The Effects of Regularization and Data Augmentation are Class Dependent
Authors: Randall Balestriero
Leon Bottou
Yann LeCun
Venue: NeurIPS 2022 Accept (2022)
Keywords: data augmentation
class dependent bias
regularization
fairness
cross validation
risk minimization

Abstract: Regularization is a fundamental technique to prevent over-fitting and to improve generalization performances by constraining a model's complexity. Current Deep Networks heavily rely on regularizers such as Data-Augmentation (DA) or weight-decay, and employ structural risk minimization, i.e. cross-validation, to select the optimal regularization hyper-parameters. In this study, we demonstrate that techniques such as DA or weight decay produce a model with a reduced complexity that is unfair across classes. The optimal amount of DA or weight decay found from cross-validation over all classes leads to disastrous model performances on some classes e.g. on Imagenet with a resnet50, the ``barn spider'' classification test accuracy falls from $68\%$ to $46\%$ only by introducing random crop DA during training. Even more surprising, such performance drop also appears when introducing uninformative regularization techniques such as weight decay. Those results demonstrate that our search for ever increasing generalization performance ---averaged over all classes and samples--- has left us with models and regularizers that silently sacrifice performances on some classes. This scenario can become dangerous when deploying a model on downstream tasks e.g. an Imagenet pre-trained resnet50 deployed on INaturalist sees its performances fall from $70\%$ to $30\%$ on class \#8889 when introducing random crop DA during the Imagenet pre-training phase. Those results demonstrate that finding a correct measure of a model's complexity without class-dependent preference remains an open research question.

The Effects of Regularization and Data Augmentation are Class Dependent
Randall Balestriero Meta Al Research, FAIR NYC, USA rbalestrrerr@mett...
Leon Bottou Meta AI Research, FAIR NYC, USA leonb@meta.c com
Yann LeCun
Meta Al Research, FAIR, NYU NYC, USA
ylecun com
Abstract
Regularization is a fundamental technique improve model's generalization per- formances by limiting its complexity. Deep Neural Networks (DNNs), which tend overfit their training data, heavily rely regularizers such as Data-Augmentation (DA) or weight-decay with spper-parameters found from structural risk minimiza- tion, i.e., cross-validation. In this study, demonstrate that the optimal regular- ization's hyper-parameters found from cross-validation over all classes leads to disastrous model performances trained on Imagenet sees its "barn spider" test accuracy falls from 68% to 46% only by introducing random crop DA during training. Even more surprising, such unfair impact of regularization also appears when introducing uninformative regularizers such as weight decay dropout. Those results demonstrate that our search for ever increasing generalization performance averaged over all classes and samples has left us with models and regularizers that silently sacrifice performances on some classes. This scenario can become dangerous when deploying a model on downstream tasks e-g. an Imagenet pre-trained resnet50 deployed on INaturalist sees its performances fall from 70% to 30% on class #8889 when introducing ran- dom crop DA during the Imagenet pre-training phase. Those results demonstrate that finding correct measure of model's complexity without class-dependent preference remains an open research question.
a minority of classes. For example, resnet50
Introduction
Machine learning and deep learning aim learning systems to solve accurately as possible given task at hand [LeCun al. 1998, Bishop and Nasrabadi] 2006 Jordan and Mitchell 2015). This process often takes the form of (i) being given a finite dataset, a (differentiable) loss function, and a performance measure, (ii) splitting the dataset into train/valid/test sets to optimizing the system's 's parameters c.g. from gradient updates the loss on the train set while cross- cooss-vlid pperrpparameters using the valid set, and (iii) assessing the system's 's performance the training set is finite, and the optimal design of the system is unknown, it is common to employ regularization during the optimization phase reduce over-fitting [Tikhonov] 1943] Tihonov] 1963) iee. to decrease the system's 's performance gap between train set and test set samples (Simard et al. 1991 Chapelle et al.) 2000 [Bottou] 2012 Neyshabur et al., (0144) Central to study is the fact that selection is done via cross-vast cross-validttion by maximizing the valid set performance with ad-hoe statistics e.g. the average accuracy over all samples for classes in classification tasks.
the test set. As
Cross-validation commonly involves many different types of regularization along with their "strengths" Goodffllow et al.) [0066] He al. (2021]. Most variants of regularization take of two forms:
Data-Augment Augmentation (DA) and weight-decay DA is a data-driven and informed regularization strategy that artificially increase the number of training samples [Shorten and Khoshgoltaar 2019). As
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Current Deep Learning
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Figure 1: Structural risk minimization minimizes the empirical risk of several models of varying complexity, and
selects the one offering the best compromise between under-fitting and over-fitting [Vapnik and Chervonenkis 1974]. In deep learning, one commonly control the model's s complexity by picking different DN architectures and/or by applying different levels and flavors of regularization. The key observation of our study is that when the model complexity is calibrated by DA (see Figs. alndd or weight-decay (see Fig. 3 the class-conditonall empirical risks do not align between classes i.e. croxs-validee voss-valiaation produces models that perform well on the mujority of classes but arbitrarily poorly on a few of them depicted on the left-hand-side. In an ideal setting where the control of the model's complexity is well aligned with the task and model, one would observe the right-hand side ideal scenario where the same model complexity is optimal for all classes.
opposed most explicit regularizers e.g. Tikhonov regularization (Krogh and Hertz 1991 J. also denoted as weight decay, DA's regularization is implicir as it is not a function of a model 's parameter, and König 2018 LeJeune et al. 2019] ; although some DA strategies can be turned into explicit regularizers [Blestriero) iet al. [022]. J Nevertheless, a key distinction between DA and weight decay is that DA tends to
but a function of the training samples [Neyshabur et al., 2014 |Hernández-
require more domain knowledge to be successful than weight decay. Most if not all- of current
state-of-the-er employ such regularizers [Huang et al. 2018 Chen et al. 2020b Liu et al. 2021 Tan
and Le 2021 Liu et al.) 2022]
In this paper, we will demonstrate that when cross-validation is employed to select the regularization settings maximizing the validation performance, significant bias is introduced into the trained model: the regularized model exhibits strong per-class favoritism i.e. while the average test performance is improved, it is at the cost of producing a model with significant performance drop on some of the classes as illustrated in the schematic Fig. [T] For readers familiar with statistical estimation results e.g. the bias-variancs trade-off [Kohavi et al. 1996, Von Luxburg and Schölkopf. [001] or bayesian estimation e.g. Tikhonov regularization [Box and Tiao [ 2ruberr 2017], it should not be surprising that regularization produces bias (more details and background provided in Appendix (A). In fact, it is beneficial to introduce bias through regularization it results in a significant reduction of the estimator variance- -when one minimizes the average empirical risk. However, the potentially dangerous effect of regularization that this study brings forward is that the bias introduced by regularization is elass-dep,ndent, including on transfer learning tasks.
To thoroughly validate this observation, we propose a variety of controlled experiments in Section First, we carefully quantify the impact of DA, weight decay and dropout on the per-class performance of model in Section 2.1 demonstrating that current deep learning finds itself in the scenario depicted on the left of Fig. | Then, we consider the task of transfer learning in Section 2.2 where it is again possible to identify again a per-class bias on the target dataset even though the regularization was applied on a different (source) training set. This latter scenario is particularly relevant in current times where it is common to deploy a large pre-trained model on a variety of tasks and raise an important issue: selecting the -on average- best performing pre-trained model can lead to catastrophic individual class performance even on for different downstream tasks.
Our next Section 3 will aim at exploring possible explanations and solutions. First, we will provide a brief theoretical justification on why and when DA can the cause of model bias (Section 33.) regardless of the task and data at hand. This will shed light to first possible issue: the DA parameters that make the transformed input preserve its label information vary depending on the class underlying statistics. In short, DA silently introduces inss-immalance in the training set. We propose dedicated

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pickop
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jeney sungrass overall avg.
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random crop lower bound 171 used during training
Figure 2: Varying the random crop DA lower bound (x-axis) from 100% to 8% provides greater average test accuracy (blue) but makes the per-class performance fall for some of the classes. Images of each class are provided in Fig. [2] in the appendix. See Fig- B for the convnext and ViT experiments. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained Imagenet with horizontal flip and varying random crop lower bound DA.
analysis of the label-preserving property of DA different classes and models in Section 3.2 We then take on the task of searching for a possible solution by first reviewing known theoretical studies quantifying the interplay between regularization and bias in Section 3.3 Lastly,
propose some solutions of our own in Section 3.44buill from the gained insights of Section 3.2 using label-distillation and adaptive DA. All the codebase used train the various models and to generate the figures in the supplementary files.
2 Maximizing the Average Model Performance by Cross- Validation Silently Produce Poor Final Performances on a Minority of the Classes
We now turn to the empirical validation of Fig. Ti.. quantifying the amount of class-dependent bias caused by DA, weight decay and dropout in various realistic scenarios (Section 2.1 We then demonstrate how the bias introduced by regularization transfers to downstream tasks e.g. when deploying an Imagenet (source) trained model on the INaturalist (targer) dataset in Section (..2] that scenario is key as it demonstrates the potential harm of selecting the best performing model on the source dataset which could turn out to also be the most biased model against the target dataset class of interest.
2.1 Precisely Measuring the Per-Class Effect of Data-Augmeentttin and Uninformed Regularization with Controlled Experiments
This section aims at quantifying precisely the amount of downward or upward per-class performance shift that came as result from using DA or uninformed regularization c.g. weight-decay or dropout. In fact, it is crucial to remember that regularization, or any other form of structural risk minimization, improves generalization performances by increasing the bias of the estimator so that the estimator's variance is decreased by greater amount. However, nothing guarantees the fairness of this bias i.c. for it to be equally distributed amongst the dataset classes. We thus propose a sensitivity analysis by training a large collection of models with varying regularization policies to precisely assess their impact on the class-dependent model bias.
Data- Antaeeuummettiinn. DA samples have been known to sometimes disregard the semantic information of the original samples (Krizhevsky al. 2012). Nevertheless, DA remains applied universally across tasks and datasets [Shorten and Khoshgoftaar [2019] as it provides significant performance improvements, even in semi-supervised and unsupervised settings Guo al. (2018), Xie et al. 22000). Misra and Maaten (022)). To measure the impact of DA onto per-class performances, we propose in Fig. 2|a sensitivity analysis by training the same architecture on Imagenet with varying DA policies. In particular, we consider a given DA (random crop in this case) and we vary the support of the parameter a which represents how much of the original image is kept in the crop (examples at the top of Fig. 5). We train multiple Deep Neural Networks (DNN)s using a E [100, 7] with 7 varying from 100 to and for each case, we report our metries averaged over 20 trained models. We observe a clear relation between increase in the strength of the DA, increase in the average test accuracy overall classes, and decrease some per-class test accuracies. For example, on a resnet50 Imagenet setting, the accuracy on the "academic gown" class goes from 62% to 40% steadily decreases.
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nemraiure poudil
wnre fucred fax larien black and gold garden solder ovaraali avg.
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Figure 3: Varying the amount of weight decay, an uninformed regularizer employed throughout, surprisingly exhibits similar class dependent bias as the DA scenario of Fig. 2] Images for each class are provided in Fig. [1] in the appendix. Results obtained by averaging over 20 runs, official PyTorch resnet50 implementation trained on Imagenet with varying weight decay, see Fig. 16 DenseNet121 results with the same trend and Fig. for the convnext and ViT experiments.
Uninformed Regularization. As per the arguments given in Sections 3.1 and 3.2 it would be natural to assume that what makes DA responsible for creating class-dependent bias in DNNs is our misfortune in defining correct augmentation policies. Hence, uninformed weight decay or dropout should behave differently and more fairly. We demonstrate here that such regularizers also unfair between classes. We thus propose to train multiple models with varying weight-decay and dropout parameters. In our setting, weight-decay is applied to all the DNN parameters except for the ones of nalah-normallzztionn layers, as commonly done [Hastie et al. 2009 Leclerc et al. 2022 We report in Fig. ttee the per-class performance of resnet50 trained on Imagenet with varying weight decay coefficient (as was done for DA in Fig. 2 and we observe that different classes have different test accuracy sensitivities to variations in y. Some will see their generalization performance increase, while others will have decreasing generalization performances. We further confirm such findings in Figs. 9 and 10 for dropout where the same per-class trend is observed. In short, even for uninformative regularizers such as weight decay or dropout, a per-class bias is introduced, reducing performances for some of the classes. Although weight-decay one of the most popular regularizer that is uninformed on the data and task at hand, recent studies have demonstrated that techniques such as model pruning -which can be seen as a post-training model complexity reduction i.e. regularization- also produce increased bias towards under-representee features [Hooker et al. 2019 2020]. More recently, [Balestriero et al. [2022] obtained the close-form explicit regularizer DA from which it is possible to quantify the sample-dependent aspect of DA's regularization. From findings, it seems that classes sharing the same type of features are thus impacted different by DAs.
Formal Statistical Test. To further convey our claim, we now propose formal statistical test [Neyaan and Pearson 1933 Fisher 1955) on the hypothesis that the per-class accuracy is significantly higher when DA is applied for each class (details provided in Appendix A.2). We obtain that there is enough evidence to reject the hypothesis with 95% confidence for 4.5% all the classes, and with 99% confidence for 2.6% of all the classes. Hence there is sufficient evidence to say that the per-class test accuracies not increased when introducing DA for 4.5% of the 1000 Imagenet classes. We provide in Table T the same statistical test but applied on variety of settings including different architectures (resnet50, densenet121, ViT-small and ConvNext- ConvNext-Tiny) and across the random crop DA and the weight decay controlled experiments. We should highlight however that this is not necessarily a meaningful measure since for example one regularization might not have any negative impact on the classes, but can provide beneficial gain that is drastically different between classes. Hence, although the per-class performance does not drop by introduce the regularizer, the inter-class performance gap can be increased by it, which is an equally harmful impact.
The next Section 2.2 proposes to study the scenario of introducing a pre-trained model, on a different downstream task to show that the class-dependent effect of regularization remains present and unfair towards specific downstream classes
2.2 The Class-Dependent Bias Transfers to Other Downstream Tasks
The last experiment we propose is to quantify the amount of class-dependent bias that transfers to other downstream tasks, a common situation in transfer learning and in system deployment to the real world [Pan and Yang 2009]. We thus want to measure how regularization applied during the

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Table 1: Percentage of Imagenet classes for which the test set performance (per-class) is statistically not greater when applying random crop DA (or weight decay) measured by the statistical test from Section We observe that although this measure is highly conservative since a regularizer (DA or weight decay) might now have a negative impact on a per-class performance but still increase the performance gap between difference classes, already, a nonzero proportion of classes are negatively impact by introduce random crop DA or weight decay.
confidence
resmet50
resnet50
convnext-tiny convnext ViT random crop
densenet121 weight decay
random crop weight decay
random crop weight decay random crop
90%
5.9
39.4
0.1
85.5
1.2
33.8
95%
4.5
30.3
0.1
78.8
1.0
17.1
99%
2.6
15.3
0.0
63.9
0.7
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bost accuracy In % on INaturral aall
clase 0716
clase 300
class 1228 class 9354
class 5350 class DE09
class 1700 Class 9113
N Ant Mah
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$3.00
bound (in during Imagenet training
Figure 4: Varying the random crop lower-bound during pre-training (Imagenet) also produces unfair per-class bi- ases on the downstream task (INaturalist). Hence, selecting the pre-trained model with best average test accuracy on the source datasct might result in deploying model with the worst performance the classes of interest in the target task. Images for cach class are provided Fig. [7, in the appendix. Results obtained by averaging over 20 runs, official Py Torch resnet50 implementation trained on Imagenet with varying random crop lower bound and transferred to INaturalist with frozen backbone parameters.
pre-training phase on source dataset impacts the per-class accuracy of that model on the target dataset.
In order to keep the setting similar to Section 2.1 we adopt a resnet50 model with random crop DA. That model is pre-trained on Imagenet dataset (source) with varying value of 7 (random crop lower bound) and then, the trained model is transferred to the INaturalist dataset [Van Horn et al., 2018 (target) that consists of 10,000 classes. When transferring the model INaturalist, the parameters are kept frozen, and only linear classifier is trained on top of it. We report in Fig. the performance of the trained models with varying r on different INaturalist classes. We observe once again that the best resnet50 -on average- is not necessarily the one that should be deployed as there exists a strong per-class bias that varies with т. As a result, picking the best performing model from a source dataset to a target dataset, might leave the pipeline to perform poorly since that model might also be the one that is the most biased against the class of interest in the target dataset.
This result should motivate the design of novel regularizers that do not reduce performances between classes different regimes. Additionally. due the cost of training multiple models with varying regularization settings, one might wonder on the possible alternative solutions to detect trends such as shown in Fig. 4 Honly when given single pre-trained model.
3 Understanding Why and When Regularization Produces Models With Class-Dependent Preferences
The first part of our study (Section 2 empirically validated that DNN regularization produces unfair model complexity control over different classes, resulting in a model performing poorly on few of the classes although being highly performing on average. We now provide in Sections [31 and 3.2 some intuition on why DA can be source of bias regardless of the task, dataset and model at hand. Then, Section 3.3 reviews existing works trying to confront regularization and model bias, and as we will see, an out-of-the-box solution does not seem to exist when there are only a few classes suffering from regularization (Section 3.4
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3.1 A Data-Augmennattion Policy That is Not Label-Preserving For All Classes Will Create Class-Imbalance Performances
To provide simple explanation on how DA causes bias in a trained model, we propose the following derivation that holds for any signal e.g. timeseries, images, videos. Without loss of generality, we will consider here the & loss which was shown to perform as well as the cross-entropy even on Imagenet (Hui and Belkin 2020
Dataset notations. Given a sample a: E X' with X' C RD, we consider y f (a: to be the ground-truth target value. Hence our hope is to learn an approximator fe that is as close as possible to I" everywhere in X, although we only observe a finite training set X _ {(x1, 9 ) ......,... (2 yN))- Given an output vector u we also define the level-set of mapping / to be { E X f(x)) u). Aata-Auumentation notations. Additionally, one employs DA policy T RD K. + RD such that given a transformation parameter a E K. (x) produces the transformed view of æ. Often, one also defines a density p on K. that helps in sampling transformation parameters that are priori known to be the most useful.
Theorem I. Whenever the transformations produced by To Va do not respect the level-set of ʃ" and whenever the model has enough capacity to minimize the training loss, the DA will create irreducible bias in fo as in
Σ E. [Iv (x-y)Ex
> O and Σ
E. [1lv- So 0
biased fo- (1)
(m.
O iff the DAs of in are on the same level-set of f
zero training error
The main idea of the proof, provided Appendix B to show that if transformation does not move samples on the level-set of the true function (left-hand-side of Eq. (I). then fa will learn a different level-set (since it has 0 training error), and thus |f foll 0 so is biased.
Whenever the seft-hand-siee of Eq. (T) is O, the DA is denoted as gabel-preserving [Cui et al. 2015 Taylor and Nitschke 2018). From the above,
see that unless the target y associated to Ta x is modified accordingly to encode the shift in the larget function level-sel produced by Ta any DA that subel-preserving will introduce bias. Some DAs propose to incorporate label transformation i.e. not only x: but also y is augmented to better inform on the uncertainty that has been added into (a(x). This is for example the case for MixUp (Zhang et al. 2017], ManifoldMixUp /eema et al.
2019]. CutMix Yunn et al. 2019 and their extensions.
Our goal in the next Section 3.2 is to demonstrate how DAs such as random crop, color jittering, or CutOut are only label preserving for some values of a that vary with the sample class. As a consequence, while the use of the DA improves the average test performance, it is at the cost of significant reduction in performance for some of the classes.
3.2 The Same Data-Augmentattion can be Preeervies or Not Between Different Classes
In the previous Section 3.1 provided general argument on the sufficient conditions for DA to produce a biased model. We hope in this section to provide a more concrete example that applies to current DNN training. To that end, will demonstrate that a DA can be labee-preserving or not depending on the sample's s class, hence, since the same DA policy is employed for all classes, the augmented dataset will exhibit dlss-imballane in favor of the classes for which the DA is most label-preserving.
To measure by how much given DA, To. is label-preserving, we propose to take 6 popular architec- tures that are pre-trained on Imagenet [Deng et al.) 2009) from the official PyTorch [Paskke et al.) 2019) repository, and to evaluate their performances for varying DA settings (top of Fig. 5. We observe that considering the dataset as a whole is not a good indicator of the optimal DA value to employ since per-class accuracy performance (bottom of Fig. 5) vary drastically, For example, for some classes, any level of transformation a can produce augmented samples with enough information to be correctly classified, while for other classes, even small amount of DA makes the samples unpredictable.
To further ensure that the observed relation between jabel-preservation, sample class, and amount of transformation a sound, we provide Fig. (6 the per-class test accuracy on different models, all exhibit the same trends. In short, we identify that when creating augmented dataset by applying the

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DA samples
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40 % original Image kept the crop Figure 5: Top: examples of an augmented image of class "bird".. Middle: average accuracy (train set in dashed line and test set in plain) on Imagenet, using 6 popular architectures. Bottom: per-class performances from the middle scenario along with 9 images of the corresponding classes. We observe that the random crop DA seems loose paese-pre property on average when less than 30% the image is kept the crop but looking at the per-class performance, we observe that such DA can tabel-preserving with only 8% of the original image for some classes, while for other classes the label information starts to reduce at around 50%. Results obtained from the official Imagenet pre-tre pre-trained Py Torch models. CutOut and ColorJitter cases provided in Figs. and and exhibit the same trend.
random crop
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Figure 6: Reprise of the bottom left of Fig. for three different DAs (each column) and using the same 6 popular architectures (different lines). We observe that across DAs, different architectures agrec the ensemble of model would not reduce the class-dependent bias of the
prese-inn regimes for To i.e. final prediction. Results obtained from the official Imagenet pre-trained Py Torch models.
same DA across classes, the number of per-class samples that actually contain enough information about their true labels will become largely imbalance between classes, even if the original dataset was balanced. Any model trained on the augmented dataset will thus focus on the classes for which the DA the most grree-pres..iing.
Beyond the above intuitive and natural understand we obtained in term of data-augmentation. there exists theoretical studies that we propose to summarize below
3.3 Do Existing Studies Provide Answers Into the Inter-Play Between Regularization and Class-Dependent Bias?
We explored in Section 3.2 a possible explanation of the class-dependent bias we observed in Section hinting at the need to use class-dependent DA. We propose here to summarize existing studies that

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academic gown
sunglass sungless ass wio DA
Figure 7: Reprise of Fig. but now implement- ing the class-dependent DA as prescribed Sec- tion we only apply DA (random crop) to the classes that see their per-class accuracy im- prove when this DA is employed during training.. We observe here that the hias of the model intro- duced from random crop on the majority of the classes spills-over to the minority of the classes that do not benefit from that DA, even though such classes never received that DA during train- ing. Results are averaged over 5 runs and employ the official resner50 implementation trained on Imagenct with horizontal flip but per-class ran- dom crop DA.
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are attempting better understand the impact of DA and regularization in general onto model's performance and bias. Then, Section 3.4 will explore their application.
Input independent DA exacerbates the bias already present in dataset. Especially relevant to our results is a recent result Xu et al. 2220). In this work, it was theorized that when the underlying dataset contains inherent biases, training on the original data is more effective than employing an i.i.d. DA policy, i.e. applying the same random augmentation to all samples/classes, to produce an unbiased model. In short, the DA exacerbates the already present biases and makes the trained model further away from the unbiased optimum. The difficulty of this result lies in defining bias for real images. As per our experiments from Section we observe that bias can take many form e.g. one class might naturally represent its object always under the same angle. This is particularly true say for boats which are rarely captured upside-down. Hence, simply having classes with different natural statistics could be enough for Xu et al. [2020) to prohibit the use of DA in current datasets. Furthermore, Raghunathan et al. [2020] obtained surprising result combining both DA and regularization. In that case, it was found that the minimum norm interpolant on the original + DA dataset could have a larger standard error than the minimum norm interpolant on the original dataset alone, even when using gabbel-preseringg DA. However, this phenomenon only occurs as long as the model remains over- over-parametrized, even when considering the original+DA dataset.
Input dependent DA can reduce sample/class bias. Recall that Section 3.2 brought forward one possible explanation on how DA can be the source of training-sel by introduce class-imbalance due to the same DA being pabee--g for some classes and not for others. From this, direct solution would to adapt the "strength". This solution, formalized in Xu et al. [2020), consists in measuring the bias of a model and adapt the DA policy accordingly to correct it. This has been done in different flavors e.g. in McLaughlin et al. (2015), Iosifidis and Ntoutsi 2018). Jaipuria et al. 2220). One limitation of this direction is that it requires to estimate model bias and adapt the DA accordingly which can be challenging for large scale models.
Learned DA e.g. from GAN can produce even more class-dependent bias. One natural extension of the hands-on adaptivity of a DA to a measured model bias would be to learn a DA to maximize a model's ' performance, for example. This line of work has led to many learn DA policies e.g. using Generative Adversarial Networks [Hu and Li 2019]. Yet, it has been shown that learning a DA to maximize some aggregated measure of performance will produce even more bias in model (uu and 2019]. In fact, and as per the controlled experiments from Section 2 the learned DA will entirely disregard minority of the classes if it means that the produced DA can drastically improve performances on all others, effectively maximizing the average performances.
Inherent tradeoff between DA and model robustness. In addition the implication of DA into bias, there exists an intertwined relationship between DA and model robustness. In fact, even assuming the use of perfectly adapted DAs, there exists an inherent tradeoff between accuracy and robustness that holds even in the infinite data limit Tiipras et al.) 2018 Fawzi et al., (2188) Zhang et al. 2019). For example, Min et al. [202T) proved in the robust linear classification regime that (i) more data improves generalization in weak adversary regime, (ii) more data can improve generalization up to point where additional data starts to hurt generalization in a medium adversary regime, and that (iii) more data immediately decreases generalization error in strong adversary regime.
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3.4 Class-Dependent Data-Angmentation Seems Insufficient for Performance Recovery
We observed in Section 2 that DA could lead to dísastrous per-class performances on a minority of classes. We now propose to implement one solution from Section 3.3 that consists in simply not applying the harmful DA to the classes suffering from it. As will become clear, applying the DA on all other classes will be enough to skew the training of the model preventing any performing recovery on those classes
To motivate the need for a better control of the per-class performance of a model, we first present an illustrative argument. A standard resnet50 on Imagenet reaches 77.14% top-1 with random crop lower bound of 8%. Using precise cross-validation, one could reach 77.29 by using a random crop lower bound of 10% instead of 8% on all classes. Yet, and most interestingly. if one were able to get the best per-class performance -as per varying the lower-bound as in Fig. 2 and picking for each class the best per-class performance of any of the models one could reach 79.37. Beyond pure average test performance, controlling the worst-case per-class performance is of crucial importance for fairness [Du et al. 2020 Veitch et al. [021]. We thus explore one of the solution that reviewed in Section 3.3 that consists in only applying the random crop DA to the classes whose test performances increased with the DA's level. We obtain in Fig. 7 that such class-specific strategy is not sufficient to guarantee that the classes negatively impacted by the random crop DA see their performance to be constant across the DA level applied to all the other classes. This finding is also supported by our weight decay experiment in Figs. and 16 in which the regularization of the DN impacted classes differently. In fact, first notice that applying weight decay only when seeing some specific classes would simply amount (on average) to reducing the weight decay hyper-parameter proportionally to how many classes are considered to be without regularization. In similar fashion, DA produces an implicit regularizer [Balestriero et al.) [022] and applying class-specific DA level reduces the impact of the implicit regularizer. But since the amount of classes for which we do not apply random crop is quite small compared to the total number of classes (between 1% and 5%), means that the DA'S implicit regularizer remains nearly the same and thus the model's 's bias is nearly the same regardless if that DA is applied or not onto those classes. This is what we observe, applying the random crop DA to the classes that benefit from it is enough to bias the model and degrade the performances some classes at the same pace than when applying the DA to all classes unconditionally (compare Figs. 2 and 77
As result, we observe that no readily and easily implemented solution provides us with a strategy to prevent deep learning to fall into the scenario depicted on the left of Fig. [ Those findings however motivate the search of novel model complexity controls that is fair among classes. From a more theoretical viewpoint, might also be possible to better understand if even such a fair per-class model complexity could exist, which is not clear as natural image classes tend to have inherently different statistics.
4 Conclusions and Limitations
We proposed in this study to understand the impact of regularization, in particular data-augmentation and weight decay, into the final performances of a deep network. We obtained that the use of regularization increases the average test performances at the cost of significant performance drops on some specific classes. By focusing on maximizing aggregate performance statisties we have produced learning mechanisms that can be potentially harmful, especially in transfer learning tasks. In fact, we have also observed that varying the amount of regularization employed during pre-training of a specific dataset impacts the per-class performances of that pre- pre-trained model on different downstream tasks e.g. going from Imagenet to INaturalist. Lastly, commonly prescribed solutions e.g. class- dependent autaaaaugmenttionnn do not seem to help indicating that the sole use of an augmentation on some classes is enough to bias the model on all classes. Hence, there remains a vast research area to explore in order to turn deep learning model selection from the current regime more ideal one (left to right of Fig. II.
The main limitation of this work is its focus on computer vision datasets and models. It is possible that our observation will be further conveyed in other regimes or not, and we leave such analysis for future work.

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [N/A] We do not propose novel method that requires such discussion
(c) Did you discuss any potential negative societal impacts of your work? [Yes] We carefully described why the observations we have made can be dangerous for real world applications
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] Our only "theoretical result" consists a formal statistical test for which we precisely describe our settings and statistics
(b) Did you include complete proofs of all theoretical results? [N/A] No theoretical result requiring proofs was provided
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or as a URI.)? [Yes] We include summary statistics in the supplementary material which is enough to validate claims. The full codebase and all the saved models will be released upon completion of the review process (this includes almost a thousand pre-trained resnet50s)
(b) Did you specify all the training details (e.g.. data splits, hyperparameters, how they were chosen)? [Yes] We specify within each caption the key experimental setups, additional training details are provided in the appendix
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] we averaged all of our runs over significant amount of realizations (20) and perform formal statistical test on the significance of our results (d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, cloud provider)? [Yes] We provide this in the beginning of the appendix. Note that we will be providing the saved models and summary statistics to make the results easily reprodueible even without GPU or computation ressources.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes] For Imagenct and INaturalist
(b) Did you mention the license of the assets? [Yes] the full code will be released with NC by NC license on GitHub
(c) Did you include any new assets either in the supplemental material or as URL? [Yes] we include summary result files that contain all the statistics to reproduce the main claims of the paper in the supplementary material. The full codebase along with pre-trained models will be released upon completion of the review process.
(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] We are only using the Imagenet and INaturalist datasets
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(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We are only using the Imagenet and INaturalist datasets
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
14


Reviews:
[{"review_id": "ob2VMEmWu-C", "reviewer": "Reviewer_aqbS", "summary": "The paper investigates the per-class effect of some common regularization techniques including data augmentation (DA), weight decay, and dropout on neural classifiers.  Experiments across several models and two classification tasks are conducted to show that the popular approach of maximizing mean model performance by cross-validation has heterogeneous effects on different classes. The paper combines intuitive theoretical and empirical evidence to show that the class-dependent label-preserving regime may be the cause of the harmful effects on some classes by inducing class-imbalance for augmented data.", "questions": "Questions and suggestions along with the contexts can be found in the “weaknesses” section.", "limitations": "The authors discussed the limitations of the work. Please see weakness 1 for some suggested discussion.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n\n1. Overall the paper is well written and easy to follow.\n\n2. The question of interest is well motivated as regularization techniques are a crucial part of the success of modern neural models which are usually overparameterized.\n\n3. The authors provide a good discussion of several lines of related work.\n\n4. There is a good combination of intuition, math, and experimental validation which offers a well-balanced package.\n\n5. Experimental designs are simple yet well controlled to demonstrate the class-dependent effects of regularization in a clear way.\n\n6. Experiments are conducted with a variety of realistic neural models on large and popular datasets/tasks including ImageNet (supervised classification) and iNaturalist (supervised transfer learning).\n\n7. A simple fix with class-dependent DA is evaluated in an attempt to recover the performance.\n\nWeaknesses:\n\nMinor:\n\n1. Despite the fact that pre-trained models of a diversity of architectures are used to observe the label preserving regimes (and they seem to agree qualitatively), I believe that it should be more clearly stated that the observed label-preserving regimes represent those of a CNN family, which may deviate the “true” regime. \nSome newer models with patchified input such as ConvNext and Swin Transformers may show different behaviors under the same DA transforms. Additionally, one potentially better empirical estimate may involve repeating the experiments with human observers which are arguably closer to “ideal” observers.\n\n2. The paper can be further strengthened if a practical strategy that recovers the harmful effects of DA on certain classes was proposed.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "bPheGcmST2v", "reviewer": "Reviewer_fQNv", "summary": "The paper set up a hypothesis that implicit regularisation via data  augmentation (in the sense of Poggio's stability) and explicit Tiknonov regularisation (L2 is investigated) penalise different classes by different degrees. This implies that the average test accuracy is not a reliable measure of performance across all class alternatives. It also discovers that data augmentation may even not be label preserving, which it studies in the context of imbalance in the class's support in terms of labels, but it also opens up possibilities of adversarial data augmentation. ", "questions": "1. If the plot on the RHS in Fig. 1 is contrived to present an ideal case, it puts the rest of the results in question. If it is, mentioning so would help. \n\n2. Both the regularisation strategies investigated induce smoothness. It makes you wonder  if classes are being selectively regularized, what happens with a choice of regulariser that does so by design and produces sparse representations. \n", "limitations": "Although the paper keeps prescriptions out of scope, it may have built upon suggestions made in the work of Raghunath et al. on when and how much data augmentation is suitable. \n\nALso,  it does not discuss what discuss what proportion of classes are adversely affected by both forms of regularisation, if it is but 1% of classes, the solution may worsen generalisation in the vast majority of classes. Both data augmentation and L2 reguarisation are widely used methods to help with generalisation, and knowng this seems fairly consequential. \n", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The whole set of ideas being investigated esp. the  transferrability to downstream tasks seem to be original although nonetheless a natural extension to the questions posed by Xu and Raghunathan. I would have had more confidence in the hypotheses if 1.the manner in which unfavoured classes were identified was systematised. At the moment, a few hand-picked classes are demonstrated upon. 2. We had quantified support from more datasets and models to rule out bias from them. Otherwise, the paper is delightfully well-explained and well-structured. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "VItaHX_xvSV", "reviewer": "Reviewer_MY5V", "summary": "The paper studies the effects of regularization techniques — including data augmentation (DA), weight decay, and dropout — on per-class accuracies in image classification problems. They demonstrate that when we pick hyper-parameters such as the strength of regularization with regards to the average accuracy across all classes, it does not necessarily result in the fairest model; instead some classes are affected negatively. The experiments tests both in the case of directly training image classifiers as well as finetuning a pretrained classifier on a downstream task. It also explored possible explanations and showed that simple solutions such as selectively applying regularization to certain classes don't mitigate this problem. ", "questions": "Line 37: What about dropout? It is lightly touched in 2.2 but never mentioned in Introduction\n\nLine 44: requires -> require\n\nLine 78-82: consider using footnote for the promise on codebase. Should delete the second part -- it is not quire appropriate to address to reviewers directly in this style in main text.\n\nLine 147: \"all the DN parameters except for the ones of batch-normalization layers\" -> I think the author try to say only weights and not biases, but this sentence is confusing as parameters include biases.\n\nLine 206: \"respect the level-set of f*\" -> unclear what level-set means\n\nLine 218-219: since MixUp etc were brought up as possible fixes for label-preserving, should consider adding them to exps to see if they help at all\n\nLine 235-237: \"becomes insufficient to predict the correct label on average\" -> should define clearly what that means. What's the definition of \"unpredictable\"? What's the threshold? The vertical dashed line in Fig 5 is never explained. ", "limitations": "There's no Limitations section in this paper, which is somewhat reasonable as the authors only exposed the problem (effect of regularization on various classes) but didn't propose a solution. However, I still suggest adding a section about potential limitations if a hypothetical solution is proposed.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "In general the paper is well written, neatly structured, easy to follow, with comprehensive and solid experimental setup. The authors studied an important problem of class fairness: when we apply techniques with the overall average performance across all classes as a target, do we incur silent unfair and unfavorable effects on certain classes? The angle on regularization is unique and novel, and all experiments are well constructed to prove the author's point. \n\nHowever, the paper can be improved by shifting the emphasis and reconstructing the experimental presentation structure. \n\nThe experiments focus overly on data augmentation. However I think it is not too hard to give the same weight on the other two: weight decay and dropout, both only lightly experimented or mentioned. Only the first experiment (2.1) is \"scientific\" enough — with t-test results; the others only show cherry-picked classes. The author should consider at least adding t-tests to all other sections. In addition, I think the author should define a metric of \"class fairness\" (e.g. the variance or gap between best performing and worst performing classes) and measure/show it across regularization strengths. Most figures in this paper only show cherry-picked classes, and while illuminating, are not scientifically convincing.\n\nThe presentation could be tightened, e.g. Fig 1-4 have almost the same setup, and a lot of texts in 2.1 - 2.3 are repetitive; maybe consider merging 2.1 and 2.2, and bringing dropout results in from appendix. Transitions in between sections are a little redundant, and can be made succinct if needed to save space for, say, adding t-test results to other sections and adding dropout.\n\nSpecifically, the presentation of results could've been much stronger with the following suggested changes:\n\n - Consider merging 2.1 and 2.2, moving dropout results from appendix to main sections\n - t-test should be applied to all scenarios, not just DA\n - In addition to showing cherry-picked classes, define metrics for class fairness and show them across regularization strength \n - should also explain how the class pairs are picked for figs 1-4\n - For each figure, try starting with a succinct title, instead of details on how results are obtained. Like Figure 2:  “Effect of DA on per-class accuracies” Fig 3 “Effect of weight decaly on per-class accuracies.”\n - annotating in figures what every line is. The blue lines in fig 2-4 are not explained\n - Overall, the text of paper used too much italic/underline/bold formatting \n - “Section” \"Figure\" etc. before numbers should be capitalized\n - undefined what \"DN\" is. If it's \"Deep Network\", I don't think that's a common abbreviation.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "A9K-Fzss5HN", "reviewer": "Reviewer_Yj6j", "summary": "The paper builds around the fact that data augmentation (DA) can harm the accuracies over certain classes despite improving the accuracies overall and offers discussions on several related topics. Some of the discussions are making importance contributions to the community, while other discussions seem fairly straightforward. ", "questions": "The overall paper is a very interesting read and easy to follow. The main message of the paper, that DA will introduce negative effects to certain classes, despite the overall improvement across all the classes, is quite important and inspiring. \n\n", "limitations": "There are a couple of generic words regarding the societal impact. The authors do not believe discussions of limitations are relevant, which I don't agree with. For example, the empirical study is limited to a couple choice of transformation functions (and datasets), which could potentially limit whether the conclusions obtained are reproduced in other settings. However, the evaluation is not based on this part. Several questions\n\n- The theorem 1 does not seem to be presented in a rigorous manner, thus very hard to evaluate. \n        \n    - there is no definition of \"level-set\" across the paper. I guess it means decision boundaries or the features related to decision boundaries.\n    - in the proof, line 599, assumes p is a constant and omits it, but the remaining proof directly assumes $p=1$ instead of omitting it. (dropping the constant does not allow the equality below line 601 to hold, only assuming it to be 1 does)  \n    - the proof concludes by saying \"last equation is always positive\" (line 603), but the entire term is always positive throughout the proof, they are always norms, which are by definition positive. \n    - I do not see how other choices of loss, such as cross-entropy, will be carried out trivially following this proof sketch, despite the authors suggest so (line 197)\n    - these concerns together, since theorem 1 is the only theoretical result of this paper, put a significant question on the formal results of this work. \n\n- While the paper appears to be dense in contents, I feel like some contents are not significant enough to be presented in the main manuscript, such as details in statistical tests, and augmented samples; some can be presented in a more compact manner, such as Figure 5. To conclude, I feel this paper does not contribute a sufficient amount of results as a paper at this tier. \n\n- Section 3.4 also offers an exciting beginning of the investigation, but the investigation ends very immaturely with a shallow attempt with the most obvious remedy. I would suggest the authors to offer substantially more results following the direction of 3.4. \n    - the authors suggested \"no readily and easily implemented solution\" available (line 311) but it seems not even all the solutions attempted as the previous sections are customized into class-dependent version and tried out yet.  \n\nOverall, I think this is an exciting read and interesting start of an important investigation, but the results do not seem mature enough yet. ", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "- strengths\n   - the discussion on the effectiveness of class-dependent DA is relevant and potentially will inspire the development of methods for a long way. \n   - the main message that DA can harm the performances for certain classes while improving the overall performances is fairly important. \n   - the paper is backed up with a heavy set of empirical results with seemingly no issues in reproduction. \n- weakness\n   - several other messages (other than the main messages) seem very straightforward\n   - theorem 1 (the only theorem in the main paper) is not presented rigorously enough for evaluation, which puts up some question for the theoretical rigor of the entire paper. 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Author rebuttal:

Decision: Accept