Churn Reduction via Distillation

In real-world systems , models are frequently updated as more data becomes available , and in addition to achieving high accuracy , the goal is to also maintain a low difference in predictions compared to the base model ( i.e . predictive “ churn ” ) . If model retraining results in vastly different behavior , then it could cause negative effects in downstream systems , especially if this churn can be avoided with limited impact on model accuracy . In this paper , we show an equivalence between training with distillation using the base model as the teacher and training with an explicit constraint on the predictive churn . We then show that distillation performs strongly for low churn training against a number of recent baselines on a wide range of datasets and model architectures , including fully-connected networks , convolutional networks , and transformers . 1 INTRODUCTION Deep neural networks ( DNNs ) have had profound success at solving some of the most challenging machine learning problems . While much of the focus has been spent towards attaining state-of-art predictive performance , comparatively there has been little effort towards improving other aspects . One such important practical aspect is reducing unnecessary predictive churn with respect to a base model . We define predictive churn as the difference in the prediction of a model relative to a base model on the same datapoints . In a production system , models are often continuously released through an iterative improvement process which cycles through launching a model , collecting additional data and researching ways to improve the current model , and proposing a candidate model to replace the current version of the model serving in production . In order to validate a candidate model , it often needs to be compared to the production model through live A/B tests ( it ’ s known that offline performance alone isn ’ t a sufficient , especially if these models are used as part of a larger system where the offline and online metrics may not perfectly align ( Deng et al. , 2013 ; Beel et al. , 2013 ) ) . Live experiments are costly : they often require human evaluations when the candidate and production model disagree to know which model was correct ( Theocharous et al. , 2015 ; Deng & Shi , 2016 ) . Therefore , minimizing the unnecessary predictive churn can have a significant impact to the cost of the launch cycle . It ’ s been observed that training DNN can be very noisy due to a variety of factors including random initialization ( Glorot & Bengio , 2010 ) , mini-batch ordering ( Loshchilov & Hutter , 2015 ) , data augmentation and processing ( Santurkar et al. , 2018 ; Shorten & Khoshgoftaar , 2019 ) , and hardware ( Turner & Nowotny , 2015 ; Bhojanapalli et al. , 2021 ) – in other words running the same procedure multiple times can lead to models with surprisingly amount of disagreeing predictions even though all can have very high accuracies ( Bahri & Jiang , 2021 ) . While the stability of the training procedure is a separate problem from lowering predictive churn , such instability can further exacerbate the issue and underscores the difficulty of the problem . Knowledge distillation ( Hinton et al. , 2015 ) , which involves having a teacher model and mixing its predictions with the original labels has proved to be a useful tool in deep learning . In this paper , we show that this surprisingly is not only an effective tool for churn reduction by using the base model as the teacher , it is also mathematically aligned with learning under a constraint on the churn . Thus , in addition to providing a strong method for low churn training , we also provide insight into the distillation . Our contributions are as follows : • We show theoretically an equivalence between the low churn training objective ( i.e . minimize a loss function subject to a churn constraint on the base model ) and using knowledge distillation with the base model as the teacher . • We show that distillation performs strongly in a wide range of experiments against a number baselines that have been considered for churn reduction . • Our distillation approach is similar to a previous method called “ anchor ” ( Fard et al. , 2016 ) , which trains on the true labels instead of distilled labels for the incorrectly predicted examples by the base model . Our proposal outperforms it by a surprising amount . We present both theoretical and experimental results showing that the modification of anchor relative to distillation actually hurts performance . 2 RELATED WORKS Prediction Churn . There are few works that address low churn training with respect to a base model . Fard et al . ( 2016 ) proposed an anchor loss which is similar to distillation when the base model ’ s prediction agrees with the original label , and uses a scaled version of the original label otherwise . In our empirical evaluation , we find that this procedure performs considerably worse than distillation . Cotter et al . ( 2019a ) ; Goh et al . ( 2016 ) use constrained optimization by adding a constraint on the churn . We use some of the theoretical insights found in that work to show an equivalence between distillation and the constrained optimization problem . Thus , we are able to bypass the added complexity of using constrained optimization ( Cotter et al. , 2019b ) in favor of distillation , which is a simpler and more robust method . A related but different notion of churn that has been studied is where the goal is to reduce the training instability . Anil et al . ( 2018 ) noted that co-distillation is an effective method . Bahri & Jiang ( 2021 ) proposes a locally adaptive variant of label smoothing and Bhojanapalli et al . ( 2021 ) propose entropy regularizers and a variant of co-distillation . We tested many of the baselines proposed in these papers and adapt them to our notion of churn and showed that they were not effective at reducing predictive churn w.r.t . a base model . Distillation . Distillation ( Ba & Caruana , 2013 ; Hinton et al. , 2015 ) , first proposed to transfer knowledge from larger networks to smaller ones , has become immensely popular . Applications include learning from noisy labels ( Li et al. , 2017 ) , model compression ( Polino et al. , 2018 ) , adversarial robustness ( Papernot et al. , 2016 ) , DNNs with logic rules ( Hu et al. , 2016 ) , visual relationship detection ( Yu et al. , 2017 ) , reinforcement learning ( Rusu et al. , 2015 ) , domain adaptation ( Asami et al. , 2017 ) and privacy ( Lopez-Paz et al. , 2015 ) . Our work adds to the list of applications in which distillation is effective . The theoretical motivation of distillation however is less established . Lopez-Paz et al . ( 2015 ) studied distillation as learning using privileged information ( Vapnik & Izmailov , 2015 ) . Phuong & Lampert ( 2019 ) establishes fast convergence of the expected risk of a distillation-trained linear classifier . Foster et al . ( 2019 ) provides a generalization bound for the student with an assumption that it learns a model close to the teacher . Dong et al . ( 2019 ) argued distillation has a similar effect as that of early stopping . Mobahi et al . ( 2020 ) showed an equivalence to increasing the regularization strength for kernel methods . Menon et al . ( 2020 ) establish a bias-variance trade-off for the student . Our analysis provides a new theoretical perspective of its relationship to churn reduction . 3 DISTILLATION FOR CONSTRAINING CHURN We are interested in a multiclass classification problem with an instance space X and a label space rms “ t1 , . . . , mu . Let D denote the underlying data distribution over instances and labels , and DX denote the corresponding marginal distribution over X . Let ∆m denote the pm ´ 1q-dimensional simplex with m coordinates . We will use p : XÑ∆m to denote the underlying conditional-class probabilities , where pypxq “ PpY “ y|X “ xq . We assume that we are provided a base classifier g : XÑ∆m that predicts a vector of probabilities gpxq P ∆m for any instance x . Our goal is to then learn a new classifier h : XÑ∆m , constraining it to have low predictive churn against g. We will measure the classification performance of a classifier h using a loss function ` : rms ˆ ∆mÑR ` that maps a label y P rms and prediction hpxq P ∆m to a non-negative number ` py , hpxqq , and denote the classification risk by Rphq : “ Epx , yq „ D r ` py , hpxqqs . We would ideally like to define predictive churn as the fraction of examples on which h and g disagree . For the purpose of designing a tractable algorithm , we will instead work with a softer notion of churn , which evaluates the divergence between their output distributions . To this end , we use a measure of divergence d : ∆m ˆ∆mÑ R ` , and denote the expected churn between h and g by Cphq : “ Ex „ DX rdpgpxq , hpxqqs . We then seek to minimize the classification risk for h , subject to the expected churn being within an allowed limit ą 0 : min h : XÑ∆m Rphq s.t . Cphq ď . ( 1 ) We consider loss and divergence functions that are defined in terms of a scoring function φ : ∆mÑRm ` that maps a distribution to a m-dimensional score . Specifically , we will consider scoring functions φ that are strictly proper ( Gneiting & Raftery , 2007 ; Williamson et al. , 2016 ) , i.e . for which , given any distribution u P ∆m , the conditional risk Ey „ u rφypvqs is uniquely minimized by v “ u . The following are general loss and divergence functions derived from φ : ` φpy , vq : “ φypvq ; dφpu , vq : “ ÿ yPrms uypφypvq ´ φypuqq . ( 2 ) e.g . cross-entropy , KL-divergence , and squared loss are special cases of this formulation . 3.1 BAYES-OPTIMAL CLASSIFIER We show below that for the loss and divergence functions defined in ( 2 ) , the optimal-feasible classifier for the constrained problem in ( 1 ) is a convex combination of the class probability function p and the base classifier g. Proposition 1 . Let p ` , dq be defined as in ( 2 ) for a strictly proper scoring function φ . Suppose φpuq is strictly convex in u . Then there exists λ˚ P r0 , 1s such that the following is an optimal-feasible classifier for ( 1 ) : h˚pxq “ λ˚ppxq ` p1´ λ˚qgpxq . Furthermore , if u ¨ φpuq is α-strongly concave over u P ∆m w.r.t . the Lq-norm , then λ˚ ď b 2 { ` αEx “ } ppxq ´ gpxq } 2q ‰˘ . The strong concavity condition in Proposition 1 is satisfied by the cross-entropy loss and KLdivergence for α “ 1 with the L1-norm , and by the squared loss and L2-distance for α “ 2 with the L2-norm . The bound suggests that the mixing coefficient λ˚ depends on how close the base classifier is to the class probability function p. Algorithm 1 Distillation-based Churn Reduction 1 : Inputs : Training sample S “ tpx1 , y1q , . . . , pxn , ynqu , Grid of mixing coefficients Λ “ tλ1 , . . . , λLu , Base classifier g , Constraint slack ą 0 2 : Train a classifier hk for each λk P Λ by minimizing the distilled loss in ( 4 ) : hk P argminhPH pLλkphq 3 : Find a convex combination of h1 , . . . , hL by solving following convex program in L variables : min hPcoph1 , ... , hLq pRphq s.t . pCphq ď and return the solution ph 3.2 DISTILLATION-BASED APPROACH Proposition 1 directly motivates the use of a distillation-based approach for solving the churnconstrained optimization problem in ( 1 ) . We propose treating the base classifier g as a teacher model , mixing the training labels y with scores from the teacher gpxq , and minimizing a classification loss against the transformed labels : Lλphq “ Epx , yq „ D rpλey ` p1´ λqgpxqq ¨ φphpxqqs , ( 3 ) where ey P t0 , 1um denotes a one-hot encoding of the label y P rms and φ is a strictly proper scoring function . It is straight-forward to show that when λ “ λ˚ , the optimal classifier for the above distillation loss takes the same form in Proposition 1 , i.e . h˚pxq “ λ˚ppxq ` p1´ λ˚qgpxq . While the optimal mixing parameter λ˚ is unknown , we propose treating this as a hyper-parameter and tuning it to reach the desired level of churn . In practice , we do not have direct access to the distribution D and will need to work with a sample S “ tpx1 , y1q , . . . , pxn , ynqu drawn from D. To this end , we define the empirical risk and the empirical churn as follows : pRphq “ 1 n n ÿ i “ 1 ` φpyi , hpxiqq ; pCphq “ 1 n n ÿ i “ 1 dφpgpxiq , hpxiqq , where ` φ and dφ are defined as in ( 2 ) for a scoring function φ . Our proposal is to then solve the following empirical risk minimization problem over a hypothesis class H Ă th : XÑ∆mu for different values of coefficient λk chosen from a finite grid tλ1 , . . . , λLu Ă r0 , 1s : hk P argminhPH pLλkphq : “ 1 n n ÿ i “ 1 pλkeyi ` p1´ λkqgpxiqq ¨ φphpxiqq . ( 4 ) To construct the final classifier , we find a convex combination of the L classifiers h1 , . . . , hL that minimizes pRphq while satisfying the constraint pCphq ď , and return an ensemble of the L classifiers . The overall procedure is outlined in Algorithm 1 , where we denote the set of convex combinations of classifiers h1 , . . . , hL by coph1 , . . . , hLq “ th : x ÞÑ řL j “ 1 αjhjpxq |α P ∆Lu . The post-processing step in Algorithm 1 amounts to solving a simple convex program in L variables . This is needed for technical reasons in our theoretical results , specifically , to translate a solution to a dual-optimal solution to ( 1 ) to a primal-feasible solution . In practice , however , we do not construct an ensemble , and instead simply return a single classifier that achieves the least empirical risk while satisfying the churn constraint . In our experiments , we use the cross-entropy loss for training , i.e . set φypuq “ ´ logpuyq . 4 THEORETICAL GUARANTEES We provide optimality and feasibility guarantees for the proposed algorithm and also explain why our approach is better-suited for optimizing accuracy ( subject to a churn constraint ) compared to the previous churn-reduction method of Fard et al . ( 2016 ) . 4.1 OPTIMALITY AND FEASIBILITY GUARANTEES We now show that the classifier ph returned by Algorithm 1 approximately satisfies the churn constraint , while achieving a risk close to that of the optimal-feasible classifier inH . This result assumes that we are provided with generalization bounds for the classification risk and churn . Theorem 2 . Let the scoring function φ : ∆mÑRm ` be convex , and } φpzq } 8 ă B , @ z P ∆m . Let the set of classifiersH be convex , with the base classifier g P H. Suppose C and R enjoy the following generalization bounds : for any δ P p0 , 1q , w.p . ě 1´ δ over draw of S „ Dn , for any h P H , |Rphq ´ pRphq| ď ∆Rpn , δq ; |Cphq ´ pCphq| ď ∆Cpn , δq , for some ∆Rpn , δq and ∆Cpn , δq that is decreasing in n and approaches 0 as nÑ8 . Let rh be an optimal-feasible classifier in H , i.e . Cprhq ď and Rprhq ď Rphq for all classifiers h for which Cphq ď . Let ph be the classifier returned by Algorithm 1 with Λ “ maxt ` 2B , uu ˇ ˇu P t 1L , 2 L , . . . , 1u ( for some L P N ` . For any δ P p0 , 1q , w.p . ě 1´ δ over draw of S „ Dn , Optimality : Rpphq ď Rprhq ` O `` 1 ` 2B ˘ ` ∆Rpn , δq ` ∆Cpn , δq ` BL ˘˘ , Feasibility : Cpphq ď ` ∆C pn , δq . In practice , we expect the churn metric to generalize better than the classification risk , i.e . for ∆Cpn , δq to be smaller than ∆Rpn , δq . This is because the classification risk is computed on “ hard ” labels y P rms from the training sample , whereas the churn metric is computed on “ soft ” labels gpxq P ∆m from the base model . The traditional view of distillation ( Hinton et al. , 2015 ) suggests that the soft labels from a teacher model come with confidence scores for each example , and thus allow the student to generalize well to unseen new examples . A similar view is also posed by Menon et al . ( 2020 ) , who argue that the soft labels from the teacher have “ lower variance ” than the hard labels from the training sample , and therefore aid in better generalization of the student . See Appendix B for explicit bounds on ∆Cpn , δq and ∆Rpn , δq . In fact , for certain base classifiers g , generalizing well on “ churn ” can have the additional benefit of improving classification performance , as shown by the proposition below . Proposition 3 . Let p ` , dq be defined as in ( 2 ) for a strictly proper scoring function φ . Suppose φpuq is strictly convex in u , Φ-Lipschitz w.r.t . L1-norm for each y P rms , and } φpzq } 8 ă B , @ z P ∆m . Let λ˚ be the optimal mixing coefficient defined in Proposition 1 . Let ∆Cpn , δq be the churn generalization bound defined in Theorem 2 . Let rh be an optimal feasible classifier inH and ph be the classifier returned by Algorithm 1 . Then for any δ P p0 , 1q , w.p . ě 1´ δ over draw of S „ Dn : Rpphq ´ Rprhq ď ` ∆Cpn , δq ` pB ` Φλ˚qEx „ DX r } ppxq ´ gpxq } 1s . This result bounds the excess classification risk in terms of the churn generalization bound and the expected difference between the base classifier g and the underlying class probability function p. When the base classifier is close to p , low values of ∆Cpn , δq result in low classification risk . 4.2 ADVANTAGE OVER ANCHOR LOSS We next compare our distillation loss in ( 3 ) with the previous anchor loss of Fard et al . ( 2016 ) , which uses the base model ’ s prediction only when it agrees with the original label , and uses a scaled version of the original label otherwise . While originally proposed for churn reduction with binary labels , we provide below an analogous version of this loss for a multiclass setup : Lancphq “ Epx , yq „ D ra ¨ φphpxqqs , ( 5 ) where a “ `` αgpxq ` p1´ αqey if y “ argmaxk gkpxq ηey otherwise , for hyper-parameters α , η P r0 , 1s and a strictly proper scoring function φ . Here , we have used argmax to denote ties being broken in favor of the larger class . While this helps us simplify the exposition , our results can be easily extended to a version of the loss which includes ties . The anchor loss does not take into account the confidence with which the base model disagrees with the sampled label y . For example , if the base model predicts near-equal probabilities for all classes , but happens to assign a slightly higher probability to a class different from y , the anchor loss would still completely ignore the base model ’ s score ( even though it might be the case that all the labels are indeed equally likely to occur ) . In some cases , this selective use of the teacher labels can result in a biased objective and may hurt the classifier ’ s accuracy . To see this , consider an ideal scenario where the base model predicts the true conditional-probabilities ppxq and the student hypothesis class is universal . In this case , minimizing the churn w.r.t . the base model has the effect of maximizing classification accuracy , i.e . a classifier that has zero churn w.r.t . the base model also produces the least classification error . However , as shown below , even in this ideal setup , minimizing the anchor loss may result in a classifier different from the base model . Proposition 4 . When gpxq “ ppxq , @ x , for any given λ P r0 , 1s , the minimizer for the distillation loss in ( 3 ) over all classifiers h is given by : h˚pxq “ ppxq , whereas the minimizer of the anchor loss in ( 5 ) is given by : h˚j pxq “ zj ř j zj where zj “ '' αp2j pxq ` p1´ αqpjpxq if j “ argmaxk pkpxq pη ` αmaxk pkpxqq pjpxq otherwise . Unless α “ 0 and η “ 1 ( which amounts to completely ignoring the base model ) or the base model makes hard predictions on all points , i.e . pjpxq P t0 , 1u , @ x , the anchor loss encourages scores that differ from the base model p. For example , when α “ η “ 1 ( and the base model makes soft predictions ) , the anchor loss has the effect of down weighting the label that the base model is most confident about , and as a result , encourages lower scores on that label and higher scores on all other labels . While one can indeed tweak the two hyper-parameters to reduce the gap between the learned classifier and the base model , our proposal requires only one hyper-parameter λ , which represents an intuitive trade-off between the one-hot and teacher labels . In fact , irrespective of the choice of λ , the classifier that minimizes our distillation loss in Proposition 4 mimics the base model p exactly , and as a result , achieves both zero churn and optimal accuracy . We shall see in the next section that even on real-world datasets , where the base classifier does not necessarily make predictions close to the true class probabilities ( and where the student hypothesis class is not necessarily universal and of limited capacity ) , our proposal performs substantially better than the anchor loss in minimizing churn at a particular accuracy . Figure 3 provides a further ablation study , effectively interpolating between the anchor and distillation methods , and provides evidence that using the true ( hard ) label instead of the teacher ( soft ) label can steadily degrade performance . 5 EXPERIMENTS We now show empirically that distillation is an effective method to train models for both accuracy and low churn . We test our method across a large number of datasets and neural network architectures . 5.1 SETUP Datasets and architectures : The following are the datasets we use in our experiments , along with the associated model architectures : • 12 OpenML datasets using fully-connected neural networks . • 10 MNIST variants , SVHN , CIFAR10 , 40 CelebA tasks using convolutional networks . • CIFAR10 and CIFAR100 with ResNet-50 , ResNet-101 , and ResNet-152 . • IMDB dataset using transformer network . For each architecture ( besides ResNet ) , we use 5 different sizes . For the fully connected network , we use a simple network with one-hidden layer of 10 , 102 , 103 , 104 , and 105 units , which we call fcn-x where x is the respective size of the hidden layer . For the convolutional neural network , we start with the LeNet5 architecture ( LeCun et al. , 1998 ) and scale the number of hidden units by a factor of x for x “ 1 , 2 , 4 , 8 , 16 , which we call ConvNet-x for the respective x . Finally , we use the basic transformer architecture from Keras tutorial ( Keras , 2020 ) and scale the number of hidden units by x for x “ 1 , 2 , 4 , 8 , 16 , which we call Transformer-x for the respective x . Code for the models in Keras can be found in the Appendix . For each dataset , we use the standard train/test split if available , otherwise , we fix a random train/test split with ratio 2:1 . Setup : For each dataset and neural network , we randomly select from the training set 1000 initial examples , 100 validation examples , and a batch of 1000 examples , and train an initial model using Adam optimizer with default settings on the initial set and early stopping ( i.e . stop when there ’ s no improvement on the validation loss after 5 epochs ) and default random initialization , and use that model as the base model . Then , for each baseline , we train on the combined initial set and batch ( 2000 datapoints ) , again using the Adam optimizer with default settings and the same early stopping scheme and calculate the accuracy and churn against the base model on the test set . We average across 100 runs and provide the error bands in the Appendix . For all the datasets except the OpenML datasets , we also have results for the case of 10000 initial examples , 1000 validation examples , and a batch 1000 . We also show results for the case of 100 initial samples , 1000 validation examples , and a batch of 1000 for all of the datasets . Due to space , we show those results in the Appendix . We ran our experiments on a cloud environment . For each run , we used a NVIDIA V100 GPU , which took up to several days to finish all 100 trials . Baselines : We test our method against the following baselines . ( 1 ) Cold start , where we train the model from scratch with the default initializer . ( 2 ) Warm start , where we initialize the model ’ s parameters to that of the base model before training . ( 3 ) Shrink-perturb ( Ash & Adams , 2019 ) , which is a method designed to improve warm-starting by initializing the model ’ s weights to α ¨θbase ` p1´αq¨θinit before training , where θbase are the weights of the base model , θinit is a randomly initialized model , and α is a hyperparameter we tune across t0.1 , 0.2 , ... , 0.9u . ( 4 ) Mixup ( Zhang et al. , 2017 ) ( a baseline suggested for a different notion of churn ( Bahri & Jiang , 2021 ) ) , which trains an convex combinations of pairs of datapoints . We search over its hyperparameter α P t0.1 , ... , 0.9u , as defined in Zhang et al . ( 2017 ) . ( 5 ) Label smoothing ( Szegedy et al. , 2016 ) , which was suggested by Bahri & Jiang ( 2021 ) for the variance notion of churn , proceeds by training on a convex combination between the original labels and the base models ’ soft prediction . We tune across the convex combination weight α P t0.1 , 0.2 , ... , 0.9u . ( 6 ) Co-distillation ( Anil et al. , 2018 ) , which was proposed for the variance notion of churn , where we train two warm-started networks that train simultaneously on a loss that is a convex combination on the original loss and a loss on the difference between their predictions . We tune across the convex combination weight α P t0.1 , 0.2 , ... , 0.9u . ( 7 ) Anchor ( Fard et al. , 2016 ) , which as noted in Section 4.2 , proceeds by optimizing the cross-entropy loss on a modified label : we use the label αgpxq ` p1´αqey when the base model g agrees with the true label y , and ηey otherwise . We tune across α P t0.1 , 0.2 , ... , 0.9u and η P t0.5 , 0.7 , 1u . For distillation , we tune the trade-off parameter λ across t0.1 , 0.2 , ... , 0.9u . Metric : All of the methods will produce a model that we evaluate for both accuracy and churn with respect to the base model on the test set . We consider the hard notion of churn , which measures the average difference in hard predictions w.r.t . the base classifier on a test set . We will see later that there is often-times a trade-off between accuracy and churn , and in an effort to produce one metric for quantitative evaluation , we propose churn at cold accuracy metric , which is defined as follows . Each baseline produces a set of models ( one for each hyperparameter setting ) . We take the averaged churn and accuracy across the 100 runs and choose the model with the lowest churn that is at least as accurate as the cold-start model ( it ’ s possible that no such model exists for that method ) . This way , we can identify the method that delivers the lowest churn but still performs at least as well as if we trained on the updated dataset in a vanilla manner . We believe this metric is practically relevant as a practitioner is unlikely to accept a reduction in accuracy to reduce churn . 5.2 RESULTS The detailed results for the following experiments can be found in the Appendix . Given space constraints , we only provide a high level summary in this section OpenML datasets with fully-connected networks : In Table 1 we show the results for the OpenML datasets using the fcn-1000 network . We see that distillation performs the well across the board , and for the other fully connected network sizes , distillation is the best in the majority of cases ( 84 % of the time for initial batch size 1000 and 52 % of time for initial batch size 100 ) . MNIST variants , SVHN , and CIFAR10 with convolutional networks : In Table 2 , we show the results for 10 MNIST variants , SVHN and CIFAR10 using convnet-4 . We see that distillation performs strongly across the board . We found that distillation performs best in 84 % of combinations between dataset and network . When we increase the initial sample size to 10000 and keep the batch size fixed at 1000 , then we found that label smoothing starts becoming competitive with distillation , where distillation is best 64 % of the time , and label smoothing wins by a small margin all other times . We only saw this phenomenon for a handful of the MNIST variants , which suggests that label smoothing may be especially effective in these situations . When we decreased the initial sample down to 100 and kept the batch size the same , we found that distillation was best 48 % of the time , with Anchor being the second best method winning 24 % of the time . For SVHN and CIFAR10 , of the 10 combinations , distillation performs the best on all 10 out of the 10 . If we increased the initial sample size to 10000 and kept the batch size fixed at 1000 , then we find that distillation still performs the best all 10 out of 10 combinations . If we decreased the initial sample size to 100 and kept the same batch size , then distillation performs the best on 8 out of the 10 combinations . CelebA with convolutional networks : Across all 200 combinations of task and network , distillation performs the best 79 % of the time . Moreover , if we increased the initial sample size to 10000 and kept the batch size fixed at 1000 , distillation is even better , performing the best 91.5 % of the time . If we decreased the initial sample size to 100 , then distillation is best 96 % of the time . CIFAR10 and CIFAR100 with ResNet : Due to the computational costs , we only run these experiments for initial sample size 1000 . In all cases ( across ResNet-50 , ResNet-101 and ResNet-152 ) , we see that distillation outperforms the other baselines . IMDB with transformer network : We experimented for initial batch size 100 , 1000 , and 10000 . We found that distillation performed the best the majority of the time , where the only notable weak performance was in some instances where no baselines were even able to reach the accuracy of the cold starting method . In Figure 2 we show the Pareto frontiers of the various baselines as well as plotting cost of each method as we vary the trade-off between accuracy and churn . We see that not only does distillation do well in churn , but it performs the best at any trade-off between churn and accuracy for the cases shown . 6 CONCLUSION We have proposed knowledge distillation as a new practical solution to churn reduction , and provided both theoretical and empirical justifications for the approach . Some directions for future work include investigating the interplay between churn and fairness . One way this connection may arise is that in trying to lower churn , some slices of the data may be disproportionately affected . Another way is if the teacher itself contains harmful biases , in which it makes sense to control what kind of churn should be minimized . Further work may also include combining distillation with other methods to achieve even stronger results . Reproducibility Statement : All details of experimental setup are in the main text , along with descriptions of the baselines and what hyperparameters were swept across . 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Hongyi Zhang , Moustapha Cisse , Yann N Dauphin , and David Lopez-Paz . mixup : Beyond empirical risk minimization . arXiv preprint arXiv:1710.09412 , 2017 . A PROOFS A.1 PROOF OF PROPOSITION 1 Proposition ( Restated ) . Let p ` , dq be defined as in ( 2 ) for a strictly proper scoring function φ . Suppose φpuq is strictly convex in u . Then there exists λ˚ P r0 , 1s such that the following is an optimal-feasible classifier for ( 1 ) : h˚pxq “ λ˚ppxq ` p1´ λ˚qgpxq . Furthermore , if u ¨ φpuq is α-strongly concave over u P ∆m w.r.t . the Lq-norm , then λ˚ ď d 2 αEx “ } ppxq ´ gpxq } 2q ‰ . Proof . Let h˚ denote an optimal feasible solution for ( 1 ) . We first note that Rphq “ Ex , y r ` py , hpxqqs “ Ex “ Ey|x r ` py , hpxqqs ‰ “ Ex ” ÿ iPrms pipxqφiphpxqq ı and Cphq “ Ex ” ÿ iPrms gipxq pφiphpxqq ´ φipgpxqqq ı . Because φi is strictly convex in its argument , both Rphq and Cphq are strictly convex in h. In other words , for any α P r0 , 1s , and classifiers h1 , h2 , Rpαh1 ` p1´ αqh2q ă αRph1q ` p1´ αqRph2q , and similarly for C. Furthermore because Cpgq “ 0 ă , the constraint is strictly feasible , and hence strong duality holds for ( 1 ) ( as a result of Slater ’ s condition being satisfied ) . Therefore ( 1 ) can be equivalently formulated as a max-min problem : max µPR ` min h Rphq ` µCphq , for which there exists a µ˚ P R ` such that pµ˚ , h˚q is a saddle point . The strict convexity of Rphq and Cphq gives us that h˚ is the unique minimizer of Rphq ` µ˚Cphq . Setting λ˚ “ 11 ` µ˚ , we equivalently have that h˚ is a unique minimizer of the weighted objective λ˚Rphq ` p1´ λ˚qCphq . We next show that the minimizer h˚ is of the required form . Expanding the R and C , we have : λ˚Rphq ` p1´ λ˚qCphq “ Ex ” ÿ iPrms ` λ˚pipxq ` p1´ λ˚qgipxq ˘ φiphpxqq ´ p1´ λ˚qgipxqφipgpxqq ı “ Ex ” ÿ iPrms ` λ˚pipxq ` p1´ λ˚qgipxq ˘ φiphpxqq ı ` a term independent of h “ Ex ” ÿ iPrms p̄ipxqφiphpxqq ı ` a term independent of h , ( 6 ) where p̄pxq “ λ˚ppxq ` p1´ λ˚qgpxq . Note that it suffices to minimize ( 6 ) point-wise , i.e . to choose h˚ so that the term within the expectation ř iPrms p̄ipxqφiphpxqq is minimized for each x . For a fixed x , the inner term is minimized when h˚pxq “ p̄pxq . This is because of our assumption that φ is a strictly proper scoring function , i.e . for any distribution u , the weighted loss ř i uiφipvq is uniquely minimized by v “ u . Therefore ( 6 ) is minimized by h˚pxq “ p̄pxq “ λ˚ppxq ` p1´ λ˚qgpxq . To bound λ˚ , we use a result from Williamson et al . ( 2016 ) ; Agarwal ( 2014 ) to lower bound Cphq in terms of the norm difference } hpxq ´ gpxq } q . Define Qpuq “ infvP∆m u ¨ φpvq . Because φ is a proper scoring function , the infimum is attained at v “ u . Therefore Qpuq “ u ¨ φpuq , which recall is assumed to be strongly concave . Also , note thatQpuq “ infvP∆m u ¨φpvq is an infimum of “ linear ” functions in u , and therefore ∇Qpuq “ φpuq is a super-differential for Q at u . See Proposition 7 in Williamson et al . ( 2016 ) for more details . We now re-write Cphq in terms of Q and lower bound it using the strong concavity property : Cphq “ Ex ” gpxq ¨ pφphpxqq ´ φpgpxqqq ı “ Ex ” hpxq ¨ φphpxqq ` pgpxq ´ hpxqq ¨ φphpxqq ´ gpxq ¨ φpgpxqq ı “ Ex ” Qphpxqq ` pgpxq ´ hpxqq ¨∇Qphpxqq ´Qpgpxqq ı ě Ex ” α 2 } hpxq ´ gpxq } 2q ı , where the last step uses the fact that Q is α-strongly concave over u P ∆m w.r.t . the Lq-norm . Since the optimal scorer h˚ satisfies the coverage constraint Cph˚q ď , we have from the above bound Ex ” α 2 } h˚pxq ´ gpxq } 2q ı ď . Substituting for h˚ , we have : Ex „ pλ˚q2α 2 } ppxq ´ gpxq } 2q ď , or pλ˚q2 ď 2 αEx “ } ppxq ´ gpxq } 2q ‰ , which gives us the desired bound on λ˚ . A.2 PROOF OF THEOREM 2 Theorem ( Restated ) . Let the scoring function φ : ∆mÑRm ` be convex , and } φpzq } 8 ă B , @ z P ∆m . Let the set of classifiers H be convex , with the base classifier g P H. Suppose C and R enjoy the following generalization bounds : for any δ P p0 , 1q , w.p . ě 1 ´ δ over draw of S „ Dn , for any h P H , |Rphq ´ pRphq| ď ∆Rpn , δq ; |Cphq ´ pCphq| ď ∆Cpn , δq , for some ∆Rpn , δq and ∆Cpn , δq that is decreasing in n and approaches 0 as nÑ8 . Let rh be an optimal-feasible classifier in H , i.e . Cprhq ď and Rprhq ď Rphq for all classifiers h for which Cphq ď . Let ph be the classifier returned by Algorithm 1 with Λ “ maxt ` 2B , uu ˇ ˇu P t 1L , 2 L , . . . , 1u ( for some L P N ` . For any δ P p0 , 1q , w.p . ě 1´ δ over draw of S „ Dn , Optimality : Rpphq ď Rprhq ` O `` 1 ` 2B ˘ ` ∆Rpn , δq ` ∆Cpn , δq ` BL ˘˘ , Feasibility : Cpphq ď ` ∆C pn , δq . We first note that because } φpzq } 8 ă B , @ z P ∆m , both pRphq ă B and pCphq ă B . Also , because φi is convex , both pRphq and pCphq are convex in h. In other words , for any α P r0 , 1s , and classifiers h1 , h2 , pRpαh1 ` p1 ´ αqh2q ď α pRph1q ` p1 ´ αq pRph2q , and similarly for pC . Furthermore , the objective in ( 4 ) can be decomposed into a convex combination of the empirical risk and churn : pLλphq “ 1 n n ÿ i “ 1 pλeyi ` p1´ λqgpxiqq ¨ φphpxiqq “ λ pRphq ` p1´ λq pCphq ` 1´ λ n n ÿ i “ 1 gpxiq ¨ φpgpxiqq . Therefore minimizing pLλphq is equivalent to minimizing the Lagrangian function rLλphq “ λ pRphq ` p1´ λqp pCphq ´ q ( 7 ) over h. Moreover , each hk minimizes rLλkphq . We also note that the churn-constrained optimization problem in ( 1 ) can be posed as a Lagrangian game between a player that seeks to minimize the above Lagrangian over h and a player that seeks to maximize the Lagrangian over λ . The next two lemmas show that Algorithm 1 can be seen as finding an approximate equilibrium of this two-player game . Lemma 5 . Let the assumptions on φ and H in Theorem 2 hold . Let ph be the classifier returned by Algorithm 1 when Λ is set to Λ “ maxt ` 2B , uu |u P t 1 L , . . . , 1u ( of the range r ` 2B , 1s for some L P N ` . Then there exists a bounded Lagrange multiplier λ̄ P r ` 2B , 1s such that pph , λ̄q forms an equilibrium of the Lagrangian min-max game : λ̄ pRpphq ` p1´ λ̄qp pCpphq ´ q “ min hPcoph1 , ... , hLq λ̄ pRphq ` p1´ λ̄qp pCphq ´ q max λPr0,1s p1´ λqp pCpphq ´ q “ p1´ λ̄qp pCpphq ´ q . Proof . The classifier ph returned by Algorithm 1 is a solution to the following constrained optimization problem over the convex-hull of the classifiers h1 , . . . , hL : min hPcoph1 , ... , hLq pRphq s.t . pCphq ď . Consequently , there exists a λ̄ P r0 , 1s such that : λ̄ pRpphq ` p1´ λ̄qp pCpphq ´ q “ min hPcoph1 , ... , hLq λ̄ pRphq ` p1´ λ̄qp pCphq ´ q ( 8 ) max λPr0,1s p1´ λqp pCpphq ´ q “ p1´ λ̄qp pCpphq ´ q . ( 9 ) To see this , note that the KKT conditions ( along with the convexity of R and C ) give us that there exists a Lagrange multiplier µ̄ ě 0 such that ph P argmin hPcoph1 , ... , hLq pRphq ` µ̄p pCphq ´ q ( stationarity ) µ̄p pCpphq ´ q “ 0 ( complementary slackness ) . When pCpphq ď , µ̄ “ 0 , and so ( 8 ) and ( 9 ) are satisfied for λ̄ “ 1 . When pCpphq “ , then ( 8 ) and ( 9 ) are satisfied for λ̄ “ 11 ` µ̄ . It remains to show that that λ̄ P r ` 2B , 1s . For this , we first show that there exists a h 1 P coph1 , . . . , hLq such that pCph1q ď { 2 . To see why , pick h1 to be the minimizer of the Lagrangian rLλphq over all h P H for λ “ ` 2B . Because rLλph 1q ď rLλpgq ď λB ´ p1´ λq , where g is the base classifier that we have assumed is inH , it follows that pCph1q ď λ1´λB ď { 2 . Next , by combining ( 8 ) and ( 9 ) , we have λ̄ pRpphq ` max λPr0,1s p1´ λqp pCpphq ´ q “ min hPcoph1 , ... , hLq λ̄ pRphq ` p1´ λ̄qp pCphq ´ q . Lower bounding the LHS by setting λ “ 1 and upper bounding the RHS by setting h “ h1 , we get : λ̄ pRpphq ď λ̄ pRph1q ´ p1´ λ̄q 2 , which gives us : { 2 ď λ̄p { 2 ` pRph1q ´ pRpphqq ď λ̄p { 2 ` Bq . Hence λ̄ ě ` 2B , which completes the proof . Lemma 6 . Let ph be the classifier returned by Algorithm 1 when Λ is set to Λ “ maxt ` 2B , uu |u P t 1L , . . . , 1u ( of the range r ` 2B , 1s for some L P N ` . Fix δ P p0 , 1q . Suppose R and C satisfy the generalization bounds in Theorem 2 with error bounds ∆Rpn , δq and ∆Cpn , δq respectively . Then there exists a bounded Lagrange multiplier pλ P r ` 2B , 1s such that pph , pµq forms an approximate equilibrium for the Lagrangian min-max game , i.e . w.p . ě 1´ δ over draw of sample S „ Dn , pλRpphq ` p1´ pλqpCpphq ´ q ď min hPH pλRphq ` p1´ pλqpCphq ´ q ` O p∆Rpn , δq ` ∆Cpn , δq ` B { Lq ( 10 ) and max λPr0,1s p1´ λqpCpphq ´ q ď p1´ pλqpCpphq ´ q ` O p∆Cpn , δq ` B { Lq . ( 11 ) Proof . We have from Lemma 5 that there exists λ̄ P r ` 2B , 1s such that λ̄ pRpphq ` p1´ λ̄qp pCpphq ´ q “ min hPcoph1 , ... , hLq λ̄ pRphq ` p1´ λ̄qp pCphq ´ q ( 12 ) max λPr0,1s p1´ λqp pCpphq ´ q “ p1´ λ̄qp pCpphq ´ q . ( 13 ) Algorithm 1 works with a discretization Λ “ maxt ` 2B , uu |u P t 1 L , . . . , 1u ( of the range r ` 2B , 1s . Allowing pλ to denote the closest value to λ̄ in this set , we have from ( 12 ) : pλ pRpphq ` p1´ pλqp pCpphq ´ q ď min hPcoph1 , ... , hLq pλ pRphq ` p1´ pλqp pCphq ´ q ` 4B L “ min hPH pλ pRphq ` p1´ pλqp pCphq ´ q ` 4B L , ( 14 ) where the last step follows from the fact that coph1 , . . . , hLq Ď H and each hk was chosen to minimize p1´ λkq pRphq ` λkp pCphq ´ q for λk P Λ . Similarly , we have from ( 13 ) , max λPr0,1s p1´ λqpCpphq ´ q ď p1´ pλqpCpphq ´ q ` B L . ( 15 ) What remains is to apply the generalization bounds for R and C to ( 14 ) and ( 15 ) . We first bound the LHS of ( 14 ) . We have with probability at least 1´ δ over draw of S „ Dn : pλ pRpphq ` p1´ pλqp pCpphq ´ q ě pλRpphq ` p1´ pλqpCpphq ´ q ´ pλ∆R pn , δq ´ p1´ pλq∆C pn , δq ě pλRpphq ` p1´ pλqpCpphq ´ q ´ ∆R pn , δq ´ ∆C pn , δq , ( 16 ) where the last step uses the fact that 0 ď pλ ď 1 . For the RHS , we have with the same probability : min hPH ! pλ pRphq ` p1´ pλqp pCphq ´ q ) ` 4B { L ď min hPH ! pλRphq ` p1´ pλqpCphq ´ q ` 4B { L ` pλ∆R pn , δq ` p1´ pλq∆C pn , δq ) ď min hPH ! pλRphq ` p1´ pλqpCphq ´ q ) ` 4B { L ` ∆R pn , δq ` ∆C pn , δq , where we again use 0 ď pλ ď 1 . Combining ( 14 ) with ( 16 ) and ( 17 ) completes the proof for the first part of the lemma . Applying the generalization bounds to ( 15 ) , we have with the same probability : B { L ě max λPr0,1s p1´ λqp pCpphq ´ q ´ p1´ pλqp pCpphq ´ q ě max λPr0,1s ! p1´ λqpCpphq ´ q ´ p1´ λq∆C pn , δq ) ´ p1´ pλqpCpphq ´ q ´ p1´ pλq∆C pn , δq ě max λPr0,1s p1´ λqpCpphq ´ q ´ p1´ pλqpCpphq ´ q ´ 2∆C pn , δq , which completes the proof for the second part of the lemma . We are now ready to prove Theorem 2 . Proof of Theorem 2 . To show optimality , we combine ( 10 ) and ( 11 ) and get : pλ pRpphq ` max λPr0,1s p1´ λqpCpphq ´ q ď min hPH pλ pRphq ` p1´ pλqp pCphq ´ q ` O ´ ∆Rpn , δq ` ∆Cpn , δq ` B { L ¯ . ( 17 ) We then lower bound the LHS in ( 17 ) by setting λ “ 1 and upper bound the RHS by setting h to the optimal feasible solution rh , giving us : pλRpphq ď pλRprhq ` p1´ pλqp0q ` O ´ ∆R pn , δq ` ∆C pn , δq ` BL ¯ . Dividing both sides by pλ , Rpphq ď Rprhq ` 1 pλ O ´ ∆R pn , δq ` ∆C pn , δq ` BL ¯ . Lower bounding pλ by ` 2B gives us the desired optimality result . The feasibility result directly follows from the fact that Algorithm 1 chooses a ph that satisfies the empirical churn constraint pCpphq ď , and from the generalization bound for C. A.3 PROOF OF PROPOSITION 3 Proposition ( Restated ) . Let p ` , dq be defined as in ( 2 ) for a strictly proper scoring function φ . Suppose φpuq is strictly convex in u , Φ-Lipschitz w.r.t . L1-norm for each y P rms , and } φpzq } 8 ă B , @ z P ∆m . Let λ˚ be the optimal mixing coefficient defined in Proposition 1 . Let ∆Cpn , δq be the churn generalization bound defined in Theorem 2 . Let rh be an optimal feasible classifier inH and ph be the classifier returned by Algorithm 1 . Then for any δ P p0 , 1q , w.p . ě 1´ δ over draw of S „ Dn : Rpphq ´ Rprhq ď ` ∆Cpn , δq ` pB ` Φλ˚qEx „ DX r } ppxq ´ gpxq } 1s . Proof . Let h˚ be the Bayes optimal classifier , i.e . the optimal-feasible classifier over all classifiers ( not just those inH ) . We have : Rpphq ´ Rprhq ď Rpphq ´ Rph˚q “ Ex „ DX ” Ey|x ” ey ¨ φpphpxqq ıı ´ Ex „ DX “ Ey|x rey ¨ φph˚pxqqs ‰ “ Ex „ DX ” ppxq ¨ φpphpxqq ı ´ Ex „ DX rppxq ¨ φph˚pxqqs “ Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ´ Ex „ DX rppxq ¨ pφph˚pxqq ´ φpgpxqqqs ď Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` ˇ ˇ ˇ ˇ ˇ ˇ Ex „ DX » – ÿ yPrms pypxq pφyph˚pxqq ´ φypgpxqqq fi fl ˇ ˇ ˇ ˇ ˇ ˇ ď Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` Ex „ DX » – ÿ yPrms pypxq ˇ ˇφyph˚pxqq ´ φypgpxqq ˇ ˇ fi fl ď Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` ΦEx „ DX » – ÿ yPrms pypxq } h˚pxq ´ gpxq } 1 fi fl ď Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` ΦEx „ DX r } h˚pxq ´ gpxq } 1s , where the second-last step follows from Jensen ’ s inequality , and the last step uses the Lipschitz assumption on φy . We further have : Rpphq ´ Rprhq ď Ex „ DX ” gpxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` Ex „ DX ” pppxq ´ gpxqq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` ΦEx „ DX r } h˚pxq ´ gpxq } 1s ď Ex „ DX ” gpxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` Ex „ DX ” } ppxq ´ gpxq } 1 } φpphpxqq ´ φpgpxqq } 8 ı ` ΦEx „ DX r } h˚pxq ´ gpxq } 1s ď Ex „ DX ” ppxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` BEx „ DX r } ppxq ´ gpxq } 1s ` ΦEx „ DX r } h˚pxq ´ gpxq } 1s ď Ex „ DX ” gpxq ¨ ´ φpphpxqq ´ φpgpxqq ¯ı ` BEx „ DX r } ppxq ´ gpxq } 1s ` λ˚ΦEx „ DX r } ppxq ´ gpxq } 1s “ Cpphq ` pB ` λ˚ΦqEx „ DX r } ppxq ´ gpxq } 1s , where the second step applies Hölder ’ s inequality to each x , the third step follows from the boundedness assumption on φ , and the fourth step uses the characterization h˚pxq “ λ˚ppxq ` p1´λ˚qgpxq , for λ˚ P r0 , 1s from Proposition 1 . Applying Theorem 2 to the churn Cpphq completes the proof . A.4 PROOF OF PROPOSITION 4 Proposition ( Restated ) . When gpxq “ ppxq , @ x , for any given λ P r0 , 1s , the minimizer for the distillation loss in ( 3 ) over all classifiers h is given by : h˚pxq “ ppxq , whereas the minimizer of the anchor loss in ( 5 ) is given by : h˚j pxq “ zj ř j zj where zj “ '' αp2j pxq ` p1´ αqpjpxq if j “ argmaxk pkpxq p ` αmaxk pkpxqq pjpxq otherwise . Proof . For the first part , we expand ( 3 ) with gpxq “ ppxq , and have for any λ P r0 , 1s , Lλphq “ Epx , yq „ D rpλey ` p1´ λqppxqq ¨ φphpxqqs ( 18 ) “ λEpx , yq „ D rey ¨ φphpxqqs ` p1´ λqEx „ DX rppxq ¨ φphpxqqs “ λEx „ DX “ Ey|x reys ¨ φphpxqq ‰ ` p1´ λqEx „ DX rppxq ¨ φphpxqqs “ λEx „ DX rppxq ¨ φphpxqqs ` p1´ λqEx „ DX rppxq ¨ φphpxqqs “ Ex „ DX rppxq ¨ φphpxqqs . ( 19 ) For a fixed x , the inner term in ( 19 ) is minimized when h˚pxq “ ppxq . This is because of our assumption that φ is a strictly proper scoring function , i.e . for any distribution u , the weighted loss ř i uiφipvq is uniquely minimized by v “ u . Therefore ( 19 ) is minimized by h˚pxq “ ppxq , @ x . For the second part , we expand ( 5 ) with gpxq “ ppxq , and have : Lancphq “ Epx , yq „ D ra ¨ φphpxqqs , where a “ `` αppxq ` p1´ αqey if y “ argmaxk pkpxq ey otherwise , For a given x , let us denote jx “ argmaxk pkpxq . We then have : Lancphq “ Epx , yq „ D rp1py “ jxq pαppxq ` p1´ αqeyq ` 1py ‰ jxqeyq ¨ φphpxqqs “ Ex „ DX “ Ey|x rp1py “ jxq pαppxq ` p1´ αqeyq ` 1py ‰ jxqeyq ¨ φphpxqqs ‰ “ Ex „ DX « ÿ k pkpxq pp1pk “ jxq pαppxq ` p1´ αqekq ` 1pk ‰ jxqekq ¨ φphpxqqq ff “ Ex „ DX « pjxpxq pαppxq ` p1´ αqejxq ¨ φphpxq ` ÿ k‰jx pkpxqφkphpxqq ff “ Ex „ DX „ pjxpxq pαpjxpxq ` p1´ αqqφjxphpxqq ` pjxpxq ÿ k‰jx αpkpxqφkphpxqq ` ÿ k‰jx pkpxqφkphpxqq “ Ex „ DX « pjxpxq pαpjxpxq ` p1´ αqqφjxphpxqq ` pαpjxpxq ` q ÿ k‰jx pkpxqφkphpxqq ff “ Ex „ DX rrppxq ¨ φphpxqqs , ( 20 ) where rpspxq “ `` αp2spxq ` p1´ αqpspxq if s “ jx pαpjxpxq ` qpspxq otherwise “ `` αp2spxq ` p1´ αqpspxq if s “ argmaxk pkpxq pαmaxk pkpxq ` qpspxq otherwise . For a fixed x , the inner term in ( 20 ) is minimized when h˚pxq “ 1Zpxqrppxq , where Zpxq “ ř k rpkpxq . This follows from the fact that for a fixed x , the minimizer of the inner term rppxq ¨ φphpxqq is the same as the the minimizer of the scaled term 1Zpxqrppxq ¨ φphpxqq , and from φ being a strictly proper scoring function . This completes the proof . B GENERALIZATION BOUNDS We adapt a result from Menon et al . ( 2020 ) to provide generalization bounds for the classification risk and churn in terms of the empirical variance of the loss and churn values . Proposition 7 ( Generalization bound for classification risk ) . Let the scoring function φ : ∆mÑRm ` be bounded . Let Vφ Ď RX denote the class of loss functions vpx , yq “ ` φpy , hpxqq “ φyphpxqq induced by classifiers h P H. LetMRn “ N8p 1n , Vφ , 2nq denote the uniform L8 covering number for Vφ . Fix δ P p0 , 1q . Then with probability ě 1´ δ over draw of S „ Dn , for any h P H : Rphq ď pRphq ` O ˜ c VRn phq logpMRn { δq n ` logpM R n { δq n ¸ . where VRn phq denotes the empirical variance of the loss computed on n examples teJyiφphpxiqqu n i “ 1 . Proposition 8 ( Generalization bound for churn ) . Let the scoring function φ : ∆mÑRm ` be bounded . For base classifier g , let Uφ Ď RX denote the corresponding class of divergence functions upxq “ dφphpxq , gpxqq “ gpxqJ ` φphpxqq ´ φpgpxqq ˘ induced by classifiers h P H. Let MCn “ N8p 1n , Uφ , 2nq denote the uniform L8 covering number for Uφ . Fix δ P p0 , 1q . Then with probability ě 1´ δ over draw of S „ Dn , for any h P H : Cphq ď pCphq ` O ˜ c VCn phq logpMCn { δq n ` logpM C n { δq n ¸ . where VCn phq denotes the empirical variance of the churn values computed on n examples tgpxiqJ ` φphpxiqq ´ φpgpxiqq ˘ uni “ 1 . The term VCn captures the variance in the labels provided by the base model g. When this term is low ( which is what we would expect in practice from a base model that makes soft-predictions ) , the churn metric enjoys a tight generalization bound . In contrast , the classification risk is evaluated on one-hot training labels and the variance term VRn there is not impacted by the base model ’ s scores . C DEFINITIONS OF NETWORK ARCHITECTURES USED C.1 FULLY CONNECTED NETWORK FCN-x refers to the following model with size set to `` x '' . In other words , it ’ s a simple fully connected network with one hidden layer with x units . def get_fcn ( n_columns , num_classes=10 , size=100 , weight_init=None ) : model = None model = tf.keras.Sequential ( [ tf.keras.layers.Input ( shape= ( n_columns , ) ) , tf.keras.layers.Dense ( size , activation=tf.nn.relu ) , tf.keras.layers.Dense ( num_classes , activation= '' softmax '' ) , ] ) model.compile ( optimizer=tf.keras.optimizers.Adam ( ) , loss=tf.keras.losses.CategoricalCrossentropy ( ) , metrics= [ tf.keras.metrics.categorical_accuracy ] ) return model C.2 CONVOLUTIONAL NETWORK Convnet-x refers to the following model with size set to `` x '' . Convnet-1 is based on the lenet5 architecture LeCun et al . ( 1998 ) . def get_convnet ( input_shape= ( 28 , 28 , 3 ) , size=1 , num_classes=2 , weight_init=None ) : model = tf.keras.Sequential ( ) model.add ( tf.keras.layers.Conv2D ( filters=16 * size , kernel_size= ( 5 , 5 ) , padding= '' same '' , activation= '' relu '' , input_shape=input_shape ) ) model.add ( tf.keras.layers.MaxPool2D ( strides=2 ) ) model.add ( tf.keras.layers.Conv2D ( filters=24 * size , kernel_size= ( 5 , 5 ) , padding= '' valid '' , activation= '' relu '' ) ) model.add ( tf.keras.layers.MaxPool2D ( strides=2 ) ) model.add ( tf.keras.layers.Flatten ( ) ) model.add ( tf.keras.layers.Dense ( 128 * size , activation= '' relu '' ) ) model.add ( tf.keras.layers.Dense ( 84 , activation= '' relu '' ) ) model.add ( tf.keras.layers.Dense ( num_classes , activation= '' softmax '' ) ) model.compile ( optimizer=tf.keras.optimizers.Adam ( ) , loss=tf.keras.losses.CategoricalCrossentropy ( ) , metrics= [ tf.keras.metrics.categorical_accuracy ] ) return model C.3 TRANSFORMER Transformer-x refers to the following with size set to `` x '' . It is based on keras tutorial on text classification ( https : //keras.io/examples/nlp/text_classification_ with_transformer/ licensed under the Apache License , Version 2.0 ) . def get_transformer ( maxlen , size=1 , num_classes=2 , weight_init=None ) : model = None class TransformerBlock ( tf.keras.layers.Layer ) : def __init__ ( self , embed_dim , num_heads , ff_dim , rate=0.1 , weight_init=None ) : super ( TransformerBlock , self ) .__init__ ( ) self.att = tf.keras.layers.MultiHeadAttention ( num_heads=num_heads , key_dim=embed_dim ) self.ffn = tf.keras.Sequential ( [ tf.keras.layers.Dense ( ff_dim , activation= '' relu '' ) , tf.keras.layers.Dense ( embed_dim ) , ] ) self.layernorm1 = tf.keras.layers.LayerNormalization ( epsilon=1e-6 ) self.layernorm2 = tf.keras.layers.LayerNormalization ( epsilon=1e-6 ) def call ( self , inputs , training ) : attn_output = self.att ( inputs , inputs ) # attn_output = self.dropout1 ( attn_output , training=training ) out1 = self.layernorm1 ( inputs + attn_output ) ffn_output = self.ffn ( out1 ) return self.layernorm2 ( out1 + ffn_output ) class TokenAndPositionEmbedding ( tf.keras.layers.Layer ) : def __init__ ( self , maxlen , vocab_size , embed_dim , ) : super ( TokenAndPositionEmbedding , self ) .__init__ ( ) self.token_emb = tf.keras.layers.Embedding ( input_dim=vocab_size , output_dim=embed_dim ) self.pos_emb = tf.keras.layers.Embedding ( input_dim=maxlen , output_dim=embed_dim ) def call ( self , x ) : maxlen = tf.shape ( x ) [ -1 ] positions = tf.range ( start=0 , limit=maxlen , delta=1 ) positions = self.pos_emb ( positions ) x = self.token_emb ( x ) return x + positions embed_dim = 32 * size # Embedding size for each token num_heads = 2 * size # Number of attention heads ff_dim = 32 * size # Hidden layer size in feed forward network inside transformer inputs = tf.keras.layers.Input ( shape= ( maxlen , ) ) embedding_layer = TokenAndPositionEmbedding ( maxlen , 20000 , embed_dim ) x = embedding_layer ( inputs ) transformer_block = TransformerBlock ( embed_dim , num_heads , ff_dim , weight_init ) x = transformer_block ( x ) x = tf.keras.layers.GlobalAveragePooling1D ( ) ( x ) outputs = tf.keras.layers.Dense ( num_classes , activation= '' softmax '' ) ( x ) model = tf.keras.Model ( inputs=inputs , outputs=outputs ) model.compile ( optimizer=tf.keras.optimizers.Adam ( ) , loss=tf.keras.losses.CategoricalCrossentropy ( ) , metrics= [ tf.keras.metrics.categorical_accuracy ] ) return model D MODEL TRAINING CODE def model_trainer ( get_model , X_train , y_train , X_test , y_test , weight_init=None , validation_data=None , warm=True , mixup_alpha=-1 , codistill_alpha=-1 , distill_alpha=-1 , anchor_alpha=-1 , anchor_eps=-1 ) : model = get_model ( ) if weight_init is not None and warm : model.set_weights ( weight_init ) if FLAGS.loss == `` squared '' : model.compile ( optimizer=tf.keras.optimizers.Adam ( ) , loss=tf.keras.losses.MeanSquaredError ( ) , metrics= [ tf.keras.metrics.categorical_accuracy ] ) callback = tf.keras.callbacks.EarlyStopping ( monitor= '' val_loss '' , patience=3 ) history = None if distill_alpha > = 0 : original_model = get_model ( ) original_model.set_weights ( weight_init ) y_pred = original_model.predict ( X_train ) y_use = distill_alpha * y_pred + ( 1 - distill_alpha ) * y_train history = model.fit ( x=X_train , y=y_use , epochs=FLAGS.n_epochs , callbacks= [ callback ] , validation_data=validation_data ) elif anchor_alpha > = 0 and anchor_eps > = 0 : original_model = get_model ( ) original_model.set_weights ( weight_init ) y_pred = original_model.predict ( X_train ) y_pred_hard = np.argmax ( y_pred , axis=1 ) y_hard = np.argmax ( y_train , axis=1 ) correct = ( y_pred_hard == y_hard ) correct = np.tile ( correct , ( y_train.shape [ 1 ] , 1 ) ) correct = np.transpose ( correct ) correct = correct.reshape ( y_train.shape ) y_use = np.where ( correct , anchor_alpha * y_pred + ( 1 - anchor_alpha ) * y_train , y_train * anchor_eps ) history = model.fit ( x=X_train , y=y_use , epochs=FLAGS.n_epochs , callbacks= [ callback ] , validation_data=validation_data ) elif mixup_alpha > = 0 : training_generator = deep_utils.MixupGenerator ( X_train , y_train , alpha=mixup_alpha ) ( ) history = model.fit ( x=training_generator , validation_data=validation_data , steps_per_epoch=int ( X_train.shape [ 0 ] / 32 ) , epochs=FLAGS.n_epochs , callbacks= [ callback ] ) elif codistill_alpha > = 0 : teacher_model = get_model ( ) if weight_init is not None and warm : teacher_model.set_weights ( weight_init ) val_losses = [ ] optimizer = tf.keras.optimizers.Adam ( ) global_step = 0 alpha = 0 codistillation_warmup_steps = 0 for epoch in range ( FLAGS.n_epochs ) : X_train_ , y_train_ = sklearn.utils.shuffle ( X_train , y_train ) batch_size = 32 for i in range ( int ( X_train_.shape [ 0 ] / batch_size ) ) : if global_step > = codistillation_warmup_steps : alpha = codistill_alpha else : alpha = 0. with tf.GradientTape ( ) as tape : X_batch = X_train_ [ i * 32 : ( i + 1 ) * 32 , : ] y_batch = y_train_ [ i * 32 : ( i + 1 ) * 32 , : ] prob_student = model ( X_batch , training=True ) prob_teacher = teacher_model ( X_batch , training=True ) loss = deep_utils.compute_loss ( prob_student , prob_teacher , y_batch , alpha ) trainable_weights = model.trainable_weights + teacher_model.trainable_weights grads = tape.gradient ( loss , trainable_weights ) optimizer.apply_gradients ( zip ( grads , trainable_weights ) ) global_step += 1 val_preds = model.predict ( validation_data [ 0 ] ) val_loss = np.sum ( deep_utils.cross_entropy ( validation_data [ 1 ] .astype ( `` float32 '' ) , val_preds ) ) val_losses.append ( val_loss ) if len ( val_losses ) > 3 and min ( val_losses [ -3 : ] ) > val_losses [ -4 ] : break else : history = model.fit ( X_train , y_train , epochs=FLAGS.n_epochs , callbacks= [ callback ] , validation_data=validation_data ) y_pred_train = model.predict ( X_train ) y_pred_test = model.predict ( X_test ) return y_pred_train , y_pred_test , model.get_weights ( ) E ADDITIONAL EXPERIMENTAL RESULTS E.1 ADDITIONAL OPENML RESULTS E.1.1 INITIAL SAMPLE 100 , BATCH SIZE 1000 , VALIDATION SIZE 100 In Tables 3 and 4 , we show the churn at cold accuracy metric across network sizes ( fcn-10 , fcn-100 , fcn-1000 , fcn-10000 , fcn-100000 ) . Table 5 shows the standard error bars . They are obtained by fixing the dataset and model , and taking the 100 accuracy and churn results from each baseline and calculating the standard error , which is the standard deviation of the mean . We then report the average standard error across the baselines We see that distillation is the best 52 % of the time . E.1.2 INITIAL SAMPLE 1000 , BATCH SIZE 1000 , VALIDATION SIZE 100 In Tables 6 and 7 , we show the churn at cold accuracy metric across network sizes ( fcn-10 , fcn-100 , fcn-1000 , fcn-10000 , fcn-100000 ) . We see that distillation consistently performs strongly across datasets and sizes of networks . Table 8 shows the standard error bars . We see that distillation is the best 84 % of the time . E.2 ADDITIONAL MNIST VARIANT RESULTS E.2.1 INITIAL SAMPLE SIZE 100 , BATCH SIZE 1000 , VALIDATION SIZE 100 We show full results in Table 9 . We see that distillation is the best for 24 out of the 50 combinations of dataset and network . Error bands can be found in Table 10 . E.2.2 INITIAL SAMPLE SIZE 1000 , BATCH SIZE 1000 , VALIDATION SIZE 100 We show full results in Table 11 . We see that distillation is the best for 42 out of the 50 combinations of dataset and network . Error bands can be found in Table 12 . E.2.3 INITIAL SAMPLE SIZE 10000 , BATCH SIZE 1000 , VALIDATION SIZE 1000 We show full results in Table 13 . We see that in this situation , label smoothing starts becoming competitive with distillation with either of them being the best . Distillation is the best for 32 out of the 50 combinations of dataset and network , and losing marginally to label smoothing in other cases . See Table 14 for error bands . E.3 ADDITIONAL SVHN AND CIFAR RESULTS E.3.1 INITIAL SAMPLE 100 , BATCH SIZE 1000 , VALIDATION SIZE 100 Results are in Table 15 , where we see that distillation is best on 8 out of 10 combinations of dataset and network . Error bands can be found in Table 16 . E.3.2 INITIAL SAMPLE 1000 , BATCH SIZE 1000 , VALIDATION SIZE 100 The results can be found in Table 17 . We include the error bands here in Table 18 . Distillation is best in all combinations . E.3.3 INITIAL SAMPLE 10000 , BATCH SIZE 1000 , VALIDATION SIZE 1000 Results are in Table 19 , where we see that distillation is best on all combinations of dataset and network . Error bands can be found in Table 20 . E.4 ADDITIONAL CELEBA RESULTS E.4.1 INITIAL SAMPLE 100 , BATCH SIZE 1000 , VALIDATION SIZE 100 Tables 21 , 22 , 23 , and 24 show the performance of CelebA tasks when we instead use an initial sample size of 100 . We see that across the 200 combinations of task and network , distillation is the best 192 of time , or 96 % of the time . The error bands can be found in Table 25 . E.4.2 INITIAL SAMPLE 1000 , BATCH SIZE 1000 , VALIDATION SIZE 100 We show some additional CelebA results for initial sample 1000 and batch size 1000 in Tables 26 , 27 , 28 , and 29 which show performance for each dataset across convnet-1 , convnet-2 , convnet-4 , convent-8 , convnet-16 . This gives us 40 ¨ 5 “ 200 results , of which distillation performs the best 158 out of those settings , or 79 % of the time . The error bands can be found in Table 30 . E.4.3 INITIAL SAMPLE SIZE 10000 , BATCH SIZE 1000 , VALIDATION SIZE 1000 Tables 31 , 32 , 33 , and 34 show the performance of CelebA tasks when we instead use an initial sample size of 10000 . We see that across the 200 combinations of task and network , distillation is the best 183 of time , or 91.5 % of the time . The error bands can be found in Table 35 . E.5 CIFAR10 AND CIFAR100 ON RESNET Results can be found in Table 36 . We see that distillation outperforms in every case . E.6 ADDITIONAL IMDB RESULTS In Table 37 , we show the results for the IMDB dataset and transformer networks for initial batch sizes of 100 , 1000 and 10000 with the batch size fixed at 1000 . The error bands can be found in Table 38 . We see that for initial sample size of 100 , distillation performs poorly for the smaller networks as the process of distillation hurts the performance with a weak teacher trained on only 100 examples , but performs well for the larger networks . For initial sample size of 1000 and 10000 , distillation is the clear winner losing in only one instance . We show the full Pareto frontiers and cost curves in Figure 4 .

Summary: This paper explores the relationship between the low-churn problem and distillation. The authors show that there is an equivalence between those two methods and that distillation performs particularly well on low-churn dataset tasks. The authors propose a novel churn reduction algorithm based on distillation which involves the training of a classifier by minimizing a distilled loss and solving a convex program. The authors  provide theoretical guarantees for the proposed algorithm and explain the advantages of the proposed approach compared to the anchor loss, another churn-reduction method.   The authors validate empirically their approach on 12 OpenML datasets, 10 MNIST variants, CIFAR10, CIFAR100 and IMDB.