Take One Gram of Neural Features, Get Enhanced Group Robustness
Authors: Simon Roburin
Charles Corbière
Gilles Puy
Nicolas THOME
Mathieu Aubry
Renaud Marlet
Patrick Perez
Venue: Submitted to ICLR 2023 (2023)
Keywords: group robustness
distribution shift
spurious correlations
Gram matrices

Abstract: Predictive performance of machine learning models trained with empirical risk minimization (ERM) can degrade considerably under distribution shifts. In particular, the presence of spurious correlations in training datasets leads ERM-trained models to display high loss when evaluated on minority groups not presenting such correlations in test sets. Extensive attempts have been made to develop methods improving worst-group robustness. However, they require group information for each training input or at least, a validation set with group labels to tune their hyperparameters, which may be expensive to get or unknown a priori. In this paper, we address the challenge of improving group robustness without group annotations during training. To this end, we propose to partition automatically the training dataset into groups based on Gram matrices of features extracted from an identification model and to apply robust optimization based on these pseudo-groups. In the realistic context where no group labels are available, our experiments show that our approach not only improves group robustness over ERM but also outperforms all recent baselines.

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TAKE ONE GRAM OF NEURAL FEATURES, GET ENHANCED GROUP ROBUSTNESS
Anonymous authors Paper under double-bling review
ABSTRACT
Predictive performance of machine learning models trained with empirical risk minimization (ERM) can degrade considerably under distribution shifts. In partic- ular, the presence of spurious correlations in training datasets leads -RM-trained models to display high loss when evaluated on minority groups not presenting such correlations in test sets. Extensive attempts have been made to develop meth- ods improving worst-group robustness. However, they require group information for each training input at least, validation set with group labels to tune their hyperparameters, which may be expensive to get or unknown a priori. In this paper, address the challenge of improving group robustness without group notations during training. To this end, we propose to partition automatically the training dataset into groups based Gram matrices of features extracted from an identification model and to apply robust optimization based on these pseudo- groups. In the realistic context where no group labels are available. our experi- ments show that our approach not only improves group robustness over ERM but also outperforms all recent baselines.
INTRODUCTION
Empirical Risk Minimization (ERM) is the most standard machine learning formulation, which as- sumes that training and testing samples are independent and identically distributed (Vapnik, 1991). While academic datasets are mainly built respect this assumption, practical settings display more challenging configurations with distribution shifts. Among different types of shifts, training data can be affected by selection biases and confounding factors, also called spurious correlations (Wood- ward, 2005; Duchi et al., 2019)
Imagine growd-sourchts an image dataset of camels and cows (Beery al., 2018). Due to selection biases, large majority of cows stand in front of grass environment and camels in the desert. A sim- ple way to differentiate cows from camels would to classify the background, an undesirable short- cut that ERM will naturally exploit. Consequently, ERM may perform poorly on minority groups that do not display such spurious correlations (Hashimoto et al., 2018; Tatman, 2017; Duchi et al., 2019), e.g.,
standing in the desert. To overcome this issue, recent works (Creager al., 2021; Bao & Barzilay, 2022; Sohoni et al., 2020; Liu et al., 2021; Ahmed et al., 2021; Kirichenko et al., 2022) rely on two-stage schemes: first, automatic environment discovery (e.g.. based on deep feature clustering); then, robust optimization based on environment pseudo-labels. Environment here refers recurring setting, not intrinsic the object of interest, that may affect its classification, such as background, object color or object pose. However, all these approaches require the availability of ground -ruth environment labels on validation set to properly tune their hyperparameters.
This paper addresses the problem of learning a robust classifier, which, for instance, would not confuse a cow standing in the desert with camel although not given any annotation about grass or desert. In computer vision, many identified spurious correlations are closely related to visual aspects, such as background (Beery et al., 2018), texture (Geirhos et al., 2019), image style (Hendrycks et al., 2021), physic attributes (Liu et al., 2015) or camera characteristies (Koh et al., 2021). In this work, we assume that relevant environment labels can be inferred from visual feature statistics, and demonstrate they lead to meaningful environments and robust classifiers for standard datasets used to evaluate robust classification. We propose a two-stage approach, GRAMCLUST, which first assigns group label, i.e., class-environment pair label, by partitioning training dataset into clusters of

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K-Means clustering
Training
Neural feature Gram matrices extraction
Standard truining (ERM)
Robust training with gpeudo-groupe labels
Accuraty
Accursiy
Groups
Coos graa Oxherizso
Canels on radd Other corads
Test data
Figure I: Overview of the proposed GRAMCLUST approach for robust classification with un- supervised group discovery. (1) We first extract deep image features using an identification model and (2) we cluster the training dataset based on Gram matrices of images features; (3) Then, we train the targeted classifier with robust optimization that exploits the assigned pseudo-group labels. Consequently, GRAMCLUST properly classifies samples in minority groups, c.g. cows and camels in unusual environments in contrast to standard Empirical Risk Minimization (ERM) training.
images with similar visual statistics and then trains a robust classifier based on these pseudo-grus group labels. Our approach is summarized in Fig. 1. We use Gram matrices visual descriptive statistics, which are second-order moments of neural activation. Gram matrices are well known for displaying impressive results in style transfer techniques (Gatys et al., 2016), but more importantly for the interpretation of our approach, Li et al. (2017) demonstrate that matching Gram matrices between two groups of images is equivalent to aligning the respective distribution of each group, minimizing the Maximum Mean Discrepancy. Therefore, our method can be interpreted as grouping images into clusters of similar feature distributions that are sensible candidates for environments.
Our main contributions are as follows: (1) We introduce an easy-to-scale method to split train- ing images among distinct nseudo-environments based on feature Gram matrices extracted by specifically-tranee identification model; (2) GRAMCL.ST alleviates the need of ground-truth group labels altogether, even in the validation set, as hyperparameters are set based on validation perfor- mance computed from our pscudo-gutt pscudo-groups; groups; (3) Extensive experiments various image classifica- tion datasets with spurious correlations show that GRAMCLUST outperforms all recent baselines addressing robustness without group annotation. In particular, the realistic large-scale CelebA dataset (Liu et al., 2015), we improve worst worst-group test accuracy by +24.3 points.
2 RELATED WORK
Robustness to distribution shift (Rusak et al., 2020; Hendrycks* et al., 2020; Guirajani & Lopez- Pazz, 2021: Geirhos al., 2019) has recently been an increasingly popular topic among machine learning researchers. Koh et al. (2021) distinguish two types of distribution shifts: domain generalization, where test samples come from different distribution than training datasets, and subpopulation shift, where train and test distributions overlap but their relative proportion differs. With subpopulation shifts, the goal to perform well even on the minority group, also referred to as group robustness. In this study, focus the latter form of distribution shift.
Group robustness with group annotations. Recent approaches propose to leverage group annota- tions during training to improve group robustness. IRM (Arjovsky et al.. 2020) augments the stan- dard ERM term with invariance penalties across data from different groups. Ahmed et al. (2021) pro- mote, through simple penalty, identical prediction behaviour across groups. Other works (Sagawa
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et al., 2020a; Zhang al., 2021) minimize explicitly the worst-group loss during training. Finally, Sagawa et al. (2020b) re-balance majority and minority groups via re-weighting and sub-sampling.
Group robustness without group annotations. The focus here is more realistic setting in which group annotations are not available on the training data. Creager et al. (2021) derive a group infer- ence objective from trained identification model that maximizes variability across environments, and is differentiable w.r.t a distribution over group assignments. Liu et al. (2021) introduce a sim- ple method in which environments are defined by images on which trained identification model performs poorly. Sohoni et al. (2020) propose an unsupervised clustering algorithm in the feature space of a trained identification model. But these methods require implicitly or explicitly small validation set with ground-truth group annotation: Liu et al. (2021) explicitly tunes its hyperparam- eters on small validation set with group annotation while Creager al. (2021) and Sohoni et al. (2020) use best hyperparameters used in the GroupDRO paper (Sagawa et al., 2020a) to minimize the worst- vorst-group accuracy on their inferred groups. These best hyperparameters were actually found using a validation set with true-group labels in the original study.
Gram matrices. The original work of Gatys et al. (2016) demonstrated impressive results to gener- ate images with the style of an existing image. The style of a first image is transferred to a second one by matching Gram matrices of features extracted by a convolutional neural network. Sastry & Oore (2020) also used Gram matrices in out disttiibiitiitto detection to identify anomaly by compar- ing their values to the respective range observed over the training data. Interestingly, Li et al. (2017) demonstrate formal equivalence between matching Gram matrices of neural activations with L2 norm and the MMD with the second-order polynomial kernel. This shows that Gram matrices are also implicitly used in the process of distribution alignment between images. This finding motivates our approach, which consists in discovering pseudo-groups using Gram matrices.
3 GRAMCLUST: A CLUSTERING APPROACH FOR ROBUST OPTIMIZATION
Our method, GRAMCLUST, consists of two main steps. First, we discover gpeudo-environeen among the images of given dataset (see Section 3.2). Second, we train a robust classifier that leverages the inferred pseudo-enviromment labels to reduce classification errors due to spurious en- vironment correlations (see Section 3.3). To discover environments, we train during a few iterations an exogenous "identification model".. Then, using this model, we compute for each image its Gram matrix representation from different layers and apply random projections to reduce dimension. The resulting concatenated features are then fed to unsupervised clustering algorithm (k-means) to produce
labels. This allows us to define pseudo-groups as the intersection of pseudo-environments and classes. Last, we train the target classifier by minimizing the standard cross-entropy classification loss on the worst pseudo-group with GroupDRO (Sagawa et al., 2020a).
3.1 PROBLEM FORMULATION AND NOTATIONS
Let us consider a dataset D = {xx,, 3 E X' x y of N samples where X' is the input space and y [1, K] set of labels. We assume the data is sampled from random variables (Xe, Y.) in X' x y with probability law (x) for all e E [[1, F.. where F is the number of environments. The full dataset can then be
the union of subsets associated each random variable, i.e., D U De where each D. is composed of i.i.d. realisations of random variable with joint probability law P ()) For notation purposes, we actually choose the following equivalent formulation for the dataset D = {(,, yr. e E X x y x [1, E] where refers to the environment from which 2; and yi were sampled.
Our goal is to find model m in a given hypothesis space M that minimizes the error on the worst group. A group is defned as a set of samples both from the same class and in the same environment. Formally, we introduce group distributions:
P PG.. PP = ... P.G. = P(X = K).
(I)
The purpose is then to solve the following objective minimization problem:
arg min MEM
max
E(n.y)-aa, [((m(x).))
(2)
gE[1, [1
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fi
fta.
"", f,
Training of identification model Feature Gram matrices extraction
Projected Gram matrices
clustering
Cg
D
41
4/
Grave matris compuiation randow projection
Cross Entropy
(P(,,w)
Identification model
Figure 2: Illustration of the proposed dataset partition. An identification model 4 with parame- ters w is trained for limited number 7 of epochs with ERM to groups with easy-to-learn spurious correlations. Then, for each image pute their Gram matrix G, with random projection. These projected Gram matrix representations used as features to cluster the training dataset Duain in E" environments.
X' we extract intermediate features Φi at layer / and com-
where x'''y- R is the cross-entropy loss between the model's prediction and the true label. Note that we have access to any environment labels. To circumvent this issue, we first discover pseudo-enviromment labels, then estimate the geeudo-gropp distributions to be used in Eq. (2).
3.2 DATASET PARTITION
In this section, we describe the first stage of GRAMCLUST, which aims at environment discovery. The method is illustrated in Fig. 2.
Identification model. Our approach starts by initializing a convolutional neural network + for the classification task hand; composed of layers with parameters w and is pre-trained on ImageNet (Deng et al., 2009). Liu et al. (2021) observed that ERM tends to fit models on data presenting easy-to-learn spurious correlations the beginning of the learning process. after a significant number of epochs that the model starts to learn more difficult patterns. Hence, we only train & during a few iterations, minimizing w.r.t. the following empirical loss function:
only
N
1
El (D) W;)
N
where e 3 R+ is the cross-entropy loss between the model's predicted label ((x;, w) and the true label y associated with sample æ.
In the following, we call & the identification model as clustering is based on features extracted from this model. The idea is to leverage the biases learned by to identify relevant environments and partition the training dataset into groups of images presenting spurious correlations, hand, and groups of images free from these correlations on the other hand. Hence, after this initial training and in the rest of the paper. the parameters of the identification model & are frozen.
the one
Features Gram matrices. We denote the feature map of an image æ at layer M where C is the number of channels and M, the spatial size of the feature map. For each image 2; E X, we extract its feature maps at S & different and fixed layers {li,-.. Is). and compute the Gram matrices defined as:
&b by Φ((x) e
(x = Mi 0i (1(2;) E Raxa, LE 1l.h
(4)
Given input x; and identification model Φ the Gram matrix of its feature (x;;) encodes visual correlations via inner product between each pair of vectorized feature maps. In visual style transfer (Gatys et al., 2016), these Gram matrices have been shown to encode the "style" of an image, that is, loosely speaking, its textures and color palette, by contrast with its "structure".
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Clustering with k-means. For each image 24, vectorize and normalize its S associated Gram Rci, The normalization permits to balance the contributions of the different Gram matrices in the clustering loss. Each image æ; is thus en- coded by the vector fi .f........ fi.s E RRC where C E. Cf Relying on the assumption that environments can be inferred from visual feature statistics, we propose to discover E' environ- ments, E" being a parameter as E' is unknown, by clustering the N training images into E" clusters (Ci.... Cry) via k-means clustering, i.e., by computing a solution to:
matrices: f.; vec( (x,))/ |v
min [CI,..Cg) go)
Σ 2 Cel
Σ
IIf. - 1f.-f5112
(5)
where 1 Ei [if f311l
Scaling with random projections. Storing all these vectors and computing distances between them in a high-dimensional space is computationally and memory expensive on large datasets. We overcome this difficulty by projecting the vectors fa, in lower-dimensional space as proposed by Achlioptas (2003). Given size lo, build matrix P R&xx whose entries P Pnn the realisation of independent random variables: Pmn- = or P Pmn -1 with probability 1/2. Then compute
f.. = 1 1P VEo
(6)
and substitute fú, for ft, in Eq. (5). We justify this choice by the fact that this projection preserves the distances fi, frall involved in the k-means objective of Eq. (5). Indeed, let EJ0, 1[ and to log(N), then with high probability.
(1 (1-6) llfu foall2 & Ilft, f,oll2 & (1+6) Iff. ftall2
for all (i,j) [1. In practice, choose to [10 log([V] which yields dimensions for f(! much lower than typical values of Ct. We remark that this choice of projection is independent of all fi, and thus be defined and fixed before any feature extraction.
3.3 ROBUST OPTIMIZATION WITH PSEUDO-GROUP LABELS
Given these estimated environments, we define their intersection with classes as "pseudo-groups". Formally, given the predicted environment é, E E'], image its pseud-group label is g: (ê,, y:) [1, E"T x [1, K].
Going back Eq. (2), the distributions over the groups PG,, are estimated by for 9 E [1, FT [1, K].
PG 5(G())
where 5 the Dirac distribution, and
((1, 1) = ((2,;, [1,N] = 1, E C1}
(9)
G(E', (1) e [1, y1 = K, E C
(10)
are the sets of images and labels associated with the pscudo-group group labels.
Each training point æ; E X' is now associated with class label y, and a pscudo-grout annotation ÿ. We train a robust classifier h with parameters f by minimizing the worst-group risk on the training dataset (Sagawa et al., 2020a):
1
Ô E arg min
max GE
(((h(2,)),))}}
(11)
((
(a.9)=9(g)
where the loss é: y x the robust classifier and the true label y associated with sample
) y - + remains the cross-entropy between the predicted label l( 0) of
We let the reader refer to Theorem 1.1 in (Achlioptas, 2003) for the exact expression of this probability function of e, . and Eo-
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3.4 MODEL SELECTION VIA CROSS- VALIDATION ON VALIDATION DATA
Setting relevant hyperparameters is important in optimization algorithms to ensure proper con- vergence. Hyperparameters tuning performed with roos-validation using a held-out subset of training data. With robust optimization, worst-group accuracy of the final classifier is the go-to met- ric for model selection. Previous approaches rely on true group labels of the validation set define and assess performance the worst group. In contrast, we do not rely on such prior information. We partition the validation set using the clusters found on the training set and we conduct cross- validation based on the resulting pseudo groups. In our experiments, we observe that this type of model selection is effective to achieve proper group robustness.
4 EXPERIMENTS
In this section, we evaluate the capacity of GRAMCLUST to improve group robustness on image classification datasets with spurious correlations. In Section 4.2, we empirically show that it out- performs other baselines addressing robustness without group annotation on three datasets. We then present in Section 4.3 an empirical analysis of our approach, including the importance of using Gram matrices as visual features, the impact of the choice of layers to extract features from, and the impact of the number of clusters. The code is available with the supplementary material.
4.1 SETUP
Datasets. We experiment with three image classification datasets on which previous works evalu- ate worst-group performance. Waterbirds (Sagawa et al., 2020a) is a dataset composed of images combining bird photographs from the CUB dataset (Welinder et al.. 2010) with background scenes taken from the Places365 dataset (Zhou et al., 2018). The target labels ("landbirds") h/trodds//wwtttbbrrrl ") are spuriously correlated with the background images w"ater"" The train set is composed of 4,795 images and the validation and test set are respectively composed of 1,199 and 2,897 images. CelebA (Liu et al., 2015) a celebrity face dataset with 202,599 images. Sagawa et al. (2020a) con- sidered the task of classifying the hair color of the individual as "blond" or "not blond".. The authors observed that there exists a spurious correlation between the hair color and the gender ("male" or "female") of person. In fact, in the dataset, only 2% of blond people is a dataset of 10 segmented MS scenes from the Places365 dataset. The train set has 7,200 training images - 800 images per category and validation and test sets are composed of 900 images - 100 images per category. Unlike Waterbirds, multiple backgrounds have spurious correlations with the object classification (Ahmed et al., 2021). We rebuilt this dataset based on the code provided by Ahmed et al. (2021)² but with images resized to 224 x 224 (instead of 64 x 64 in the original paper, which considerably degrades visual features of objects and background).
male. We use the official
train-val val-tst split from Liu et al. (2015). COCO (Lin et al., 2014) objects superimposed PICO-on-Places--
Baselines. We compare our approach against the standard ERM baseline and recent methods that aim at robust predictions across groups without the use of train group annotations: EIIL (Creager et al., 2021), GEORGE (Sohoni et al., 2020), and JTT (Liu et al., 2021). We also include robust methods that use train group annotations: IRM (Arjovsky et al., 2020), importance weighting and GroupDRO (Sagawa et al., 2020a). The latter methods and ERM were already implemented and we took care to reproduce results for all methods. Our results with baselines are in line with those reported respectively in each original paper. Note that our approach and GroupDRO share the same robust optimization objective (Eq. (2)). Hence, GRAMCLUST would boil down to GroupDRO if discovered pseudo-groups were to match exactly the ones annotated in the dataset.
Training details. All methods, including ours, use ResNet-50 (He et al., 2016) architecture pre- trained on Image as the robust classifier. Models are optimized using SGD with momentum. For GroupDRO and ERM, we use the hyperparameters reported by Sagawa et al. (2020a) on Waterbirds and CelebA datasets. Further training details are available in Appendix A. Note that hyperparameters have been selected with the use of a validation set with group labels. Regarding our approach, we select VGG-19 (Simonyan & Zisserman, 2015) architecture for the identification model T. Although dating from 2015, VGG-19 is still the go-to architecture for applications involving Gram
httpp::///t thub. com F'a ruk Ahmed/ /predd c re_grrop_invarre

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Table 1: Comparative results on Waterbirds, CelebA and COCO-on-Places-224, -orst-group (W-g) and average (arg) test accuracies (% mean and std.) for Waterbirds and CelebA datasets; systematically-shitee (shift) and in-distribution (in-dis) test-set accuracies (% mean and std.) for SOCCO-on-Places-221 dataset. Experiments are with ResNet-50 models. Underlined and bold type indicate respectively best and per-block best performance (with significance p < 0.05 according to paired t-test on five runs). Methods are grouped according to their need for ground-truth group labels on train and/or val set: proposed GRAMCLUST-CV is the only one requiring none.
Group labels train
Waterbirds
CelebA
COCO-on-P
Method
val
(w-g)
(avg)
(W-g)
(avg)
(shift)
(in-dis)
ERM
v
65, 25.07
97.3+0.1
42.4
94.8=b.1
71.9
95,5+a.1
IRM (Arjovsky et al., 2020)
77.4
97.3+0.1
+5.1+0.
94,5+0.1
78.8+0.a
95.1+aa2
Importance weighting
74.4
97.4
72.4
94.4
71.720.5
93.7
GroupDRO (Sagawa al., 2020a)
83.9 v
96.8t..
85.7.ta
93.7
79.0
95.2102
FIIL (Creager et al., 2021)
78.7
96.910.1
68.5
94.8103
GEORGE (Sohoni et al., 2020)
76.2+2.0
95.7+a.s
53.7+1.a
94.6+0.2
71.6+0.3
95.1+aa
JTT' (Liu et al., 2021)
82,92n.a
96.420.
56.02n.7
93.6gne
69. 2
94.72ma
GRAMCLUST-orig (Ours)
/
85.3 11.1
96.610.1
77.9
94.2
72.4
95.01u.2
GRAMCLUST-cv (Ours)
85.3+1.1
96.6+0.1
80.3+1.
93.4+0.1
73.2+0.a
95.3+0.3
matrices such as image style transfer (Zhang et al., 2022; Höllein et al., 2022; Xie et al., 2022).
Results of our approach include two types of model selection via cross-validation (i) based on validation set with true-group annotations (GRAMCI.UST-orig)) and (ii) based on pseudo-group labels (GRAMCLuST-cV) predicted by our clustering (see Section 3.4).
Metrics. We report worst-group and average test accuracy for Waterbirds and CelebA datasets. On COCO-on-Places-2244 we follow the evaluation protocol proposed by Ahmed et al. (2021) and report predictive performance on the in-distribution test set, which follows the same distribution as the training set, and on the systemaatically-shitted test set, where the spurious correlations have been removed and COCO objects are composed with mniformly-sampler random backgrounds.
4.2 COMPARATIVE RESULTS
We report quantitative comparisons Waterbirds, CelebA and COCO-on-Places-- in Table 1. We observe that GRAMCLUST improves worst-group test accuracy over ERM baseline on Water- birds and CelebA and systematic generalisation on COCC-on-Placce2224 More importantly, GRAM- CLUST-CV achieves state-of-the-
performance on group robustness compared to all methods that do not use group labels on the training set. This results show empirically that proposed approach, using Gram matrices of feature to discover pseudo-grooe groups, which are then used for robust optimiza- tion and hyperparameter cross-validation, is very effective for group robustness. It also supports that Gram matrices are well suited to capture various types of dataset biases (background for Water- birds, physical attribute in CelebA, multiple backgrounds in -OCCO-on-Places-2 224). For instance, on Waterbirds, GRAMCLUST-cr achieves 85.3% worst-group accuracy compared to the second- best method, JTT, which reaches 82.9%. The gap even more pronounced on CelebA where our approach outperforms JTT by 24.3 pts. CelebA constitutes an interesting dataset to evaluate the scalability of methods as the training dataset composed of 200k images. For instance, we were not able to scale EIIL on this dataset. Note that GRAMCLUST-ore uses the same hyperparameters as EIIL, GEORGE and JTT for robust training of the target classifier from predicted group labels, and still displays significant improvements Liu et al. (2021) reported results that were obtained with early stopping thanks to a small validation set annotated with group labels. The authors selected models before convergence (around epoch 3) with low average accuracy on the test set but high worst-group accuracy. We argue that it is not a suitable property for a model and prefer models with high accuracy both in average and on the worst test set. Surprisingly, GRAMCLUST-CV and GRAMCLUST-oris outperform GroupDRO on Waterbirds with 85.3% vs. 83.9%, while the latter method uses true-group labels during train- ing. Our intuition is that it may be due to the ambiguity of the background in some Waterbirds
the three datasets in terms of group accuracy.
group of the
'Rssults with JTT differ from the original paper as the scores that we report correspond to models trained without early-stopping.
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Table 2: Study of the clustering features. Results in worst-group (Waterbirds, CelebA) and systematically-shnnn (COCO-on-P) test-set accuracies (%). Gram matrices show to be the most effective type of information obtain improved group robustness.
Visual features
Architecture
Layer
Waterbirds
CelebA
COCO-on-P
Standard
ResNet-50
ArgPool
76.2+2.0
53.7
71.6+0.8
Mean Var
VGG-19
Comv5 I
85.3g12
69.8
71.420.
Gram matrix
VGG-19
Comv5_1
85.3a1.1
77.9
72.4
Can
155
0994
Figure 3: Impact of the layer choice to extract fea- tures. Results in matching accuracy the validation set for GRAMCLUST on Waterbirds.
Figure 4: Impact of the number of clus- ters. Results in worst-grn gorst val accura- cies of GRAMCLUST on Waterbirds.
images. We further discuss this result in Section 4.5. Overall, these results show that our pseudo- groups the validation set are relevant to select good hyperparameters and more importantly, that GRAMCLUST does not require any group labels during training to achieve group robustness.
4.3 STUDY OF THE CLUSTERING FEATURES
In this section, we compare the performance obtained when clustering images with different visual features. In neural style transfer, Huang & Belongie (2017) proposed the channel-wise mean and variance of image features, instead of Gram matrices as in (Gatys et al., 2016). We thus compare the use of such features ('MeanVar") against our use Gram matrices. We also compared our use of VGG-19 features with the direct use of the penultimate representation of a ResNet-50 identification model ('AvgPool"). Recall that although our features for group identification are extracted using a VGG-19, our robust classifier is a ResNet-50. One may wonder if using directly the deepest features before the classification head in ResNet-50 ('Standard")) could be better than using VGG- 19 features. For a fair comparison, we trained the robust classifier with the same hyperparameters for each method, which are consistent with those found by Sagawa et al. (2020a) with GroupDRO.
The results are available in Table 2 for Waterbirds, CelebA and COCO-on- OOCO-on-Places-222 penultimate layer of a ResNet-50 as visual features for the clustering produces poorer performance than Gram matrices of VGG-19 19 features in every configuration. Mean reaches test worst-group accuracy on-par with Gram matrix on Waterbirds but degrades significantly performance on CelebA: 69.8% in average compared 77.9% with Gram matrix. Gram matrices provide more information than Mean Var as their diagonals already contain the information about the channel-wise mean and variance of the deep features (see Eq. (4)). This show that when scaling large datasets such as CelebA, keeping all the correlations between different channels is important for group robustness.
Using the
4.4 CLUSTERING ANALYSIS
Effeet of the selected layers for features. We evaluate the impact of the selection of VGG-19 layers to extract the features in the clustering stage. To this end, we study the matching of the pre- dicted environments to the true environment labels on the validation set. The assignment problem is solved via Hungarian matching (Kuhn, 1955) and we measure the global matching accuracy across all validation samples, where matching accuracy is the percentage of samples whose predicted group corresponds to its true group. In Fig. 3, we compare results on Waterbirds using either one of the five layers commonly used in neural style transfer (convl_l, cony2.1, conv3_l, comv4_1, conv5_1 or using all layers together. Experiments show that: (i) Features from deeper layers correlate with better matching accuracy; (ii) Our approach is robust the choice of deep layers either taken to-
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(a) True land background predicted as water background by GRAMCLUST
(b) True water background predicted as land background by GRAMCLUST
Figure 5: Example of confusing samples in Waterbirds dataset, wrongly predicted by GRAM- CLUST. (a) Samples of confusing band-background images predicted as water background; (b) Samples confusing land- backkgudd images predicted as water background. In each case, the ac- tual image background is confusing due the joint presence of elements reflecting land background (forest, heavy vegetation, sand) and water background (water surface, rainfalls, mist).
gether (allconvX_I) /) or individually such conv4_/ and conv5_/; (iii) Using conv5_/ outperforms selecting all traditional style layers and results in the highest score of 93. 43.41%. We found consistent conclusions on the CelebA dataset. Results are reported in Appendix B.
Impact of the number of clusters. We study the impact of the number of clusters as hyperparameeer in the clustering algorithm. Worst-group accuracy on the validation set for E' E {2. 4, 8, 16, 32) clusters are reported in Fig. 4 for Waterbirds datasets. Overall, our method is robust to variation the number of clusters: GRAMCLUST with higher numbers of clusters produces slight drop in performance but still outperforms ERM. It also has on-par performance with GroupDRO.
4.5 DISCUSSION ABOUT RESULTS OF GRAMCLUST vS. GROUPDRO ON WATERBIRDS
Comparative results in Table 1 show that GRAMCLUST-orig outperforms GroupDRO on the Wa- terbirds dataset. The difference between the approaches lies in the usage of true-group labels on the training dataset for GroupDRO while GR AMCLUUSTTorii leverages its predicted pseudo-groups. This results might be surprising given that the evaluation performed on true test group labels and that the two methods share the same robust optimization algorithm and hyperparameters. We intuit that this behavior, which occurs only on Waterbirds, is related the group labels in the dataset. In Fig. 5, we show some examples of confusing images that were not correctly assigned with our predicted group labels with GRAMCLUST. These images are taken from the set of mismatches be- tween true-group labels and our pseudo-group labels after the Hungarian matching. We can that some of these samples present dominant characteristic elements from land background, such heavy vegetation and sand, while being labeled as water background. Conversely, some samples labeled as land background display a high percentage of water surfaces in the image. As mentioned in Section 4.1, the Waterbird dataset created by combining bird photographs with background scenes taken from the Places365 dataset. But the latter dataset is composed very diverse images which might not reflect the expected background for a category. This unwanted behavior outlines the difficulty of manually annotating groups and raises the need for creating benchmarks including datasets with spurious correlations from non-artificial, weal-wrrdd data, such hair color/gender bias observed in CelebA. It also motivates further research on the automatic discovery of groups in data, as proposed in our method.
CONCLUSION
In this paper, we introduce GRAMCLUST, two-stage method that first partitions a training dataset into clusters via k-means clustering based on Gram matrices computed from image features, which are extracted from a identification model trained to catch spurious correlations in a biased dataset. This first stage is then followed by learning robust classifier which minimizes the error on the worst pseudo-group labels previously discovered. GRAMCLUST demonstrates to be an effective approach to tackle group robustness and outperforms every baseline standard datasets with spurious corre- lations. The usage of Gram matrices of features crucial to capture pertinent visual statistics of the image and enables a relevant partition for robust training. Our approach also alleviates the need to label a validation set of images with group information and able to tune its hyperparameters in an unsupervised fashion by applying its clustering algorithm the validation set.

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A IMPLEMENTATION DETAILS
This section focuses on implementation details used to produce the results in the main text of our paper. The code that we used provided along with this appendix. Our implementation builds upon the WILDS framework released wby Koh et al. (2021).
A.1 CONSTRUCTION COCO-ON-PLACES:
We generated the dataset using the code² of Ahmed et al. (2021) but, as explained in the main paper, we modified it to produce images of size 224 x 224 instead of x 64. The reader can refer to the appendix of (Ahmed et al., 2021) for more details regarding the generation of the COCO-on-Places, dataset.
A.2 DETAILS ABOUT ROBUST OPTIMIZATION
We trained all models on one NVIDIA @ V100 Tensor Core with 16GB of memory, using Py PyTorch 1.10 and CUDA 10.2.
We used the implementations of IRM (Arjovsky et al., 2020), Importance Weighting and GroupDRO (Sagawa al., 2020a) available in WILDS (Koh al., 2021), et al., 2021) 1) and GEORGE (Sohoni et al., 2020) (while making sure that we could reproduce the original performance on Waterbirds and CelebA), and the official implementation" of EIIL (Creager et al., 2021). Concerning EIIL, we recall that we were not able to make this method scale to large datasets such CelebA.
implementations of JTT (Liu
For all methods, we used a ResNet-50 (He et al., 2016) architecture trained using stochastic gradient descent with momentum (SGD-M) and L2 regularization, but without any learning rate scheduler. We used a momentum of 0.9 and a batch size of 128 for all datasets and all methods. The learning rate 17 and L2 regularization parameters À are set as detailed below.
JTT, GEORGE, EIIL, GRAMCLUST all use GroupDRO (Sagawa et al., 2020a) as robust optimiza- tion step. On Waterbirds and CelebA, we did not redo any grid search and used the hyperparameters found in (Sagawa et al., 2020a). These hyperparameters were optimized using a small validation set annotated with true group labels. To produce the results on grid search using the annotated validation set. We considered values of 1 and À close to those used in (Sagawa et al., 2020a): The best hyperparameters for GroupDRO summarized in Table 3.
Ce----on-Paacc-2221 -244, we performed
{10- 10 10-2, 2, 10 1, 1} and 17 E {10-5 5 10 -5 10 10-4).
To ensure fair comparisons, we also performed the same grid search over and À for ERM, IRM and Importance Weighting. The best hyperparameters for ERM and IRM are summarized for each dataset in Table 4 and Table 5, respectively. Note that they correspond to those reported in (Sagawa al., 2020a) for Waterbirds and CelebA.
Table 3: SGD-M hyperparameters for GroupDRO training.
SGD-M hyperparameters
Waterbirds
CelebA
COCO-on CiCC--on-PPacss--- 224
Learning rate
10-5
10-5
5 10-5
Le regularization À
1.0
0.1
10
A.3 GROUP DISCOVERY DETAILS
For GRAMCLUST, we follow standard practice of neural style transfer (Gatys et al., 2016) and use the VGG-19 (Simonyan & Zisserman, 2015) architecture for the identification model. This model trained during epoch on the training dataset with ERM using a batch size of 128 and SGD-M.
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Table 4: SGD-M hyperparameters for ERM training.
SGD-M hyperparameters
Waterbirds
CelebA
COCO-on Pioo-elaccs--- 224
Learning rate
10-4
10
10-4
L2 regularization À
10
10 44
10
Table 5: SGD-M hyperparameters for IRM training.
SGD-M hyperparameters
Waterbirds
CelebA
PiC0--on-Placss-
Learning rate !)
10
10-5
5 10
L2 regularization À
10
0.1
0.1
Among usual layers used to compute representations in neural style transfer, observed improved performance by selecting deeper layers in the network (see Section 4.4). Consequently, for each dataset, we consistently extract features from the conv5_/ layer, i.e., the first convolutional layer of block 5. Following results of Fig, 4, we set the number of clusters to 2 in our experiments.
For EIIL and GEORGE, the identification model is ResNet-50 used in the original methods. We train the model for epoch with ERM using SGD-M, as for GRAMCLUST. Note that the activation at the output the last layer a sigmoid in EIIL while it is softmax in GEORGE. As for GRAMCLUST, the best results were obtained when using 2 clusters for EllL and GEORGE. We refer the reader (Creager et al., 2021) and (Sohoni al., 2020) for other implementation details specific to EIIL and GEORGE, respectively.
A.4 CROSS VALIDATION ON PSEUDO-GROUS ANNOTATIONS
We report in Table 6 the results of our grid search on the validation set of each dataset using the psendo-annotations discovered with our method, i.e. using our discovered environments instead of ground-truth ones. Hence, the average and worst group accuracies Table 6 are computed using the discovered pseudo-groups The hyperparameters used in GRAMCLUST-cv correspond to those which yield the best worst-group accuracy in this table.
Table 6: Grid search results on the validation sets of Waterbirds, CelebA and COCO-on- Places-224 with pseudo-group labels. We report the worst-group ('w-g') and average ('avg') accu- racies for Waterbirds and CelebA datasets, and the systematically-shiteee ('shift') and in-distribution ('ind') accuracies for COCO-on-Placee ('COCO-on-P') dataset.
Hyperparam.
Waterbirds w-g avg
CelebA w-g
COCO-on-P ind
Method
X
7
avg
sys
GRAMCLUST-eV
0.01 1.10-
74.6
82.4
86.0
93.2
62.8
92.3
0.01 5.10-5
69.2
79.9
53.5
94.6
70.7
76.5
0.01 1.10-4
70.0
80.6
78.5
82.7
0.1 1.10-5
75.4
82.6
85.6
93.7
78.7
83.3
0.1 5.10-5
73.8
82.4
85.0
89.1
70.4
76.4
0.1 1.10-4
76.9
85.8
76.2
81.2
1.10-5
80.8
86.4
65.5
72.6
5.10-5
0.0
23.1
0.1
11.1
1.
0.0
23.1
0.2
11.1
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B CLUSTERING ANALYSIS ON CELEBA
We present, Figure 6, the matching accuracy between the ground-truth environments and the environments discovered with our method on the validation of CelebA for different layers of the VGG-19. As on Waterbirds, we notice that the best result is obtained when using the layer conv5_1.
Matching Accuracy
75
70
65
60
55
convl_ 1_
conv2_1 1
conv3_1
conv4_
conv5_ _
allconvX 1
Figure 6: Impact of the layer choice to extract features on CelebA. We show the matching accu- racy between the ground-truth environments the validation set CelebA and the discovered ones with GRAMCLUST when using different VGG-19 layers. The result denoted allcomxX_ obtained when using all the layers convl_l, comv2_1, conv3 conv4_,, conv5 our method.
15


Reviews:
[{"review_id": "u2SGrFAQxM6", "reviewer": "Reviewer_4Q7E", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "One of the challenges in training over-parameterized neural networks is that they may overfit to \"spurious features\", resulting in poor generalization performance on minority subgroups. A popular approach to alleviate such biases is the use of group distributionally robust optimization (DRO). However, prior DRO approaches required group labels to be available either in the training or validation set. This paper proposes a method to automatically discover the underlying subgroups, targeting practical use cases where the underlying spurious correlations are unknown. \n\nThe proposed method first trains an embedding model, and clusters the data points into groups based by computing (dot product) similarity between embeddings. Group DRO is then applied treating the identified clusters as subgroups. The proposed method uses no group label information, yet is competitive with methods that use group labelings in a validation set.\n\n", "summary_of_the_review": "Overall, intuitive idea, strong experiments, but limited novelty. The paper would benefit from some minimal theoretical justification.", "strength_and_weaknesses": "Pros:\n- The idea of clustering to identify group information is a natural one to try out.\n- The paper shows that one can get competitive performance on common benchmarks even with no knowledge of the underlying groups, even against baselines that do make use of group label info in the validation set. This I think is a significant empirical contribution.\n\nCons:\n- Unfortunately, the idea lacks sufficient formalism. While it is intuitive that clustering the data based on learned embeddings could help discover subgroups, I think the paper would be more compelling if the authors are able to provide formal framework for characterizing spurious correlations and identifying scenarios where applying a simple k-means clustering to the training data would uncover the right subgroups.\n- Determining the right number of clusters remains an important challenge. The authors show robustness to the number of clusters on one of the datasets (do these trends hold for the others too?), but in general, I worry that mis-specifying the number of groups could adversely impact the subsequent DRO step. Since DRO optimizes for worst-case accuracy, doing so with a larger number of groups than needed may adversely impact average accuracy. Could you show a similar plot as Fig 4 for average accuracy too?\n\nOther questions:\n- In the experiments, some times GramClust-cv (no group labels) outperforms GramClust-orig (groups labels in val set). This happens not only in Waterbirds (where the authors point to possible noise in group annotations), but also in CelebA. Is there a reason you've identified for this?\n-  In your implementation of Group DRO from Sagawa et al., did you implement the per-group regularized updates in eq (5) in their paper (which I think is what they use in their experiments), or the online version in Algorithm 1 in their paper?\n\nAdditional related literature: Kirichenko et al. Last Layer Re-Training is Sufficient for Robustness to Spurious Correlations. ArXiv:2204.02937.\n\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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "QVDuQfx_y8w", "reviewer": "Reviewer_9Mcs", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "Recently, it has been observed that although neural networks achieve very good accuracy on average over the dataset but they can have very low accuracy on certain subgroups of the distribution. There have been many works on trying to fix this gap. This paper proposes an algorithm to improve the accuracy on different subgroups of the distribution and show improved performance on many benchmark datasets as compared to many existing baseline methods. The benefit to their approach is they do not require information about the group labels and learn them using unsupervised clustering. The unsupervised clustering has been used previously but they propose to use clustering over the Gram matrices of the activations of the intermediate layers which are shown to capture the texture and style patterns of images and hence can capture the groups divisions well. They use these clusters to perform group distributionally robust optimization in the second stage. Moreover, they propose to use the learned clusters of the validation set for tuning the hyper parameters as well and hence, do not even require knowledge of these subgroups over the validation set.", "summary_of_the_review": "I like the overall idea of the paper. This work also improves performance on many benchmark datasets. The only concern is whether the gain is coming from using Gram matrices or the early stopped model. ", "strength_and_weaknesses": "The idea of using Gram matrices which are supposed to capture the style and texture features of the images and hence, can capture the spurious features is a simple and interesting idea. \n\nThey propose to use the groups learned for even the validation set but I wanted to check with the authors that even with other previous methods, we can do the same thing? \n\nAlso, is there any guarantee that we will indeed learn the natural groups of the data. Are there any examples of the dataset where the groups contain divisions based on non-style and texture based features and hence, won’t be captured well by the Gram matrices based clustering?  \n\nThe authors also compare their method to using last layer features for clustering. In their identifier model, they have also trained the model for only a few iterations so that the model only capture easy features. Did the authors use the same early stopped model for the last layer features while doing this experiment? It would be good to separate out the gains coming from early stopping vs. using Gram matrices for clustering. \n\nIt is quite interesting to see that increasing the number of clusters does not hurt their performance. I would imagine that increasing the number of clusters can maybe form some clusters which only contain hard data points and having low loss on them can hurt the overall performance. Do the authors have some intuition on what kind of clusters does this method find when the set the number of clusters to be large and why it does not hurt the performance?", "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": "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": "PajzuWDL4K", "reviewer": "Reviewer_nkU5", "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": "In this paper, the author combines the existing gram matrices into the group robustness classification. The whole pipeline can be summarized into two steps: the author first adopts the clustering techniques to cluster the dataset samples into several groups with the gram matrices features; then,  cluster groups with the pseudo-group labels are integrated into the existing group robustness optimization framework.\nThe proposed method is validated on three datasets. The proposed method only can achieve higher test accuracy on worst-group (w-g) but not achieve the best performance on average (avg) test accuracy.", "summary_of_the_review": "\nOverall, novelty is weak. The experiment results only can support part of the claims or the proposed method.", "strength_and_weaknesses": "Strength:\nThe proposed method can achieve best worst-group (w-g) test accuracy.\n\n\nWeaknesses:\nThe novelty is weak. The proposed method is just a combination of gram matrices and group robustness optimization essentially.\nThe proposed method can't achieve the best performance on average accuracy.\nThe proposed method is not validated on the large-scale dataset.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "BSUVSUwM3u", "reviewer": "Reviewer_Nwqa", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "The paper proposes a new method for improving the robustness of learning neural networks with respect to group correlations using the Gram matrix that is extracted from the output of different layers in a learned neural network.", "summary_of_the_review": "The paper's contributions are marginal without adding much to the fundamental understanding of why the method works better than other methods (this part is questionable based on the lower than expected numbers on JTT), why it does not need validation group labels and what are the limitations of it (e.g. are there any spurious correlations that this method cannot capture)?", "strength_and_weaknesses": "Strength:\n- This is an important problem.\n- The proposed method seems to be mostly competitive with the other baselines\n- Their proposed method does not need group labels for the validation set (although the paper does not provide any clear answer as to why this is the case)\nWeaknesses:\n- The proposed method is computationally much more expensive than the baselines. I strongly recommend the authors to give a more comprehensive comparison between their methods and the baselines with respect to the computation burden.\n- The fact that we know unsupervised clustering and two step training procedures with GDRO could result in better worst case performance has already been observed. As a result the overall results of the paper are really incremental. Unfortunately, the paper does not shed much light on why their method is superior or what are the limitations of their approaches? For example, at one point the authors mention that gram matrices seem to encode \"textures and color palette\" rather than “structure” in the image. So, does this mean that the proposed method works for cases where group labels or spurious correlations relate to these? I think these are more important questions to answer rather than just showing that the method works better than very similar previously proposed methods on some datasets. \n- The proposed method, and specifically the hp tuning for it (for the first and second step) are not very well elaborated. This makes it very difficult to judge if their claims about fair comparisons are correct.\n- Why is it that the method does not require labeled validation? Could it be true that other methods could also be applied without labeled validation? If not, what is so different about the approach that enables that? In fact, on all datasets GRAMCLUST-cv is better than GRAMCLUST-orig which is a bit counter-intuitive. I suspect this might be due to some counter-intuitive hp tuning. Note that the explanations around the mismatch in the labels might be true for water-birds, but is not very applicable to CelebA or Coco dataset. \n- What is so special about VGG-19 and why should it be sued for Gram matrix? Would the result still hold if one uses other network architectures, e.g. resnet? other pre-trainings? This is so mysterious that all results are only presented with VGG-19 without any intuition/understanding.\n- The JTT numbers are much lower than what they should be (reported in other papers). I suspect it is due to footnote 3. But I am not sure why footnote 3 is applicable and fair. With those numbers I do not believe that gram matrix approaches can show much gain over the baselines. I also noted that the gram matrix performances have relatively larger std compared to the baselines, which might mean that the methods are not very stable and their performance depends on the random seeds.\n- Compared to meanvar, the proposed gram matrix solution is very computationally intensive with not significant gains over waterbirds and coco on p. \n- The definition of matching accuracy is not provided (at least for the sake of completeness). Also, could you provide more details on the correlation between matching accuracy and the final performance? \n- For figure 4 only the results for water-birds are provided while in that case the ground truth E=2 (which is the smallest possible number fo clusters). Could you please provide evidence on cases where the ground truth E is different than the minimum, e.g. Coco on P data, to better illustrate the trade-off?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject