes from the difference between M networks . During the warming-up process , M networks are trained independently . During the filtering process , M networks are trained on the intersection of ( 100- ) % small-loss examples of M networks . • Self-Ensemble Consensus ( LSEC ) : We focus on the relationship between Morcos et al . ( 2018 ) and Lakshminarayanan et al . ( 2017 ) : network predictions for memorized features are uncertain and those for generalized features are certain . Since the uncertainty of predictions also can be captured by multiple stochastic predictions ( Gal & Ghahramani , 2016 ) , the perturbation δ comes from the difference between M stochastic predictions of a single network.1 During the filtering process , the network is trained on the intersection of ( 100- ) % small-loss examples obtained with M stochastic predictions . • Temporal-Ensemble Consensus ( LTEC ) : Inspired by the observation that during training , atypical features are more easily forgetful compared to typical features ( Toneva et al. , 2018 ) , the perturbation δ comes from the difference between networks at current and preceding epochs . During the filtering process , the network is trained on the intersection of ( 100- ) % small-loss examples at the current epoch t and preceding min ( M − 1 , t − 1 ) epochs . We collect ( 100- ) % small-loss examples at the preceding epochs , rather than network parameters to reduce memory usage . 1As in Gal & Ghahramani ( 2016 ) , the stochasticity of predictions is caused by stochastic operations such as dropout ( Srivastava et al. , 2014 ) .

Summary: In this paper, the authors proposed to identify noisy training examples using ensemble consensus. The authors argued and demonstrated through numeric studies that, to the contrary of some earlier work, training examples with low training loss are not necessarily mislabeled. Rather, the authors hypothesized that examples with high noise require memorization, which is sensitive to perturbations. Thus, the authors proposed to identify and subsequently remove those examples from training by looking at the loss after small perturbations to the model parameters. Examples with consistently low training loss are retained for training. The authors also provided several alternatives of perturbations, including examining the consensus between an ensemble of networks, between multiple stochastic predictions, or between predictions from prior training epochs. Finally, the authors demonstrated the performance of their procedures using numerical studies.