# churn_reduction_via_distillation__7d4e0fd9.pdf

Published as a conference paper at ICLR 2022

CHURN REDUCTION VIA DISTILLATION

Heinrich Jiang, Harikrishna Narasimhan, Dara Bahri, Andrew Cotter, Afshin Rostamizadeh Google Research {heinrichj, hnarasimhan, dbahri, acotter, rostami}@google.com

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:

Published as a conference paper at ICLR 2022

Figure 1: Illustration of our proposal: We propose using knowledge distillation with the base model as the teacher (with some mixing parameter λ) and then training on the distilled label. We show theoretically that training our loss with the distilled label yields approximately the same solution as the original churn constrained optimization problem for some slack ϵ that depends on λ (or vice versa). The significance is that the simple and popular distillation procedure yields the same solution as the original churn problem, without having to deal with the additional complexity that comes with solving constrained optimization problems.

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, but outperform this method 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.

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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 Pp Y 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 its churn to be within an acceptable limit.

We will measure the classification performance of a classifier h using a loss function ℓ: rms ˆ mÑR that maps a label y P rms and prediction hpxq P m to a non-negative number ℓpy, hpxqq, and denote the classification risk by Rphq : Epx,yq D rℓpy, 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 ϵ ą 0: min h:XÑ m Rphq s.t. Cphq ď ϵ. (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 forms of the loss and divergence functions employed in this paper:

ℓφpy, vq : φypvq; dφpu, vq : ÿ

y Prms uypφypvq φypuqq. (2)

The cross-entropy loss and KL-divergence are a special case of this formulation when φypvq logpvyq, and the squared loss and the squared L2 distance can be recovered by setting φypvq ř

i Prms p1pi yq viq2.

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 pℓ, 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

2ϵ{ α Ex }ppxq gpxq}2q .

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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 ϵ ą 0 2: Train a classifier hk for each λk P Λ by minimizing the distilled loss in (4):

hk P argminh PH p Lλkphq

3: Find a convex combination of h1, . . . , h L by solving following convex program in L variables:

min h Pcoph1,...,h Lq p Rphq s.t. p Cphq ď ϵ

and return the solution ph

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.

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:

i 1 ℓφpyi, hpxiqq; p Cphq 1

i 1 dφpgpxiq, hpxiqq,

where ℓφ 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 argminh PH p Lλkphq : 1

i 1 pλkeyi p1 λkqgpxiqq φphpxiqq. (4)

To construct the final classifier, we find a convex combination of the L classifiers h1, . . . , h L that minimizes p Rphq while satisfying the constraint p Cphq ď ϵ, 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, . . . , h L by coph1, . . . , h Lq 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.

Published as a conference paper at ICLR 2022

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 in H. 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 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 p Rphq| ď Rpn, δq; |Cphq p Cphq| 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 ϵ and Rprhq ď Rphq for all classifiers h for which Cphq ď ϵ. Let ph be the classifier returned by Algorithm 1 with Λ maxt ϵ ϵ 2B , uu ˇˇ u P t 1

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

ϵ Rpn, δq Cpn, δq B

Feasibility : Cpphq ď ϵ 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. To this end, we apply the generalization bound from (Menon et al., 2020, Proposition 2) to the student s churn. Proposition 3 (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 gpxq J φphpxqq φpgpxqq induced by classifiers h P H. Let MC n N8p 1

n, 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 p Cphq O

VC n phqlogp MC n {δq n logp MC n {δq n

where VC n phq denotes the empirical variance of the divergence values computed on n examples tgpxiq J φphpxiqq φpgpxiqq un i 1; the lower the variance, the tighter is the bound.

In fact, for certain base classifiers g, generalizing well on churn can have the additional benefit of improving classification performance, as shown in Proposition 7 in Appendix B.

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)

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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 "αp2 jpxq p1 αqpjpxq if j argmaxk pkpxq pη α maxk 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 predicts soft probabilities), 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.

Datasets and architectures: The following are the datasets we use in our experiments, along with the associated model architectures:

12 Open ML datasets using fully-connected neural networks. 10 MNIST variants, SVHN, CIFAR10, 40 Celeb A tasks using convolutional networks.

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Dataset cold warm s-perturb mixup ls co-dist anchor distill

adult 6.27 N/A 6.05 6.57 N/A 5.78 6.62 4.39 bank 10.04 8.43 7.8 8.25 8.89 7.55 8.77 5.58 magic04 27.56 27.41 24.37 24.68 27.79 23.67 25.22 18.51 phonemes 10.45 10.66 10.09 N/A 9.02 9.3 11.14 7.4 electricity 18.16 17.53 17.23 15.69 16.19 14.94 18.22 8.99 eeg 48.02 42.96 42.04 39.98 49.98 54.99 26.99 2.0 churn 27.15 25.58 22.19 20.49 N/A 18.71 17.59 5.51 elevators 33.34 35.87 30.41 31.47 32.91 30.53 34.38 10.44 pollen 44.03 N/A 42.63 44.6 42.06 41.78 40.44 35.15 phishing 4.43 4.2 3.97 4.01 4.1 3.74 4.08 2.91 wilt 9.55 7.27 7.27 6.67 N/A 7.0 7.58 4.93 letters 23.01 23.15 23.47 23.86 23.06 23.44 22.04 16.92

Table 1: Results for Open ML datasets under churn at cold accuracy metric.

CIFAR10 and CIFAR100 with Res Net-50, Res Net-101, and Res Net-152.

IMDB dataset using transformer network.

For each architecture (besides Res Net), 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 Le Net5 architecture (Le Cun 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 Conv Net-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 Open ML 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

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Dataset cold warm s-perturb mixup ls co-dist anchor distill

mnist 6.68 N/A 6.78 5.93 5.02 N/A 5.21 4.81 fashion mnist 18.48 N/A 17.08 16.93 16.52 16.53 15.75 11.9 emnist balanced 42.21 N/A 37.46 37.12 35.53 N/A 33.64 29.41 emnist byclass 36.4 N/A 32.33 31.74 30.79 31.96 30.42 24.5 emnist bymerge 34.17 30.62 30.38 29.86 28.72 29.59 27.07 21.28 emnist letters 29.58 N/A 26.99 26.2 24.61 N/A 23.22 20.16 emnist digits 6.81 N/A 6.95 6.09 5.29 N/A 5.42 4.81 emnist mnist 6.42 N/A 6.28 5.67 4.9 N/A 5.21 4.49 kmnist 15.95 N/A 14.08 13.41 12.08 N/A 12.0 9.9 k49 mnist 46.35 N/A 39.48 39.46 37.33 39.99 35.24 29.46 svhn 32.12 26.88 27.39 29.2 29.21 26.01 25.43 22.64 cifar10 52.01 47.57 46.36 47.17 47.92 44.61 45.75 29.13

Table 2: Results for MNIST variants, SVHN and CIFAR10 under churn at cold accuracy metric.

Figure 2: IMDB dataset with Transformer-1, Transformer-4 and Transformer-16. We show the Pareto frontier for each of the baselines. We see that distillation is able to obtain solutions that dominate the other baselines in both churn and accuracy.

Figure 3: Distillation vs Anchor Ablation: We provide an ablation study further showing that using the true labels for wrongly predicted examples by the base model (as done in anchor method) is worse than using distillation for all the examples. We show the performance as we vary the number of wrongly predicted examples that we use the true label instead of the distilled label. The x-axis is the fraction of the most (sorted by softmax score) wrongly predicted examples (i.e. 0 is distillation and 1 is anchor method) and y-axis is the churn at cold accuracy metric. We show the results for phishing dataset using fcn-1000 and celeb A dataset predicting attractiveness using convnet-1, where the average accuracies across the runs of the base model were 93.3% and 69.2%, respectively.

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

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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

Open ML datasets with fully-connected networks: In Table 1 we show the results for the Open ML 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.

Celeb A 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 Res Net: Due to the computational costs, we only run these experiments for initial sample size 1000. In all cases (across Res Net-50, Res Net-101 and Res Net-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.

Conclusion: We have proposed knowledge distillation as a new practical solution to churn reduction, and provided both theoretical and empirical justifications for the approach.

Published as a conference paper at ICLR 2022

Reproducibility Statement: All details of experimental setup are in the main text, along with descriptions of the baselines and what hyperparameters were swept across. Code can be found in the Appendix. All proofs are in the Appendix.

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A.1 PROOF OF PROPOSITION 1

Proposition (Restated). Let pℓ, 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

2ϵ α Ex }ppxq gpxq}2q .

Proof. Let h denote an optimal feasible solution for (1). We first note that

Rphq Ex,y rℓpy, hpxqqs Ex Ey|x rℓpy, hpxqqs Ex ÿ

i Prms pipxqφiphpxqq ı

and Cphq Ex ÿ

i Prms 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 αq Rph2q, and similarly for C. Furthermore because Cpgq 0 ă ϵ, 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 1 1 µ , we equivalently have that h is a unique minimizer of the weighted objective λ Rphq p1 λ q Cphq.

We next show that the minimizer h is of the required form. Expanding the R and C, we have:

λ Rphq p1 λ q Cphq

λ pipxq p1 λ qgipxq φiphpxqq p1 λ qgipxqφipgpxqq ı

λ pipxq p1 λ qgipxq φiphpxqq ı a term independent of h

i Prms pipxq φiphpxqq ı a term independent of h, (6)

where ppxq 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 i Prms pipxq φiphpxqq is minimized for each x. For a fixed x, the inner term 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 (6) is minimized by h pxq ppxq 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 infv P 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 that Qpuq infv P 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.

Published as a conference paper at ICLR 2022

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

2 }hpxq gpxq}2 q 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 ϵ, we have from the above bound Ex α

2 }h pxq gpxq}2 q ı ď ϵ.

Substituting for h , we have:

2 }ppxq gpxq}2 q

or pλ q2 ď 2ϵ 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 p Rphq| ď Rpn, δq; |Cphq p Cphq| 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 ϵ and Rprhq ď Rphq for all classifiers h for which Cphq ď ϵ. Let ph be the classifier returned by Algorithm 1 with Λ maxt ϵ ϵ 2B , uu ˇˇ u P t 1

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

ϵ Rpn, δq Cpn, δq B

Feasibility : Cpphq ď ϵ C pn, δq .

We first note that because }φpzq}8 ă B, @z P m, both p Rphq ă B and p Cphq ă B. Also, because φi is convex, both p Rphq and p Cphq are convex in h. In other words, for any α P r0, 1s, and classifiers h1, h2, p Rpαh1 p1 αqh2q ď α p Rph1q p1 αq p Rph2q, and similarly for p C. Furthermore, the objective in (4) can be decomposed into a convex combination of the empirical risk and churn:

p Lλphq 1 n

i 1 pλeyi p1 λqgpxiqq φphpxiqq

λ p Rphq p1 λq p Cphq 1 λ

i 1 gpxiq φpgpxiqq.

Therefore minimizing p Lλphq is equivalent to minimizing the Lagrangian function r Lλphq λ p Rphq p1 λqp p Cphq ϵq (7)

over h. Moreover, each hk minimizes r Lλkphq.

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.

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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 ϵ ϵ 2B , uu | u P t 1

L, . . . , 1u ( of the range r ϵ ϵ 2B , 1s for

some L P N . Then there exists a bounded Lagrange multiplier λ P r ϵ ϵ 2B , 1s such that pph, λq forms an equilibrium of the Lagrangian min-max game: λ p Rpphq p1 λqp p Cpphq ϵq min h Pcoph1,...,h Lq λ p Rphq p1 λqp p Cphq ϵq

max λPr0,1sp1 λqp p Cpphq ϵq p1 λqp p Cpphq ϵq.

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, . . . , h L:

min h Pcoph1,...,h Lq p Rphq s.t. p Cphq ď ϵ.

Consequently, there exists a λ P r0, 1s such that: λ p Rpphq p1 λqp p Cpphq ϵq min h Pcoph1,...,h Lq λ p Rphq p1 λqp p Cphq ϵq (8)

max λPr0,1sp1 λqp p Cpphq ϵq p1 λqp p Cpphq ϵq. (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 µ ě 0 such that ph P argmin h Pcoph1,...,h Lq p Rphq µp p Cphq ϵq (stationarity)

µp p Cpphq ϵq 0 (complementary slackness).

When p Cpphq ď ϵ, µ 0, and so (8) and (9) are satisfied for λ 1. When p Cpphq ϵ, then (8) and (9) are satisfied for λ 1 1 µ.

It remains to show that that λ P r ϵ ϵ 2B , 1s. For this, we first show that there exists a h1 P coph1, . . . , h Lq such that p Cph1q ď ϵ{2. To see why, pick h1 to be the minimizer of the Lagrangian r Lλphq over all h P H for λ ϵ ϵ 2B . Because r Lλph1q ď r Lλpgq ď λB p1 λqϵ, where g is the base classifier that we have assumed is in H, it follows that p Cph1q ď λ 1 λB ď ϵ{2.

Next, by combining (8) and (9), we have λ p Rpphq max λPr0,1sp1 λqp p Cpphq ϵq min h Pcoph1,...,h Lq λ p Rphq p1 λqp p Cphq ϵq.

Lower bounding the LHS by setting λ 1 and upper bounding the RHS by setting h h1, we get: λ p Rpphq ď λ p Rph1q p1 λq ϵ

which gives us: ϵ{2 ď λpϵ{2 p Rph1q p Rpphqq ď λpϵ{2 Bq. Hence λ ě ϵ ϵ 2B , which completes the proof.

Lemma 6. Let ph be the classifier returned by Algorithm 1 when Λ is set to Λ maxt ϵ ϵ 2B , uu | u P t 1

L, . . . , 1u ( of the range r ϵ ϵ 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 ϵ ϵ 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 pλqp Cpphq ϵq ď min h PH pλRphq p1 pλqp Cphq ϵq

O p Rpn, δq Cpn, δq B{Lq (10) and max λPr0,1s p1 λqp Cpphq ϵq ď p1 pλqp Cpphq ϵq O p Cpn, δq B{Lq . (11)

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Proof. We have from Lemma 5 that there exists λ P r ϵ ϵ 2B , 1s such that

λ p Rpphq p1 λqp p Cpphq ϵq min h Pcoph1,...,h Lq λ p Rphq p1 λqp p Cphq ϵq (12)

max λPr0,1sp1 λqp p Cpphq ϵq p1 λqp p Cpphq ϵq. (13)

Algorithm 1 works with a discretization Λ maxt ϵ ϵ 2B , uu | u P t 1

L, . . . , 1u ( of the range

r ϵ ϵ 2B , 1s. Allowing pλ to denote the closest value to λ in this set, we have from (12):

pλ p Rpphq p1 pλqp p Cpphq ϵq ď min h Pcoph1,...,h Lq pλ p Rphq p1 pλqp p Cphq ϵq 4B

min h PH pλ p Rphq p1 pλqp p Cphq ϵq 4B

where the last step follows from the fact that coph1, . . . , h Lq Ď H and each hk was chosen to minimize p1 λkq p Rphq λkp p Cphq ϵq for λk P Λ. Similarly, we have from (13),

max λPr0,1s p1 λqp Cpphq ϵq ď p1 pλqp Cpphq ϵq B

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 p Rpphq p1 pλqp p Cpphq ϵq

ě pλRpphq p1 pλqp Cpphq ϵq pλ R pn, δq p1 pλq C pn, δq

ě pλRpphq p1 pλqp Cpphq ϵq 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:

! pλ p Rphq p1 pλqp p Cphq ϵq ) 4B{L

! pλRphq p1 pλqp Cphq ϵq 4B{L pλ R pn, δq p1 pλq C pn, δq )

! pλRphq p1 pλqp Cphq ϵq ) 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:

ě max λPr0,1s p1 λqp p Cpphq ϵq p1 pλqp p Cpphq ϵq

ě max λPr0,1s

! p1 λqp Cpphq ϵq p1 λq C pn, δq ) p1 pλqp Cpphq ϵq p1 pλq C pn, δq

ě max λPr0,1sp1 λqp Cpphq ϵq p1 pλqp Cpphq ϵq 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 p Rpphq max λPr0,1s p1 λqp Cpphq ϵq ď min h PH pλ p Rphq p1 pλqp p Cphq ϵq

O Rpn, δq Cpn, δq B{L . (17)

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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 B

Dividing both sides by pλ,

Rpphq ď Rprhq 1

pλO R pn, δq C pn, δq B

Lower bounding pλ by ϵ ϵ 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 p Cpphq ď ϵ, and from the generalization bound for C.

A.3 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 "αp2 jpxq p1 αqpjpxq if j argmaxk pkpxq pϵ α maxk 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 λq Ex DX rppxq φphpxqqs

λEx DX Ey|x reys φphpxqq p1 λq Ex DX rppxq φphpxqqs

λEx DX rppxq φphpxqqs p1 λq Ex 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 ,

a "αppxq p1 αqey if y argmaxk pkpxq ϵey otherwise ,

For a given x, let us denote jx argmaxk pkpxq. We then have:

Lancphq Epx,yq D rp1py jxq pαppxq p1 αqeyq ϵ1py jxqeyq φphpxqqs

Ex DX Ey|x rp1py jxq pαppxq p1 αqeyq ϵ1py jxqeyq φphpxqqs

k pkpxq pp1pk jxq pαppxq p1 αqekq ϵ1pk jxqekq φphpxqqq

pjxpxq pαppxq p1 αqejxq φphpxq ϵ ÿ

k jx pkpxqφkphpxqq

pjxpxq pαpjxpxq p1 αqq φjxphpxqq

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k jx αpkpxqφkphpxqq ϵ ÿ

k jx pkpxqφkphpxqq

pjxpxq pαpjxpxq p1 αqq φjxphpxqq pαpjxpxq ϵq ÿ

k jx pkpxqφkphpxqq

Ex DX rrppxq φphpxqqs , (20)

rpspxq "αp2 spxq p1 αqpspxq if s jx pαpjxpxq ϵqpspxq otherwise

"αp2 spxq p1 αqpspxq if s argmaxk pkpxq pα maxk pkpxq ϵqpspxq otherwise .

For a fixed x, the inner term in (20) is minimized when h pxq 1 Zpxq rppxq, 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 1 Zpxq rppxq φphpxqq, and from φ being a strictly proper scoring function. This completes the proof.

B ADDITIONAL THEORETICAL RESULTS

B.1 RELATIONSHIP BETWEEN CHURN AND CLASSIFICATION RISK

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 7. Let pℓ, 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 in H 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 ϵ Cpn, δq p B Φλ q Ex 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 in H). 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

ˇˇˇˇˇˇ Ex DX

y Prms pypxq pφyph pxqq φypgpxqqq

ď Ex DX ppxq φpphpxqq φpgpxqq ı Ex DX

y Prms pypxq ˇˇφyph pxqq φypgpxqq ˇˇ

ď Ex DX ppxq φpphpxqq φpgpxqq ı ΦEx DX

y Prms pypxq}h pxq gpxq}1

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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 p B λ Φq Ex 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.

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.

B.2 GENERALIZATION BOUND FOR CLASSIFICATION RISK

As a follow-up to Proposition 3, we also provide generalization bounds for the classification risk in terms of the empirical variance of the loss values based on a result from (Menon et al., 2020, Proposition 2). Proposition 8 (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 ℓφpy, hpxqq φyphpxqq induced by classifiers h P H. Let MR n N8p 1

n, 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 p Rphq O

VR n phqlogp MR n {δq n logp MR n {δq n

where VR n phq denotes the empirical variance of the loss computed on n examples tφyiphpxiqqun i 1.

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

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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.Categorical Crossentropy(), 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 Le Cun 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.Max Pool2D(strides=2)) model.add( tf.keras.layers.Conv2D( filters=24 * size, kernel_size=(5, 5), padding="valid", activation="relu")) model.add(tf.keras.layers.Max Pool2D(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.Categorical Crossentropy(), 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).

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def get_transformer(maxlen, size=1, num_classes=2, weight_init=None): model = None

class Transformer Block(tf.keras.layers.Layer):

def __init__(self, embed_dim, num_heads, ff_dim, rate=0.1, weight_init=None): super(Transformer Block, self).__init__() self.att = tf.keras.layers.Multi Head Attention( 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.Layer Normalization(epsilon=1e-6) self.layernorm2 = tf.keras.layers.Layer Normalization(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 Token And Position Embedding(tf.keras.layers.Layer):

def __init__( self, maxlen, vocab_size, embed_dim, ): super(Token And Position Embedding, 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,))

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embedding_layer = Token And Position Embedding(maxlen, 20000, embed_dim) x = embedding_layer(inputs) transformer_block = Transformer Block(embed_dim, num_heads, ff_dim, weight_init) x = transformer_block(x) x = tf.keras.layers.Global Average Pooling1D()(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.Categorical Crossentropy(), 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.Mean Squared Error(), metrics=[tf.keras.metrics.categorical_accuracy]) callback = tf.keras.callbacks.Early Stopping(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)

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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.Mixup Generator( 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.Gradient Tape() 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_weigh 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)

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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.

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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 Celeb A 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 Celeb A 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 Celeb A 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.

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Dataset network cold warm s-perturb mixup ls co-dist anchor distill

fcn-10 12.75 N/A 11.55 12.3 N/A 10.85 N/A N/A fcn-100 11.8 N/A 10.86 11.14 11.11 10.0 N/A N/A fcn-1000 11.78 12.2 11.45 11.84 12.39 10.67 12.16 8.55 fcn-10000 14.03 13.18 13.28 13.1 12.8 13.1 14.22 9.7 fcn-100000 14.13 13.28 13.45 13.49 13.74 13.32 N/A 8.43

fcn-10 10.24 9.03 9.17 9.07 8.48 8.17 6.28 6.72 fcn-100 8.19 N/A 6.79 6.81 8.28 6.45 7.85 3.09 fcn-1000 10.1 10.71 9.96 9.59 10.29 8.93 10.38 4.35 fcn-10000 12.81 N/A 11.04 11.76 12.97 10.86 11.42 7.4 fcn-100000 10.78 10.97 9.79 9.78 8.36 9.21 10.33 5.48

fcn-10 21.71 18.29 18.19 17.98 19.06 14.32 20.76 N/A fcn-100 17.59 16.34 16.42 16.17 16.85 14.44 14.03 12.0 fcn-1000 19.08 18.37 17.69 17.57 18.64 16.78 N/A 11.76 fcn-10000 23.13 23.02 22.23 21.83 N/A 21.6 N/A N/A fcn-100000 24.79 N/A 24.12 24.2 N/A 24.84 N/A N/A

fcn-10 30.42 25.96 27.12 26.58 26.88 25.71 25.88 23.58 fcn-100 32.15 N/A 28.3 28.22 N/A 25.61 N/A N/A fcn-1000 32.35 N/A 29.75 29.64 N/A 28.59 31.94 20.93 fcn-10000 30.84 31.0 28.5 29.57 29.28 27.07 29.09 27.49 fcn-100000 27.56 27.75 25.81 26.37 25.25 25.12 26.73 23.65

fcn-10 18.64 16.77 16.73 N/A N/A N/A 18.0 N/A fcn-100 18.15 N/A 17.33 N/A N/A N/A N/A 16.23 fcn-1000 18.97 19.36 18.25 N/A N/A 18.46 20.76 13.24 fcn-10000 20.6 20.56 20.32 19.68 19.71 20.35 22.28 16.5 fcn-100000 19.66 19.9 18.97 18.3 18.58 18.93 20.93 12.82

electricity

fcn-10 38.8 39.6 36.8 35.8 39.8 33.8 N/A 30.4 fcn-100 33.45 N/A N/A N/A N/A N/A N/A N/A fcn-1000 40.29 N/A 33.43 29.76 33.14 35.14 42.52 27.52 fcn-10000 35.78 N/A 35.81 28.63 32.81 34.11 38.63 26.33 fcn-100000 33.92 N/A N/A 32.96 36.46 37.71 N/A N/A

fcn-10 45.92 N/A N/A N/A N/A N/A N/A N/A fcn-100 52.8 47.31 46.88 47.56 N/A 41.82 32.66 25.33 fcn-1000 54.73 N/A 44.22 48.4 57.85 N/A 27.38 2.12 fcn-10000 50.59 N/A N/A N/A N/A N/A N/A N/A fcn-100000 46.57 50.91 44.54 43.34 40.63 44.43 45.86 29.3

Table 3: Results for Open ML datasets for initial sample size 100 under churn at cold accuracy metric across different sizes of fully connected networks. Part 1 of 2.

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Dataset network cold warm s-perturb mixup ls co-dist anchor distill

fcn-10 19.57 N/A 16.9 18.82 N/A N/A N/A 7.61 fcn-100 23.51 16.9 15.33 17.67 N/A 13.66 15.0 3.16 fcn-1000 22.2 N/A 18.2 18.6 N/A 14.68 14.5 4.48 fcn-10000 24.8 24.1 22.33 24.74 N/A 20.79 18.97 20.69 fcn-100000 23.15 18.68 18.52 16.93 20.82 16.85 18.88 18.03

fcn-10 35.52 N/A 32.43 30.78 N/A 29.12 N/A 29.5 fcn-100 34.06 N/A 32.25 30.54 N/A 29.27 23.89 21.37 fcn-1000 39.06 39.64 35.92 37.35 N/A 34.22 30.56 21.32 fcn-10000 39.8 39.68 38.0 35.04 41.94 35.19 39.02 33.96 fcn-100000 33.84 N/A 34.13 N/A N/A 35.67 N/A N/A

fcn-10 48.06 36.85 46.15 34.7 36.74 33.85 18.54 2.07 fcn-100 46.97 N/A 44.94 41.89 41.4 40.91 28.01 36.61 fcn-1000 47.06 N/A N/A N/A N/A N/A 36.93 5.37 fcn-10000 45.85 N/A 45.65 46.06 47.11 45.81 39.53 N/A fcn-100000 45.77 N/A 46.53 48.12 N/A 48.91 43.12 40.57

fcn-10 9.74 N/A 8.97 8.76 9.18 8.65 N/A 8.12 fcn-100 7.44 N/A 7.48 6.91 N/A N/A 7.28 6.69 fcn-1000 8.25 N/A N/A 7.9 7.85 8.11 N/A N/A fcn-10000 9.21 9.45 8.91 8.7 8.56 8.61 8.53 6.48 fcn-100000 10.2 N/A 9.95 9.74 8.85 9.76 9.89 N/A

fcn-10 7.44 N/A 6.48 N/A N/A N/A N/A N/A fcn-100 3.85 N/A 3.39 3.15 N/A 2.42 3.81 3.05 fcn-1000 6.45 4.98 3.83 3.41 N/A 0.88 1.61 0.15 fcn-10000 5.08 4.22 3.21 1.58 N/A 0.56 1.89 0.01 fcn-100000 7.69 3.98 4.67 3.45 4.19 3.11 3.7 0.22

fcn-10 91.44 91.67 91.33 N/A 92.0 90.89 92.22 90.56 fcn-100 63.1 N/A 63.6 N/A 63.5 N/A N/A N/A fcn-1000 59.6 60.1 59.1 58.57 58.9 N/A 59.13 54.43 fcn-10000 61.67 N/A N/A N/A 60.05 N/A N/A N/A fcn-100000 61.62 N/A 61.78 60.53 60.78 61.88 61.72 N/A

Table 4: Results for Open ML datasets for initial sample size 100 under churn at cold accuracy metric across different sizes of fully connected networks. Part 2 of 2.

fcn-10 fcn-100 fcn-1000 fcn-10000 fcn-100000 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn adult 0.35 0.36 0.32 0.39 0.38 0.41 0.43 0.6 0.53 0.61 bank 0.25 0.64 0.25 0.48 0.42 0.78 0.57 0.96 0.53 0.81 COMPAS 0.48 0.77 0.45 0.68 0.47 0.67 0.54 0.91 0.52 0.94 magic04 0.54 0.99 0.63 1.05 0.88 1.37 0.9 1.68 0.71 1.4 phonemes 0.42 0.54 0.36 0.48 0.41 0.57 0.41 0.66 0.44 0.74 electricity 0.94 2.06 0.67 1.49 0.55 1.43 0.62 1.62 0.62 1.82 eeg 0.65 3.23 0.59 4.59 0.59 4.71 0.59 4.69 0.47 4.64 churn 1.17 1.49 1.72 2.29 2.02 2.93 2.13 3.22 1.34 2.72 elevators 0.54 1.06 0.74 1.48 0.93 1.89 0.97 1.97 0.81 1.77 pollen 0.51 0.75 0.46 0.89 0.44 1.16 0.45 1.25 0.42 1.4 phishing 0.27 0.32 0.28 0.27 0.31 0.31 0.37 0.44 0.45 0.51 wilt 0.45 0.62 0.57 0.83 1.12 1.43 1.2 1.41 0.94 1.63 letters 0.62 0.78 0.53 0.69 0.53 0.66 0.51 0.69 0.51 0.59

Table 5: Open ML Error Bands for initial sample size 100: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

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Dataset network cold warm s-perturb mixup ls co-dist anchor distill

fcn-10 4.96 N/A 4.58 N/A N/A N/A N/A N/A fcn-100 5.49 N/A 4.87 N/A 5.3 4.39 4.51 3.53 fcn-1000 6.27 N/A 6.05 6.57 N/A 5.78 6.62 4.39 fcn-10000 8.8 N/A 8.71 8.72 N/A N/A N/A 4.68 fcn-100000 10.36 9.47 9.38 9.28 9.29 9.1 N/A 3.13

fcn-10 4.29 N/A 3.99 4.23 3.35 2.57 4.19 2.39 fcn-100 6.23 N/A 5.32 5.72 6.32 4.87 N/A 1.48 fcn-1000 10.04 8.43 7.8 8.25 8.89 7.55 8.77 5.58 fcn-10000 10.04 9.19 9.15 8.72 8.75 8.68 9.25 3.75 fcn-100000 7.81 8.02 7.86 8.28 6.86 7.35 8.51 7.29

fcn-10 17.97 13.42 12.59 12.95 13.69 11.37 13.04 5.34 fcn-100 22.4 21.2 19.47 20.4 N/A 18.9 20.7 10.94 fcn-1000 27.56 27.41 24.37 24.68 27.79 23.67 25.22 18.51 fcn-10000 26.83 23.97 23.01 23.19 25.72 22.85 24.0 19.97 fcn-100000 18.04 18.89 16.15 17.49 16.08 16.68 17.76 8.73

fcn-10 12.05 10.03 10.41 N/A N/A 10.1 10.5 7.26 fcn-100 9.37 8.79 8.69 N/A N/A 8.91 9.28 7.11 fcn-1000 10.45 10.66 10.09 N/A 9.02 9.3 11.14 7.4 fcn-10000 13.04 13.16 13.26 N/A 12.62 12.45 14.3 8.14 fcn-100000 14.08 14.1 14.0 13.16 12.97 12.91 14.79 8.58

electricity

fcn-10 16.27 14.56 14.97 13.54 14.24 14.16 15.77 10.36 fcn-100 17.11 15.42 15.73 14.63 15.39 13.98 17.16 15.25 fcn-1000 18.16 17.53 17.23 15.69 16.19 14.94 18.22 8.99 fcn-10000 19.94 19.47 18.64 17.38 18.15 17.01 20.53 10.18 fcn-100000 20.68 20.23 19.14 18.2 19.44 18.47 19.53 5.21

fcn-10 47.44 35.23 36.92 33.12 38.18 33.34 28.04 13.54 fcn-100 41.01 N/A N/A 39.82 N/A 44.6 33.45 N/A fcn-1000 48.02 42.96 42.04 39.98 49.98 54.99 26.99 2.0 fcn-10000 41.02 50.38 44.65 37.09 49.09 38.15 30.02 1.01 fcn-100000 27.73 20.25 19.75 19.67 24.75 19.72 22.67 17.89

Table 6: Results for Open ML datasets with initial sample size 1000 under churn at cold accuracy metric across different sizes of fully connected networks. Part 1 of 2.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

fcn-10 21.61 17.23 17.85 15.69 N/A 14.79 17.52 15.5 fcn-100 26.42 N/A 20.34 21.44 26.07 16.68 18.32 4.13 fcn-1000 27.15 25.58 22.19 20.49 N/A 18.71 17.59 5.51 fcn-10000 27.84 29.72 22.21 21.26 N/A 20.22 22.92 18.39 fcn-100000 14.51 11.64 11.27 10.72 10.96 11.0 11.53 8.57

fcn-10 24.34 19.75 20.38 18.83 19.88 16.72 21.77 11.64 fcn-100 30.83 29.56 29.18 28.81 29.97 26.77 30.48 13.68 fcn-1000 33.34 35.87 30.41 31.47 32.91 30.53 34.38 10.44 fcn-10000 34.79 34.36 29.77 30.9 32.76 28.95 31.85 11.29 fcn-100000 23.23 N/A 22.86 N/A 22.57 24.38 N/A N/A

fcn-10 46.05 N/A 23.42 N/A 31.2 N/A 33.21 N/A fcn-100 42.82 35.15 36.11 33.8 35.65 34.58 39.95 10.93 fcn-1000 44.03 N/A 42.63 44.6 42.06 41.78 40.44 35.15 fcn-10000 45.94 N/A 41.64 41.04 40.78 42.25 41.95 6.74 fcn-100000 45.72 N/A 43.77 43.16 43.31 41.33 43.03 13.41

fcn-10 3.25 N/A N/A N/A N/A N/A N/A N/A fcn-100 3.95 N/A 3.68 3.42 3.2 3.21 3.45 2.52 fcn-1000 4.43 4.2 3.97 4.01 4.1 3.74 4.08 2.91 fcn-10000 5.29 5.2 5.15 5.09 4.69 5.07 5.07 4.53 fcn-100000 5.93 5.79 5.38 5.51 5.03 5.12 5.38 3.49

fcn-10 4.1 2.61 2.87 2.76 N/A 2.14 2.9 1.53 fcn-100 4.5 4.68 3.89 3.96 4.93 3.49 3.96 3.02 fcn-1000 9.55 7.27 7.27 6.67 N/A 7.0 7.58 4.93 fcn-10000 11.56 10.07 9.67 9.51 N/A 9.2 10.13 9.68 fcn-100000 5.42 5.22 5.0 4.53 4.63 4.43 4.64 3.43

fcn-10 38.44 N/A 25.92 N/A N/A N/A N/A 23.56 fcn-100 22.74 20.92 21.31 N/A N/A N/A 20.81 19.05 fcn-1000 23.01 23.15 23.47 23.86 23.06 23.44 22.04 16.92 fcn-10000 27.44 26.48 26.29 26.1 24.79 24.86 24.73 18.97 fcn-100000 30.33 29.57 28.76 28.23 26.96 27.89 27.82 20.71

Table 7: Results for Open ML datasets with initial sample size 1000 under churn at cold accuracy metric across different sizes of fully connected networks. Part 2 of 2.

fcn-10 fcn-100 fcn-1000 fcn-10000 fcn-100000 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn adult 0.35 0.22 0.36 0.28 0.35 0.35 0.44 0.51 0.46 0.55 bank 0.2 0.26 0.28 0.38 0.37 0.6 0.52 0.77 0.51 0.69 COMPAS 0.45 0.36 0.44 0.46 0.48 0.66 0.55 0.73 0.56 0.76 magic04 0.45 0.67 0.6 1.09 0.85 1.58 0.81 1.51 0.67 0.99 phonemes 0.43 0.37 0.38 0.33 0.41 0.43 0.41 0.54 0.42 0.55 electricity 0.59 0.83 0.47 0.84 0.51 1.04 0.58 1.31 0.57 1.38 eeg 0.63 2.89 0.59 4.57 0.59 4.71 0.59 4.62 0.4 4.02 churn 1.08 1.73 1.74 2.58 2.02 2.98 1.91 3.0 0.87 2.19 elevators 0.5 1.0 0.74 1.55 0.89 1.79 0.85 1.8 0.82 1.5 pollen 0.48 0.73 0.45 0.94 0.45 1.34 0.42 1.35 0.43 1.41 phishing 0.26 0.16 0.26 0.2 0.26 0.24 0.32 0.34 0.39 0.41 wilt 0.32 0.39 0.57 0.72 1.12 1.55 1.08 1.94 0.72 1.2 letters 0.67 0.74 0.47 0.5 0.51 0.56 0.54 0.61 0.58 0.69

Table 8: Open ML Error Bands with initial sample size 1000: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 38.0 36.1 36.3 36.4 N/A 36.6 34.5 N/A convnet-2 25.4 N/A 24.0 24.5 24.8 N/A N/A N/A convnet-4 20.7 N/A 19.4 19.0 18.9 N/A 17.5 17.3 convnet-8 23.4 N/A 23.0 22.7 22.1 N/A 21.2 N/A convnet-16 27.0 N/A 27.1 25.8 25.8 N/A 25.4 N/A

fashion mnist

convnet-1 36.9 34.4 34.7 32.8 33.9 33.7 32.8 28.9 convnet-2 34.0 32.9 33.0 31.5 31.0 31.2 31.8 28.0 convnet-4 31.8 N/A N/A 31.1 30.8 30.8 30.6 N/A convnet-8 28.9 N/A 29.7 27.9 27.3 27.5 27.9 24.1 convnet-16 35.0 N/A 32.5 33.0 32.7 32.4 33.5 26.2

emnist balanced

convnet-1 93.4 91.7 92.0 91.6 N/A 91.0 91.7 N/A convnet-2 87.0 84.2 84.4 84.5 84.9 84.1 84.4 N/A convnet-4 85.9 82.6 82.0 81.7 82.0 82.2 82.0 76.4 convnet-8 84.8 82.0 82.2 82.1 82.2 82.0 81.7 74.3 convnet-16 88.6 N/A 87.5 87.4 87.3 N/A 87.4 82.2

emnist byclass

convnet-1 73.5 71.75 70.5 69.75 70.25 69.25 70.0 65.75 convnet-2 68.8 64.6 65.8 63.6 63.2 64.2 63.6 N/A convnet-4 64.8 62.6 62.0 60.0 59.2 61.2 61.4 52.6 convnet-8 67.5 63.25 64.25 64.5 63.25 64.25 61.25 51.5 convnet-16 63.0 63.33 59.67 58.67 57.67 61.33 57.33 50.67

emnist bymerge

convnet-1 76.5 77.5 75.0 75.5 77.25 75.5 75.0 N/A convnet-2 71.8 N/A 67.4 66.0 67.4 67.8 67.6 62.0 convnet-4 61.4 N/A 58.0 59.0 59.0 61.4 58.2 N/A convnet-8 65.25 61.75 58.75 60.75 60.0 59.75 59.75 57.25 convnet-16 65.33 59.67 60.33 59.67 59.33 60.33 57.67 50.67

emnist letters

convnet-1 77.4 75.9 76.4 75.3 N/A 74.5 75.7 N/A convnet-2 68.8 67.7 66.4 66.8 66.6 66.6 65.6 N/A convnet-4 61.9 N/A 59.9 59.8 59.4 60.5 58.9 55.2 convnet-8 63.4 N/A 62.7 62.0 61.0 N/A 60.5 N/A convnet-16 66.5 N/A 66.1 66.2 65.2 N/A 65.2 N/A

emnist digits

convnet-1 33.8 32.0 32.1 31.9 31.5 31.6 30.9 29.6 convnet-2 23.0 N/A 22.9 N/A 22.8 N/A N/A N/A convnet-4 23.3 22.2 22.9 21.7 21.4 N/A 20.2 N/A convnet-8 18.8 N/A 19.7 19.1 18.3 N/A 16.9 16.8 convnet-16 21.89 N/A 23.44 22.56 21.56 N/A 20.67 19.56

emnist mnist

convnet-1 33.3 31.4 31.5 31.0 N/A 32.6 N/A N/A convnet-2 22.6 22.2 22.2 22.3 22.1 N/A 21.4 N/A convnet-4 19.6 N/A N/A 19.0 19.5 N/A N/A N/A convnet-8 21.6 N/A 20.9 21.8 20.4 N/A 20.0 N/A convnet-16 22.8 N/A N/A 22.1 21.9 N/A 21.1 N/A

convnet-1 53.4 49.8 50.6 50.4 51.2 49.2 47.5 N/A convnet-2 42.7 40.4 40.9 40.9 41.1 40.7 37.9 37.1 convnet-4 40.4 N/A 37.5 38.7 37.8 N/A 37.3 35.5 convnet-8 39.9 N/A 40.3 38.1 38.3 N/A 37.2 34.2 convnet-16 41.2 N/A N/A 40.3 39.5 N/A 38.8 N/A

convnet-1 93.5 89.8 89.7 89.1 89.9 87.9 88.6 N/A convnet-2 86.2 83.7 83.8 83.7 83.9 83.1 83.3 76.8 convnet-4 83.4 N/A 82.6 81.4 81.4 81.1 80.9 72.5 convnet-8 76.5 73.8 74.1 73.3 73.0 71.8 70.8 62.1 convnet-16 79.44 78.11 76.22 76.11 75.89 76.11 77.0 65.11

Table 9: Results for MNIST variants with initial sample 100 under churn at cold accuracy metric across different sizes of convolutional networks.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn mnist 0.48 0.52 0.38 0.41 0.37 0.39 0.41 0.43 0.43 0.45 fashion mnist 0.52 0.58 0.56 0.59 0.55 0.59 0.56 0.59 0.59 0.62 emnist balanced 0.59 0.61 0.69 0.7 0.73 0.73 0.77 0.76 0.68 0.69 emnist byclass 0.58 0.65 0.68 0.72 0.71 0.73 0.6 0.64 0.59 0.72 emnist bymerge 0.56 0.62 0.66 0.68 0.7 0.76 0.58 0.64 0.55 0.58 emnist letters 0.59 0.63 0.69 0.69 0.75 0.74 0.73 0.74 0.68 0.69 emnist digits 0.44 0.47 0.34 0.37 0.43 0.47 0.44 0.48 0.45 0.49 emnist mnist 0.44 0.46 0.35 0.36 0.39 0.41 0.41 0.46 0.4 0.43 kmnist 0.54 0.59 0.56 0.6 0.59 0.62 0.62 0.67 0.66 0.71 k49 mnist 0.63 0.62 0.72 0.68 0.74 0.75 0.76 0.75 0.69 0.68

Table 10: MNIST Error Bands with initial sample size 100: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 11.18 N/A 10.33 9.75 9.67 10.05 8.85 8.57 convnet-2 6.96 N/A 6.86 6.62 5.82 N/A 5.81 5.67 convnet-4 6.68 N/A 6.78 5.93 5.02 N/A 5.21 4.81 convnet-8 7.13 N/A N/A 5.71 4.95 N/A N/A 5.22 convnet-16 7.91 N/A 8.25 6.39 5.35 N/A 6.95 6.35

fashion mnist

convnet-1 22.17 18.72 19.4 18.76 18.07 18.13 17.51 11.88 convnet-2 19.85 18.17 17.93 17.44 17.49 16.86 16.48 12.16 convnet-4 18.48 N/A 17.08 16.93 16.52 16.53 15.75 11.9 convnet-8 17.57 N/A 16.35 16.07 14.62 15.36 15.1 9.05 convnet-16 17.98 N/A 17.23 16.88 15.9 16.21 15.87 11.11

emnist balanced

convnet-1 52.34 41.04 41.86 43.79 41.68 40.97 38.62 N/A convnet-2 45.04 N/A 39.29 39.45 37.78 38.43 35.3 32.44 convnet-4 42.21 N/A 37.46 37.12 35.53 N/A 33.64 29.41 convnet-8 40.95 N/A 35.82 35.66 33.96 N/A 32.5 26.39 convnet-16 42.74 N/A 39.11 37.3 35.13 N/A N/A 29.27

emnist byclass

convnet-1 44.18 36.18 36.35 36.15 34.9 34.96 35.62 27.07 convnet-2 38.39 33.21 34.04 33.68 32.56 33.2 31.41 24.17 convnet-4 36.4 N/A 32.33 31.74 30.79 31.96 30.42 24.5 convnet-8 36.72 N/A 32.9 31.7 30.17 N/A N/A 25.51 convnet-16 36.81 33.06 33.11 31.59 29.64 31.21 30.68 21.81

emnist bymerge

convnet-1 42.59 N/A 34.44 34.76 34.12 33.76 32.58 N/A convnet-2 37.35 N/A 32.87 32.52 31.43 N/A 30.74 N/A convnet-4 34.17 30.62 30.38 29.86 28.72 29.59 27.07 21.28 convnet-8 33.59 N/A 30.56 29.89 27.63 N/A N/A N/A convnet-16 35.46 N/A 31.65 30.77 28.83 N/A 28.27 22.9

emnist letters

convnet-1 38.03 30.14 30.95 31.53 30.39 30.26 27.41 24.84 convnet-2 31.65 N/A 28.36 27.72 26.37 N/A 24.26 21.88 convnet-4 29.58 N/A 26.99 26.2 24.61 N/A 23.22 20.16 convnet-8 29.52 N/A 28.08 26.19 24.39 N/A 22.73 20.94 convnet-16 30.15 N/A 27.36 26.39 24.28 N/A 23.77 20.67

emnist digits

convnet-1 9.16 N/A 8.71 8.29 8.04 N/A 7.62 7.31 convnet-2 6.43 N/A 6.38 5.98 5.17 N/A 5.12 5.02 convnet-4 6.81 N/A 6.95 6.09 5.29 N/A 5.42 4.81 convnet-8 6.74 N/A N/A 6.14 5.11 N/A 5.9 4.91 convnet-16 6.93 N/A N/A 5.72 4.72 N/A N/A 5.61

emnist mnist

convnet-1 9.16 N/A 8.28 8.3 7.71 8.62 7.39 7.34 convnet-2 5.7 N/A 6.02 5.4 4.94 N/A 4.95 4.59 convnet-4 6.42 N/A 6.28 5.67 4.9 N/A 5.21 4.49 convnet-8 6.39 N/A 6.64 5.16 4.48 N/A 5.28 4.23 convnet-16 7.09 N/A 7.14 6.03 5.15 N/A 6.11 4.97

convnet-1 23.31 18.22 18.25 19.1 18.97 18.56 16.68 16.08 convnet-2 16.54 N/A 15.35 14.98 13.79 N/A 13.12 12.18 convnet-4 15.95 N/A 14.08 13.41 12.08 N/A 12.0 9.9 convnet-8 16.89 N/A 15.17 14.19 12.68 N/A 12.72 10.67 convnet-16 18.0 N/A 16.57 15.37 13.49 N/A 13.64 11.97

convnet-1 56.23 44.13 44.89 46.74 46.12 43.13 41.79 40.3 convnet-2 48.22 40.43 41.02 40.89 38.97 40.5 36.31 31.52 convnet-4 46.35 N/A 39.48 39.46 37.33 39.99 35.24 29.46 convnet-8 47.84 N/A 41.35 40.98 38.23 N/A 36.39 29.13 convnet-16 49.02 42.1 42.1 41.44 38.45 41.59 38.16 30.74

Table 11: Results for MNIST variants under churn at cold accuracy metric across different sizes of convolutional networks with initial sample size 1000..

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn mnist 0.48 0.52 0.38 0.41 0.37 0.39 0.41 0.43 0.43 0.45 fashion mnist 0.52 0.58 0.56 0.59 0.55 0.59 0.56 0.59 0.59 0.62 emnist balanced 0.59 0.61 0.69 0.7 0.73 0.73 0.77 0.76 0.68 0.69 emnist byclass 0.58 0.65 0.68 0.72 0.71 0.73 0.6 0.64 0.62 0.76 emnist bymerge 0.56 0.62 0.66 0.68 0.7 0.76 0.58 0.64 0.59 0.61 emnist letters 0.59 0.63 0.69 0.69 0.75 0.74 0.73 0.74 0.68 0.69 emnist digits 0.44 0.47 0.34 0.37 0.43 0.47 0.44 0.48 0.45 0.49 emnist mnist 0.44 0.46 0.35 0.36 0.39 0.41 0.41 0.46 0.4 0.43 kmnist 0.54 0.59 0.56 0.6 0.59 0.62 0.62 0.67 0.66 0.71 k49 mnist 0.63 0.62 0.72 0.68 0.74 0.75 0.76 0.75 0.69 0.68

Table 12: MNIST Error Bands with initial sample size 1000: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 4.84 4.22 4.01 3.73 3.34 3.9 3.81 3.32 convnet-2 4.16 3.78 3.61 3.22 2.86 3.46 3.3 2.96 convnet-4 4.06 3.66 3.49 3.09 2.72 3.4 3.2 2.93 convnet-8 4.29 4.01 3.81 3.35 2.99 3.79 3.54 3.19 convnet-16 4.47 4.04 3.97 3.44 3.08 3.84 3.65 3.2

fashion mnist

convnet-1 14.91 12.32 12.5 12.26 11.98 11.65 11.46 7.81 convnet-2 13.82 11.96 11.9 11.77 11.41 11.29 11.12 7.5 convnet-4 13.04 11.52 11.42 11.3 10.93 10.77 10.64 6.78 convnet-8 12.92 11.44 11.52 11.18 10.98 10.8 10.85 7.06 convnet-16 13.56 12.13 12.0 11.81 11.38 11.55 11.29 7.61

emnist balanced

convnet-1 25.34 20.86 20.83 20.8 19.89 19.97 19.48 14.26 convnet-2 23.33 20.18 20.09 19.45 18.38 19.11 18.67 13.62 convnet-4 24.71 19.93 19.76 19.07 17.5 19.0 18.07 13.0 convnet-8 22.49 20.17 20.12 19.21 17.62 19.35 18.48 13.22 convnet-16 22.48 20.09 20.3 19.35 17.48 19.52 18.35 13.4

emnist byclass

convnet-1 23.66 N/A 19.92 19.46 18.93 18.69 N/A 13.34 convnet-2 21.7 N/A 19.25 18.73 17.64 18.18 N/A 12.32 convnet-4 21.45 N/A 19.01 18.71 17.41 18.09 18.39 11.76 convnet-8 21.47 N/A 19.42 18.72 17.08 18.13 N/A 11.9 convnet-16 21.83 N/A 19.63 18.82 17.28 18.68 18.78 12.71

emnist bymerge

convnet-1 21.1 17.78 17.35 17.44 16.67 16.61 16.98 12.09 convnet-2 19.27 17.02 16.74 16.33 15.41 15.84 16.21 11.23 convnet-4 18.37 16.72 16.41 15.85 14.65 15.42 15.91 10.74 convnet-8 19.01 17.08 17.12 16.31 15.05 16.18 16.44 10.95 convnet-16 18.91 N/A 17.49 16.56 15.08 16.5 16.66 11.11

emnist letters

convnet-1 17.17 14.29 14.25 13.82 13.03 13.56 13.09 10.47 convnet-2 15.44 13.62 13.51 12.9 11.74 12.96 12.39 9.82 convnet-4 15.25 13.58 13.32 12.67 11.45 12.99 12.27 9.33 convnet-8 15.16 13.25 13.33 12.52 11.32 12.76 12.17 9.14 convnet-16 15.19 13.63 13.33 12.55 11.18 12.99 12.42 9.4

emnist digits

convnet-1 3.98 3.43 3.28 3.0 2.64 3.21 3.06 2.82 convnet-2 3.64 3.36 3.13 2.91 2.53 3.1 2.95 2.7 convnet-4 3.59 3.24 3.17 2.77 2.37 3.07 2.88 2.56 convnet-8 3.89 3.41 3.33 2.88 2.58 3.3 3.16 2.73 convnet-16 4.0 3.56 3.52 2.96 2.63 3.34 3.24 2.74

emnist mnist

convnet-1 4.09 3.5 3.39 3.14 2.84 3.29 3.17 2.93 convnet-2 3.69 3.33 3.11 2.78 2.42 3.02 2.91 2.65 convnet-4 3.64 3.36 3.22 2.86 2.5 3.15 3.04 2.74 convnet-8 3.76 3.4 3.41 2.92 2.55 3.35 3.13 2.73 convnet-16 3.94 3.6 3.51 2.98 2.63 3.41 3.23 2.93

convnet-1 8.99 7.31 7.2 6.76 6.18 7.07 6.64 6.17 convnet-2 7.91 6.96 6.73 6.17 5.48 6.63 6.28 5.62 convnet-4 7.69 6.8 6.66 6.07 5.24 6.57 6.22 5.42 convnet-8 7.93 6.71 6.74 6.03 5.26 6.6 6.23 5.52 convnet-16 8.07 6.99 6.89 6.02 5.35 6.72 6.29 5.81

convnet-1 27.34 21.79 21.76 21.62 20.83 20.8 19.27 17.11 convnet-2 23.82 19.97 20.03 19.27 17.75 19.42 18.02 15.99 convnet-4 22.73 19.42 19.34 18.48 16.57 18.93 17.43 15.4 convnet-8 22.33 19.22 19.29 18.21 16.21 18.69 17.1 15.14 convnet-16 22.37 19.47 19.23 18.02 15.91 18.94 17.19 15.46

Table 13: Results for MNIST variants with initial sample 10000 under churn at cold accuracy metric across different sizes of convolutional networks.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn mnist 0.34 0.36 0.31 0.32 0.38 0.39 0.43 0.44 0.46 0.47 fashion mnist 0.27 0.3 0.27 0.3 0.36 0.39 0.37 0.39 0.37 0.49 emnist balanced 0.23 0.25 0.31 0.34 0.34 0.37 0.36 0.39 0.36 0.38 emnist byclass 0.23 0.3 0.26 0.32 0.29 0.35 0.29 0.33 0.31 0.35 emnist bymerge 0.25 0.29 0.3 0.34 0.32 0.37 0.29 0.34 0.31 0.35 emnist letters 0.31 0.34 0.42 0.44 0.45 0.47 0.46 0.49 0.47 0.48 emnist digits 0.26 0.27 0.29 0.3 0.36 0.37 0.49 0.5 0.44 0.45 emnist mnist 0.29 0.31 0.28 0.29 0.41 0.42 0.48 0.49 0.43 0.44 kmnist 0.39 0.4 0.49 0.51 0.53 0.54 0.6 0.61 0.62 0.63 k49 mnist 0.41 0.42 0.45 0.46 0.54 0.55 0.54 0.54 0.57 0.56

Table 14: MNIST Error Bands with initial sample size 10000: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 79.2 N/A N/A 77.2 78.4 N/A 80.8 70.8 convnet-2 81.2 79.9 N/A 80.3 81.2 82.6 83.4 74.1 convnet-4 80.3 81.5 74.6 79.1 80.2 72.4 84.7 64.6 convnet-8 70.22 80.78 72.78 70.22 62.56 69.33 71.33 59.89 convnet-16 41.33 70.83 41.0 52.5 52.33 53.67 42.83 34.5

convnet-1 77.9 N/A 76.3 N/A 76.0 75.5 N/A N/A convnet-2 74.7 77.1 73.3 74.6 73.0 72.3 74.1 70.1 convnet-4 71.7 70.4 70.8 73.6 70.9 69.0 N/A 61.5 convnet-8 75.5 N/A N/A N/A N/A N/A N/A N/A convnet-16 79.4 79.5 79.9 76.7 78.2 78.1 82.8 69.9

Table 15: Results for SVHN and CIFAR datasets with initial sample size 100 under churn at cold accuracy metric across different sizes of convolutional networks..

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn svhn 0.67 1.33 0.73 1.37 0.88 1.59 1.35 2.18 1.87 3.25 cifar10 0.4 0.89 0.39 0.85 0.41 0.9 0.4 1.0 0.43 1.1 cifar100 0.39 0.93 0.4 0.92 0.4 0.9 0.41 1.02 0.45 1.09

Table 16: SVHN and CIFAR with initial sample size 100 Error Bands: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 31.1 N/A 24.44 25.97 27.33 23.3 23.54 21.26 convnet-2 29.49 24.26 24.05 25.88 26.48 23.23 21.41 16.73 convnet-4 32.12 26.88 27.39 29.2 29.21 26.01 25.43 22.64 convnet-8 42.22 36.14 34.78 37.41 36.91 34.82 35.46 28.55 convnet-16 50.94 46.12 37.26 37.87 42.62 44.44 41.01 29.65

convnet-1 50.82 N/A 43.71 44.4 45.66 41.53 44.37 29.45 convnet-2 52.16 N/A 51.11 47.64 48.83 44.72 N/A 39.89 convnet-4 52.01 47.57 46.36 47.17 47.92 44.61 45.75 29.13 convnet-8 52.65 N/A 47.42 47.39 48.29 44.34 47.07 34.17 convnet-16 53.24 46.97 48.43 48.2 48.79 44.83 47.59 34.54

Table 17: Results for SVHN and CIFAR under churn at cold accuracy metric across network sizes.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn svhn 0.53 0.77 0.61 0.97 0.6 1.45 0.73 2.19 1.4 2.67 cifar10 0.41 0.65 0.41 0.62 0.39 0.62 0.39 0.64 0.41 0.66

Table 18: SVHN and CIFAR with initial sample size 1000 Error Bands: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 15.43 N/A 11.29 12.08 12.01 10.63 9.94 6.37 convnet-2 14.29 N/A 12.43 12.15 11.43 10.97 N/A 6.99 convnet-4 14.25 N/A 13.27 12.16 11.38 11.16 9.34 7.43 convnet-8 14.61 N/A 11.9 11.92 11.04 11.08 9.09 7.09 convnet-16 24.35 16.97 16.81 16.0 15.89 17.04 15.39 9.68

convnet-1 41.28 N/A 28.82 29.9 29.61 27.47 28.5 14.08 convnet-2 39.5 N/A 31.67 31.23 31.96 28.75 N/A 16.03 convnet-4 39.47 N/A 33.4 32.9 31.18 N/A N/A 18.7 convnet-8 39.67 N/A 32.78 31.53 30.87 30.56 25.42 16.87 convnet-16 40.75 N/A 33.94 N/A 31.6 N/A N/A 20.12

Table 19: Results for SVHN and CIFAR datasets with initial sample size 10000 under churn at cold accuracy metric across different sizes of convolutional networks..

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn svhn 0.28 0.29 0.33 0.35 0.43 0.45 0.49 0.51 0.86 1.8 cifar10 0.16 0.26 0.16 0.26 0.17 0.26 0.18 0.24 0.39 0.67

Table 20: SVHN and CIFAR with initial sample size 10000 Error Bands: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

5 o Clock Shadow

convnet-1 5.08 N/A 3.41 4.45 1.11 1.9 5.48 0.04 convnet-2 5.57 N/A 3.29 4.28 2.23 2.86 N/A 0.16 convnet-4 4.97 N/A 3.69 4.05 2.11 2.44 N/A 0.08 convnet-8 4.84 N/A 2.95 2.86 1.03 2.08 N/A 0.01 convnet-16 4.49 3.81 2.32 2.02 0.37 1.74 4.57 0.06

Arched Eyebrows

convnet-1 19.39 N/A 18.04 14.6 N/A 13.77 N/A 1.97 convnet-2 20.83 20.17 18.86 17.21 15.36 14.97 21.2 1.32 convnet-4 17.66 N/A 15.71 13.74 15.2 13.17 N/A 0.97 convnet-8 16.99 N/A 14.54 11.22 11.68 12.7 N/A 0.79 convnet-16 16.96 15.49 12.35 8.59 6.26 9.77 N/A 0.34

convnet-1 36.39 35.27 35.45 34.43 37.04 32.97 36.42 30.17 convnet-2 37.0 36.5 34.84 36.24 36.7 34.59 38.84 27.0 convnet-4 37.9 37.46 36.38 36.73 37.22 35.93 40.48 29.28 convnet-8 40.86 N/A 40.14 39.71 39.36 38.49 41.6 32.93 convnet-16 44.17 41.65 40.81 42.31 42.15 41.38 43.67 37.37

Bags Under Eyes

convnet-1 11.22 N/A 7.77 9.09 5.04 4.2 N/A 0.13 convnet-2 11.46 11.6 9.13 9.67 6.2 6.13 11.69 0.25 convnet-4 12.14 10.92 8.06 7.75 4.39 4.69 10.57 0.16 convnet-8 10.01 N/A 6.74 7.91 3.21 4.16 N/A 0.2 convnet-16 7.36 8.4 5.18 5.8 2.31 2.63 N/A 0.01

convnet-1 0.62 N/A 0.41 0.43 0.0 0.26 0.0 0.0 convnet-2 0.58 N/A 0.5 N/A 0.06 0.31 0.6 0.04 convnet-4 0.71 N/A 0.45 N/A 0.04 0.39 0.75 0.02 convnet-8 0.74 N/A 0.25 0.5 0.03 0.33 0.53 0.0 convnet-16 0.78 N/A 0.3 0.35 0.0 0.24 0.53 0.0

convnet-1 11.62 N/A N/A 10.06 10.09 9.88 N/A N/A convnet-2 12.77 N/A 12.02 9.5 10.32 10.65 12.16 6.97 convnet-4 12.11 11.35 11.4 9.55 9.64 10.7 11.63 7.34 convnet-8 12.67 N/A 11.79 10.02 9.66 11.08 12.41 5.84 convnet-16 13.61 N/A 12.92 10.69 11.7 11.51 12.76 8.21

convnet-1 3.89 4.1 0.78 1.36 N/A 1.11 N/A 0.06 convnet-2 5.01 N/A 1.99 2.0 N/A 2.65 N/A 0.31 convnet-4 3.79 N/A 2.19 2.28 N/A 2.33 N/A 0.21 convnet-8 2.1 N/A 1.36 0.75 N/A 1.45 N/A 0.05 convnet-16 2.56 N/A 0.89 0.37 0.94 0.79 N/A 0.01

convnet-1 14.24 N/A 11.52 12.31 10.3 6.85 N/A 0.36 convnet-2 15.33 14.56 11.85 13.72 11.49 10.19 14.97 0.73 convnet-4 11.32 N/A N/A N/A N/A 10.14 N/A 2.4 convnet-8 13.76 12.49 10.95 11.26 8.83 7.81 14.23 0.29 convnet-16 11.7 12.16 9.26 9.43 7.12 6.84 12.15 0.4

convnet-1 19.43 20.03 20.82 N/A 19.7 18.78 N/A N/A convnet-2 21.93 20.7 20.07 20.92 20.48 18.96 21.73 16.49 convnet-4 22.01 20.72 20.07 20.71 20.8 20.15 22.05 15.07 convnet-8 23.02 21.72 20.8 21.92 20.93 19.46 20.49 15.59 convnet-16 21.57 21.94 20.44 21.33 19.86 18.84 22.31 15.74

convnet-1 11.79 11.13 11.38 9.73 11.27 10.67 12.26 7.31 convnet-2 12.76 N/A 12.23 10.69 11.86 11.43 13.67 6.62 convnet-4 11.69 N/A 11.4 10.44 11.51 10.81 12.37 6.97 convnet-8 12.18 N/A N/A 10.13 11.38 11.36 N/A 8.66 convnet-16 13.51 11.94 11.85 9.83 12.26 11.8 12.64 7.26

Table 21: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 100. Part 1 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 0.03 N/A 0.03 0.01 N/A N/A 0.01 0.0 convnet-2 0.0 N/A 0.0 0.0 0.0 0.0 N/A 0.0 convnet-4 0.03 N/A 0.0 0.0 N/A 0.0 N/A 0.0 convnet-8 0.14 0.07 0.0 0.0 0.01 0.0 0.06 0.0 convnet-16 0.02 N/A 0.02 0.02 0.02 0.02 N/A 0.01

convnet-1 14.11 N/A 12.46 6.42 9.48 8.39 N/A 0.46 convnet-2 14.79 12.54 11.45 5.93 9.5 8.38 15.33 0.27 convnet-4 13.88 14.21 12.12 6.01 9.22 8.71 16.76 0.47 convnet-8 14.21 N/A 12.64 5.19 7.52 8.54 N/A 0.4 convnet-16 13.14 13.45 9.99 4.18 3.1 7.23 N/A 0.07

Bushy Eyebrows

convnet-1 5.5 4.55 2.81 3.27 2.66 2.33 6.1 0.06 convnet-2 5.05 N/A 4.44 4.89 3.4 3.27 N/A 0.16 convnet-4 5.15 N/A 3.95 3.7 3.5 2.94 N/A 0.17 convnet-8 6.17 N/A 3.42 3.14 1.83 2.45 N/A 0.01 convnet-16 4.2 4.03 2.54 1.59 1.62 1.99 N/A 0.08

convnet-1 1.01 N/A 0.66 1.03 0.15 0.39 N/A 0.0 convnet-2 1.49 1.2 0.58 0.89 0.11 0.46 1.5 0.0 convnet-4 1.31 1.52 0.89 1.16 0.21 0.63 1.6 0.02 convnet-8 1.35 1.6 0.93 0.97 0.3 0.59 N/A 0.03 convnet-16 0.94 N/A 0.38 0.58 0.06 0.32 N/A 0.0

Double Chin

convnet-1 0.81 N/A 0.41 0.7 0.1 0.27 N/A 0.0 convnet-2 0.91 N/A 0.48 0.98 0.08 0.21 N/A 0.0 convnet-4 1.14 N/A 0.66 0.9 0.21 0.53 1.12 0.08 convnet-8 0.82 N/A 0.37 0.48 0.04 0.38 0.85 0.01 convnet-16 0.7 N/A 0.17 0.43 0.07 0.18 N/A 0.0

convnet-1 4.21 4.1 4.07 3.82 2.6 3.48 4.41 2.1 convnet-2 4.33 N/A 3.95 3.82 3.5 3.57 4.49 2.43 convnet-4 4.22 N/A 4.03 3.76 2.93 3.34 4.34 2.26 convnet-8 4.48 3.97 3.78 3.96 2.91 3.38 4.09 2.38 convnet-16 4.76 N/A 3.64 3.91 2.34 2.86 4.1 2.25

convnet-1 1.83 N/A 1.24 1.78 0.18 0.38 2.18 0.02 convnet-2 1.67 N/A 1.15 1.35 0.24 0.72 N/A 0.01 convnet-4 2.23 1.6 0.9 1.43 0.38 0.46 1.61 0.07 convnet-8 1.48 N/A 0.82 1.08 0.14 0.33 1.74 0.01 convnet-16 1.01 1.03 0.59 0.48 0.09 0.29 1.04 0.0

convnet-1 2.42 2.04 1.66 1.82 0.17 1.5 2.08 0.02 convnet-2 2.41 2.24 2.07 1.92 0.25 1.75 2.14 0.03 convnet-4 2.82 N/A 2.32 2.47 0.55 1.77 2.33 0.05 convnet-8 2.54 N/A 1.83 2.04 0.33 1.3 2.27 0.15 convnet-16 2.58 N/A 1.99 1.97 0.16 1.59 2.37 0.04

Heavy Makeup

convnet-1 33.24 N/A 32.88 32.09 33.07 31.89 34.89 28.94 convnet-2 33.33 N/A N/A 31.73 33.04 N/A N/A 30.77 convnet-4 34.25 N/A 32.64 31.62 32.73 31.38 N/A 27.21 convnet-8 34.66 36.17 33.25 33.08 33.03 33.12 36.2 26.42 convnet-16 37.36 35.43 34.73 32.62 34.98 35.18 35.91 29.18

High Cheekbones

convnet-1 42.29 N/A 43.53 41.0 N/A 41.86 N/A N/A convnet-2 43.96 43.86 41.62 41.11 42.38 40.82 43.29 37.8 convnet-4 42.79 N/A N/A 43.1 44.1 43.45 47.4 41.11 convnet-8 42.15 N/A 40.87 40.02 41.99 41.39 N/A 37.68 convnet-16 44.06 42.6 41.34 39.23 42.28 42.03 43.09 35.13

Table 22: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 100. Part 2 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 32.43 N/A 32.08 32.34 31.49 N/A 33.36 27.92 convnet-2 32.7 32.02 32.23 32.63 32.03 31.82 33.32 28.17 convnet-4 31.72 N/A N/A 31.34 30.87 30.75 32.75 25.78 convnet-8 34.85 32.86 33.4 34.14 33.96 32.84 34.52 26.24 convnet-16 37.02 N/A 35.57 36.9 35.91 35.21 36.84 29.26

Mouth Slightly Open

convnet-1 44.93 46.3 44.95 45.8 N/A 44.73 47.42 46.37 convnet-2 46.1 45.38 46.17 46.32 46.9 45.13 N/A 42.84 convnet-4 45.31 N/A 45.69 44.16 45.62 44.84 47.76 42.58 convnet-8 45.74 48.62 N/A N/A N/A 46.68 N/A 47.03 convnet-16 51.37 N/A N/A N/A N/A N/A N/A N/A

convnet-1 0.3 0.41 0.04 0.22 0.02 0.06 N/A 0.01 convnet-2 0.27 N/A 0.07 0.07 0.01 0.03 0.3 0.0 convnet-4 0.6 0.45 0.18 0.28 0.09 0.17 0.69 0.03 convnet-8 0.39 0.5 0.13 0.19 0.09 0.17 0.56 0.05 convnet-16 0.21 0.2 0.07 0.08 0.04 0.05 0.19 0.02

Narrow Eyes

convnet-1 0.08 0.17 0.02 0.03 N/A 0.07 N/A 0.02 convnet-2 0.11 N/A 0.01 0.01 N/A 0.03 0.07 0.0 convnet-4 0.57 0.14 0.06 0.07 0.39 0.15 0.26 0.05 convnet-8 0.09 N/A 0.03 0.03 N/A N/A N/A 0.0 convnet-16 0.04 N/A 0.01 0.01 0.02 0.05 N/A 0.01

convnet-1 11.04 N/A 10.86 11.74 12.13 7.67 13.68 0.95 convnet-2 12.31 11.91 12.61 12.16 9.93 7.81 12.0 2.54 convnet-4 13.4 13.82 10.83 11.8 6.17 8.72 13.28 0.49 convnet-8 12.85 N/A 10.39 11.29 7.11 7.29 10.95 0.05 convnet-16 11.34 10.31 8.75 9.88 4.16 5.53 10.26 0.15

convnet-1 9.89 N/A 7.95 8.49 N/A 5.3 N/A 0.58 convnet-2 13.65 13.6 7.52 9.02 10.89 7.86 14.03 0.63 convnet-4 11.1 N/A 7.48 6.62 N/A 6.42 N/A 0.63 convnet-8 7.54 N/A 4.49 3.66 6.6 4.44 N/A 0.18 convnet-16 8.97 7.4 3.53 2.63 3.73 4.29 7.81 0.2

convnet-1 1.51 N/A 1.15 0.74 0.03 0.74 1.2 0.0 convnet-2 1.42 N/A N/A N/A 0.23 0.92 1.76 0.01 convnet-4 1.25 1.32 0.96 0.76 0.12 0.52 N/A 0.03 convnet-8 1.21 N/A 0.73 0.84 0.11 0.55 N/A 0.05 convnet-16 1.9 1.19 0.69 0.45 0.11 0.41 1.05 0.01

Pointy Nose

convnet-1 13.35 13.06 8.18 8.8 11.93 7.32 N/A 0.82 convnet-2 14.22 N/A 8.32 8.34 N/A 7.52 N/A 0.78 convnet-4 10.51 N/A 6.62 6.75 N/A 7.34 N/A 0.79 convnet-8 10.21 10.3 4.12 2.65 6.38 5.53 N/A 0.28 convnet-16 8.4 6.03 3.13 2.26 3.13 4.33 N/A 0.35

Receding Hairline

convnet-1 2.44 N/A 2.24 2.13 0.31 1.09 N/A 0.0 convnet-2 3.15 3.49 2.33 2.14 0.66 1.69 2.74 0.03 convnet-4 3.2 N/A 2.3 2.43 0.87 1.54 N/A 0.03 convnet-8 2.83 N/A 2.22 1.64 0.34 1.64 N/A 0.01 convnet-16 3.03 2.64 2.0 1.45 0.37 1.22 3.2 0.04

Rosy Cheeks

convnet-1 1.66 N/A 0.63 0.94 0.16 0.41 N/A 0.02 convnet-2 1.8 1.67 0.48 0.58 0.22 0.41 1.33 0.01 convnet-4 1.28 1.03 0.43 0.54 0.31 0.28 N/A 0.02 convnet-8 0.98 N/A 0.35 0.27 0.1 0.31 N/A 0.0 convnet-16 0.65 0.59 0.18 0.27 0.06 0.19 N/A 0.0

Table 23: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 100. Part 3 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 1.48 N/A 0.84 1.18 0.07 0.2 1.28 0.01 convnet-2 1.87 N/A 0.99 1.7 0.2 0.53 1.88 0.0 convnet-4 1.72 N/A 1.04 1.29 0.15 0.5 1.63 0.04 convnet-8 1.44 N/A 0.79 1.25 0.26 0.39 N/A 0.05 convnet-16 0.77 N/A 0.5 0.66 0.0 0.22 N/A 0.0

convnet-1 42.83 42.46 42.17 41.71 N/A 40.72 N/A 40.15 convnet-2 42.82 N/A 41.86 42.56 42.63 41.52 N/A N/A convnet-4 44.26 N/A N/A 43.51 N/A 43.41 N/A N/A convnet-8 45.69 45.97 45.27 44.87 45.01 44.48 46.89 41.71 convnet-16 47.36 49.32 46.42 47.13 47.14 46.85 50.39 43.22

Straight Hair

convnet-1 2.94 2.91 1.43 1.61 2.63 1.15 N/A 0.12 convnet-2 3.86 N/A 1.85 2.18 4.14 1.84 N/A 0.19 convnet-4 3.77 N/A 1.35 1.72 N/A 1.87 N/A 0.21 convnet-8 2.19 N/A 1.51 1.23 N/A 1.48 N/A 0.07 convnet-16 2.33 N/A 0.84 0.62 0.8 1.17 N/A 0.03

convnet-1 27.48 24.54 23.44 20.77 24.68 21.24 25.77 17.54 convnet-2 24.67 N/A 25.29 23.02 N/A 22.82 N/A 22.78 convnet-4 26.69 N/A 25.02 21.85 N/A 22.59 N/A N/A convnet-8 24.65 N/A 24.16 21.01 N/A 23.29 N/A 23.5 convnet-16 25.68 26.74 23.53 17.7 22.7 21.47 25.47 16.2

Wearing Earrings

convnet-1 6.61 N/A 3.82 4.14 3.84 2.06 N/A 0.01 convnet-2 7.24 N/A 4.44 5.05 4.0 2.86 N/A 0.21 convnet-4 5.52 N/A 3.83 3.36 4.28 1.92 N/A 0.08 convnet-8 3.85 N/A 2.91 2.3 2.52 2.56 N/A 0.07 convnet-16 4.41 4.65 1.68 1.61 1.61 2.12 N/A 0.13

Wearing Hat

convnet-1 3.52 3.61 3.43 3.38 2.33 2.92 3.35 1.94 convnet-2 3.82 3.89 3.58 3.58 2.88 3.23 3.54 2.36 convnet-4 3.33 3.64 3.63 3.86 2.46 3.35 3.39 1.79 convnet-8 3.95 N/A N/A N/A 3.01 N/A N/A 2.53 convnet-16 3.93 3.59 3.1 3.67 1.75 2.83 3.43 1.97

Wearing Lipstick

convnet-1 33.01 33.37 32.15 32.32 32.06 31.24 32.8 27.09 convnet-2 32.99 N/A 32.23 32.45 32.59 31.67 33.89 28.27 convnet-4 34.36 N/A N/A 33.77 33.88 N/A N/A 29.93 convnet-8 36.84 N/A N/A 36.27 35.6 35.16 38.95 29.11 convnet-16 38.4 N/A 37.44 37.24 37.47 36.44 37.14 30.81

Wearing Necklace

convnet-1 1.03 N/A 0.21 0.3 0.3 0.16 N/A 0.0 convnet-2 1.39 N/A 0.37 0.54 1.25 0.45 N/A 0.02 convnet-4 0.94 N/A 0.28 0.15 0.75 0.47 N/A 0.0 convnet-8 1.2 N/A 0.59 0.21 0.31 0.49 N/A 0.05 convnet-16 0.9 0.69 0.49 0.15 0.19 0.27 N/A 0.04

Wearing Necktie

convnet-1 5.1 5.48 5.12 4.88 3.14 4.54 5.62 2.08 convnet-2 5.11 N/A 4.99 5.07 3.23 4.54 5.09 1.96 convnet-4 5.64 N/A N/A N/A 3.94 4.96 5.61 2.53 convnet-8 5.51 N/A N/A 5.15 3.85 4.78 N/A 1.51 convnet-16 5.25 N/A 5.11 4.98 2.39 4.4 5.08 1.59

convnet-1 15.03 14.34 12.91 13.25 12.65 10.12 16.0 2.63 convnet-2 15.15 15.94 13.72 12.88 10.28 10.82 15.26 1.05 convnet-4 15.53 N/A 13.24 13.1 14.28 11.15 14.29 0.72 convnet-8 14.18 15.34 11.96 11.72 9.47 10.25 14.33 0.51 convnet-16 13.18 N/A 12.24 11.56 10.2 10.22 N/A 0.36

Table 24: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 100. Part 4 of 4.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn 5 o Clock Shadow 0.8 1.1 0.77 1.07 0.82 1.14 0.81 1.13 0.83 1.13 Arched Eyebrows 0.83 1.73 0.82 1.73 0.82 1.78 0.82 1.89 0.84 1.88 Attractive 0.4 1.3 0.41 1.36 0.39 1.57 0.42 1.86 0.42 1.97 Bags Under Eyes 0.85 1.54 0.88 1.55 0.87 1.55 0.86 1.54 0.87 1.55 Bald 0.41 0.43 0.44 0.49 0.47 0.51 0.53 0.57 0.48 0.51 Bangs 0.79 1.13 0.83 1.15 0.82 1.14 0.84 1.18 0.86 1.26 Big Lips 0.74 1.46 0.77 1.54 0.79 1.6 0.79 1.55 0.74 1.46 Big Nose 0.82 1.6 0.83 1.63 0.82 1.65 0.82 1.68 0.84 1.71 Black Hair 0.82 1.42 0.8 1.39 0.82 1.48 0.83 1.56 0.81 1.52 Blond Hair 0.83 1.17 0.85 1.15 0.84 1.18 0.84 1.18 0.85 1.27 Blurry 0.5 0.57 0.53 0.58 0.5 0.56 0.55 0.6 0.59 0.66 Brown Hair 0.84 1.51 0.84 1.53 0.85 1.55 0.87 1.63 0.86 1.62 Bushy Eyebrows 0.84 1.25 0.83 1.23 0.82 1.25 0.84 1.27 0.79 1.21 Chubby 0.63 0.73 0.64 0.75 0.63 0.74 0.69 0.81 0.63 0.73 Double Chin 0.58 0.66 0.61 0.7 0.62 0.74 0.58 0.67 0.59 0.67 Eyeglasses 0.7 0.81 0.68 0.81 0.65 0.78 0.71 0.84 0.72 0.86 Goatee 0.69 0.83 0.66 0.8 0.68 0.81 0.73 0.85 0.65 0.77 Gray Hair 0.55 0.63 0.61 0.69 0.6 0.7 0.66 0.75 0.66 0.75 Heavy Makeup 0.62 1.38 0.62 1.42 0.63 1.46 0.64 1.67 0.68 1.8 High Cheekbones 0.46 1.35 0.48 1.43 0.51 1.58 0.59 1.82 0.7 2.1 Male 0.49 1.22 0.49 1.3 0.5 1.31 0.53 1.53 0.58 1.65 Mouth Slightly Open 0.4 1.39 0.43 1.51 0.47 1.56 0.63 2.04 0.78 2.82 Mustache 0.49 0.54 0.53 0.58 0.58 0.64 0.57 0.64 0.51 0.56 Narrow Eyes 0.65 0.87 0.65 0.86 0.71 0.94 0.69 0.92 0.7 0.91 No Beard 0.84 1.35 0.87 1.37 0.86 1.4 0.89 1.48 0.87 1.44 Oval Face 0.76 1.83 0.78 1.84 0.79 1.85 0.78 1.86 0.76 1.99 Pale Skin 0.6 0.68 0.62 0.71 0.65 0.74 0.57 0.68 0.6 0.69 Pointy Nose 0.76 1.84 0.78 1.87 0.79 1.9 0.78 1.88 0.75 1.86 Receding Hairline 0.69 0.87 0.73 0.91 0.73 0.92 0.76 0.97 0.75 0.97 Rosy Cheeks 0.64 0.77 0.62 0.74 0.62 0.74 0.68 0.8 0.67 0.78 Sideburns 0.6 0.71 0.64 0.78 0.63 0.76 0.71 0.85 0.65 0.78 Smiling 0.39 1.27 0.39 1.39 0.41 1.44 0.48 1.66 0.71 2.03 Straight Hair 0.76 1.37 0.79 1.41 0.82 1.47 0.8 1.5 0.74 1.34 Wavy Hair 0.73 1.53 0.74 1.55 0.73 1.69 0.75 1.78 0.73 1.83 Wearing Earrings 0.81 1.42 0.85 1.48 0.82 1.43 0.83 1.43 0.82 1.42 Wearing Hat 0.6 0.68 0.64 0.73 0.62 0.7 0.66 0.76 0.65 0.76 Wearing Lipstick 0.4 1.1 0.39 1.16 0.41 1.32 0.43 1.51 0.44 1.67 Wearing Necklace 0.75 1.01 0.77 1.05 0.77 1.02 0.8 1.07 0.74 0.99 Wearing Necktie 0.7 0.86 0.67 0.83 0.72 0.87 0.77 0.93 0.75 0.93 Young 0.82 1.46 0.85 1.5 0.88 1.57 0.85 1.54 0.87 1.66

Table 25: Celeb A Error Bands with initial sample 100: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

5 o Clock Shadow

convnet-1 7.17 N/A 5.75 5.27 4.75 5.39 N/A 1.0 convnet-2 7.26 N/A N/A 6.28 4.75 5.4 N/A 1.11 convnet-4 7.64 N/A 6.71 6.29 5.41 5.79 N/A 1.29 convnet-8 6.52 N/A N/A 6.0 5.51 5.86 N/A 2.16 convnet-16 5.81 N/A 5.09 5.15 4.29 4.96 N/A 1.19

Arched Eyebrows

convnet-1 19.42 N/A N/A 16.14 N/A 15.52 N/A 12.26 convnet-2 19.72 17.63 17.96 16.18 19.56 16.19 N/A 6.66 convnet-4 21.16 N/A N/A 18.0 N/A 17.15 N/A 12.34 convnet-8 20.23 N/A 18.22 16.77 16.93 17.3 N/A 3.84 convnet-16 18.21 N/A 17.59 16.43 16.83 16.37 N/A 11.43

convnet-1 22.41 N/A 20.62 N/A N/A N/A N/A N/A convnet-2 25.48 22.66 22.71 23.22 N/A 20.95 23.52 N/A convnet-4 23.51 N/A 21.25 N/A 22.17 20.04 N/A 13.73 convnet-8 24.05 22.27 22.04 22.29 22.73 21.09 22.41 9.39 convnet-16 24.17 N/A 21.85 21.22 22.03 20.32 21.95 12.02

Bags Under Eyes

convnet-1 13.52 N/A 11.88 10.28 N/A 10.07 N/A 2.11 convnet-2 14.87 N/A 12.15 12.12 13.11 11.32 N/A 3.73 convnet-4 13.15 N/A 12.19 11.77 11.56 10.52 N/A 2.44 convnet-8 12.87 N/A 11.75 11.18 9.97 10.96 N/A 2.27 convnet-16 12.13 N/A 10.53 9.29 7.24 8.47 N/A 1.88

convnet-1 1.25 1.32 1.05 0.99 0.63 0.9 0.71 0.27 convnet-2 1.23 N/A 0.98 1.06 0.63 0.95 0.69 0.15 convnet-4 1.31 N/A 1.34 N/A 0.92 1.22 N/A 0.34 convnet-8 1.24 1.25 0.91 0.96 0.59 0.86 0.72 0.3 convnet-16 1.03 N/A 0.74 0.86 0.56 0.82 0.56 0.25

convnet-1 6.92 6.31 6.28 6.08 6.28 5.74 5.89 3.23 convnet-2 7.44 6.76 6.52 6.28 6.34 6.21 6.46 2.92 convnet-4 7.78 7.21 7.08 6.87 7.04 6.38 6.63 3.48 convnet-8 8.36 8.07 7.7 7.84 7.65 7.39 7.53 4.0 convnet-16 7.45 N/A 7.45 6.74 N/A 5.75 N/A 5.42

convnet-1 7.46 N/A 5.89 N/A N/A N/A N/A 1.24 convnet-2 7.39 N/A 5.86 N/A N/A N/A N/A 1.44 convnet-4 6.08 N/A 4.7 N/A N/A N/A N/A 1.34 convnet-8 6.27 N/A N/A N/A N/A N/A N/A N/A convnet-16 4.68 N/A 4.04 4.32 3.77 N/A N/A N/A

convnet-1 14.84 N/A 12.46 13.11 N/A 12.06 N/A 2.66 convnet-2 15.35 N/A N/A N/A N/A N/A N/A N/A convnet-4 15.2 N/A 13.56 13.64 N/A 12.48 N/A 2.6 convnet-8 14.0 N/A 13.31 13.45 12.86 12.73 N/A 2.74 convnet-16 13.81 N/A 13.19 13.04 12.65 12.5 N/A 3.02

convnet-1 15.1 N/A 12.96 N/A 13.72 12.45 13.88 6.63 convnet-2 14.7 N/A N/A N/A N/A 13.08 N/A 10.36 convnet-4 15.32 N/A N/A N/A 14.68 N/A N/A N/A convnet-8 14.2 N/A 14.35 N/A N/A 13.16 13.28 8.52 convnet-16 14.41 14.52 14.02 N/A 14.04 12.81 14.12 8.58

convnet-1 7.95 6.95 6.57 6.47 6.57 6.07 5.69 3.28 convnet-2 8.36 7.89 7.3 7.24 7.59 6.93 6.89 2.99 convnet-4 8.33 7.97 7.81 7.21 7.71 7.11 7.71 3.37 convnet-8 9.08 8.69 8.57 8.14 8.58 7.8 8.16 3.18 convnet-16 9.37 N/A 9.0 8.56 8.99 8.1 8.73 5.74

Table 26: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks. Part 1 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 0.03 N/A 0.03 N/A N/A N/A N/A N/A convnet-2 0.22 N/A 0.22 N/A N/A N/A N/A N/A convnet-4 0.21 N/A 0.21 0.27 N/A N/A N/A N/A convnet-8 0.28 N/A 0.28 N/A N/A N/A N/A N/A convnet-16 0.22 N/A 0.2 N/A 0.2 N/A N/A N/A

convnet-1 14.06 N/A 13.06 10.9 13.23 11.82 N/A 6.16 convnet-2 13.06 N/A N/A N/A N/A N/A N/A 6.37 convnet-4 13.46 N/A 13.05 11.23 N/A 11.31 N/A 7.81 convnet-8 13.2 12.61 N/A 11.05 13.2 11.86 N/A 3.66 convnet-16 13.79 N/A 12.79 10.76 12.9 11.53 N/A 3.63

Bushy Eyebrows

convnet-1 7.31 N/A 6.19 6.07 5.52 5.51 N/A 1.02 convnet-2 7.02 N/A 6.23 6.38 5.65 6.32 7.4 1.0 convnet-4 7.01 N/A 6.4 6.48 N/A 6.56 N/A 1.34 convnet-8 6.71 N/A 6.09 6.44 5.8 N/A N/A 2.16 convnet-16 5.15 N/A 5.24 N/A 4.49 N/A N/A N/A

convnet-1 2.21 N/A 1.87 1.91 1.22 1.78 N/A 0.4 convnet-2 2.33 N/A 2.11 2.23 1.56 1.94 N/A 0.43 convnet-4 2.57 N/A 2.12 2.17 1.62 2.18 N/A 0.41 convnet-8 2.04 N/A 1.6 2.08 1.32 1.7 N/A 0.49 convnet-16 1.81 N/A 1.66 1.7 1.12 1.58 N/A 0.43

Double Chin

convnet-1 2.04 N/A 1.43 1.6 1.0 1.55 N/A 0.37 convnet-2 1.39 N/A 1.45 N/A 0.64 1.22 N/A 0.18 convnet-4 2.04 N/A 1.57 2.14 1.15 1.63 N/A 0.47 convnet-8 2.38 N/A 1.7 2.08 1.38 1.86 N/A 0.55 convnet-16 1.26 N/A 1.22 N/A 0.7 N/A N/A 0.35

convnet-1 3.14 N/A 2.94 2.6 2.5 2.42 2.67 1.96 convnet-2 2.91 N/A 2.78 2.73 2.46 2.49 2.71 1.46 convnet-4 3.07 N/A 2.75 2.65 2.44 2.39 2.73 0.77 convnet-8 3.28 N/A 2.91 2.92 2.83 2.61 2.85 N/A convnet-16 3.11 N/A N/A N/A 2.58 2.51 N/A N/A

convnet-1 3.19 N/A N/A N/A 1.93 N/A N/A 1.01 convnet-2 3.46 N/A 3.28 3.48 2.07 2.46 N/A 0.58 convnet-4 3.22 N/A 2.79 3.01 1.53 2.39 N/A 0.47 convnet-8 3.05 N/A 2.54 2.69 1.24 1.93 2.97 0.39 convnet-16 2.66 N/A 2.46 2.45 1.13 1.95 N/A 0.4

convnet-1 2.63 N/A 2.25 2.32 1.89 2.04 N/A 0.76 convnet-2 2.35 N/A 2.31 2.04 2.03 2.23 2.51 0.49 convnet-4 2.67 N/A 2.43 2.34 2.18 2.2 2.75 0.88 convnet-8 3.06 N/A 3.18 2.95 2.83 2.89 2.85 1.42 convnet-16 3.21 2.66 2.71 2.63 1.98 2.19 1.93 0.67

Heavy Makeup

convnet-1 17.61 N/A 16.25 16.27 N/A 14.45 N/A N/A convnet-2 17.56 N/A 16.22 15.81 N/A 14.73 N/A N/A convnet-4 19.8 N/A 18.69 17.95 N/A 16.71 18.03 N/A convnet-8 20.46 N/A 20.03 20.16 N/A 17.65 N/A N/A convnet-16 22.4 N/A 20.09 19.48 21.11 18.73 20.07 12.52

High Cheekbones

convnet-1 20.08 N/A 17.11 N/A N/A N/A N/A N/A convnet-2 20.35 N/A 18.38 N/A N/A 16.79 N/A N/A convnet-4 23.14 N/A 20.56 N/A N/A 18.68 N/A N/A convnet-8 24.9 N/A 22.56 N/A N/A N/A N/A N/A convnet-16 29.46 N/A 27.22 26.76 N/A 25.42 N/A N/A

Table 27: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks. Part 2 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 14.08 N/A 10.88 11.53 10.88 10.68 10.74 N/A convnet-2 13.93 N/A 12.23 12.7 N/A N/A N/A N/A convnet-4 15.16 N/A 13.89 13.32 12.99 12.48 12.5 10.07 convnet-8 16.31 N/A 15.26 15.33 14.92 14.62 N/A 12.87 convnet-16 16.86 15.56 15.79 15.77 15.91 14.98 14.92 10.73

Mouth Slightly Open

convnet-1 22.17 N/A 17.62 N/A N/A 16.96 N/A N/A convnet-2 22.29 N/A 20.38 N/A N/A N/A N/A N/A convnet-4 23.01 N/A N/A N/A N/A N/A N/A N/A convnet-8 26.52 N/A N/A N/A N/A N/A N/A N/A convnet-16 31.25 29.07 29.08 28.74 28.95 27.64 30.03 23.06

convnet-1 0.7 N/A 0.44 0.64 0.21 0.52 N/A 0.07 convnet-2 0.61 N/A 0.66 N/A 0.42 0.8 N/A 0.17 convnet-4 0.77 N/A 0.4 0.86 0.38 0.68 N/A 0.11 convnet-8 0.62 N/A 0.5 N/A 0.41 0.65 0.41 0.18 convnet-16 0.54 N/A 0.38 N/A 0.34 N/A N/A 0.14

Narrow Eyes

convnet-1 0.97 N/A 0.36 0.54 0.74 0.66 N/A 0.09 convnet-2 1.0 N/A 0.75 1.07 N/A N/A N/A 0.34 convnet-4 0.48 N/A 0.32 N/A 0.56 N/A N/A N/A convnet-8 0.48 N/A N/A N/A N/A N/A N/A N/A convnet-16 0.19 N/A 0.18 N/A N/A N/A N/A N/A

convnet-1 12.32 N/A 10.85 11.5 N/A 9.95 11.34 7.4 convnet-2 12.82 11.52 10.9 11.05 10.91 9.95 11.34 4.02 convnet-4 12.21 N/A 11.92 11.64 N/A 10.59 N/A 6.4 convnet-8 13.59 N/A 12.55 11.99 12.61 10.91 13.0 6.16 convnet-16 13.75 12.59 12.35 12.1 12.19 11.45 12.77 6.25

convnet-1 14.15 N/A 12.26 N/A N/A N/A N/A 2.43 convnet-2 16.24 N/A 14.91 14.84 N/A N/A N/A 3.21 convnet-4 13.96 N/A 12.65 13.23 N/A 14.23 N/A 2.77 convnet-8 12.14 N/A 11.65 10.94 10.82 N/A N/A 2.54 convnet-16 10.69 N/A N/A 8.81 N/A N/A N/A 1.91

convnet-1 2.09 N/A 2.08 1.87 1.5 1.98 N/A 1.26 convnet-2 1.93 2.08 1.51 1.66 1.04 1.58 2.27 0.34 convnet-4 2.13 N/A 1.87 1.94 1.42 1.9 N/A 0.46 convnet-8 1.98 1.88 1.4 1.41 0.83 1.52 1.89 0.29 convnet-16 1.44 N/A N/A N/A 1.0 1.56 N/A N/A

Pointy Nose

convnet-1 15.42 N/A 13.99 N/A N/A N/A N/A 2.65 convnet-2 16.07 N/A 14.29 13.62 N/A N/A N/A N/A convnet-4 13.81 N/A 12.23 12.73 N/A N/A N/A 3.06 convnet-8 11.64 N/A 9.99 N/A 11.61 N/A N/A 2.3 convnet-16 9.84 N/A 10.12 9.1 9.44 N/A N/A 3.94

Receding Hairline

convnet-1 4.18 N/A 3.37 3.07 2.55 2.96 3.44 0.63 convnet-2 4.42 N/A 4.32 4.37 3.7 3.89 N/A 1.08 convnet-4 4.62 N/A 4.17 N/A 3.31 3.76 N/A 1.44 convnet-8 4.5 N/A 4.42 4.38 3.83 4.24 N/A 1.83 convnet-16 3.76 N/A 3.38 2.97 2.5 N/A N/A 0.89

Rosy Cheeks

convnet-1 2.54 N/A 2.09 2.05 1.16 1.97 N/A 0.38 convnet-2 2.38 N/A 2.3 2.14 1.25 2.28 N/A 0.35 convnet-4 3.0 N/A 2.43 2.51 1.9 2.41 N/A 0.57 convnet-8 2.23 N/A 1.86 1.86 1.44 N/A N/A 0.45 convnet-16 1.58 N/A 1.37 1.48 0.84 N/A N/A 0.33

Table 28: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks. Part 3 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 2.94 N/A 2.86 2.44 1.1 2.26 N/A 0.39 convnet-2 2.81 2.55 2.72 2.28 1.15 2.03 2.85 0.4 convnet-4 3.64 N/A 2.43 2.78 1.39 1.98 2.89 0.38 convnet-8 2.74 N/A 2.48 2.56 1.12 1.72 N/A 0.24 convnet-16 3.68 3.45 3.0 3.01 1.97 2.68 3.34 0.73

convnet-1 15.69 N/A 13.19 N/A N/A 12.1 11.68 7.9 convnet-2 14.6 N/A 12.49 N/A N/A 12.1 N/A N/A convnet-4 15.51 N/A 13.95 N/A N/A 12.93 12.93 10.51 convnet-8 18.75 N/A 15.82 16.83 N/A 15.42 N/A N/A convnet-16 23.22 N/A 20.93 20.69 20.7 19.72 21.27 N/A

Straight Hair

convnet-1 5.75 N/A 3.71 4.29 5.56 4.46 N/A 0.85 convnet-2 6.46 N/A 5.28 6.14 N/A N/A N/A 1.08 convnet-4 6.05 N/A 5.02 N/A N/A N/A N/A 1.28 convnet-8 4.52 N/A 4.55 N/A N/A N/A N/A 1.2 convnet-16 5.47 N/A 5.58 N/A 4.68 N/A N/A N/A

convnet-1 19.16 N/A 17.36 17.53 N/A 16.15 N/A 9.73 convnet-2 20.05 N/A 19.23 19.77 N/A 18.29 N/A N/A convnet-4 19.38 N/A 17.93 17.01 20.82 16.92 N/A 7.53 convnet-8 19.13 N/A 17.86 17.5 18.6 17.5 18.9 12.31 convnet-16 19.23 N/A 19.7 17.36 18.97 16.75 N/A 11.37

Wearing Earrings

convnet-1 12.07 10.52 9.43 8.34 7.36 9.14 11.0 1.78 convnet-2 10.98 N/A 9.78 9.8 N/A 8.77 N/A 1.97 convnet-4 10.41 N/A 9.21 9.22 8.72 8.79 N/A 2.15 convnet-8 7.05 N/A 7.31 7.54 6.44 N/A N/A 1.71 convnet-16 7.73 N/A 7.58 6.49 5.52 N/A N/A 1.53

Wearing Hat

convnet-1 2.73 2.35 2.26 2.3 2.11 2.1 2.09 1.23 convnet-2 2.33 N/A 2.25 2.39 2.06 2.22 2.18 1.44 convnet-4 2.57 2.58 2.42 2.57 2.15 2.2 2.17 0.93 convnet-8 3.0 2.83 2.71 2.89 2.49 2.32 2.54 1.01 convnet-16 3.09 N/A 3.03 N/A 2.83 2.81 2.73 2.42

Wearing Lipstick

convnet-1 15.83 N/A 14.95 N/A N/A N/A N/A N/A convnet-2 14.95 13.66 13.89 13.55 13.95 13.03 12.26 8.63 convnet-4 17.23 N/A 15.16 15.08 15.28 14.37 N/A 11.92 convnet-8 18.3 N/A 16.93 16.46 N/A 15.27 N/A N/A convnet-16 20.15 17.99 17.87 17.28 18.2 16.93 17.86 12.74

Wearing Necklace

convnet-1 2.08 N/A 1.62 N/A 1.67 N/A N/A N/A convnet-2 2.37 N/A 1.87 N/A 2.21 N/A N/A N/A convnet-4 1.81 N/A 1.63 N/A N/A N/A N/A N/A convnet-8 2.04 N/A N/A N/A 1.76 N/A N/A N/A convnet-16 1.46 N/A 1.42 N/A 1.06 N/A N/A N/A

Wearing Necktie

convnet-1 4.21 N/A 3.76 3.83 3.45 3.66 3.57 3.26 convnet-2 4.03 N/A 3.71 3.8 3.82 3.39 3.64 1.06 convnet-4 4.45 N/A 4.28 N/A 4.24 4.03 4.37 N/A convnet-8 4.75 4.04 3.88 4.02 3.94 3.51 4.13 2.01 convnet-16 4.78 4.06 4.03 3.96 4.2 3.95 4.0 1.72

convnet-1 13.17 N/A N/A N/A N/A 10.53 N/A N/A convnet-2 13.9 N/A 12.1 12.29 N/A 11.5 N/A 4.24 convnet-4 13.59 N/A 12.1 11.88 N/A N/A N/A 5.21 convnet-8 13.73 N/A 12.73 12.24 N/A 11.89 N/A N/A convnet-16 12.62 N/A 12.06 11.93 N/A 11.22 N/A 6.54

Table 29: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks. Part 4 of 4.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn 5 o Clock Shadow 0.79 0.98 0.78 0.98 0.78 0.98 0.83 1.06 0.83 1.07 Arched Eyebrows 0.82 1.2 0.81 1.23 0.84 1.21 0.84 1.38 0.81 1.36 Attractive 0.37 0.79 0.36 0.92 0.37 0.83 0.38 0.97 0.4 0.85 Bags Under Eyes 0.84 1.2 0.87 1.25 0.86 1.24 0.86 1.32 0.87 1.38 Bald 0.38 0.42 0.43 0.47 0.48 0.54 0.44 0.48 0.54 0.58 Bangs 0.77 0.83 0.8 0.88 0.8 0.85 0.83 0.93 0.82 0.91 Big Lips 0.8 1.51 0.82 1.55 0.81 1.53 0.8 1.53 0.76 1.56 Big Nose 0.81 1.23 0.82 1.18 0.82 1.25 0.82 1.36 0.81 1.37 Black Hair 0.78 0.96 0.79 1.01 0.78 0.97 0.78 0.98 0.8 1.04 Blond Hair 0.8 0.86 0.8 0.91 0.82 0.95 0.82 0.94 0.83 0.96 Blurry 0.59 0.66 0.55 0.66 0.57 0.67 0.58 0.67 0.55 0.63 Brown Hair 0.83 1.2 0.83 1.14 0.84 1.17 0.87 1.21 0.83 1.3 Bushy Eyebrows 0.83 1.09 0.82 1.12 0.86 1.16 0.85 1.22 0.82 1.14 Chubby 0.62 0.72 0.65 0.77 0.65 0.78 0.64 0.76 0.64 0.77 Double Chin 0.58 0.68 0.6 0.69 0.59 0.69 0.6 0.72 0.59 0.69 Eyeglasses 0.6 0.68 0.64 0.7 0.63 0.68 0.61 0.69 0.66 0.73 Goatee 0.67 0.8 0.66 0.78 0.69 0.81 0.69 0.82 0.66 0.8 Gray Hair 0.54 0.59 0.54 0.59 0.61 0.68 0.62 0.72 0.59 0.67 Heavy Makeup 0.56 0.74 0.58 0.75 0.59 0.83 0.6 0.92 0.59 0.92 High Cheekbones 0.45 0.76 0.45 0.77 0.49 1.09 0.47 1.05 0.5 1.31 Male 0.45 0.54 0.47 0.61 0.45 0.65 0.48 0.64 0.47 0.71 Mouth Slightly Open 0.39 0.67 0.4 0.83 0.38 0.92 0.45 1.44 0.57 1.78 Mustache 0.5 0.55 0.54 0.61 0.55 0.62 0.5 0.57 0.54 0.61 Narrow Eyes 0.73 0.97 0.76 1.06 0.76 1.02 0.76 1.04 0.73 0.97 No Beard 0.83 1.01 0.83 1.07 0.84 1.11 0.85 1.11 0.86 1.19 Oval Face 0.79 1.54 0.83 1.51 0.79 1.57 0.77 1.59 0.79 1.85 Pale Skin 0.63 0.71 0.59 0.68 0.63 0.74 0.61 0.72 0.61 0.73 Pointy Nose 0.79 1.46 0.79 1.45 0.78 1.55 0.77 1.66 0.75 1.62 Receding Hairline 0.66 0.78 0.7 0.85 0.75 0.92 0.71 0.87 0.72 0.91 Rosy Cheeks 0.67 0.8 0.69 0.83 0.69 0.89 0.69 0.86 0.66 0.79 Sideburns 0.66 0.77 0.62 0.73 0.65 0.77 0.63 0.78 0.62 0.96 Smiling 0.35 0.59 0.34 0.51 0.35 0.69 0.39 0.85 0.42 1.35 Straight Hair 0.81 1.38 0.83 1.48 0.83 1.42 0.81 1.39 0.82 1.76 Wavy Hair 0.72 0.98 0.73 1.06 0.72 1.06 0.71 1.09 0.72 1.14 Wearing Earrings 0.87 1.32 0.84 1.27 0.87 1.41 0.84 1.36 0.85 1.38 Wearing Hat 0.53 0.55 0.57 0.58 0.62 0.65 0.59 0.63 0.63 0.67 Wearing Lipstick 0.4 0.56 0.39 0.61 0.4 0.66 0.38 0.62 0.39 0.7 Wearing Necklace 0.75 1.05 0.78 1.06 0.8 1.11 0.78 1.08 0.76 1.06 Wearing Necktie 0.66 0.7 0.71 0.75 0.71 0.78 0.73 0.78 0.71 0.78 Young 0.85 1.11 0.85 1.18 0.85 1.19 0.87 1.21 0.82 1.17

Table 30: Celeb A Error Bands: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

5 o Clock Shadow

convnet-1 6.65 N/A 6.51 5.47 N/A N/A N/A 1.65 convnet-2 6.5 N/A 6.29 N/A N/A N/A N/A 1.87 convnet-4 6.33 N/A 5.85 5.6 N/A N/A N/A 1.79 convnet-8 6.1 N/A N/A N/A N/A N/A N/A 1.61 convnet-16 6.67 N/A 6.32 5.52 N/A 5.55 N/A 1.67

Arched Eyebrows

convnet-1 15.6 N/A 12.64 12.44 N/A N/A N/A 3.73 convnet-2 15.8 N/A 14.01 N/A N/A N/A N/A 5.24 convnet-4 14.87 N/A 13.34 N/A N/A N/A N/A 3.92 convnet-8 15.52 N/A N/A N/A N/A N/A N/A 5.16 convnet-16 15.72 N/A 12.99 12.91 N/A 12.17 N/A 3.8

convnet-1 16.61 N/A N/A N/A N/A N/A N/A 5.33 convnet-2 16.67 N/A 14.58 N/A N/A N/A N/A 4.25 convnet-4 16.01 N/A 15.36 N/A N/A N/A N/A 4.15 convnet-8 16.63 N/A 14.49 N/A N/A N/A N/A 4.02 convnet-16 16.45 N/A 14.96 N/A N/A N/A N/A 4.05

Bags Under Eyes

convnet-1 10.22 N/A 8.62 N/A N/A N/A N/A 2.48 convnet-2 10.07 N/A 8.58 N/A N/A N/A N/A 2.5 convnet-4 9.14 N/A N/A N/A N/A N/A N/A 3.54 convnet-8 9.5 N/A 8.52 N/A N/A N/A N/A 2.4 convnet-16 9.47 N/A N/A N/A N/A N/A N/A N/A

convnet-1 1.61 N/A 1.3 1.32 1.15 1.2 1.13 0.44 convnet-2 1.58 N/A 1.35 1.3 1.24 1.29 1.44 0.5 convnet-4 1.38 N/A 1.24 1.27 1.1 1.22 1.02 0.39 convnet-8 1.42 N/A 1.33 1.25 1.17 1.21 1.1 0.45 convnet-16 1.14 N/A 1.17 1.15 0.93 1.06 0.79 0.33

convnet-1 5.69 N/A 4.91 N/A N/A 4.21 N/A 1.76 convnet-2 5.64 N/A N/A N/A N/A 4.38 N/A 1.82 convnet-4 5.51 N/A N/A N/A N/A N/A N/A 2.58 convnet-8 5.48 N/A 5.28 N/A N/A 4.58 N/A 2.21 convnet-16 5.52 N/A N/A N/A N/A N/A N/A 2.02

convnet-1 8.07 N/A 7.58 N/A N/A N/A N/A 1.98 convnet-2 7.91 N/A 7.31 N/A N/A N/A N/A 1.79 convnet-4 7.72 N/A N/A N/A N/A N/A N/A N/A convnet-8 6.37 N/A N/A N/A N/A N/A N/A 1.51 convnet-16 4.48 N/A N/A N/A N/A N/A N/A 1.0

convnet-1 12.13 N/A 10.63 N/A N/A N/A N/A 4.06 convnet-2 12.06 N/A 10.75 N/A N/A N/A N/A 2.86 convnet-4 11.45 N/A N/A N/A N/A N/A N/A N/A convnet-8 11.16 N/A N/A N/A N/A N/A N/A 2.8 convnet-16 11.82 N/A 10.64 N/A N/A N/A N/A 2.81

convnet-1 11.61 N/A 10.19 N/A N/A 8.78 N/A 2.98 convnet-2 10.9 N/A 10.05 N/A N/A N/A N/A 3.02 convnet-4 10.37 N/A 10.09 N/A N/A N/A N/A 2.96 convnet-8 10.61 N/A 10.57 N/A N/A N/A N/A 2.97 convnet-16 10.68 N/A 10.27 N/A N/A N/A N/A 2.9

convnet-1 5.74 N/A N/A N/A N/A N/A N/A 2.09 convnet-2 5.62 N/A 5.37 N/A N/A 4.52 N/A 1.88 convnet-4 5.51 N/A 5.39 4.9 N/A 4.49 N/A 1.97 convnet-8 5.62 N/A 4.95 4.88 4.77 4.52 4.54 1.93 convnet-16 5.48 N/A 5.07 5.04 4.86 4.64 4.73 1.94

Table 31: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 10000. Part 1 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 0.08 N/A 0.06 N/A N/A N/A N/A N/A convnet-2 0.08 N/A 0.09 N/A N/A N/A N/A N/A convnet-4 0.04 N/A N/A N/A N/A N/A N/A N/A convnet-8 0.08 N/A N/A N/A N/A N/A N/A N/A convnet-16 0.04 N/A N/A N/A N/A N/A N/A N/A

convnet-1 11.37 N/A 10.55 N/A N/A N/A N/A 3.63 convnet-2 11.18 N/A 11.02 N/A N/A N/A N/A 2.88 convnet-4 10.91 N/A N/A N/A N/A N/A N/A 3.0 convnet-8 10.87 N/A 10.56 N/A N/A N/A N/A 2.85 convnet-16 10.93 N/A 10.55 N/A N/A N/A N/A 2.64

Bushy Eyebrows

convnet-1 7.34 N/A 6.85 N/A N/A N/A N/A 2.38 convnet-2 7.24 N/A 6.19 N/A N/A N/A N/A 1.77 convnet-4 7.18 N/A 7.07 N/A N/A N/A N/A 2.38 convnet-8 6.96 N/A 6.86 N/A N/A N/A N/A 1.77 convnet-16 6.79 N/A 6.65 N/A N/A N/A N/A 1.69

convnet-1 3.01 N/A 2.7 N/A 2.28 N/A N/A 0.73 convnet-2 2.99 N/A 2.78 N/A N/A N/A N/A 0.85 convnet-4 2.82 N/A 2.77 N/A N/A N/A N/A 0.83 convnet-8 2.47 N/A 2.49 N/A N/A N/A N/A 0.74 convnet-16 2.36 N/A N/A N/A N/A N/A N/A 0.66

Double Chin

convnet-1 2.4 N/A 2.18 N/A 1.99 N/A N/A 0.62 convnet-2 2.13 N/A N/A N/A 1.95 N/A N/A 0.63 convnet-4 2.07 N/A 1.94 N/A N/A N/A N/A 0.63 convnet-8 2.11 N/A 1.9 N/A N/A N/A N/A 0.58 convnet-16 1.94 N/A 2.02 N/A N/A N/A N/A 0.59

convnet-1 2.52 2.0 2.09 1.98 1.87 1.8 1.76 0.9 convnet-2 2.44 N/A 2.25 2.01 1.92 1.83 N/A 0.84 convnet-4 2.33 N/A 1.97 2.0 1.87 1.79 N/A 0.81 convnet-8 2.34 N/A 2.24 2.2 N/A 1.81 N/A 1.08 convnet-16 2.25 2.01 1.93 2.05 1.82 1.75 1.93 0.79

convnet-1 3.55 N/A 3.38 N/A N/A N/A N/A 1.28 convnet-2 3.46 N/A N/A N/A N/A N/A N/A 1.7 convnet-4 3.56 N/A N/A N/A N/A N/A N/A 1.56 convnet-8 3.27 N/A 3.07 N/A N/A N/A N/A 0.95 convnet-16 3.23 N/A 3.09 N/A N/A N/A N/A 0.96

convnet-1 2.62 N/A 2.41 2.04 N/A 2.05 N/A 0.85 convnet-2 2.57 N/A 2.39 2.16 2.3 2.04 N/A 0.87 convnet-4 2.49 N/A 2.45 2.2 N/A 2.12 N/A 0.82 convnet-8 2.5 N/A 2.37 2.29 N/A 2.2 2.32 0.85 convnet-16 2.4 N/A 2.22 2.01 2.16 1.98 2.14 0.77

Heavy Makeup

convnet-1 11.2 N/A 9.59 8.79 N/A 7.8 N/A 3.04 convnet-2 10.72 N/A 9.53 N/A N/A N/A N/A 3.92 convnet-4 10.53 N/A 9.35 N/A N/A 8.12 N/A 3.24 convnet-8 10.67 N/A 9.93 N/A N/A 8.18 N/A 3.17 convnet-16 10.96 N/A 9.76 N/A N/A 8.07 N/A 3.4

High Cheekbones

convnet-1 12.86 N/A 10.43 N/A N/A 9.04 N/A 3.04 convnet-2 12.42 N/A 10.41 N/A N/A N/A N/A 3.04 convnet-4 12.1 N/A 10.4 N/A N/A N/A N/A 3.04 convnet-8 12.37 N/A 9.83 N/A N/A 8.65 N/A 2.87 convnet-16 13.01 N/A 11.45 N/A N/A N/A N/A 4.27

Table 32: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 10000. Part 2 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 8.42 6.21 6.31 6.2 6.18 5.79 5.5 3.08 convnet-2 8.11 N/A 6.39 6.43 6.14 6.03 5.51 3.27 convnet-4 7.65 6.08 6.15 6.03 6.03 5.79 5.37 3.18 convnet-8 8.04 6.52 6.61 6.34 6.38 6.17 5.67 3.09 convnet-16 8.06 6.45 6.6 6.5 6.49 6.25 5.83 3.21

Mouth Slightly Open

convnet-1 12.43 N/A 10.22 N/A N/A N/A N/A N/A convnet-2 12.16 N/A 9.61 N/A N/A 7.94 N/A 2.86 convnet-4 12.54 N/A 10.41 N/A N/A N/A N/A 2.82 convnet-8 12.31 N/A 9.84 N/A N/A 7.82 N/A 3.36 convnet-16 14.93 N/A 10.6 N/A N/A N/A N/A N/A

convnet-1 1.94 N/A N/A N/A 1.49 N/A N/A 0.51 convnet-2 1.86 N/A N/A N/A 1.49 N/A N/A 0.52 convnet-4 2.09 N/A 1.88 N/A 1.83 N/A N/A 0.58 convnet-8 1.98 N/A N/A N/A 1.56 N/A N/A N/A convnet-16 1.64 N/A 1.56 N/A 1.33 N/A N/A 0.49

Narrow Eyes

convnet-1 4.11 N/A 3.43 N/A N/A N/A N/A 0.94 convnet-2 3.58 N/A N/A N/A N/A N/A N/A 0.97 convnet-4 3.14 N/A 2.63 N/A N/A N/A N/A 0.69 convnet-8 2.0 N/A N/A N/A N/A N/A N/A 0.44 convnet-16 1.06 N/A 0.9 N/A N/A N/A N/A 0.18

convnet-1 8.44 N/A 6.59 N/A N/A 6.04 N/A 2.22 convnet-2 8.27 N/A 7.63 N/A N/A 6.26 N/A 2.27 convnet-4 7.97 N/A 6.93 N/A N/A 6.18 N/A 2.36 convnet-8 8.0 N/A 7.26 N/A N/A 6.26 N/A 2.31 convnet-16 8.34 N/A 7.29 7.2 N/A 6.2 6.79 2.25

convnet-1 14.2 N/A 13.96 N/A N/A N/A N/A 3.5 convnet-2 14.39 N/A 13.74 N/A N/A N/A N/A 3.66 convnet-4 14.24 N/A N/A N/A N/A N/A N/A N/A convnet-8 14.14 N/A 12.29 N/A N/A N/A N/A 3.09 convnet-16 12.36 N/A 12.19 N/A N/A N/A N/A 2.75

convnet-1 2.74 N/A 2.17 2.3 2.25 2.03 N/A 0.65 convnet-2 2.29 N/A 2.3 N/A N/A N/A N/A 0.58 convnet-4 2.44 N/A 2.22 N/A 2.32 2.13 N/A 0.61 convnet-8 2.38 N/A 2.17 2.2 N/A 2.05 N/A 0.75 convnet-16 2.23 N/A 2.3 N/A N/A 2.11 N/A 0.58

Pointy Nose

convnet-1 15.58 N/A 15.39 N/A N/A N/A N/A 3.7 convnet-2 15.4 N/A 15.14 N/A N/A N/A N/A 3.68 convnet-4 15.23 N/A 13.25 N/A N/A N/A N/A 3.44 convnet-8 13.53 N/A N/A N/A N/A N/A N/A N/A convnet-16 11.39 N/A N/A N/A N/A N/A N/A 2.86

Receding Hairline

convnet-1 4.57 N/A 4.2 N/A N/A N/A N/A 1.25 convnet-2 4.28 N/A 4.1 N/A N/A N/A N/A 1.15 convnet-4 4.4 N/A 4.56 N/A N/A N/A N/A 1.67 convnet-8 4.41 N/A 4.26 N/A N/A N/A N/A 1.63 convnet-16 4.18 N/A N/A N/A N/A N/A N/A 1.21

Rosy Cheeks

convnet-1 4.15 N/A 3.84 3.23 3.69 N/A N/A 1.23 convnet-2 4.49 N/A 4.26 3.66 3.88 3.61 N/A 1.23 convnet-4 4.28 N/A 4.0 3.37 N/A 3.46 N/A 1.16 convnet-8 3.86 N/A 3.71 3.31 3.44 N/A N/A 1.05 convnet-16 3.83 N/A 3.65 3.16 N/A 3.38 N/A 0.94

Table 33: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 10000. Part 3 of 4.

Published as a conference paper at ICLR 2022

Dataset network cold warm s-perturb mixup ls co-dist anchor distill

convnet-1 3.19 N/A N/A 2.52 N/A 2.36 N/A 0.9 convnet-2 3.07 N/A 2.95 N/A N/A N/A N/A 1.14 convnet-4 3.13 N/A 2.74 2.62 N/A 2.48 N/A 0.96 convnet-8 2.93 N/A 2.75 2.64 N/A 2.38 N/A 0.89 convnet-16 2.91 N/A 2.71 2.62 N/A 2.37 N/A 0.79

convnet-1 9.61 N/A 7.58 7.66 N/A 7.18 N/A 4.44 convnet-2 9.43 N/A 8.35 N/A N/A 7.0 N/A 3.04 convnet-4 9.4 N/A 9.14 N/A N/A 6.91 N/A 3.09 convnet-8 9.81 N/A 7.69 7.93 N/A 6.84 7.44 2.58 convnet-16 10.19 7.42 7.73 7.9 7.58 7.18 7.4 2.81

Straight Hair

convnet-1 6.64 N/A 6.34 N/A N/A N/A N/A 1.58 convnet-2 6.96 N/A N/A N/A N/A N/A N/A 1.62 convnet-4 7.33 N/A N/A N/A N/A N/A N/A N/A convnet-8 6.47 N/A N/A N/A N/A N/A N/A 1.54 convnet-16 5.95 N/A N/A N/A N/A N/A N/A 1.24

convnet-1 16.13 N/A N/A N/A N/A N/A N/A N/A convnet-2 16.31 N/A N/A N/A N/A N/A N/A 5.63 convnet-4 15.8 N/A 14.23 14.27 N/A 13.68 N/A 3.97 convnet-8 15.76 N/A N/A N/A N/A N/A N/A 3.84 convnet-16 15.25 N/A 13.89 N/A N/A N/A N/A 3.81

Wearing Earrings

convnet-1 12.37 N/A 10.47 9.11 N/A N/A N/A 2.94 convnet-2 12.09 N/A 11.23 N/A N/A N/A N/A 3.23 convnet-4 12.05 N/A 10.72 9.95 N/A N/A N/A 3.37 convnet-8 11.55 N/A 10.6 9.83 N/A 9.68 N/A 3.35 convnet-16 11.33 N/A 10.42 9.61 N/A 9.44 N/A 2.95

Wearing Hat

convnet-1 1.9 N/A 1.58 1.61 1.52 1.43 1.39 0.64 convnet-2 1.95 N/A 1.81 1.82 1.71 1.66 1.58 0.73 convnet-4 1.86 N/A N/A N/A N/A 1.53 N/A 0.9 convnet-8 1.86 N/A N/A N/A N/A 1.54 N/A 0.85 convnet-16 1.88 1.66 1.77 1.78 1.6 1.6 1.59 0.71

Wearing Lipstick

convnet-1 9.1 N/A 7.6 7.14 N/A 6.47 N/A 3.29 convnet-2 9.07 N/A 7.96 7.55 N/A 6.77 N/A 2.83 convnet-4 8.46 N/A N/A 7.54 N/A 6.84 N/A 2.86 convnet-8 8.67 N/A N/A N/A N/A 6.8 N/A 2.93 convnet-16 9.18 N/A 7.83 N/A N/A 6.97 N/A 2.99

Wearing Necklace

convnet-1 0.94 N/A 0.75 N/A N/A N/A N/A N/A convnet-2 1.64 N/A 1.45 N/A N/A N/A N/A N/A convnet-4 1.36 N/A 1.14 N/A N/A N/A N/A 0.41 convnet-8 1.51 N/A 1.45 N/A N/A N/A N/A 0.41 convnet-16 0.99 N/A N/A N/A N/A N/A N/A 0.25

Wearing Necktie

convnet-1 3.34 N/A 3.03 2.86 N/A 2.8 N/A 0.98 convnet-2 3.04 N/A N/A N/A N/A N/A N/A 0.97 convnet-4 3.43 N/A 2.99 2.97 3.02 2.85 N/A 1.05 convnet-8 3.28 N/A 3.06 3.07 N/A 2.75 N/A 1.04 convnet-16 3.26 N/A N/A N/A N/A 2.87 N/A 1.53

convnet-1 10.65 N/A 9.59 8.32 N/A N/A N/A 2.52 convnet-2 10.51 N/A 9.54 N/A N/A N/A N/A 2.9 convnet-4 10.08 N/A 9.21 N/A N/A N/A N/A 2.81 convnet-8 10.02 N/A 9.42 N/A N/A N/A N/A 2.79 convnet-16 9.67 N/A N/A N/A N/A 8.3 N/A 2.63

Table 34: Results for Celeb A tasks under churn at cold accuracy metric across different sizes of convolutional networks with initial sample 10000. Part 4 of 4.

Published as a conference paper at ICLR 2022

convnet-1 convnet-2 convnet-4 convnet-8 convnet-16 Dataset Error Churn Error Churn Error Churn Error Churn Error Churn 5 o Clock Shadow 0.67 0.77 0.68 0.78 0.65 0.75 0.67 0.79 0.7 0.82 Arched Eyebrows 0.59 0.75 0.61 0.79 0.61 0.78 0.61 0.77 0.61 0.79 Attractive 0.14 0.36 0.14 0.35 0.15 0.38 0.14 0.37 0.14 0.4 Bags Under Eyes 0.67 0.92 0.69 0.94 0.69 0.95 0.69 0.94 0.68 0.95 Bald 0.33 0.35 0.36 0.39 0.39 0.42 0.34 0.37 0.37 0.4 Bangs 0.66 0.7 0.67 0.73 0.64 0.68 0.65 0.71 0.68 0.73 Big Lips 0.67 1.18 0.65 1.19 0.68 1.23 0.67 1.27 0.67 1.31 Big Nose 0.64 0.9 0.65 0.92 0.65 0.9 0.65 0.94 0.65 0.94 Black Hair 0.6 0.67 0.61 0.69 0.61 0.69 0.59 0.68 0.6 0.67 Blond Hair 0.72 0.77 0.7 0.74 0.7 0.74 0.68 0.71 0.69 0.72 Blurry 0.54 0.61 0.56 0.64 0.55 0.62 0.51 0.58 0.54 0.6 Brown Hair 0.65 0.82 0.66 0.81 0.67 0.83 0.66 0.82 0.67 0.84 Bushy Eyebrows 0.67 0.82 0.69 0.84 0.7 0.84 0.68 0.84 0.73 0.9 Chubby 0.57 0.64 0.58 0.65 0.54 0.6 0.58 0.65 0.56 0.64 Double Chin 0.53 0.58 0.53 0.58 0.53 0.58 0.55 0.61 0.49 0.54 Eyeglasses 0.56 0.58 0.57 0.59 0.53 0.55 0.54 0.57 0.55 0.58 Goatee 0.59 0.63 0.56 0.6 0.57 0.61 0.57 0.61 0.58 0.63 Gray Hair 0.48 0.51 0.52 0.54 0.54 0.57 0.52 0.55 0.5 0.53 Heavy Makeup 0.36 0.39 0.36 0.38 0.37 0.37 0.36 0.4 0.37 0.38 High Cheekbones 0.21 0.32 0.21 0.35 0.22 0.34 0.21 0.33 0.23 0.49 Male 0.26 0.29 0.26 0.3 0.26 0.29 0.26 0.3 0.27 0.3 Mouth Slightly Open 0.16 0.28 0.15 0.29 0.16 0.29 0.15 0.28 0.57 0.77 Mustache 0.51 0.56 0.45 0.49 0.47 0.52 0.45 0.51 0.45 0.5 Narrow Eyes 0.71 0.9 0.71 0.9 0.72 0.94 0.69 0.93 0.71 0.96 No Beard 0.68 0.77 0.67 0.76 0.66 0.74 0.66 0.74 0.66 0.77 Oval Face 0.6 1.05 0.61 1.03 0.61 1.06 0.61 1.12 0.61 1.2 Pale Skin 0.55 0.6 0.48 0.55 0.54 0.61 0.56 0.64 0.51 0.58 Pointy Nose 0.61 1.09 0.61 1.05 0.62 1.11 0.61 1.11 0.63 1.26 Receding Hairline 0.66 0.76 0.63 0.72 0.64 0.73 0.62 0.72 0.65 0.73 Rosy Cheeks 0.55 0.61 0.58 0.65 0.63 0.7 0.6 0.67 0.58 0.67 Sideburns 0.58 0.62 0.56 0.61 0.55 0.59 0.52 0.57 0.53 0.58 Smiling 0.13 0.25 0.14 0.23 0.13 0.23 0.14 0.26 0.16 0.49 Straight Hair 0.71 1.14 0.71 1.13 0.72 1.14 0.72 1.16 0.72 1.19 Wavy Hair 0.52 0.69 0.52 0.68 0.52 0.69 0.52 0.73 0.51 0.71 Wearing Earrings 0.69 0.89 0.69 0.89 0.69 0.9 0.7 0.91 0.71 0.94 Wearing Hat 0.5 0.51 0.56 0.6 0.54 0.56 0.51 0.53 0.52 0.54 Wearing Lipstick 0.17 0.21 0.16 0.21 0.16 0.21 0.16 0.2 0.16 0.21 Wearing Necklace 0.7 0.97 0.73 1.0 0.73 0.99 0.72 0.98 0.73 1.0 Wearing Necktie 0.62 0.65 0.58 0.61 0.59 0.62 0.59 0.61 0.61 0.7 Young 0.65 0.87 0.66 0.88 0.66 0.88 0.67 0.9 0.66 0.9

Table 35: Celeb A Error Bands with initial sample 10000: Average standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Dataset Architecture cold warm sperturb mixup ls codist anchor distill (ours) cifar10 Res Net-50 49.85 46.4 46.6 46.45 44.4 N/A 45.35 42.85 cifar100 Res Net-50 89.05 85.9 83.7 85.05 81.95 N/A 84.45 72.05 cifar10 Res Net-101 50.65 48.2 46.25 48.45 N/A N/A 46.8 43.85 cifar100 Res Net-101 88.9 87.8 86.75 87.75 86.4 N/A 86.25 77.75 cifar10 Res Net-152 46.9 47.7 47.7 47.1 50.0 N/A 46.4 43.4 cifar100 Res Net-152 89.4 86.9 85.9 86.3 84.1 N/A 83.8 78.6

Table 36: Results for CIFAR10 and CIFAR100 under churn at cold accuracy metric across Res Net-50, Res Net-101 and Res Net-152. Initial sample size and batch size is fixed at 1000..

Initial Sample network cold warm s-perturb mixup ls co-dist anchor distill

transformer-1 38.34 36.72 37.36 N/A 36.35 N/A N/A N/A transformer-2 37.33 N/A N/A N/A 35.99 N/A N/A N/A transformer-4 40.78 N/A 39.48 N/A 38.07 N/A N/A 34.38 transformer-8 44.32 N/A 45.67 N/A 42.65 N/A 43.9 38.96 transformer-16 50.87 46.87 46.99 N/A 47.27 45.49 45.38 40.05

transformer-1 15.69 N/A 15.31 N/A N/A 13.41 10.33 7.78 transformer-2 16.62 14.35 15.1 N/A 13.0 15.1 13.11 7.0 transformer-4 18.79 N/A N/A N/A N/A N/A 16.72 14.24 transformer-8 20.11 N/A 19.79 N/A 17.56 18.3 17.51 12.93 transformer-16 24.06 N/A 22.4 N/A 20.27 21.36 20.43 15.75

transformer-1 8.87 N/A N/A N/A N/A N/A 8.48 6.4 transformer-2 9.0 N/A N/A N/A N/A N/A N/A 6.83 transformer-4 9.24 N/A N/A N/A N/A N/A N/A 6.55 transformer-8 9.95 N/A N/A N/A 6.74 N/A N/A 6.74 transformer-16 12.73 N/A N/A N/A N/A N/A N/A N/A

Table 37: Results for IMDB under churn at cold accuracy metric across different sizes of transformer networks and initial sample sizes. Batch size is fixed at 1000.

transformer-1 transformer-2 transformer-4 transformer-8 transformer-16 Iniitial Sample Error Churn Error Churn Error Churn Error Churn Error Churn 100 0.51 1.31 0.53 1.44 0.57 1.36 0.62 1.74 0.71 2.27 1000 0.45 0.61 0.48 0.67 0.52 0.9 0.61 1.0 0.74 1.57 10000 0.18 0.33 0.19 0.31 0.23 0.32 0.3 0.4 0.84 1.56

Table 38: IMDB Error Bands: Mean standard errors for error and churn across baselines for each dataset and network across 100 runs.

Published as a conference paper at ICLR 2022

Figure 4: IMDB dataset with transformer. Pareto frontier for each baseline and costs of each method, where the cost is a convex combination between the error and the churn, as we vary the weight between churn and accuracy. Top two: Initial batch size 100. Middle: Initial batch size 1000. Bottom: Initial batch size 10000.