Improving group robustness under noisy labels using predictive uncertainty
Authors: Dongpin Oh
Dae Lee
Jeunghyun Byun
Bonggun Shin
Venue: Submitted to ICLR 2023 (2023)
Keywords: Spurious-cue
group robustness
noisy label robustness
uncertainty

Abstract: The standard empirical risk minimization (ERM) can underperform on certain minority groups (i.e., waterbirds in lands or landbirds in water) due to the spurious correlation between the input and its label. Several studies have improved the worst-group accuracy by focusing on the high-loss samples. The hypothesis behind this is that such high-loss samples are spurious-cue-free (SCF) samples. However, these approaches can be problematic since the high-loss samples may also be samples with noisy labels in the real-world scenarios. To resolve this issue, we utilize the predictive uncertainty of a model to improve the worst-group accuracy under noisy labels. To motivate this, we theoretically show that the high-uncertainty samples are the SCF samples in the binary classification problem. This theoretical result implies that the predictive uncertainty is an adequate indicator to identify SCF samples in a noisy label setting. Motivated from this, we propose a novel Entropy based Debiasing (END) framework that prevents models from learning the spurious cues while being robust to the noisy labels. In the END framework, we first train the \textit{identification model} to obtain the SCF samples from a training set using its predictive uncertainty. Then, another model is trained on the dataset augmented with an oversampled SCF set. The experimental results show that our END framework outperforms other strong baselines on several real-world benchmarks that consider both the noisy labels and the spurious-cues.

Under review as a conference paper at ICLR 2023
IMPROVING GROUP ROBUSTNESS UNDER NOISY LABELS USING PREDICTIVE UNCERTAINTY
Anonymous authors Paper under double-blind review
ABSTRACT
The standard empirical risk minimization (ERM) can underperform on certain minority groups (i.e., waterbirds in lands or landbirds in water) due to the spu- rious correlation between the input and its label. Several studies have improved the worst-group accuracy by focusing on the high-loss samples. The hypothesis behind this is that such high-loss samples are spurious-cue-fiee (SCF) samples. However, these approaches can be problematic since the high-loss samples may also be samples with noisy labels in the real-world scenarios. To resolve this is- sue, we utilize the predictive uncertainty of model to improve the worst-group accuracy under noisy labels. To motivate this, we theoretically show that the high- uncertainty samples are the SCF samples in the binary classification problem. This theoretical result implies that the predictive uncertainty is an adequate indicator identify SCF samples in a noisy label setting. Motivated from this, we propose novel ENtropy based Debiasing (END) framework that prevents models from learning the spurious cues while being robust to the noisy labels. In the END framework, we first train the identification model to obtain the SCF samples from a training set using its predictive uncertainty. Then, another model is trained on the dataset augmented with an oversampled SCF set. The experimental results show that our END framework outperforms other strong baselines on several real-world benchmarks that consider both the noisy labels and the spurious-cues.
INTRODUCTION
The standard Empirical Risk Minimization (ERM) has shown a high error on specific groups of data although it achieves the low test error on the in-distribution datasets. One of the reasons accounting for such degradation is the presence of spurious-cues The spurious cue refers to the feature which is highly correlated with labels on certain training groups-thus, easy to learn-but not correlated with other groups in the test set (Nagarajan et al., 2020; Wiles et al. 2022). This spurious-cus is problematic especially occurs when the model cannot classify the minority samples although the model can correctly classify the majority of the training samples using the spurious cue. In practice, deep neural networks tend to fit easy-to-learn simple statistical correlations like the spurious-cues (Geirhos et al.) 2020). This problem arises in the real-world scenarios due to various factors such as an observation bias and environmental factors (Beery al. 2018 Wiles et al. 2022). For in- stance, an object detection model can predict an identical object differently simply because of the differences in the background
al. 2016 Dixon et al. 2018 Xiao et al. 2020).
In nutshell, there is a low accuracy problem caused by the spurious-cues being present in a certain group of data. In that sense, importance weighting (IW) is one of the classical techniques to resolve this problem. Recently, several deep learning methods related to IW (Sagawa et al.) 2019 2020 Liu jet af. 2021 Nam et al., 2020) have shown a remarkable empirical success. The main idea of those IW-related methods is to train model with using data oversampled with hard (high-loss) samples. The assumption behind such approaches is that the high-loss samples are free from the spurious cues because these shorteut features generally reside mostly in the low-loss samples Geirhos et al. (2020). For instance, Just- Train- Twice (JTT) trains a model using an oversampled training set containing the error set generated by the identification model.
On the other hand, noisy labels are another factor of performance degradation in the real-world scenario. Noisy labels commonly occur in massive-scale human annotation data, biology and chem-

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istry data with inevitable observation noise (Looyd al. 2004 Ladbury & Arold 2012 Zhang et al. 2016). In practice, the proportions of incorrectly labeled samples in the real-world human-annnonn annotated image datasets can be up 40% (Wei et al. 2021). Moreover, the presence of noisy labels can lead to the failure of the high-loss-based IW approaches, since a large value of the loss indicates not only minority group but also that the label may be noisy (Ghosh al.
that the sample may belong 2017). In practice, we observed that even a relatively small noise ratio (10%) can impair the high- loss-based methods on the benchmarks with spurious-cues, such as Waterbirds and CelebA. This is because the high loss-based approaches tend to focus on the noisy samples without focusing on the minority group with spurious cues.
Our observation motivates the principal goal of this paper: how can we better select only spurious- cue-free (SCF) samples while excluding the noisy samples? As an answer to this question, we pro- pose the predictive uncertainty-based sampling as an oversampling criterion, which outperforms the error-set- set--eee sampling. The predictive uncertainty has been used to discover the minority or unseen samples (Liang et al. 2017: Van Amersfoort et al.) 2020). We utilize such uncertainty to detect the SCF samples. In practice, we train the identification model via the noise-robust loss and the Bayesian neural network framework to obtain reliable uncertainty for the minority group samples. By doing so, the proposed identification model is capable of properly identifying the SCF sample while preventing the noisy labels from being focused on. After training the identification model, similar to JTT, the debiased model is trained with the SCF set oversampled dataset. Our novel framework, ENtropy-based Debiasing (END), shows an impressive worst-group accuracy on several benchmarks with various degrees of symmetric label noise. Furthermore, as a theoretical motivation, we demonstrate that the predictive uncertainty (entropy) is a proper indicator for iden- tifying the SCF set regardless of the existence of the noisy labels in the simple binary classification problem setting.
To summarize, our key contributions are three folds:
1. We propose novel predictive uncertainty- oversampling method that effectively selects the SCF samples while minimizing the selection of noisy samples.
2. We rigorously prove that the predictive uncertainty is an appropriate indicator for identifying SCF set in the presence of the noisy labels, which well supports the proposed method.
3. We propose additional model considerations for real-world applications in both classification and regression tasks. The overall framework shows superior worst-group accuracy compared to recent strong baselines in various benchmarks.
RELATED WORKS
Noisy label robustness: small loss samples In this paper, we focus on two types of the noisy label robustness studies: (1) sample re-weighting based approach and (2) a robust loss functions based approach. First, the sample re-weighting methods assign sample weights during model training to achieve the robustness against the noisy label (Han et al. 2018 Ren et al. 2018 Wei et al. 2020 Yao jet al. 2021). Alternatively, the robust loss function based approaches design the loss function which implicitly focuses on the clean label (Reed al. 2015 Zhang & Sabuncu 2018 Thulasidasan al., 2019 Ma et al.] 2020).. The common premise of the sample re-weighting and robust loss function methods are that the low-loss samples are likely to be the clean samples. For instance, Co-teaching uses two models which select the clean sample for each model by choosing samples of small losses (Han et al. 2018). Similarly, (Zhang & Sabuncu 2018) design the generalized cross entropy loss function to have less emphasis on the samples of large loss than the vanilla cross entropy.
large loss samples The model with the group robustness should yield a low test error regardless of the group specific information of samples (i.e., groups by background im- ages). This group robustness can be improved if the model does not focus on the spurious cues (i.e., the background). The common assumption of prior works on the group robustness that the large loss samples are spurious-cue-free. (Sagawa et al. 22019) Zhang et al. (2020) propose the Distri- butionally Robust Optimization (DRO) methods which directly minimize the worst-grout loss via group information of the training datasets given priori. On the other hand, the group information- free approaches (Namkoong & Duchi (2017); Arjovsky et al.) (2019); Oren et al. (2019) have been proposed due the non-negligibl cost of group information. These approaches aim achieving the
Group robustness:
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group robustness by training the model via implicit worst-case loss (e.g., maximum loss around the current loss). Nam et al.] (2020) trains the debiased model by focusing on the samples with a large-loss with respect to the biased model. Notably, Just-Train-Twin is simple but effective framework. In JTT, an identification model is trained to build an error set, which consists of the wrongly predicted samples by the identification model. Then, the final model is trained after adding the oversampled error set into the original training dataset.
(JTT) by Liu et al. (202I)
3 PRELIMINARY: WORST GROUP PERFORMANCE AND SPURIOUS CUE
We consider supervised learning task with inputs x()) E X, cor- rupted labels y(u y. and true labels 20 E V. We let latent attribute of corresponding (11 as (9 E A (e.g., different back- grounds in Figure D. We assume each triplet (x(), , 200 be-
Background attribute dj
1, green pasture
desert
2- 1, cow
True label z
longs to a corresponding group, go E G (e.g., group by a back-
ground attribute and its true class (z(0) ())) in Figure D. We also denote a training dataset as D = {(x(0, ] where each pair (x19), g()) is sampled from a data distribution D on X' X' Impor- tantly, an attribute (*) can be spuriously correlated with a label on certain dataset D. For instance, in Figure 1 the label (cow/camel) can be highly correlated with the background feature (green pas- ture/desert), implying false causal relationship. In this case, the background feature is a spurioos cue.
Minority groupa
2m-1 camel
Figure 1: The example case of the minority groups and its at- tribute the cow-camel clas- sification problem.
Ideally, we aim to minimize the worst-group risk of model fo X' - I (c is the number of class), parameterized by 0, with an unknown data distribution D and true labels as follows:
0" = arg min(may mma RD; (0)),
RD: (0) B (T
(1)
9E9
where D is the data distribution for the group g and I, : RC x Y - R loss function. To achieve the above goal of improving the worst group accuracy, the prediction of the model fo should not depend on the spurious cues. We instantiate this with the cow or camel classification dataset (Beery et al. 2018) which includes the background features as the spurious cue. When this relationship is abused, a model can easily classify the majority groups while failing to classify the minority groups such group, leading to a high worst-group error.
cow in the desert. Thus, the model has poor accuracy on the minority
To improve the worst-group accuracy, is ideal to directly optimize the model via Eq However, in the real-world, we assume that only the training samples are given, with no information about groups g, attributes a, and true labels during training. Therefore, ERM with the corrupted label is the alternative solution for the parameter 0:
O arg min (0),
RD = E ((fo(()) g())
Our goal in this paper is resolve the problem caused by the unavoidable alternative, ERM. The problem lies in the poor accuracy on a specific group due to the model's reliance on the spurious cues (Sagawa et 2019). To address the problem of the poor worst-group accuracy, the common assumption in the literature has been that the model training should focus more on the particular samples that are not classifiable using the spurious cues (e.g.. oversampling the cow on the desert samples) (Sagawa et al. 2019 Xu et al. 2020 Liu et al. 2021). In this paper, we call such samples the Spurious- Opurious-Cue-Free Cue-Free (SCF) samples.
Given that we lack information to determine which samples are SCF, the remaining question is "how can we identify which samples belong to a SCF ser?". The previous studies (Liu et al. 202T) hypoth- esize that the samples with large loss values correspond to the SCF set (Sec These approaches, however, have limitations because large loss values can be attributed to both the SCF and the noisy samples. Going step further than other approaches, our primary strategy for identifying SCF set is obtaining samples with a high predictive uncertainty. By doing so, our approach allows a more careful selection of SCF samples while excluding noisy ones as much possible. As theoret- ical support, we rigorously show in the following section that the predictive uncertainty is a proper indicator to identify SCF set in the presence of the noisy labels.
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4 GROUP ROBUSTNESS UNDER NOISY LABELS VIA UNCERTAINTY
In this section, we primarily show that the predictive uncertainty is a proper indicator to identify the SCF samples under the noisy label environment. Not only that, we can verify that utilizing the loss values as an oversampling metric can fail to distinguish SCF samples from the noisy samples. To rigorously prove this, we theoretically analyze the binary classification problem including both the spurious cues and the noise.
Problem setup: data distribution and model hypothesis Consider a d-dimensional input x = (I1...., :ta and its features and labels are binary: Ii, y, E {-1,1), A data generation procedure for triplets (x, y, 2) is defined as following: first, 2 is uniformly sampled over (-1, 1}. Then, if the true label is positive (z 1), x Bp where Bp is a distribution of independent Bernoulli random samples and p (P1..... ............ .Pa) [0, 1]d. Here, p. represents the probability that the i-th feature has a value of 1. On the contrary, if = -1, x is sampled from a different distribution: x ~ Bp where P'a) E [0, 1] Furthermore, with the probability 1), there is a label noise: = -2.
p' p'= (p/. Otherwise, y = 2. Finally,
consider linear model with parameters 3 = (Su.... 3a), which are optimized via risk minimization over the joint distribution of the features and noisy labels (x, y). This problem setup is inspired by the problem definition of Nagarajan al. (020) and Sagawa al. 2020). representing the spurious features.
Next, the concept of the spurious cue in the defined classification task is demonstrated using the cow and camel images shown in Figure [ Let's assume that the j'th feature represents background attribute (zy = a). which either green pasture (x, 1) or a desert (ay = -)). Suppose 98% of cows have green pasture background (thus, P, = 0.98) while only 5% of camels have the green pasture background (p', = 0.05). In this case, the majority of the data could likely be classified only using the vj feature (e.g., only utilizing 8,). However, abusing this spurious feature could hinder a model from accurately classifying the minority groups such as cows in a desert. Note that in practice, there can be many spurious features that correspond to given latent attribute (a).
To quantify the spuriousness, we define the Spurious-CHe Score (SCS) function.
Definition 1 (Spurious cue score function) we define the spurious cue score function V pppp' : Rd + IR with any function s(;))) which satisfies following::
( v() 0,
If > pl 1f p
L PpPp' (x) Σ s(p..p)),,
(3)
(pp.. S O,
Intuitively, if the SCS function value is low, the model will struggle to correctly predict the label of a sample x by relying solely on features having a high correlation with the label. A simple example of s(:-)) can be spp) p/z) = Pj p/z- In the cow/camel classification problem, the cow in the desert sample has the lower V "p.p' value due to the term s(p,, pT),,, (0.98 0.05) = -0.93. In contrast, the majority of cow has 0.93. For the negative class samples (camel), y pp'p can be used instead. Importantly, the SCS function is only determined by the true labels (z) and the input features (x) but not the noisy labels (y).
With Definition 1, the following result formalizes our goal: to find the SCF samples (having a low SCS function values) by obtaining samples with the high predictive uncertainty.
Given any 0 < < 1/2, for a sufficiently small 8, a sufficiently large d and any P. E [c, - da with (p, p41 2 c c1 S i d), the following holds for any 0 S 1 1/2:
Theorem 1
P Px.. R(=, x) < é F(H+- 2 2
(4)
R(x, 2) = 1,-1 1==11 ( p.. (x)) + +1,=-1Fx-8, (V'''p()))
where 3" is the risk minimization solution of the linear regression on the distribution on (x, y) and Hg is the predictive entropy of the model with its parameter 3*. F,, Fx and Fx-B,, are the cumulative distribution functions with respect to the data distribution, B, and Bp respectively.
Theorem 1 states that the highly uncertain samples (F (x)) 2 1 1-6 5) are more likely to have low SCS function values among samples belonging the same class (R(x, =) S e). Thus, this
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Spurious feature
Spurious feature
Spurious feature
Spurious feature
(b)
Training set samples
Tra Training set samples
Invariant feature
Invariant feature
Invariant feature
Invariant feature
Model-A
Model-B
Figure 2: 2-D classification results of identification models on synthetic data. The blue/red represent classes. The deeper the background, the more confident the prediction. The dots and translucent stars represent training and test data, respectively. Here, frue SCF samples are in dotted circles (cannot be classified via spurious feature). Two classification results ((a) and (b)) shows that Model- A (proposed) well identifies the SCF set (large overlaps between yellow circles in (b) and true SCF samples) while Model-B (without regularization) fails due to its overconfident uncertainty for the true SCF samples.
statement implies that utilizing the uncertainty (predictive entropy) can be useful in identifying the SCF samples. Additionally, can observed that the selectivity of the predictive entropy (Hx-) is independent of the presence of the label noise (whether y= -1). Particularly, it's worth noting that the probability of the samples with high predictive uncertainty to have a label noise is not greater than 1), the original label noise ratio. On the contrary, the probability of the label noise to be present in the samples with high loss exceeds (Theorem 2 in Appendix A.6). We empirically demonstrate that the proposed framework with the predictive uncertainty outperforms the existing loss value-based ones the real-world benchmarks with the noisy labels (). The formal form of this theorem and its proof are in Appendix A.
Entropy based debiasing for neural networks trained via ERM We theoretically show that uti- lizing the predictive uncertainty can allow us to obtain the SCF samples under the ideal condition. However, there are two obstacles in applying this to the real-world scenario. Firstly, an overparam- eterized neural network trained via ERM (with finite number of samples) can perfectly memorize the label noises (Zhang al.) 2017). This memorization could cause samples with the noisy labels to have high entropy during training. For instance, the model reduces the loss of the noisy samples once after it has fitted to the noise-free samples, the predictive entropy those noisy sam- ples increases (Xia al. 2020). Secondly, since the neural network models generally tend to be overconfident (uu et al.
2017 Hein et al.] (2019), the proposed framework could potentially be unreliable in entropy-based acquisition of the SCF samples.
To summarize, utilizing the predictive uncertainty with the neural networks necessitates two require- ments for neural network: (1: robustness to label noise) the model should not memorize samples with noisy labels; (2: reliable uncertainty) the model should not be overconfident when identifying the SCF samples. As an illustrative example, Figure presents how well two different models iden- tify the SCF samples. Model-A (proposed) is successful because it satisfies these two requirements. In contrast, model-B fails due to the overconfident prediction (low predictive uncertainty) for the true SCF samples. In the following section, we describe the proposed framework along with the training process that designed to satisfy these two requirements.
5 ENTROPY BASED DEBIASING
Overview The proposed ENtropy-based Debiasing (END) framework focuses on training the sam- ples with high predictive uncertainty. The END framework, in particular, is made up of two models with identical architectures. First, the identification model uses the predictive uncertainty to identify the SCF samples. Second, the debiased model is trained on newly constructed training data set by oversampling the SCF set. As result, END achieves group and noise label robustness. In addition, framework also be extended to regression problems, whereas other baselines cannot.
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5.1 IDENTIHICATION MODEL
Robustness the label noise and reliable uncertainty are two major requirements for using the predictive uncertainty with neural networks. To achieve these goals, we design the proposed iden- tification model with the loss function robust to noisy labels and the overconfidence regularizer. In addition, we employ a Bayesian Neural Network (BNN) to obtain reliable uncertainty. Additionally, we explain how modify the proposed identification model to fit regression task.
The noise-robust loss function and the overconfidence regularizer For the loss function, we use the mean absolute error (MAE) loss instead of the typical cross-entropy loss because the noise- label robustness of MAE has been well demonstrated in the literature (Ghosh et al. 2017 Zhang & Sabuncu 2018). The MAE loss L AMA is defined as the following:
1 (ff(
20, (fa()))
(5)
where i" is the index at which y:- = 1 in the one-hot encoded ÿ and a,(() is the i-th value of the softmax function. A Although a model with the MAE loss alone could generally be noise-robust, may occasionally be overconfident in predicting the SCF samples, meaning that the model produces low uncertainties its predictions of those samples. As a result, the framework decides to exclude them from the SCF set. This problem is visually demonstrated in Model-B of Figure 2 which is trained with the MAE alone. To resolve this, we employ the confidence regularization. Importantly, the role of this regul reguuarizationni is to prevent overconfident prediction for the SCF samples (Liang et al.; 2018 Müller et al. 2019; [utaaa et al. 2020). Specifically, this confidence regularization (Pereyra al. 2017) penalizes the predictive entropy. The regularizer Rent is defined follows:
Rent (f(())) = Ei= ((())())) (fo(
(6)
In practice, we empirically show that combining MAE with the confidence regularization yields a better set of SCF samples (Model-A in Figure 2 which enhances the worst-group accuracy (Sec
and 6.22). In addition, the ablation study (Appendix C..I) shows that the contribution their combi- nation significant on the classification benchmarks.
Bayesian neural network We chose BNN as network architecture to ensure that the identifi- cation model's uncertainty reliable. This identification model is trained using the widely used stochastic gradient Markov-chain Monte Carlo sampling algorithm, Stochastic Gradient Langevin Dynamies (SGLD) (Welling & Teh [011]. The SGLD updates the parameter O, with the batch at step via the following equation:
($(0) (( Ji=ı
Oc+1 0, -
Et (Vo Voloog(0.)
E (vo
N
Σ Vo 10gg0("0000000000 + P4
where e is the step size and Pe NO Et). The negative log-likelihood term (- lolog p (1) (0,x(())) x)) can be interpreted as a loss function. The prior term (log p(f.) is equivalent to the 1.2 regularization we use the Gaussian prior over the During the parameter updates, we obtain the snapshots of the parameters at every K steps. The final predictive value of the identification model is the empirical mean of the predictions: f(x) 1 S sown (x) where M is the number of parameter snapshots, (6) is the parameters at the j'th snapshot.
Extension regression Another benefit of the END framework is that can be easily extended to fit regression task with minor modifications. Since we utilize the BNN, the entropy can be calculated over gaussian distribution with a predictive mean and a variance of SGLDD weight samples N (x),,, [frn (x)]) (Kendall & Gal, 2017). Instead of the classification-MAA loss (Eqq 5). the regression version of the identification model uses the common regression-MAE loss (L(x. y) = (x) y/) similar to the classification task. Another change in the regression version is that the confidence regularization is no longer used because it is not defined in the regression task. The regression version of END also improves worst-group performance in the regression task as shown in Sec 6.3

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Spurious feature
Spurious feature
Spurious feature
Invariant feature
Invariant feature
Invariant feature (c) END
(a) ERM
(b) JTT
2-D classification results on the synthetic data. The blue/red represents the classes. The dots and translucent stars represents training and test data respectively. The minority (SCF) groups are in the dotted circles. The ideal decision boundary is the vertical purple dotted lines. The decision boundary of END is the closest to the ideal boundary.
Figure 3:
5.2 DEBIASED MODEL.
Once the identification model is trained, we build new training dataset by oversampling the SCF samples to train the debiased model. First, we assume that the predictive entropy of the identi- fication model follows the Gamma distribution because it is common assumption that the vari- ance (uncertainty) of the Gaussian distribution follows the Gamma distribution (Bernardo & Smith, 2009). Then, obtain the SCF samples based on the p-value cut-off in fitted Gamma distribution. Formally, the SCF set (D&CF) obtained by the identification model is as follows:
DCcr= =D, DDD.DU.... U...DD D. = ((((0, (H))) a", "")))))""""))))))""""))))"" 8->1117
(8)
times
where H-)) is entropy; 4 is the CDF of the Gamma distribution; T is the p-value threshold & is the hyperparameter to represent the degree of oversampling. The parameters of the gamma distribution, a" and 3", are fitted via the moment estimation method (Hansen 1982).
Finally, the debiased model is trained via ERM with the new dataset D U D&CF after acquisition of the SCF set via eq [ Note that training the debiased model follows a conventional ERM procedure: it does not use the confidence regularization or the MAE loss. The final prediction of the END framework is the predictive value of the trained debiased model.
EXPERIMENTS
6.1 SYNTHETIC DATASET EXPERIMENT
We begin with the 2D-classification synthetic dataset experiment to qualitatively substantiate the group and noise label robustness of the END framework. The dataset has two characteristics: it has (1) the spurious features and (2) the noisy labels. Initially, we describe the two features of the dataset. One feature the spurious feature which easy learn but exploiting this feature cannot classify the test set (Figure 3). The other one is the invariant feature. This feature is hard to learn because we manually scale down this feature value. Only a few training samples' labels solely rely on this feature. The ideal model can correctly classify both the train and the test sets only when using the invariant features. Therefore, the ideal decision boundary will be the vertical line in Figure 3 Secondly, we assign random labels to the training samples with the probability of 20% to evaluate the model's robustness to the noisy labels. The details of the dataset and the neural network model are in Appendix B.
The experimental results show that END correctly classifies both the majority and the minority groups (Figure ()). We posit that the outperformance is due to the well-identified SCF sample (Model-A in Figure 2). On the other hand, ERM is insufficient to classify the minority group (sam- ples in the dotted circles), although it perfectly fits the majority group samples(Figure a)). This poor performance of ERM is consistent with the empirical studies on the real-world datasets like Waterbirds and CelebA (Liu al.) (02T). Notably, JTT has learned wrong decision boundary in favor of the minority group while completely overlooking the majority group. This is because JTT focuses too much on the noisy labels.
Note the similarity between the definition of D, and the condition F(H (x)) 1 8 in Theorem 1,

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Table 1: Average accuracy (ACC) and worst-group accuracy (WG Ace) evaluated on the Water-
birds and CelebA dataset, with varying noise levels. END consistently outperforms other baselines, especially for the noisy datasets.
Noise level
Clean
10% noise
20% noise
30% noise
Waterbirds
WG Ace
Ace
WG Ace
Ace
WG Ace
Acc
WG Ace
Aee
ERM
0.687 (0.01)
0.969 (0.00)
0.648 (0.03)
0.945 (0.01)
0.649 (0.05)
0.913 (0.01)
0.629 (0.06)
0.893 (0.03)
ERM (GCE)
0.674 (0.01)
0,968 (0.00)
0.665 (0.03)
0.945 (0.00)
0.651 (0:04)
0.902 (0.00)
0.660 (0.07)
0.885 (0.03)
LFF
0.710 (0.03)
0.947 (0.02)
0.710 (0.00)
0.914 (0.03)
0.726 (0.03)
0.858 (0.04)
0.660 (0.04)
0.899 (0.02)
JTT
0.846 (0.03)
0.865 (0.02)
0.565 (0.08)
0.670 (0.04)
0.060 (0.03)
0.135 (0.01)
0.027 (0.01)
0.107 (0.00)
END
0.828 (0.01)
0,934 (0.01)
0.842 (0.01)
0.914 (0.01)
0.832 (0.01)
0,887 (0.02)
0.818 (0.01)
0.854 (0.01)
CelebA
WG Ace
Acc
WG Ace
Acc
WG Acc
Acc
WG Acc
Acc
ERM
0.487 (0.03)
0.952 (0.00)
0.477 (0.01)
0.927 (0.01)
0.480 (0.02)
0.891 (0.01)
0.485 (0.01)
0.858 (0.03)
ERM (GCE)
0.502 (0.03)
0,956 (0.00)
0.524 (0.01)
0.950 (0.00)
0.522 (0.02)
0.941 (0.00)
0.526 (0.04)
(0.01)
LIF
0.788 (0.03)
(0.04)
0.080 (0.06)
0.217 (0.01)
0.027 (0.02)
0.089 (0.02)
0.052 (0.06)
0.236 (0:25)
ITT
0.822 (0.02)
0.915 (0.01)
0,748 (0.02)
0.810(0.00)
0.245 (0.36)
0.357 (0.29)
0.151 (0.16)
0.258 (0. 12)
END
0.826 (0.02)
0.889 (0.00)
0.797 (0.01)
0.893 (0.01)
0.811 (0.02)
0.901 (0.01)
0.778 (0.03)
0.892 (0.01)
6.2 GROUP ROBUSTNESS BENCHMARK DATASET
In this subsection, we evaluate the END framework
Noisy-label
SCF set
Groups
on the two benchmark image datasets: the CelebA and Waterbirds, which have the spurious-cues (Wah) jet al., 2011 Liu et al. 2015). To evaluate both the group and the noise label robustness, we add sim-
MAE rag
ple symmetric label noises (uniformly flip the label)
to the datasets, shown in T
Cross-entropy
In this experiment, we use two kinds of base- lines: the group-robust and the noise-robust basc- lines. The group-robust baselines include JTT (Liu et al. 2021), ERM (Zhang& & Sabuncu, 2018)) and Learning-from-Faitue
(LIF) (Nam et al. 2020).
Figure 4: 2-D PCA projection of the latent feature of the identification model (first row) and its variant (w/o AMAE and Rent) (sec- ond row). Each dot represents the training sample. (first column) The green dots are noisy labels, and the blue dots are clean la- bels; (second column) The yellow dots are the obtained SCF samples; (third column) Each color represents corresponding groups.
There is one noise-robust baseline, ERM with GCE (Generalized Cross Entropy) (Zhang & Sabuncu 2018). We use the identical model architecture, ResNet50 He et al. (2016), for all baselines and END. For JTT, LiF, and ERM, we follow the identi- cal experimental setup as Liu et al. (202T) presented. Details of the experimental setup are in Appendix B.
The results from Table I substantiate that, unlike the other baselines, END achieves both the group and the noisy-label robustness. The primary reason for this is that END employs the predictive entropy, which is unaffected by the label noise. Specifically, we observe that the worst-group accuracy of the END framework consistently outperforms the other models in the noisy cases (Table D >= 10% noise). Not only that, END also shows the competitive worst-group accuracy on the noise free case. On the other hand, as the noise rate inereases, the performances of the group-robust baselines are harmed much more severely. We interpreted that the reason for the degradation of these baselines that they focus on the large loss samples, which are likely to be the noisy labels, as discussed in Sec The noise label robust loss, GCE, improves the ERM with noisy labels, but its group robustness is insufficient. In addition, in the ablation study in Appendix [.]oohows that (1) utilizing entropy is the major contribution of the group and the noisy label robustness, as Theorem 1 stated, and (2) cooperation between the noisy-robust loss and the overconfident regularizer has an important role in its performance.
Additionally, we qualitatively show that our identification model can identify the SCF samples while being robust to the noisy labels. Concretely, we visualize the 2-D projection of the latent features (before the last linear layer) of the identification model on the Waterbirds with 30% label noise. This experiment has two implications. Firstly, the SCF set identified by our framework corresponds with the minority group (the true SCF samples). Specifically, the first row of Figure 4 (END) shows that both the minority group (red and green dots in the third column) and the SCF set mainly
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located around the middle of the images. Quantitatively, up to 30% of the SCF set consists of the minority group, which is higher than the actual proportion (5%). Secondly, the contamination of the SCF set with the noisy labeled samples is significantly mitigated by our framework. The first row of Figure 4 (END) shows that (1) the noise labels are almost identically distributed over the space, and (2) the noise labels do not severely overlap with the SCF set. This result substantiates that the proposed identification model (Sec 5.1) effectively identify the SCF samples while including less noisy samples. In contrast, the baseline identification model (the second row) shows an overlap between the noisy samples and the SCF set. This is in line with our claim: to utilize the uncertainty with the neural networks, the model should not memorize the noise labels and should have reliable uncertainty.
6.3 REAL-WORLD REGRESSION DATASET
In this subsection, we conduct experiments on the regression datasets with non-synthetic label noise to demonstrate the followings: (1) the END framework can be extended to a regression problem; (2) END achieve the group robustness under the non-artificial label noise. Particularly, we evaluate models on two drug-targe affinity (DTA) benchmarks, Davis (Davis et al. 201T) and KIBA (Tang) et al. 2014), see Appendix B for the details. Inputs of these datasets are drug molecules and protein sequences. The target value is a physical experiment-derivel affinity values between the drug and the protein. We use the DeepDTA architecture (Öztürk et al. 2018 as the base architecture, Appendix B for the details. Similar to the classification benchmarks. the DTA benchmarks have two characteristics. Firstly, the dataset has the spurious correlation: the DTA model typically relies on single modality (e.g., predict its affinity by only leveraging the drug molecule, not considering the interaction with the target protein. or vice versa). which inconsistent with the physicochemical laws (Ozçelik et al. 2021 Yang et al. 2022). To get the worst group information of each bench- marks, we group data samples by their distinet drug molecules and target proteins, respectively. Secondly, the target values are naturally noisy due to the different environments of data acquisition DDais et al. 2011 Tang et al. 2014 2018). Thus, it can be seen as the non-synthetic label noise.
Since Lff and JTT cannot be extended to the regression problems, we propose an alternative baseline, "hard." Akin to JTT and END, the hard algorithm picks up the top-K largest loss samples after the first phase of training. Next, another model is trained with the over- sampled training dataset with those hard samples. Table 2 shows that END outperforms others in terms of worst-group MSE met- rics. We posit that it is the well-identified SCF set via the proposed uncertainty-based approach that contributes to this improvement. On the other hand, hard shows no improvement over ERM due the oversampled noise labels.
Table 2: Evaluation results on the DTA datasets.
Davis
MSE
WG MSE
ERM
0.268 (0.01)
0.860 (0.14)
Hard
0.299 (0.02)
1.196 (0.24)
END
0.262 (0.01)
0.785 (0.07)
KIBA
MSE
WG MSE
ERM
0.217 (0:04)
8,915 (0.33)
Hard
0.203 (0.02)
8.748 (0.39)
END
0.207 (0.05)
8,358 (0.53)
7 DISCUSSION
In this study, we present new approach that can significantly improve the group robustness under the label noise scenario. We theoretically show that the predictive uncertainty is the proper criterion for identifying the SCF samples. Upon this foundation, we propose the END framework consisting of two procedures.
(1) Obtaining the SCF set via predictive uncertainty of the noise-robust model with reliable uncertainty; (2) Training the debiased model on the oversampled training set with the selected SCF samples. In practice, we empirically demonstrate that END achieves both the group and the noisy label robustness.
For future works, we discuss several potential areas of improvements. Firstly, the END framework adopts simple approaches (the MAE loss and the SGL.) for the identification model. Thus, the fu- ture works can employ a more advanced approach for the identification model which (1) obtains the reliable uncertainty and (2) prevents the memorization of the noisy label. Secondly, we only con- sider the total predictive uncertainty of the model in this study. However, the predictive uncertainty can be decomposed into two different types of uncertainty: aleatoric (uncertainty arising from the data noise) and epistemic (uncertainty arising from the model parameters) Kendall & Gall (2017) Oh & Shin (2022). If we disregard the decomposed alcatoric uncertainty, we believe that it could improve the END framework.

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REFERENCES
Martin Arjovsky, Léon Bottou, Ishaan Guirajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907 00000.2.99, 2019.
Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European conference computer vision (ECCV), pp. 456-473, 2018.
Vidmantas Bentkus. A lyapunov-type bound in rd. Theory of Probability Its Applications, 49(2): 311-323, 2005.
José M Bernardo and Adrian FM Smith. Bayesian theory, volume 405, John Wiley & Sons, 2009.
Mindy Davis, Jeremy Hunt, Sanna Herrgard, Pietro Ciceri, Lisa M Wodicka, Gabriel Pallares, Michael Hocker, Daniel K Treiber, and Patrick P Zarrinkar. Comprehensive analysis of kinase inhibitor selectivity. Nature biotechnology. 29(11):1046 1051, 2011.
Lucas Dixon, John Li, Jeffrey Sorensen, Nithum Thain, and Lucy Vasserman. Measuring and miti- gating unintended bias in text classification. In Proceedings of the 2018 AAAI/ACM Conference on Al, Ethics, and Sociery, pp. 67-73, 2018.
Robert Geirhos, Jörn-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel, Matthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):6655
2020.
Aritra Ghosh, Himanshu Kumar, and P Shanti Sastry. Robust loss functions under label noise for deep neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 31, 2017.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International conference on machine learning, pp. 1321-1330. PMLR, 2017.
Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. Advances in neural information processing systems, 31, 2018
Lars Peter Hansen. Large sample properties of generalized method of moments estimators. Econo- metrica: Journal of the econometric society. pp. 1029-1054, 1982.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016.
Matthias Hein, Maksym Andriushchenko, and Julian Bitterwolf. Why relu networks yield high- confidence predictions far away from the training data and how to mitigate the problem. In Proceedings of the IEFF/CVF Conference on Computer Vision and Puttern Recognition, pp. 41- 50, 2019.
Tomas Juškevičius and Valentas Kurauskas. On littlewood-offord theory for arbitrary distributions. arXiv preprini arXiv: 1912.0870, 2019.
Alex Kendall and Yarin Gal. What uncertainties do need in bayesian deep learning for computer vision? Advances in neural information processing systems, 30, 2017.
MANJUNATH Krishnapur. Anti-concentration inequalities. Lecture notes, 2016.
John E Ladbury and Stefan T Arold. Noise in cellular signaling pathways: causes and effects. Trends in biochemical sciences, 3735711731778 2012.
Shiyu Liang, Yixuan Li, and Rayadurgam Srikant. Enhancing the reliability of out-of-distribution image detection in neural networks. arXiv preprint arXiv:1706.02690 2017.
Shiyu Liang. Yixuan Li, and R Srikant. Enhancing the reliability duitt---iittiiuutin image detec- tion in neural networks. In International Conference Learning Representations, 2018.
10

---

Under review as a conference paper at ICLR 2023
Evan Z Liu, Behzad Haghgoo, Annie S Chen, Aditi Raghunathan, Pang Wei Koh, Shiori Sagawa, Percy Liang, and Chelsea Finn. Just train twice: Improving group robustness without training group information. In International Conference on Machine Learning, pp. 6781-6792. PMLR, 2021.
Ziwei Liu, Ping Luo. Xiaogang Wang. and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE international conference on computer vision, pp. 3730-3738, 2015.
Ricardo V Lloyd, Lori A Erickson, Mary B Casey, King Y Lam, Christine M Lohse, Sylvia I, Asa, John KC Chan, Ronald A ellil H Ruben Harach, Kennichi Kakudo, et al. Observer variation in the diagnosis of follicular variant of papillary thyroid carcinoma. The American journal of surgical pathology, 28(10):1336 1330-1140, 2004.
Xingjun Ma, Hanxun Huang, Yisen Wang, Simone Romano, Sarah Erfani, and James Bailey. Nor- malized loss functions for deep learning with noisy labels. In International conference on machine learning, pp. 6543-6553. PMLR, 2020.
Rafael Müller, Simon Kornblith, and Geoffrey E Hinton. When does label smoothing help? Ad- vances in neural information processing systems, 32, 2019.
Vaishnavh Nagarajan, Anders Andreassen, and Behnam Neyshabur. Understanding the failure modes of out-of-distribution generalization. In International Conference on Learning Repre- sentations, 2020.
Junhyun Nam, Hyuntak Cha, Sungsoo Ahn, Jaeho Lee, and Jinwoo Shin. Learning from failure: De- De-biasing classifier from biased classifier. Advances in Neural Information P Processing Systems, 01-20673-206844 2020.
Hongseok Namkoong and John C Duchi. Variance- based regularization with convex objectives. Advances in neural information processing systems, 30, 2017.
Dongpin Oh and Bonggun Shin. Improving evidential deep learning via multi-task learning. Pro- ceedings of the AAAI Conference on Artificial Intelligence, 36(7):7895-7905, Jun. 2022.
Yonatan Oren, Shiori Sagawa, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust language modeling. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Process- ing (EMNLP-IJCNLP), pp. 4227-4237, 2019.
Riza Özęelik, Alperen Bag, Berk Atil, Arzucan Özgür, and Elif Özkırmhı. Debiaseddta: Model debiasing to boost drug-target affinity prediction. arXiv preprint arXiv:2107.00556, 2021.
Hakime Öztürk, Arzucan Özgür, and Elif Ozkirimli. Deepdta: deep drug-target binding affinity prediction. Bioinformatics, 34(17):1821-i82) 2018.
Gabriel Pereyra, George Tucker, Jan Chorowski, Lukasz Kaiser, and Geoffrey Hinton. Regularizing neural networks by penalizing confident output distributions. ICLR, 2017.
Scott E Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich. Training deep neural networks on noisy labels with bootstrapping. ICLR (Work- shop), 2015.
Mengye Ren, Wenyuan Zeng, Bin Yang, and Raquel Urtasun. Learning to reweight examples for robust deep learning. In International conference machine learning, pp. 4334 4343. PMLR, 2018.
Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. why should i trust you?" explaining the predictions of any classifier. In Proceedings of the 22nd ACM SIGKDD international conference knowledge discovery and data mining, pp. 1135-1144, 2016.
Shiori Sagawa, Pang Wei Koh, Tatsunori Hashimoto, and Percy Liang. Distributionally robust neural networks. In International Conference Learning Representations, 2019.
11

---

Under review as a conference paper at ICLR 2023
Shiori Sagawa, Aditi Raghunathan, Pang Wei Koh, and Percy Liang. An investigation of why overparameterization exacerbates spurious correlations. In International Conference on Machine Learning. pp. 8346-8355 PMLR, 2020.
Jing Tang, Agnieszka Szwajda, Sushil Shakyawar, Tao Xu, Petteri Hintsamen, Krister Wennerberg, and Tero Aittokallio. Making sense of large-scale kinase inhibitor bioactivity data sets: compar- ative and integrative analysis. Journal of Chemical Information and Modeling, 54(3):735-743, 2014.
Jing Tang, Balaguru Ravikumar, Zaid Alam, Anni Rebane, Markus Vähää Koskela, Gopal Peddinti, Arjan van Adrichem, Janica Wakkinen, Alok Jaiswal, Ella Karjalainen, al. Drug target com- mons: a community effort to build a consensus knowledge base for drug-target interactions. Cell chemical biology, 25(2):224-223 2018.
Sunil Thulasidasan, Tanmoy Bhattacharya, Jeff Bilmes, Gopinath Chennupati, and Jamal Mohd- Yusof. Combating label noise in deep learning using abstention. In International Conference Machine Learning, pp. 6234-6243, PMLR, 2019.
Prasetya Ajie Utama, Nafise Sadat Moosavi, and Iryna Gurevych. Mind the trade-off: Debiasing nlu models without degrading the in-distribution performance. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 8717-8729, 2020.
Joost Van Amersfoort, Lewis Smith, Yee Whye Teh, and Yarin Gal. Uncertainty estimation using a single deep deterministic neural network. In International conference on machine learning, pp. 9690-9700. PMLR, 2020.
Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011.
Hongxin Wei, Lei Feng, Xiangyu Chen, and Bo An. Combating noisy labels by agreement: A joint training method with co-regularization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13726-13735, 2020.
Jiaheng Wei, Zhaowei Zhu, Hao Cheng, Tongliang Liu, Gang Niu, and Yang Liu. Learning with noisy labels revisited: A study using real-world human annotations. In International Conference on Learning Representations, 2021.
Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pp. 68 6-8-. Citeseer, 2011.
Olivia Wiles, Sven Gowal, Florian Stimberg, Sylvestre Alvise-Rebuff, Ira Ktena, Taylan Cemgil, et al. A fine-grained analysis distribution shift. In International Conference on Learning Representations, 2022.
Xiaobo Xia, Tongliang Liu, Bo Han, Chen Gong, Nannan Wang, Zongyuan Ge, and Yi Chang-
Robust carly-leamning: Hindering the memorization of noisy labels. In International conference on learning representations, 2020.
Kai Xiao, Logan Engstrom, Andrew Ilyas, and Aleksander Madry. Noise or signal: The role of image backgrounds in object recognition. arXiv preprint wrXiv:2006.09994 2020.
Da Xu, Yuting Ye, and Chuanwei Ruan. Understanding the role of importance weighting for deep learning. In International Conference Learning Representations, 2020.
Xixi Yang, Zhangming Niu, Yuansheng Liu, Bosheng Song, Weiqiang Luins, Li Zeng, and Xi- angxiang Zeng. Modality-dta: Multimodality fusion strategy for drug-target affinity prediction. JEEF/ACM Transactions on Computational Biology and Bioinformatics, 2022.
Yazhou Yao, Zeren Sun, Chuanyi Zhang, Fumin Shen, Qi Wu, Jian Zhang, and Zhenmin Tang. Jo-src: A contrastive approach for combating noisy labels. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 5192-5201, 2021.
12

---

Under review as a conference paper at ICLR 2023
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. iclr (2016), 2017.
Jing Zhang, Xindong Wu, and Victor S Sheng. Learning from crowdsourced labeled data: survey. Artificial Intelligence Review, 46(4):543 576, 2016.
Jingzhao Zhang, Aditya Krishna Menon, Andreas Veit, Srinadh Bhojanapalli, Sanjiv Kumar, and Suvrit Sra. Coping with label shift via distributionally robust optimisation. In International Conference Learning Representations, 2020.
Zhilu Zhang and Mert Sabuncu. Generalized cross entropy loss for training deep neural networks with noisy labels. Advances in neural information processing systems, 31, 2018.
APPENDIX
A FORMAL THEOREMS AND PROOFS
In this section, we formally state and prove our main theorem and an additional theorem.
In A.
we briefly summarize our conventions on notations.
In A.2
we formalize the concept of "risk minimization" on our toy dataset, and provide the
solution.
In
we list some basic definitions and lemmas that will be used in our theorem state- ments and proofs.
A.3
An4
In
we re-state our main theorem formally.
In
A.5
we prove our main theorem.
A.6
we state the additional theorem.
A.7
we prove the additional theorem.
A.1 NOTATIONS
Vectors are written bold, while (one-dimensional) numbers are not. Random variables are always written in upper case, while plain values written in lower case except for constants. For example, X is a random vector, while w is a plain real number. Note that this a slightly different from the notations in the main paper, where random variables written in lower case.
X reads as "X follows the distribution D or "X is sampled from D IP stands for the probability, and stands for the expectation.
A.2 RISK MINIMIZATION ON THE TOY DATASET
Definition 1 (Risk-minimizing linear solution of distribution). Let n N, and let D be distri- bution on Ro+1, We call 3 = =Bo, 31.... BB,) E Ro++1 the risk-minimizing linear solution of D if it is the solution of the minimization problem
8 = arg min M........ + Σ -Y, 1m1
(9)
-1. Pn
Definition 2 (Bp). Let (...... Pa) E [0. We write B, for the joint distribution of (X1, ... X..) for independent random variables X, E {-1,1} where X, = with probability pt-
When 1, simply write B,, = ((m))
Definition (The toy dataset). Let p = (pI.... .P,), ) p' (ph.... ,ph) E (0, 1)"h and S 1 S 1.
We define distribution D = Dr.p' Ru+2 by its sampling procedure:
1. Sample Z Ba.s uniformly.
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2. If Z 1. sample X (X1 (X...., ,xn) Bp Otherwise if Z -1, sample X
(X1 ... (,)) ~ Bp
3. With probability
let Y = Z. Otherwise, ler =
4. Output (X, Y,
Note that the mathematical objects used here can be interpreted as follows:
Xx: i-th feature of the sample.
Z: The true label of the sample, either positive(1) or negative(-]]
Ps: The probability that the i-th feature has value 1 for a positive sample.
p/: The probability that the i-th feature has value 1 for a negative sample.
t.: The probability of "label noise".
Y: The post-noise label of the sample
10. and 0 S S 1, let 3 (So, ,,,...,,......... the solution of D p.P" Then we have, for some k(p, p') ) 0,
Proposition 1. Given p "Pi.-..,PM p' (f'..-., .Ph)
P: 1/1
B1 k(1 2n)
(1 1... n))
(10)
(2p; - 1)² (2pp -12
and
12
8o
Σ
(pr + 14. 1)3:- =0
(11)
Proof. All the expectations and probabilities that follow with respect to (X, Y. 2) D"
Let
7
8 = arg min L(3) = arg min E(5u + Σ 3:X; - YY2
i=1
Using
E[X.] = | = - - ppp -
E[X,Y 1] (1
-1- =
m)
1 n 1,2 -1] 11 (1 7)
(1- (-p))) + t -
(1)
E[X,YY = =-1
= -1]=
= -1,2 = =
- nB = -1, 1] =
(1- -)) - (11-))) + -pp (1-p:) 2pm7 2p/ -1) -1
For # j. E[X,X,, = [B|XXXX,1 1] E[,,,Z = =-1) ][[[X,/
1|E[X,|2 = E[X, = -1] = -1] = ((pp. - -1)(2p; - 1) + (2p/
P|y= =
- 1/2
[[x?]= =1
get
1 0 0 =
L(3)
2 203
1----
E(30
8,X, Y) = Bo Σ 3
30 +
(ps + - 1)8
i=1
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1
Ps +
p4 - 1)8; =0
(12)
i=1
(which proves equation II
and
1 0
0 203,
L(B)
8=8*
EX,
Σ 3,X;
B,EX; + B,EX,
Σ
- - B[XY
J#i
Σ 3,E[X, 1 _
- = 1| 1 2 =-1
= AEEX, + 8,EX,
= So(P: + + - 1) + 8 +
Σ
3)((2p: 1)(2p, - 1) + (2p1 1)(2p'} 1))
2
j71
27) -Pi)
1)+ B 1 Σ -1( (1),_) (1-2n)(p: -p-) (13)
8olp: + -1)
2
Letting Σm (2p; 1)3, and w' Li=1(2 (2p) 1))a, get
1 + w
8o
(14)
from equation 12 and
1 10 11 w+ (1/ (vi- w +(1 (2p:
(2p% (2 -
1
112
((1-2)))- (15)
2
2
from equation
13
Plugging equation 14 into equation
15
we get
w/
p1 + (1
2 (2p: - 11)
2 (2p) - 1)2) = =(1 -2n)(pp - -p-)
2
2
hence
Di - P/-
3g = k'
(16)
1 {2ps - 1)2 2 (2p' 1)2
where
W - 2m w'
(17)
Plugging equation 16
back into equation
17 using the definitions of w and w', we get
1- 2m
K'
(p. (p--P/)²
1
1- 12p - 1)2 (2F 112%--17 1)
we conclude
Pi P4.
Bg 1-1-k(11))) ((7))
(2ps - 1)2 2 (2p)) - 1)2
where
k= k(pppp')
0.
(p. -²
1 D 1- 12pp 11 (2pp
(2p) 11²
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A.3 BASIC LEMMAS
Definition 4 (cumulative distribution function F). Let D be a distribution, and let f be real function the domain of DD Then for y E R, we define
FY(D) = =xxeDl((X) S E
We treat "F(f(X))" as a shorthand of (x())(((X)) when the definition of 'D is clear from the context.
Definition (anti-conneentration of a discrete random variable). Let W be a discrete random vari- able. We write
AC(W) max P[W = w]
We make use of the following estimates of P(F(W) & a)] and P(F(W) 2 a] for discrete random variables W throughout our proof:
Lemma 1. For any discrete random variable W and any a E [0, 11. we have
- (CCCCW) < P(F(W) - al <a
(18)
and
AC(W) < P(F(W) 2 a} aa
(19)
Proof. It is enough to prove equation
equation
can be shown from equation 18 by letting
W = W. Let
" = max (w E F()) $ a}
We have
P[F(W) sa P[ S w"1 = F(w")
Since
F(w)) S
by the definition of w", we have the inequality the right hand side. To show the inequality on the
left hand side, suppose F(w") < a AC(W). Let w=* be the smallest real number in the range of W bigger than w". By the definition of w". we have F(w"*) > a. Then, F(w F(w"")=((w)> a- AC(W)
AC(W) 2 P[W = w"*l P(W < ='']-P[W s Ew"l=
which is contradiction.
Upper bounds on AC((W) are called anti-connentrattin inequalities the literature (Krishnauur (2166). One such inequality is Litllewood-Offon inequality, and version of it that encompasses our case was proved Juskevičius Kurauskas (2019):
Lemma 2. Let W Di= B,X,, where 3i + and X (X1, .n)) B, for p [e, e]"
Then we have
C
AC(W) <
Vn
for some C' = C(e) > 0.
Proof. This direct consequence of Corollary Juskevičius & Kurauskas 2019).
An application of Lemmal is the following.
Lemma 3. Let W be discrete random variable. Let P(W and AC(W). For any 5 > 0, we have
P[F(WW) S and F(W) > 5) S 2x +
and
P[F(W) < and F(W))) > 8 <a
In particular, for any event for any 8 > 27, we have
a
P[ | F(W)) < P F(W) < 8)
1 (2xx + 7)/(0 -7)
8-7
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Proof. We have
P[F([W]) & and F(W) > 6] 1 P/6 < F(W) - + a] + P/W <0]
P 2 0 and F(W) and F(W) > 8+ al
v 2a + + P 2 0 and F([ S and F(W) > >8+
(by Lemma
2a +
where the last equality is valid because if W 2 0, then F((W)) 8 and F( > 8 + cannot
hold simultaneously since if they do, 5 8 a < F(W) = P[W" < = & [W]] - PIW" <
+ PIJW"| < [W]] = F(]WI) - 8 + a, which is contradiction.
Similarly, we have
P[F(W) < 8 and F([W) > 6] < P(W < 0] + P(W 2 0 and F(W) S 8 and F([W)) > 6]
P[W and F(W) 5 and F([])) 8]
a
We can show the remaining statement follows:
F(W)) - 5] PiEE and F((W)) S 8)
PE
P[F( (T) S
P and F(W) - - PFF(W) < 5 and F([W]) >
P[[([W] d]
P and F(W) S 6)
a
P(F((W) 1) I 6]
5 5-7
F and F(W) S 8)
P(F(W) - + P(F(W) < 5 and F(W) > 8]
8 8-7
P[E and F(W)= <
(F) - 56+ + 20 +
7-7
[P and F(W) S
P[F(W) S 6] (2x + 7) [D/P[[F(WW s 8))
8 8-7
1
0
2 P[E F(W) s 6]
1 (2a + 76 7)
8 8-7
Also, we need a quantitative version of multi-dimensional central limit theorem:
Lemma 4 [Bentkus 2005)). Let V E Rd (i 1, n) be independent random variables with
E Ler
S
V
i=1
the covariance matrix E E(S) is invertible, for Z NO.,E), have
P U] PlZ U] < Cy for any convex U C 2
where C is an absolute constant and
=
1=1
17

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Under review as a conference paper at ICLR 2023
A.4
THE ENTROPY THEOREM
Definition 6 (entropy function). For 3 = (50, -. Br) E Bb++1, we define the entropy function
Hg: R. IR follows:
exp(30 + 2n S(tt,)
Ha(a1, ... .n) H( 1 eexpp(Bu + Li=1 Br3s))
where H(p) = -p plogp (1 -p) log(1 -p)
Note that Hg( In) can be expressed f(]30 + Ei= B,zx 1) for monotonically de- creasing function f.
and A_ Definition 7 (sign-respecting function). Let A+ = {(p, E [0. 10 {
p'p.p)) E [0, 1] [0. J'). A function A: ([0, 1| [0]] x A- will be called sign-respecting if it is continuous on each of A+ and A., positive on A_ and
negative A.. For example, f(p,p') = [p-v"
p-p' is sign-respecting.
Definition 8 (spurious cue score function). Given sign-respecting function A, under the distribu- tion defined in Definitiion 3 define the spurious score function y
"p.p' IR" - R as follows:
VA P.p X
Σ A(p.Pp)Xx
Roughly speaking, Mn of label-compaible features. Here, feature X; of a positive sample is label-compatible if either X; = and p; > P (i.e. positive samples have higher probability of having X; = 1) or X, = -1 and p4 > Pt (i.e. negative samples have higher probability of having X; Here, the "number" is weighted in terms of Mp:,P')
p.p' measures how easy a given positive sample X is in terms of the number
If we want to measure similar thing for negative sample, we can use p'.p instead.
Theorem (formal). Let A be a sign-respecting function. Given < E < 1/2, there exists do (A,A, <) > such that for any O < 5 S 50. for sufficiently large n (n 2 N for some N N (A, e, 8)), the following holds:
For any combination of
0 <
. p = (pI,
=p=( ... _Pa), = (Wi, .. ......),))))) E [e, -e"" with (p; - Al e Si n)
when 3 = (Bo, B1... BB)) is the risk-minimizing linear solution of (X, for (X, Y,
D
p.p' have
P Px,YZ>>a p.p. RA (X, 2) F(Hg- -1)
(20)
where RA p.p' is the spurious cue score rank function defined as
RA P.p'
= =2 (Bp) (V-) P.p' (X)) 11==-1 (Bpr) (Wrr A px))
That is, RA p..pp (X, Z) is the rank of the spurious cue score of X within its ground truth label Z.
This means intuitively that under our dataset generation process (as described by Definition 3 with some minor conditions (Those for p and p'), relatively high (top-o) entropy w.r.f. , implies relatively small (bottom-E within the ground truth label Z) spurious cue score with high probability (probability at least - c).
A.5 PROOF THBOREM
A.5.1
A REDUCTION
We reduce Theorem 1 to the following variant:
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Under review as a conference paper at ICLR 2023
Lemma 5. Let A be sign-respecting function. Given 0 c 1/2, there exists óy = 6o(A, c) such that for any < ômin S for sufficiently large n (n 2 N for some N NN, e, 8min)) the
following holds:
For any combination of
<
p -PI, Pa), = (1h.". " ".,,))))))))) E [e, c)"² with lps - p 2 é (1 si n)
8 [ōmin ōn]
when
8 = (Su, ...., Br) is the risk-mininizing linear solution of (X, Y) for (X, 2. ~
D
P.p"
Px-Bol F( p.p. (X)) se
(21)
and
Px-Bp1 ((UPpp()) Se F(HI (1
(22)
where lgg is the entropy function Definition 6). VA p.p' and vrA are the spurious cue score func- tions (Definition (8) and F are the cumulative distribution Deniiitttineiiiion 4 related to them with respect to Bp or Bpr.
The key differences from Theorem 1 are (1) the introduction of 8msn and (2) that equation 20 has been separated into equation 27 and equation 22
Before presenting the proof of Lemma
we will show how derive Theorem from Lemma
proof of Theorem assuming Lemma Let
5
Let A be a sign-respecting function, and let 0 < < 1/2.
5
Lemma
and take
82
8n 3
Given with
< 8 80
let
8min 8e
and take
6C A, é', 8min),
40 (5)))))) )3)
N max N Lemmed
8
where C = C(e)) is from Lemma 2
To show that N satisfies the theorem statement, let 2 N.
Since 8min I 5% by Lemma
we have
V8min 5 8' 86 PxwB,l F(VA ((AA)(())) S 2
((H
2 (23)
e
and
Vomin 5855 5g Px~B,1 ((AA))))
< - ((Hg
c 1
2 (24)
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Under review as a conference paper at ICLR 2023
Our goal is to show that
Goal: Px p.p. 1 Px,. (x, se
(25)
where
R P.D (X, = 1z=1 = Fi
p.p' (B,)) (B,) D.D' (X)) + 12=-1 D' (B, (1))) D')(()))
(1)) (B)
To simplify the equations that follow, let us make the following definitions:
a' - --mmin {a E R (D" (D) D.g (4) 20))) -
p == PXYYZND' 14 = 111 (X) 2 a*]
81 1 (Bx)
62 1 Fil.- (B, (a*)
We have the following (in (in)equalitie::
1.
Px~B, (F (X)) 2 6,] di]= 2pl2 X.Y'DD'' [F (X)))(X)) 2 1 6] (26)
and
Px~B, [F(H
21 1 52] = 2(1 p) (X)) 2 1-6) 6] (27)
Pp
Proof. We have
Px~8, [F 2 1 = P (X) a*l
=PXYY p.p [ll3((X) (X) a² =
Px.Y.ZDD P.p
11 (X) 2 a²) PX.Y,ZZD" Ho (X) 2 a*) p.p.
Px,,, p.p! 16 1]
-2pIPx D.D' |11 (X) a"l
= xp.'xx.Y.>>>" p.p.
[F (X)) 2 1 - 8)
which proves equation 26 equation
27 can be proved similarly.
2.
81, 82 < 80
(28)
Proof. We have
51 ACACCx (H (X)) - Px~B, F [F (X))
(X)) 2 1 - -8,]] (By Lemmal
= 2pPx.Y,ZZND [F(H (X)) 2 1-6) (By equation 26
DiN
< 2pô (By Lemma
< 28 < 28g < 0
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Under review as a conference paper at ICLR 2023
Since
ACx~B, (Hz. (X)) ACx-B,
3 X
S 2ACx~B,
(,
2m1
2C'
(By Lemma 2
vn.
$ (Since n N ( 6C
%
)3)
60
this proves 81 S 66- 82 6g can be proved similarly.
3.
81 po,
82 (1 p)d
(29)
Proof. We have
& 2 Px~B, (X)) 2 61] (By Lemma
2pl'x,Y,Z~DD' P.P' [F(Hg- (X)) 2 1 8] (By equation 26
2p(6
ACX,Y.Z~D' D.p
(Hz. (X)) (By Lemma
2 2p(6
-
1
(ACx
(1g (X)) + AC ACx~B, (Hg (X))))
2
1
2p10
(ACx-F,
Σ 3 + ACxxE,
Σ xx 1)
2
iml
2 2p(5 - (ACx~B,
Σ (3),) + ACx~B Σ 3 x,)))
2C Vn
2p(5
(By LpplyingLeemmal
Lemma twice)
40 )3)
pd (Since
>
which proves &1 p5. 52 2 (1 -p)ō can be proved similarly.
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Under review as a conference paper at ICLR 2023
Using these inequalities, show equation 25 follows:
P Px,xZ>'" P.p.
P.pp (x, 2)) S
F(Hx 8
2 Px,Y,2D''
R (X,
2 F(H (X)) 1 8
PX.Y, P (X, Z]) 2
Hz. (X) a" | (By definition of a²)
=p- = -XXXY,ZD''' P.p. | FV (Bp) V p.P' (X)) - 2 H3 (X) 2 2
(1-)) +1--D)-xx..2-00 Fu (B,, 4 (X)) S 2 (x) (X) 2 a',
= p. Px-BBl Fv. (B, P.P (X)) S H3. (X) a"
+ (1 1 pp) PX Fu. (B, p' (X) S Hg (X) 2a"
= p. Px-B,l Fu p.pp
F( (X)) 2 21 1 61
+ (1 p) PP Fur
(X)) S F(H (X)) 2 1 1 82
2 (1 (p1s, 26min + (1 - p)1 (By equation
23
equation
24 and equation 28
2 (1 (p1 /6 + (1 p)11--26mm..// (By equation
2 (1 (1 - -dmin/ (Case analysis based on the magnitude of p)
(1 G2
=
2
2 1 -
A.5.2 PROOF 5
Now, let us prove Lemma following lemma.
The proof relies on the calculation results of Proposition and the
Lemma 6. Given 0 < < 1/2 and M 2 1, there exists do = 8o(e, M) 0 such that for any
6min S 6g, for sufficiently large n (i.e n N for some N(e, M, Swin)), the following
holds:
For any combination of
p (pI, Pn) E 1e,1-
di .... Un bn, bi.... bn E
[8msn Sul
we have
P Px~B, [F( Σ (xxx) Se F F
(30)
Before presenting the proof of this lemma, we will show how conclude Lemma from it.
Let 0 < < < 1/2, M 2 1. It is enough to find óg that satisfies the theorem statement where only equation 21 is considered but not equation 22 This is because since then a symmetric argument that accounts for equation 22 can be made and we can take the minimum of the both dg.
Let
B(a.b)
1
1 (2a
1)2 (2h 1)
2
22

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Under review as a conference paper at ICLR 2023
Take M = (A, c) such that
1 < min 13(a, b)1, min [A(a, 6)1 and max [3(a, max [A(a, s M (a.b)EL
M (a.b)EL.
(a,b)EL.
(a,b)E,.
where I. = {(a,b) b) E 1c,1- c [c, 1 c) : la b) 2 2c.-
Let óy 1)) Hommddy c/2, M), where %g H c, MM) stands for the 6u found by using Lemma 6
Suppose 6min S 5p- Suppose ò E 8minn ool. p. p' and 7 have been chosen, and 3
(Bo.... Ban) has been determined accordingly. Then our goal is show that
Goal: Px~B,1 |
F(TA P.p' (X) < c
F(H (X) 2 1 - 5 1 1-1c
for sufficiently large Letting
Pi D2 Xi 1;=1,...,
x
pi p
a = Bp 130 + Σ
3,x, <
1=1
and
7 7= AC + Σ (,X.))
we get
Px~BBl F(UA, p.p (X)) Se F(H (X)) 2 21-8
= =PXwB,l F( p.p' (X)) S F( F(.0 + Σ 3:X, ) S
1=1
a
+ 1+ (2a + (b PxmB,l F(U (X) S F(Bo + _ B,X,) S 8
2 8
(by
L
PPx PIVA. (X)) Sel F ((X+) 2
8 y
1+ (2a + 7)/(6-)))
a
+
1 (2a t7)/(6 PPxBBp
8 7
Pt- p/
a
Pt 1/4
F
App.Pl) sc F()
3(r. ()) $8 (by equation 10)
Pi
p/
p; p/
1+ (2a 7)/(0 76) (1 --/2)
8 γ
(by 6 provided that N Lewmdd
(e/2, M, 8min))
(1 1 c/2)
8mn y 1 (2a 7)/(6)
7)
provided In the second last inequality, we could apply Lemma provided that n 2 N Lemmn e/2, M, 8mmn) because
1. 5 E [8min
< </2, M)]
2. p. p.
A(pe.1) p/) 0 (since A is sign-respecting) and 1 S [A(p. s M
3. Pi p
A(pt, (p).,p))= 0 and M ((pp. M
Pp (pi pt)
4. X' (Xi, (x,) Bp", where Df
[e, o) (i
1 Ps (p. < pl)
1. 1,...n) n).
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Under review as a conference paper at ICLR 2023
Now, it is enough to show that the last quantity is 2 1 - when 2 N' for some N' = (e, mmnn) (Then, N = max(N", . Mammmmil e/2, M, 6min)) will satisty the theorem statement) Since y - oC in the convergence rate that depends only it remains to show that the same is true for a. Using equation] 11 and equation 10. we get
(5 + B(X,) Bo +
Σ
(pp: 1)3
Σ (pi pp))a
[pi Pel18(pt.P)) k(1 22n) Σ
(31)
i=1
2 k1 (())-n))a M(c)
Therefore by Hoeffding's concentration inequality, since 3r E (1:11-210) (Q) [a,, b,] (i = 1, 72) we get
PI0o +
Σ
3,X; <
< IP
B,X; E Σ
3xXi] < -k(1 2n)
M(e)
i=1
i=1
(32)
2(k(1 2n)) M[(e
M(c n
< <exp(-
Dx (b; a.
2²
exp( (c m)
finishing the proof.
A.5.3 PROOF OF LEMMA
Now we will prove Lemma We will make a bootstrapping argument composed of two stages: (I) proving the theorem imposing certain restriction on (P1,' PN, 01 ... Ony bi, ... bn (2) proving the general case using the "restricted theorem".
First, let us prove the restricted theorem. The restriction will be clarified later on. Let < 6 < 1/2 and M 1. Define
a == a=/-((</2)
and
60 = (a
where + is the cumulative density function of the standard normal distribution. We argue that 50 satisfies the theorem statement. To show that, take arbitrary 8min (0, 6o), 8 [ōmin: 60}, (pi,-.. P..) [e, 1 - e]" and a,,-..
ans bi,.... bn E <17 MJ. We have to show that
Goal: PX~B, [F Σ (XX Se F b,X,) S 0) 2 1-6
(33)
E=1
Define
[as(X; u(p:)) bi(X, p(p))) Di= pp(pp))2
dFo(P:)²
V,
1.=1....,m))
(34)
and
S
S =
Sz
i=1
00)
where ((p)) [[[X]] and (p.) E((X, EX
Since v; are independent and E(v;]
apply Lemma provided that the covariance matrix Σ Cov(S) is invertible. Indeed,
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Under review as a conference paper at ICLR 2023
may assume that Σ is invertible because Σ has the form
Ei a.bro (pi)²
K. 1
Σ
(35)
VE (p.)) Ei=
and the conclusion of the theorem trivial when (Since then (a1. an) ~ (b1,"". bn)).
The inverse is:
1 R
Σ-
(36)
1²
K 1
Therefore, applying Lemma 4 we get, for ~ N(O, E),
JP|S E P[z 01 < Ch for any convex U C R
(37)
where C' is absolute constant and
18
V [
(38)
1=1
has the following lower bound:
E a,b.o((:))
- (39)
K Ei ( (p()) V En ( (or
max" max" b,
On the other hand, upper bound comes from our "restriction".
bootstrapping restriction: S 0.99
(40)
Note that the case where 0.99 should be, least intuitvely, easier to prove, since the two events in equation 33 become more correlated in that case. We assume this upper bound temporarily to make sure y bounded from above properly. From equation 36 and equation 40 can be easily deduced that
C" Vn
7
for some C' = (c,
Now, we're going to prove equation 33 in light of equation density function Z can be written as:
37
To do so, note that the probability
22()1122)= = 2T det Σ
exp(
1
21
T
E-1
21
2
22
22
1
exp(-
(2}
2n +2))
2T 1:2)
2(1
x2)
Also, note that from this can calculate:
S f2(21, 22)
a a-K22 K2g
$1
(41)
12x fz(21, 22) 22)
VI
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Under review as a conference paper at ICLR 2023
For sufficiently large have
Px- Px-B, B, [F
axXi) Sel F( b(X,)) [
1=1
1=1
1 [(()) S e F(S2) <
-Px(F(S)) 1 c and F(S2) 5 (by Lemma
=IPx[PXx|SS] < S S and Px S Sa] S 6] 8
2 [2x[Pz>[Z] < Si] e - C7 and Pz - So] S 6 - Cr] / (by equation 37
= Px[S| 1 (c
and 52 < x (6 (6--c)) /
2 |x|(| S a and 82 < x 1 CT)] /
2 CCy00 + Fz Sa and Z2 S $ 066 - (b) / å (by equation 37p
$ 16 (6- 03)
(1 (1)
fz z2 22)d21)d22
= -C-/0 (1 - Cr/6)
1_00 (-1) Cx)
f2(21, 22 (dd
_ (31, 22 d21
2 -C//0 + (1- CT/6)
inf
22 C2)
1 fz (21, 22 dd
-Cy/0 (1 - C7/0)
inf
/b v1 k²
K22
(by equation 41
2550 ((6-C))
a xΦ - C7)
-Cy/6 (1 - C2/6) (1-C1/6) $
vI -1-M x2
Now, let further estimate the last term:
T a K 1- C7)
Φ 0 KΦ ((p)
v1 2
V1- - R²
a xM (a 11-11111 e/22)
v1 - 62
Q (a x- (1 - c//2)
2 ②
V1 R²
(a 4(1
e/2) is negative equation 39
2 1 c/2
Therefore continuing the previous chain of inequalities,
P Px~B a.X,)) Se F E b,x,)
i=1
2 C216 (1 CT/6) (1 c/2)
2 C8-10' /
/VT/// (1 - E/2)
2 1 C² / + Co C2/Vn) (1 - e/2)
when N for some N that depends only on e, M and 8min- This finishes the proof with the restriction of equation 40
Turning to the general case, let 0 < 1/2 and M 2 1. We take
dg min (phoossuuppp (c, M), o
poosstrap ( (c/2,2M)
We claim that ón satisfies the theorem statement. Let 8min (Pi, ... .Pn) E (e, - =]" and M S ",,,".,, S M. Take
(0, 80]. 5 E [6min; do]. p
N max maboosooossu (c, M, 8min), Noboossrrpp Xrbosssra(e/22225 (c/2,2M, 8min))-
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Under review as a conference paper at ICLR 2023
We may assume that
E a.b)o(p)))
K V (p)(p Di B9o(p,)
> 0.99
since otherwise, we can rely on ā E [8min; 50 ghousttrpp M)) to conclude.
We will construct
2M S "/, . dn S 2M and IM S dd'""" : 0 S 2M such that
En (p))
K' - d'o(p:) 1 (3o(D:)
S 0.99
(42)
E aiboo
n"
S 0.99
(43)
V x= a (p:)) 2 (10(1:)2
and
3a, = d + a. (1sisn)
(44)
Then since 8 1ōmin
00
bootstrap
(c/2.2M), we get
p' = P x(X,) $ c/2 F E b,x,) 10
and
= P x F b, 2
1=1
1=1
for n 2 N. Then we can finish the proof as follows:
F
a,X,) Se | F (xx,)ss
1=1
11
F
3, Se |
F
P[F
(d) + X)) Se F( b;X,) S
i=1
2 P[F Σ d(xi) + F Σ (x) Sel |
F Σ b:Xi)) [
2=1
i=1
"
2 P(F Σ
d(X,) S <e/2 and F
d(x,)) s6/2 | 2 b,X,) s 8)
1=1
1=1
2 p' py+
2 (1- + (11=)))=-=11
In the middle, we relied on the inequality F(V,+ < F((1) + F(Y)) where Y1 === =E X and Y2 = - a.',, X,, which can be easily checked.
Now it's enough construct al, a. that satisfy equation S C (1.-.., ,n) such that
equation 43 and equation 44
If we find
Σ (aboo(p)))
Σ
(45)
1E8
i=1
and
Σ afo(p)))
~
Σe
a (px))
(46)
2 1=1
iES
.e can let
20;
(i ()
(a;
(i S)
and
a
a,
(i St)
2a;
(i E Sc)
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which leads to:
K' E (pb.)(p,)
1= (p. V (p)(1
2 2i (pi) Dies (p)))
Dies (p:)) )² Lies aff((p))² V b
Σ 1 + (p)
(4 1
+ E ) Di=bboo01
2
1.5
1.5
2.5 K S
< 0.99
V2.5
that satisfies equation 45 and equation [46 can found, for example, by using a probabilistic argu- ment. (Define a random set and show that it satisfies the desired properties with high probability using Hoeffding's concentration inequality)
A.6 THE ERROR THEOREM
Definition 9. A function Err: R (-1,1) R will be called a good error function if satisfies the following properties:
1. For any (in,2)), (02, y2) E {-1,1},
hny 0, 020/ 5 Err(ū1, Err(i2, y2)
2. For any 81. 02 E R and [-1,1}
ù1 + 02 = Frr(fn v) # Err(iz, v)
That is,
1. Predictions that have the same sign with the target label always have lower error values than those that do not.
2. Different predictions on inputs that have the same target label result in distinguishable error values.
Theorem 2. Let Err be a good error function. Given < E < 1/2 and 0 < n < 1/2, there exists C C(c,n) > 0 such that for any S for sufficiently large n, the following holds:
For any combination of p (p1. .P.) and p' = (ph..... . in ie, 1 - cl" with 1p: - p 2 (1 < i n)), when 3 (Bo, ..... Ba)) is the minimminn linear solution of (X, Y) for
(X, Y, 7) ~ D p.p' we have
PX,YZZ>' p.p' YZ = -1 F(Err(Ý,Y)) 2 1 1 - 12 min(1, ")
C Vn
where V = + Ei= B:X,
A.7 PROOF OF ['HROREM 2
In proving Theorem we tackle the different ranges (1) n. 02)0 separately. In all cases, we need the following facts:
When
E C=P <
we have
< < exp(-Cin)
(47)
for some C = Ci().
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Proof.
Use Hoeffding's s inequality in conjunction with equation 1 and equation 10 in the
same manner as in equation 31 and equation 32
When
70 P|YY S 0]
we have
No - n/ S
(48)
Proof.
|n0 1/ = & - =
s SP((Y SOAYZ < -1)] (symmetric difference)
> PIÝ2 s 0] (logical implication)
When
= AC(EET(TYY)))) Y)) (See Lemma
have
22 -
(49)
n
for some C2 = C2(e).
Proof.
7 maxP(EErrY .Y) = ø]
zER
max((PFrr(F)) ( x) P[Err((, 1) = x))
DER
S AC(Err(( 0)) AC(Err(Y 1))
I AC(Ý)+ + (Since Err is "good")
2AC Σ &x,)
C2 (equation T0
Lemma
A.7.1
THE CASE 8 >
Assume >> From equation 48
equation
47 and equation
49
we see that
0 2 +
(50)
when n is sufficiently large.
First, we argue that
Claim: PY $ 50 F(Err(f (EE)(( 2))200 2 1-
(51)
Suppose otherwise. Then it happens with nonzero probability that YY s and F(Err(Ý, Y)) 1 - 0. That is,
(1), Diy SOA F(Err((i, v1)) < 0.
(52)
On the other hand, we have
P[F(Err(& 21)2111111 1- ( |[ Y)) 2 0 170
0]
- - nu (by Lemma
(by equation
50
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Since the event F(Err(Y, Y)) 2 11-8-1 AY 0 occurs with nonzero probability, we have
(Ú2. v2). D292 > 0 A F(Err(12. 2 1
(53)
From equation 52 and equation 53
since D1 I d2y/ > but Err v1) < Err Err(02. V2). find a contradiction with the definition of "good error function".
we
Now, we have
PIYZ -1F((EEMM 21
-1 - Y)) 2 0]
P[F Y) 2 1-8
s 0]]
(by equation
P[F(Err(YY))) Y)) 2 1--1] 8]]
P|YZ = -1 PY <0]]
(by Lemma
0
P|YZ -1 A 87 2 0]]
P(YZ = -1] - PÍZ z < O] 5
7 1 10 0
IV
exp(-C1r) (by equation
A.7.2
THE CASE A <1
Assume < From equation 48 and equation 47 we see that
1J0
(54)
when n is sufficiently large.
First, we argue that
Claim: F(Err(Y, Y)) 8 YY 50
(55)
Suppose otherwise. YY > O. That is,
Then it happens with nonzero probability that F(Err(Ý, Y)) 2 1 0 and
3(11. v1). > 0
(56)
On the other hand, we have
P[F(ErT(Y, ( < P ([F(E Y)) 2 1-
nu (by Lemmal
(by equation 54
Since the event F(Err(Y, Y)) < 0 A 0 happens with a nonzero probability, we have
3(02, 2212 I A F(Err((2.... v2)) < 0.
(57)
From equation 56 and equation 57 since 91v1 > 0, J2y/ I 0 but Err 82) < Err(t v1), we find contradiction with the definition of "good error function".
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Now, we have
-VZ==)PFEC(()))21-0 PLYZ -
FYZ==IAFFEM((.V) 21
PYZ
P[F(ErT(Y, 21-0]
(equation 55
P[F(Err(Y, Y)) 21-81 8]
2 1 0 y
(by Lemma
2 0/2 exp(-Cin) (by equation 47
equation 49)
when is sufficiently large.
A.7.3 THE CASE f 1)
The remaining case is when 0 = In this case, we rely approximation to the case A < DJ (Note that the proof for the previous case f < 17 relied on that no when is sufficiently large.).
Let
= minn(n (1 exp(-n))mo)-
We get
P - -1F(EE Y)) P) 1-
2 P|YZ= - Y)) 2 1 - -m] PIP(Er((Y,Y)) Y)) 1-mh m]
P[F((Er(YY, Y)) 2 2--
2 (1 exp(-Cin)) ( Y)) 2 m
7h
([F(Err(Y) Y)) 2 1 nl
(Applying the lower bound from the previous case)
exp(-Cin))
"h 7 Lemma
71
4 2 (1 exp(p (-Cn)) (1 exp( n) -7
1)
(1-eep(-n)(n-eepp(CC exp(-C1n)) Ca v8
(1 - exp(-n)) - -xpp))))nee(((--eee
4
2 (1 exp( exp(-C(nn)
7)
"
(equation 48
equation
equation 49
21 2C2/1
Vn
when sufficiently large.
B EXPERIMENTAL DETAILS
B. SYNTHETIC DATASET
Dataset details The 2-D synthetic classification dataset consists the majority groups (can be classified with the spurious features) and the minority groups (cannot be classified with the spuri- ous features). The training set consists of 1,000 majority group samples and five minority group samples. Specifically, the majority group sample is sampled from the multivariate Gaussian dis- tributions ([5.00 5.0], 1.3I) and ([[[5.0, -5.0). 1.3I) for positive and negative classes, respec- tively. On the contrary, the minority group sample is sampled from the A(550, -5.0), 1.3I) and
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VV[-5.0,5.0), 1.3I). Notably, before we feed the training samples into the model, the second fea- ture (invariant feature) is manually scaled down 1/10 times. This downscaling leads to relatively insignificant gradients for the invariant features. Thus, the invariant feature is hard to learn.
Model details We use the simplest neural network architecture for this experiment: the fully con- nected neural network with a single hidden layer with ReLU activation. The number of the hidden neuron is 50,000. For the training, we SGD optimization for all evaluated models. For the ERM, we use le-4 as the learning rate; le-5 as the weight decay. For the END's identification model, we use 5e-4 as the learning rates; le-5 as the weight decay; 1e-3 the confidence regularization; le-2 as the p-value threshold. The SGLD weight sample is saved every 3,000 iterations. For the JTT and END's debiased model, we use le-5 as the learning rate; 5e-4 as the weight decay for both the identification model and final model. The hyperparameters for JTT and END's s debiased model are determined based on the argument from Sagawa et al. (2019) Liu et al. (202]): the importance weighting approaches should use relatively lower learning rates and higher weight decays. Every evaluated method trained 100,000 iterations with batch-size 32, excepts for the identification
model of END and JTT: 30,000 iterations. Additionally, we add the additional datasets (error set for JTT, SCF set for END) 30 times.
B.2 GROUP ROBUSTNESS BENCHMARK DATASET
Dataset details We use the two image classification datasets, the CelebA and Waterbirds. The two datasets have been used to evalulate the group robustness (worst-group accuracy). Each dataset has the group information (attributes, labels). And the attributes is the spurious-cue, For instance, the target class of the CelebA dataset is hair color, but the model can predict the class by abusing the spurious cue "gender". This abuse leads to the degradation of accuracy for certain group. If the model has groupp-obustness, this degradation insignificant.
Here, we elucidate the input features, the group information, and the target classes of the dataset:
Waterbirds (Wah et al. 2011): The input data is the image of the birds and their back- grounds. The target classes are either "waterbirds" or "landbirds". Notably, the dataset has the group information (background images, labels): "waterbirds with (water/land) back- ground" or "landbirds (water/land) background". In the training dataset, most of the wa- terbird images have the water background, and vice-versa. Here, only 5% of the training dataset has a contradictory background (e.g., landbird on the water background). We con- sider 4 groups: [waterbird, water background], [landbird, water background), [waterbird, land background], and [landbird, land background].
CelebA (Liu et al. 2015) The input data is the image of the celebrities. Similar to the Sagawa et al. (2019) and Liu et al. (202I). the target class is the hair color, "blond" or "not blond". Here, the spurious attributes are the "male" or "female." The label and spurious attributes are spuriously correlated like in the Waterbird case (e.g., most of the "male" group has the "not blond" label). We consider 4 groups: [blond, female), [not blond, female]. [blond, male], and [not blond, male].
Moreover, we add symmetric noise to the two datasets. In practice, we randomly change the target labels of each sample with probabilities of [0, 10%, 20%. 30%].
Model details In our experiments on the benchmark datasets, we follow the experimental setup of Liu et al. (202T) for LiF, ERM, and JTT. Specifically, we use the same model architecture for all evaluated methods: ResNer50 (He et al.] 2016)), pre-trained with the ImageNet. For the base- line approaches' hyperparameters, we follow the prior studies (Nam et al., 2020 Liu et al. 2021). Moreover, similar to Liu et al. (202I). the model selection is based on the validation worst-group accuracy.
For the Waterbirds dataset, all evaluated models are trained with up to 300 epochs and batch-size 64, except for the identification model of END and JTT, 50 epochs. The SGD optimizers with the 0.9 momentum are used, except for the LfF. The model selection is based on the worst-group accuracy
on the validation dataset for all methods. The ERM uses 1c-3 as the learning rate, le-4 as the L.2 regularization; The JTT and END's debiased model uses le-5 as the learning rates, 1.0 the I.2
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regularization, and adding the additional dataset for JTT (error set) 50 times and for END (SCF set) 100 times; The LfF uses the Adam optimizer, with le-4 learning rates and le-4 L2 regularization. The of the LfF is le-3; END uses 1e-3 as the learning rate, the lc-4 the L2 regularization, and 3e-1 as the confidence regularization for the identification model. We save the SGLD weight samples every 5 epochs.
For the CelebA dataset, the models are trained with up to 50 epochs and batch-size 64, except for the identification of END and JTT, 5 and epochs respectively. Similar the Waterbirds, the SGD optimizers with the 0.9 momentum are used, except for the LfF. The model selection is based on the worst-group accuracy on the validation dataset for all methods. The ERM uses lc-4 as the learning rate, le-4 as the L2 regularization; The JTT uses le-5 as the learning rates, le-1 as the L2 regularization, and adding the additional dataset (error set of JTT, SCF set END) 50 times; The L.F uses the Adam optimizer, with le-4 learning rates and le-4 1.2 regularization. The of the LfF le-3; END uses 5e-2 as the learning rate, the le-4 as the 1,2 regularization, and 1e-3 as the confidence regularization for the identification model. We save the SGLD weight samples every epochs.
B.3 DRUG-TARGET AFFINITY REGRESSION DATASET
DTA task The Drug-Targed Affinity (DTA) regression task is significant task for early-stage drug discov- cry. We use two well-known benchmark datasets: Davis (Davis et al., 2011) and KIBA (Tang al., 2014). The inputs of the datasets are the Simplified Molecular Input Line-entry System (SMILES) sequence-the sequence of the drug molecule-and the amino acid sequence-the sequence of the target protein. Thus, similar to other sequence-basee deep learning tasks, the input data are the one-hot encoded sequences. The target value is real- valued drug-target affinity. The dataset has 5 different folds, which suggested (Öztürk et al. 2018). Thus, the results in Table 2 are average and standard deviation over trials.
SMILES maleccla CUS-OINEEN 110023 HOO DOC7
Erbbdding
Trn Tmbeedlling g
Convolution layors
Convol ution layers
Concatenated features
Fully-connected layers
6. The Theliiiiyy
Figure 5: The schema of the DTA task with DeepDTTA architecture.
Davis dataset consists of clinically relevant kinase inhibitor ligands and their affinity values-dissociation constant K2. Oztürk et al. (2018) does, we rescale the target affinity value: log K2. The Davis dataset consists 68 drugs and 442 target protein sequences, 1e total of 30,056 affinity values. The additional detail of the Davis dataset is that the target affinity value is 5 if the affinity is lower than 5 (Davis et al. 2011). This could be a noisy label that does not represent the true affinity value.
KIBA dataset includes kinase protein sequences and SMILES molecule sequences, similar to the Davis. The KIBA dataset uses its own affinity score, the KIBA score. The dataset has 2,111 compounds and 229 proteins, total of 118,254 affinity values.
DeepDTA model We use the well-known simple, but effective DeepDTTA architecture et al. (2018). The DeepDTA consists of the one-dimensional convolution layers encode sequences and the fully-connected layers to output the affinity value from the concatenated latent features. Figure Fillustrates the DeepDTA architecture.
Details For the evaluated model, we use le-3 as the learning rate; 1c-4 as the weight decay; 0. as the dropout probability: 256 the batch size, except for the le-1 learning rate for the identifica- tion model. We train the models with 200 epochs. For the identification model, we train the model with 200 and 100 epochs for the Davis and KIBA, respectively. We save the SGLD samples every 20 and 10 epochs for the Davis and Kiba, respectively. The bascline "hard" pick up the top-500 high-loss samples for the oversampling.
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Table 3: Worst-group accuracy (WG ACC) and average accuracy (ACC) of END with different loss functions and the regularization.
Waterbirds 30 % noise
CelebA 30% noise
AVG ACC
WG ACC
AVG ACC
WG ACC
JTT
0.012 (0.00)
0.106 (0.01)
0.151 (0.16)
0.258 (0.12)
END (w/o both)
0.787 (0.04)
0.716 (0.04)
0.698 (0.03)
0.611 (0.06)
END (w/o reg)
0.888 (0.02)
0.570 (0.02)
0.901 (0.01)
0.757 (0.06)
END (w/o MAE)
0.932 (0.01)
0.459 (0.08)
0.725 (0.04)
0.618 (0.03)
END
0.854 (0.01)
0.818 (0.03)
0.892 (0.01)
0.778 (0.03)
Spurious feature
(b)
Spurious feature
(c)
Entropy density
Gamma Pdf
Density
p-value threhold
Training set
samples
Invariant feature
Invariant feature
Entropy
Figure 6: 2-D classification results of the identification model on synthetic data. (a) The prediction results. The colors represents the classes. The dots and translucent stars represents training and test data respectively: (b) the histogram of the predictive entropy and the corresponding Gamma distribution; (c) the detected SCF samples.
C ADDITIONAL EXPERIMENTS
C.I ABLATION STUDY
In this subsection, we conduct the experiments on the CelebA and Waterbirds with 30% label noise verify contributions of END's components (Table 3). Specifically, we train the models via the END framework with its variants: training the identification model without the MAE loss or confi- dence regularization, both.
In this experiment, we have several implications. Firstly, utilizing uncertainty is major factor in improving the worst-case group accuracy compared exploiting the error (loss). In particular, com- pared to JTT, every END variant improves the worst group accuracy by at least 0.4. This supports one of our main claims: utilizing uncertainty is a proper approach to improving the group robustness under the noisy label scenarios. Secondly, the cooperation between the noisy-label robust loss and the overeonfidence regularization significantly contributes to the group robustness in the presence of the label noise. In practice, the combination of both shows the outstanding worst group accuracy with noisy labels, as shown Table and 3 In contrast, without the overconfidence regularization (END w/o reg), only utilizing the noisy robust loss (MAB) does not show outstanding improvement compared to the proposed method (END). We interpret this degradation is due to the overconfi- dent uncertainty for the minority group, Model-B Figure 2 noisy label robust loss (END w/o reg and MAE), the worst group accuracy is degraded since the identification model memorizes the noisy labels, as stated in the last two paragraphs of See 4 Hence, we conclude that the cooperation between the noisy robust MAE loss and the overconfidence regularization is the key component the classification tasks, as we argued.
shown.
addition, without the
C.2 QUALITATIVE EXPERIMENT
We conduct qualitative experiment on the waterbird dataset to evaluate the characteristic of the SCF set. Specifically, we obtain two separate groups of images: (1) the most uncertain images that their uncertainty is estimated by identification model (Figure 7): (2) randomly chosen images (Figure
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Waterbird, water backsround Waterbird, land background
background
Waterbird, backaround
Waterbird, background Waterbird, backgraund
Waterbird, land background
Waterbird, background
Water water background
Waterbird, lani bat karkguound
Waterbid,, land bar kyround Waterbird, water beckground
Figure 7: The images of the waterbirds dataset with high- predictive uncertainty (Top- 12).
We observed several abnormal properties of the uncertain (SCF) images. For instance, the uncertain image has "land background" the group label but has water background vice versa. In addition, several images have ambiguous backgrounds: hard to determine whether the water or land background is. contrast, the randomly chosen images have relatively certain backgrounds. We believe that the ambiguous backgrounds of the SCF dataset allow that model focuses the image of the bird the true causality of the label.
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Waterblind Water background
Waterbird, water background nd
Wate Ward, background
Landbird, land background
Landbird, land beckground Waterbird, water background
Landbird, land beckground
Landbird, land
Landbird, land background
Landbird, land background Waterbird, background
Waterbird, water background
Figure 8: Randomly chosen images the waterbird dataset.
36


Reviews:
[{"review_id": "ysBa04ntvfP", "reviewer": "Reviewer_YCzi", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper aims to improve the worst-case performance in the present of label noise. Because typical high-loss-based approaches for atypical groups of the data may fail under this setting, the paper introduces an uncertainty-based approach to identify spurious-cue-free (SCF) samples, and then trains a model with the SCF set oversampled dataset. This paper tries to theoretically prove the uncertainty is a proper indicator of the SCF samples, and experiments show effectiveness.", "summary_of_the_review": "I am not sure whether the proposed method is well supported. If the concerns are properly addressed I am willing to increase my score.", "strength_and_weaknesses": "Strengths:\n1. The work focuses on addressing an important and challenging problem. The motivation behind the design choices is clear. All the related works are properly addressed.\n2. Very simple method, which can be useful for practitioners.\n\nWeaknesses and questions:\n1. This work proves the predictive uncertainty is sufficient to identify SCF samples with label noise by assuming a very specific setting, and that the proof only considers the binary classification problem, while it remains unclear, even intuitively, how to extend it to true multi-class classification case.\n2. One key to the problem is to distinguish between SCF samples and noise samples. Assume that the proof of the paper is correct and general that samples with high uncertainty are SCF samples, while samples with high uncertainty/large losses are also commonly considered to be noisy samples in noisy label learning. My question is what is the relationship between high entropy and large loss?\n3. Some of the details about theoretical formalism are not clear to me. Equation 4 is difficult to understand and needs further explanation, and why beta in Theorem 1 is the solution of the regression? \n4. For preventing overconfidency of networks, Section 5.1 proposes to use a regularizer to encourage a smaller predictive entropy, but the predictive entropy is also an indicator to discriminate SCF samples, so the reliable entropy is also desirable. How to trade off these two points? \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Tz8EUY9bR8S", "reviewer": "Reviewer_ptnQ", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "In this paper, the authors propose to leverage the predictive uncertainty to build an identification model, thereby discovering SCF samples and further improving the group robustness under the noisy label environment. They first demonstrate some theoretical justifications about how SCF samples can be obtained through computing predictive uncertainty. To eliminate the drawback of simply using ERM, the authors further propose the END framework. To identify the SCF samples, they train a Bayesian neural network based on MAE and confidence regularization. The debiased model can then be trained by considering both SCF samples and the original dataset. Experiments are conducted on a 2-d synthetic dataset, two group robustness benchmark datasets, and two DTA regression datasets. The experimental results demonstrate that the proposed framework END can obtain better WG accuracy and MSE. ", "summary_of_the_review": "In sum, I would recommend “3: reject, not good enough” because the paper is not ready for both of the framework properly using uncertainty and satisfactory experimental results.", "strength_and_weaknesses": "Strengths:\n* Practical problem settings for noisy label environments.\n* Theoretical analysis on the proposed idea of leveraging predictive uncertainty to improve group robustness.\n* Experiments on both synthetic and real-world benchmark datasets with improvements over baselines.\n\nWeaknesses:\n* The proposed method does not consider the characteristics of uncertainty like aleatoric and epistemic uncertainty. This could be important to figure out whether the uncertainty on specific predictions identifies the proper SCF samples.\n* Although the proposed framework outperforms most of the baseline methods in worst-group accuracy, it generally performs worse than baseline methods in average accuracy, especially when the noises are fewer. The authors should elaborate on this phenomenon and justify the trade-off sacrifice.\n* The paper lacks qualitative analysis. For example, the authors should demonstrate some examples of identified SCF samples.\n* The authors do not provide the material for reproducibility or mention further plans. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "NMZdr_Y5en-", "reviewer": "Reviewer_71MP", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper proposes a predictive uncertainty based method to identify spurious-cue-free samples from a training set with noisy labels. Then a new training dataset is by oversampling the SCF samples to train the debiased model. Separating minority group samples and noisy samples is indeed a trickier problem. But this paper seems to solve this problem well.", "summary_of_the_review": "Overall I think this is a good paper, but it can still be improved.", "strength_and_weaknesses": "Strengths:\nThis paper is well-written and easy to follow. The main idea and modtivation is well-presented. \nWeaknesses:\n1. I don't think some of the author's claims are accurate enough. For example, the authors argue that an advantage of the proposed model is that it can be extended to regression models. However, other methods such as GroupDRO or JTT can also be extended to regression models with simple modifications.\n2. The baseline used in the paper is much lower than that in the JTT paper. For example, on the Waterbirds dataset, the worst-case acc reported by the JTT paper is 86.7% and in this paper it is 84.6%.\n3. Some very relevant other papers have not been cited and discussed[1][2]. [1] distinguishes noisy samples in long-tail data, which is relevant to the task of this paper.\n[2] also uses uncertainty to solve the subpopulation shift problem. The way it builds uncertainty is similar to the snapshots mentioned in this paper.\n[1] Identifying Hard Noise in Long-Tailed Sample Distribution\n[2] UMIX: Improving Importance Weighting for Subpopulation Shift via Uncertainty-Aware Mixup\nMinor:\nAt the end of page ten, citations to the paper \"Enhancing the reliability of out-of-distribution image detection in neural networks\" appear twice.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "K0aCzOz7V1p", "reviewer": "Reviewer_r2Mi", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper tackles the problem of spurious correlations. Specifically, the authors argue that the assumption that high-loss samples correspond to a lack of spurious cues is not always true. To this end, the authors show theoretically that using predictive uncertainty to identify spurious-cue-free samples is a better indicator. Based on this finding, they propose Entropy based Debiasing (END) to mitigate the problem of learning spurious correlations.", "summary_of_the_review": "With the findings above, I currently give the paper a borderline reject score. ", "strength_and_weaknesses": "Strength:\n\n1. The paper is well-written and easy to follow in general.\n2. The authors provide theoretical justification for their findings. Although I did not check through the theorem-proof, the findings are interesting.\n3. The findings in Section 4 are interesting.\n\nWeaknesses and Questions:\n\n1. The comparison in Table 1 can be made more comprehensive. I suggest comparing with a few recent works such as SSA [1], and CnC [2]. Further, for CelebA, I see a large decline in the average accuracy, although the improvement in worst-group is marginal compared to JTT.\n2. Is there any table reporting the accuracies of the identification model? I also wonder if the training time of END is significantly higher than the other compared methods.\n3. It would be interesting to see the performance of END against spuriously correlated NLP datasets such as CivilComments, MultiNLI.\n\n[1] Nam, J., Kim, J., Lee, J. and Shin, J., 2022. Spread Spurious Attribute: Improving Worst-group Accuracy with Spurious Attribute Estimation. ICLR 2022.\n\n[2] Zhang, M., Sohoni, N.S., Zhang, H.R., Finn, C. and Ré, C., 2022. Correct-N-Contrast: A Contrastive Approach for Improving Robustness to Spurious Correlations. ICML 2022.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject