Uniformly Distributed Feature Representations for Fair and Robust Learning
Authors: Kiran Krishnamachari
See-Kiong Ng
Chuan-Sheng Foo
Venue: Accepted by TMLR ()

Abstract: A fundamental challenge in machine learning is training models that generalize well to distributions different from the training distribution. Empirical Risk Minimization (ERM), which is the predominant learning principle, is known to under-perform in minority sub-populations and fail to generalize well in unseen test domains. In this work, we propose a novel learning principle called Uniform Risk Minimization (URM) to alleviate these issues. We first show theoretically that uniform training data distributions and feature representations support robustness to distribution shifts. Motivated by this result, we propose an empirical method that trains deep neural networks to learn a uniformly distributed feature representation in their final activation layer for improved robustness. Our experiments on multiple datasets for sub-population shifts and domain generalization show that URM improves the generalization of deep neural networks without requiring knowledge of groups or domains during training. URM is competitive with the best existing methods designed for these tasks and can also be easily combined with them for improved performance. Our work sheds light on the importance of the distribution of learned feature representations for model robustness and fairness. Code is available at https://github.com/kiranchari/UniformRiskMinimization.

Published in Transactions on Machine Learning Research (12/2024)
Uniformly Distributed Feature Representations for Fair and Robust Learning
Kiran Krishnamachari
kinnkka. edu
Institute for Infocomm Research RI² A *,TAR, Singapore School of Computing, National Universily of Singapore, Singopore
Sco-Kiong Ng
seckiong www.ggguuuuuuu....uu..
Institute of Data Science, National University of Singapore, Singapore School of Computing, National University of Singopore, Singopore
Chuan-Sheng Foo
foo__hhwee chuan sheng@ilr.as a--tarr-ssssaaraa edu..s
Institute for Infocomam Research R1) A STAR, Singapore Centre for Frontier AT Research (CFAR), A ASTAR, Singapore
Reviewed on OpenReview: /t openreviee.n net/ formm? id- PgLbS5yp8n
Abstract
A fundamental challenge in machine learning is training models that generalize well to dis- tributions different from the training distribution. Empirical Risk Minimization (ERM), which is the predominant learning principle, is known to under-perform populations and fail to generalize well in unseen test domains. In this work, we propose a novel learning principle called Uniform Risk Minimization (URM) to alleviate these issues. We first show theoretically that uniform training data distributions and feature represen- tations support robustness to distribution shifts. Motivated by this result, we propose an empirical method that trains deep neural networks to learn uniformly distributed feature representation in their final activation layer for improved robustness. Our ex- periments on multiple datasets for sub-population shifts and domain generalization show that URM improves the generalization of deep neural networks without requiring knowl- edge of groups or domains during training. URM competitive with the best exist- ing methods designed for these tasks and can also be easily combined with them for improved performance. Our work sheds light on the importance of the distribution of learned feature representations for model robustness and fairness. Code is available at https//ww...coeeeeaaaaarannntnna inn skMinimization
minority sub-
1 Introduction
ERM has been the predominant approach to train machine learning models. Consistently achieving high accuracy on test distributions identical to the training data distribution was a challenge until the resurgent success of deep learning. However, machine learning practitioners today realize that a major challenge is robustness in test distributions different from the training data distribution. Hence, the assumption that training and test data distributions identical longer captures the challenge of modern machine learning.
In this work, we consider the challenging problem of training models that perform well in distributions different from the training distribution without assuming knowledge of the test distribution during the learning process. Specifically, we explore the question of what is the best data distribution to Irain classifiers on for improved downstream generalization. We consider two downstream generalization tasks in our work, i.c., nob-lopulation shifts and domain generalization. In the sub-population or group shift problem sotup, we have input E X' and labels y E y, with the objective learn function X - y. Morcover, there exist attributes of the data. (11 -.. 0i -. Om, ai Ai, which may not be available during training.

---

Published in Transactions on Machine Learning Research (12/2024)
Hence, distinct sub-populations groups of the data can be defined using combinations of the attribute and label, for example h A x + G. The training distribution over (x,y) is mixture of group-specific distributions Py iee. Ptr = - L TuPa: where E Aigl- The test distribution, which is not observed during training, is: Pte = Eyeg SyPg, where 8 E ودا and * B. The objective for f is to perform accuralely in the unobserved Pte. Generally, models perform most poorly when the test distribution only comprises minority sub-populations of the training distribution (Hashimoto et al., 2018 Zhang et al.) 2020). In sensitive applications of machine learning such as healthcare, this leads to fairness issues when models perform better for some groups of people than others. Multiple methods have been proposed to tackle this problem of robustness to sub-population shifts. A common approach so far has been identifying the samples where models performs poorly and up-weighting these samples during training. This adentify-andd-aa and-emphasiee paradigm has been proposed in multiple works (Liu et al., 2021 Zhang & Sabuncu, 2018 Yaghoobzadeh jet al. 2021 Sanh et al. 2021 Utama et al.) 2020). Many proposed methods also rely on the knowledge of minority sub-population attributes for training or validation to achieve good performance (Gowda et al.,
2021 Izmailov al.) 2022 Menon et al.] 2021 Nam al. 2022 Yao et. al. 2022). However, knowledge of
relevant sub- aub-populations may not always be known, especially in novel settings, so this assumption may be impractical.
In domain generalization (DG), models are provided a set of training domains, each containing examples about the same task, with the goal that they must perform well on unseen target domains (Blanchard et al. 2011 Muandet et al. 2013). While challenging, DG captures real-world scenarios where unforeseen distribution shifts between training and test distributions can be encountered by models. Both scenarios, sub-population shifts and domain generalization, require novel solutions that enable generalization while roquiring limited privileged information such as accoss to minority group annotations access to samples from the test domains.
In this work, we present a novel learning principle called Uniform Risk Minimization (URM) to alleviate these issues. We first propose a novel risk measure, uniform risk, which is defined as the expected risk of the model as the lest distribution varies over the support of the training distribution. Specifically, uniform risk is equal to the expected risk of a model over all possible test distributions under uniform prior. Our main theoretical result is follows: training classifiers on uniformly distributed training data or learned feature representations is an optimal choice to lower uniform risk and supports robustness and fairness. Motivated by this result, we propose an empirical method to encourage deep neural networks to learn uniformly distributed feature representations using an adversarial training objective. We show using experiments that encouraging models to have uniformly distributed feature representations significantly improves their robustness to sub- population shifts and domain generalization, without requiring knowledge of groups or domains.
The major contributions of our work are as follows:
We propose Uniform Risk Minimization (URM), a novel learning principle for out of distribu- lion (ood) robustness and fairness (Section 2.3
2. We first thcoretically show that training classifiers on uniformly distributed input data or feature representations in deep neural networks is optimal for lowering uniform risk and supports robustness (Propositions 2.2 2.3
3. Motivated by our theoretical analysis, we propose method to learn uniformly distributed feature representations in the final activation layer of deep neural networks using an adversarial objective (Section 2.6). We empirically demonstrate the efficacy of this method for sub-population shifts and domain generalization, without requiring knowledge of task-specific knowledge such as group or domain labels (Section
2 Uniform Risk Minimization
In this section, we present novel learning principle called Uniform Risk Minimization (URM) for robustness under distribution shifts. Motivated by real-world scenarios where we do not necessarily have knowledge of the environments models will be tested on, we propose novel risk measure called wniform risk that is the
2

---

Published in Transactions on Machine Learning Research (12/2024)
We then show the- oretically that uniformly distributed training data (Proposition 2.2) or feature represent ations (Proposition 2.3) are optimal in lowering uniform risk. We then present a practical method for deep learning models that encourages their feature representations to be uniformly distributed using adversarial distribution matching to improve their robustness in various distribution shift scenarios (Section 2.6
expected test risk over all possible test distributions under uniform prior (Section 2.3
2.1 Notation and Problem Setup
We describe the general setting of distribution shifts between the training and test distributions for the classification problem. We denote the joint training distribution as Ptr(z,)) v) and the test distribution as pre y), where z denotes the input and y denotes the label. We assume these two distributions have the same support sets X.Y. We denote labeled training dataset of size Nor sampled from the training Ner distribution (#tr Mtr )i=1, where (xff ( ~ (tr y), and a dataset of size No from the test distribution (xle) Die Jim1
) where (xte v(0)) Pro(x).
(1)
2.2 A Generalization Bound for Distribution Shifts
We first define the test loss under distribution shift. The negative log-loss on the test distribution is:
Lest Eprctt log p(y[
(1)
Next, we propose a bound of the test loss using the training loss (Itrain) and the Kullback-Leibler (KT.) divergence between train and test distributions by extending Proposition in Nguyen et al. (2022).
Proposition 2.1. If the loss - log p(y(z) is bounded by MJ and the labeling mechanism, p(y(z), is unchanged between train and test distributions (covariate- shift assumption), we have:
M KL (TT
M
lest S ltrain +
(2)
V2
4v2
Proof. Proof in Appendix
An alternate proposition based on the label or torget-shift assumption discussed in Appendix E.1
2.3 Uniform Risk
In real-world deployments of machine learning, we do not always know which test distributions a model will encounter. Hence, we proposc a novel risk mcasure that incorporates the expected risk over all possible test distributions. As we assume that we do not know the test distribution, we wish train models that perform well under various scenarios. the expected risk is defined using an uninformative uniform prior over all possible test distributions. A key fact enabling the mathematical definition of such a uniform prior is that all computer systems represent data using finite-precision floating points, making all data representations discrete in practice. This enables to define the uniform prior over all test distributions using the Dirichlet distribution, which is the natural prior distribution for any discrete distribution. Let N be the size of the support set of the data distribution, i.e. the total number of possible inputs. For N-dimensional distributions, the uniform prior can be represented using the Dirichlet distribution such that (t) ~ Dir(a = 1). Note
N
that Dir(a = 1) is the uniform prior with parameters a; = 1, Vi E [1, and ou = Σ a; N. In Eq. ==1
we define uniform risk, Ry, as the expectation of the test loss under a uniform prior over all possible test distributions. We assume that training and test distributions have the same support.
1 As suggested by (Nguyen al. 2022), in classification problem, this be enforced casily by augmenting the output softmax of the classifier such that each class probability always least exp M). For example, if we choose M exp( ( M) 0.05, and if the output softmax (pt.pe....PPl) Pc)), augment (pr K 0.05, K 0.O K 0.05), where K 0.05 and C the number classes. This ensures the bound for the loss sample, while leaving the output prediction class unchanged.
3

---

Published in Transactions on Machine Learning Research (12/2024)
Ru B
(3)
We now upper bound uniform risk using Eq.
M
M
[rest] I Arain +
v2 1
(4)
4v2
Note that the bound depends on the training loss, the expected KL divergence between training and test dis- tributions, and a constant (Equation 4). Modern deep learning systems can minimize the training loss nearly zero long as the task is well defined and rcalizable by modern architectures. Hence, the main component of Eq. to minimize would be the expected KL divergence belwseen the training and lest distribulions. This leads to our first key result:
Proposition 2.2. The expected KL divergence between the train and test distributions in Eq. is minimized when the training data distribution (zr(z) is uniform:
(1)]] = u (3t) arg min (epc(()) (KLlptc (t))) (a)
where (xx) Vi ViE fI,...,NN
N
Proof. Proof in Appendix B
Proposition 2.2 states that the training data distribution that minimises the upper bound on uniform risk is the uniform distribution, (z). This means that training on uniform data distributions may improve robustness to distribution shifts. However, most practical training datasets may not be uniformly distributed. While uniformity could in principle be achieved by re- reightgh the data distribution i.c. by down-weighting high-densily regions and up-weighting low-density regions of the distribution, this would require knowledge of the data distribution or an accurate estimation of it, which is non-trivial in practice. Modern deep learning systems are usually trained on high-dimensional image or audio datasets whose distributions are not known and are difficult to estimate. Hence, we will consider a similar proposition using the learned feature representations of deep neural networks (Section 2.5
2.4 Revisiting Groupebalanced and Class-balanced Training via URM
Rather than considering the data distribution over the input space, we can also consider sub-group and class distribution shifts i.e,, shifts in p(g) and p(y), respectively. Similar to Proposition 2.2 we can show that training with uniform sub-group training-rede reduces uniform risk as defined in these scenarios (discussion and proof in Appendix Eb. Addition- ally, in these scenarios, uniform risk can be shown to be equivalent to the balanced risk across sub-groups classes, respectively (proof in Appendix E Thus, minimizing uniform risk in these scenarios directly corre- sponds to minimizing the balanced risk across sub-groups or classes. In fact, prior work (Idrissi et al 2022) has shown that group-balanced training is an effective approach for robustness to sub-population shifts when group-attributes of the training samples are known. Prior work (Buda et al. 2018) has also demonstrated the efficacy of bass-balanced training. We provide further empirical evidence below that class balanced training is effective for label-shifts in Section 3.3 Hence, URM provides unified perspective on various algorithms that perform group class balanced training.
class distributions distributions--eqivvleett
group-balanced and class-balanced
2.5 Lowering Uniform Risk in Learned Feature Representation Spaces
We have above defined uniform risk using the input space (Eq. 3) and shown that uniform training data distributions lower uniform risk (Proposition 2.2 To implement this in practice would require training our
4

---

Published in Transactions on Machine Learning Research (12/2024)
models uniformly distributed training data distributions. However, most practical training datasets are not uniformly distributed and it is difficult to modify their distribution to match uniform distribution. Thus, we propose & re-definition of the bound on uniform risk using the learned feature representation space of deep model. Hence, the focus shifts from identifying the optimal input training data distribution to determining the optimal latent variable distribution. Note that the latent variable is not fixed as it is learned by a model and hence, can be regularized during training to attain a specific distribution. As Proposition 2.2 we can show that uniformly distributed feature represenlations also lower uniform risk. This allows to implement URM by training deep neural networks to learn uniformly distributed feature representations rather than training on an input data distribution that is already uniformly distributed.
Let G be a model parameterized by a deep neural network that outputs a latent variable 2 i.e. 2 == G(x), which has dimensionality Z and is used by a downstream task head such as linear classifier. In order to extend the propositions to the latent space, 2, we make additional assumptions about the input data x, y and the latent variable, 2, following Nguyen et al. (2022). The key assumption that [r)(2 where .( is mutual information. This is a reasonable assumption that the latent variable z is sufficient to solve the task and contains the same information about the output variable y as does the imput variable 2. As the latent reprosentation 2 is learned while optimizing the downstream task objective, it must learn to retain information about the label. The remaining assumptions are weak assumptions about the data distribution and the standard covariate-shift assumption (all assumptions are detailed in Appendix Dj. Using these assumptions, Proposition of Nguyen et al. 2022 can be re-written as follows:
(x, y)
M KL (3)
M
lest lerain +
(5)
4
We have thus replaced (input space) with (latent space). The rest of our claims in Propositions 2.1 and 2.2 can then straightforwarddy applied to the latent space rather than the input space. Uniform risk can be bounded using the latent feature representation space as follows,
M
M
Ep Epoc-Dirton))lteal S lraan +
E V2 De)c (KLl) +
(6)
4
As above, we aim minimize the expected divergence between the training and test feature distributions in Eq. 6 This leads to our second key result: uniform feature distributions márdmize the expected divergence between training and test feature distribulions (Proposition 2.3
Proposition 2.3. The expected KL- EL-divergence between train and test distributions in Eg. 6 is minimized when the training feature distribution (z) is uniform:
arg arg E De) (KLipd) (z)lptr (2) Pio(z)
1
where (z))
Vi E (1....,21 2}
z
Proof. Follows proof of Proposition 2.2
Hence, we have shown that uniformly distributed feature representations also lower uniform risk. This motivates proposed method below that encourages deep neural networks to learn feature representa- tion space that is uniformly distributed for improved robustness (Section 2.6 Note that the assumption (r, v) = (r y) ensures that the representation 2 is useful for classification while being uniformly dis- tributed. In practice, this is ensured by the joint optimization of the classifier and our proposed regularizer objectives.
5

---

Published in Transactions on Machine Learning Research (12/2024)
2.6 Encouraging Feature Representation Uniformity using Adversarial Distribution Matching
Motivated by Proposition 2.3 we perform URM by learn- ing deep models with uniformly distributed feature rep- resentations. In order to encourage the distribution of deep feature vectors to match uniform distribution, we use an adversarial training approach. Adversarial train- ing is a useful approach to match distributions as it does not require significant modifications to the output of the encoder. Alternate approaches such as minimizing KT. divergence would require limiting the expressivity of the model to oulput a parametric distribution in order to alytically compute its divergence from target distribu- tion. We train domain classifier (or discriminator) to distinguish between feature vectors produced by an en- coder and samples from a uniform noise distribution. Si- multaneously, the encoder is trained to fool the domain classifier by generating feature vectors that resemble a uniform distribution. This achieved through adversar- ial training: the domain discriminator, D, learns to dif- ferentiate encoded feature vectors from random uniform noise, while the encoder, G, is optimized to both confuse D and learn useful representations for downstream task, typically handled by a linear classifier T (see Figure
Pz
imput x ~ X
G' / Encoder
T
Task head
D
U|a.B]
P
uniform
noise
Overview of adversarial distribution matching to encourage the encoder to output uni- formly distributed representations.
Figure 1:
D for an overview). Our method resembles Adversarial Autoencoders (Makhzani et al.) 2016) which use adversarial training to align the latent space autoencoders with desired prior distributions.
During training, we want the discriminator D's predictions for uniform random noise vectors to be accurate by maximizing Ez-p, [log (E)]]- On the other hand, given encoded feature vector G(x)) the discriminator must output probability, DzG(z)), close to zero by maximizing Ex-p, [log(1 - D(G(=)))] The encoder must be trained to fool D to produce a high probability for feature vectors by minimizing Bz-p, [log(1 - ((G()))) maximising Ex-p, [log (GG())) The combined objective function to update D and G represents a minimar game with the following loss function:
min max L(D,G) Essp, [log D(2)) Egsp, [log(1 D())
where Ez-p,. log D[2)] is not used to update G. Generative adversarial networks (GAN) (Goodfellow et al. 2014) theory shows that the above objective minimizes the Jensen Shannon Divergence between the encoded feature distribution and the uniform distribution when the discriminator is optimal. Let the be the parameters of the task model, which includes the encoder as its backbone and a task head such as a lincar classifier i.c. == (G,T), T). Cr is the task-specific loss function. We can summarize the overall URM training objective for 0 and D follows:
min max C), BC + AEz-p., -
D(G(x))) -
Es-p,, (log D())]
The hyper-parameter determines the weight for training the encoder and discriminator to classify the feature outputs z G(z). A higher À increases the strength of regularization at the expense of feature learning for the downstream task and lower À weakens regularization but allows the encoder to pay more attention to the downstream task. The same À is applied to both update the discriminator and encoder so that one does not overpower the other. The choice of activation function applied to the outpul of the encoder G can be changed for each task. We choose from either the hyperbolic Tangent (TanH) or ReLU activations. In case of TanH activations, the uniform noise distribution it is 24(0, 1). We use Leaky ReLUs in the discriminator improve gradient flow the generator. We train the encoder G and discriminator D alternately.
(1,1,,) and in case of ReLU

---

Published in Transactions on Machine Learning Research (12/2024)
3 Experiments
3.1
Experimental Settings and Datasets
For group robustness, our experimental setup followed Yang et al. (2023). Specifically, we used the evaluation setting where group attributes are unknown the training set. The worst-class accuracy is used perform model selection, i.e., choice of the best checkpoint from each training run, as it was found be an effective alternative in the absence of group-attributes by Yang et al. (2023). For all methods, group attributes in the validation set were used to select hyper-parameterss as well as to fine-tune methods that require a group- balanced datasct c.g. DFR (Izmailov et al. 2022 We report worst-group and balanced (across classes) accuracies. All results were averaged across Ihree random seeds. For domain generalization experiments, our experimental setup followed Gulrajani & Lopez-Pasz (2021) using the
cross- vross-valiittion scheme. We report the average accuracy across all test domains. For both tasks, we scarched over sixteen random hyper-parameter combinations and applied adversarial distribution matching to the penultimate layer before the linear classífication head. Hyper-parametee search ranges for URM are included in the Appendix (Table 4 Adversarial training is often unstable in certain contexts, such generative modeling, However, in our experiments, we did not encounter significant instability. This is likely because we are not generating high-dimensional images, as in GANs, which involve a more complex distributionnmmtchingg task. Instead, we are generating lower-dimensional feature vectors to align with a uniform distribution. For group robustness, we evaluate on Waterbirds (Wah et al. 2011), CelebA (Liu et al. 2015), CivilComments [Borkan al., 2019) and MultiNLI (Williams et al. 2018). For Waterbirds and CelebA, we use the ResNet50 ImageNet (Russakovsky et al., 2015 For MultiNLI and
(ee et al. 2016) architecture pretrained CivilComments, we use BERT (Devlin et al.) 2019) pretrained on Book Corpus and English Wikipedia data. For domain generalization (DG), we benchmark on ColoredMNIST (Arjovsky et al. 2020), RotatedMNIST (Ghifary et al. 2015), PACS (Li et al. 2017). VLCS (Fang et al. 2013), OfficeHome (Venkateswara et al, 2017) and TerraIncoguita (Beery et al. 2018 For DG benchmarks, all methods used the ResNet50 (He) et al. 2016) architect pretrained on ImageNetlk (Russakovsky el al., 2015). Other training and dataset parameters are based on (Gulrajani & Lopez-Paz 2021). We further describe all benchmark datasets in Appendix F All experiments were run on computer with 8 A100 GPUs.
3.2 Baselines
We consider baselines including ERM and various methods proposed for sub-population shifts and domain generalization. For sub- cobb-opulttion shifts, we consider GroupDRO, LfF, JTT, LISA, DFR, Mixup, IRM, CORAL, MMD, DANN, C-DANN, ReSample, ReWeight, Focal Loss, CBLoss (Class-balancel loss), LDAM, Balanced Softmax max) CRT and Re ReWeightCRT. We follow Yang et al. (2023) for implementation of these baselines. For domain generalization, we consider Inter-domain Mixup (Mixup), MLDF, MTL, ARM, SagNet, RSC, and VREx. We follow Guirajani & Lopez-Pas (0021) for implementation of these basclines. For a description of these baselines, please refer to our Appendix G We note that URM is a generic robust representation learning method that does not require any knowledge of training domains or groups for both sub-population shifts and domain generalization.
3.3 URR Supports Class-balanced Training for Robustness under Label-shifts
As discussed in Section 2.4. when considering shifts in the class-distribution (label-shifts) alone, we demon- strate that URM, which corresponds class-balancing, produces the best balanced accuracy. We consider the binary classification tasks in the Waterbirds and ColoredMNIST datasets. We trained models with differ- ent class-distributions and compute each model's balanced accuracy, which is the average of the class-specific accuracies on the test set. Balanced accuracy is a commonly used metric in imbalanced classification tasks. We train models with different values of (p E {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9)), where corresponds to the binary class distribution parameter (Table D Note that = 0.5 corresponds to class-balanced train- ing, which corresponds to URM for label shift scenarios. The URM model (p 0.5) achieves the highest balanced accuracy on the test set (Table D. Balanced accuracy deteriorates when the the training class dis- tribution is highly imbalanced e.g. p 0.9 0.1. As discussed in Section 2.4 uniform risk corresponds to
7

---

Published in Transactions on Machine Learning Research (12/2024)
balanced accuracy across classes in the label-shift scenario, hence URM (class-balaccing) is useful strategy in dealing with unexpected label shifts. Our results further support and provide unified perspective on the training.
common-practice of
3.4 Uniformly Distributed Feature Representations Improve Sub groug Robustness
CRM learns more sub-group robust represen- tations compared to ERM on multiple datasets (Table - URM can also be easily com- bined with other methods as it is a generic represenlation learning method. For exam- ple, we combined URM with Deep Feature Reweighting (DFR) it involves fine-tuning the classifier on group balanced validation set, which URM does not require. Notably, when combined with DFR, URM achieved the best worst-group accuracy on the Waterbirds and MultiNLI datasets (URM (DFR), Table D. On MultiNLII URM (DFR) achieved 4% higher worst-group accuracy than the existing best methods. URM also achieved the high- est balanced accuracy on MultiNLI The bet- ter worst-group accuracy achieved by URM (DFR) reflects the improved representation learned by URM in the absence of any group atlribute in- formation. We note that URM a generic robust representation learning method and is not specific to sub-group robustness like many of the other baseline methods.
Table 1: Balanced accuracies (%) after training with dif- ferent class distributions. Results are averaged across three random seeds.
TRAINING
WATERBIRDS
COLOREDMNIST
0.1
85.7
55.3
= 0.2
86.0
68.0
0.3
85.3
76.5
= 0.4
86.1
76.9
p 0.5 (URM)
86.8
76.9
0.6
85.7
76.5
0.7
86.2
76.3
0.8
86.4
75.7
0.9
85.7
66.6
Table 2: Worst-group and balanced accuracies on group robustness benchmarks. Results for other methods are from Yang al. (2023
Waterbirds
CelebA
CivilComments
MultiNLI
Algorithm
worst (%)
balanced (%)
worst (%)
balanced (%)
worst (%)
balanced (%)
worst (%)
balanced (%)
ERM
69.1
83.1 +2.0
57.6
#0.8
93.0
63.2 #1.2
79.4 #0.2
66.4
=2.3
81.0 #0.3
Mixup
77.5
88.9 10.2
57.8
93.1
65.8 11.
79.7
66.8
81.7 10.1
GroupDRO
73.1
86.3 20.5
68.3 =0.9
93.9
61.5
81.3 #0.1
64.1
81.1 =0.2
CVaRDRO
75.5 12.
10.3
60.2 13.0
93.1 10.1
62.9 13.8
10.1
48.2 13.4
10.2
JTT
71.2 #0.5
86.8 20.2
48.3 1..5
92.4 20.2
51.0 21.2
77.7 20.8
65.1 11.6
81.4 20.0
LIF
75.0 #0.7
86.3 40.3
53.0 14.33
85.3 #2.9
42.2 27.2
69.7 1:4.7
57.3 =0.7
7L.4 #16
LISA
77.5 70.7
88.9 20.2
57.8 10.8
93.1 10.1
65.8 21.5
79.7 10.0
66.8 20.3
81.7 20.1
ReSample
70.0
85.0 +0.2
74.1
93.8
61.0 #0.6
80.7 #0.1
66.8
81.5 #0.0
ReWeight
71.9
86.2
69.6
94.0
59.3
81.3
64.2
79.4
Weight
71.0
87.2 +0.6
66.9
93.9
68.6 #1.1
80.6 #0.2
63.8
80.6
CBLoss
74.4 11.2
10.8
65.4 11.4
93.8 10.1
10.2
10.3
63.6 12.4
10.33
Focal
71.6 #0.8
87.1 +0.3
56.9 #3.4
92.6 20.3
61.9 #1.1
78.7 #0.3
62.4 =2.0
80.9 20.2
LDAM
70.9 +11.7
86.0 10.2
57.0 14.1
93.2 10.2
28.4 17.7
69.5 13.2
65.5 20.8
80,9 10.1
BSoftmax
74.1 #0.9
87.0 +1.0
69.6 #1.2
94.2 20.1
58.3 #1.1
81.1 20.1
63.6 22.4
80.6 20.2
DER
89.0 #0.2
91.2 40.1
73.7 10.8
93.2
64.4 40.1
79.0 #0.0
63.8
80.2 #0.0
CRT
76.3
87.9
69.6 10.7
93.6
67.8
80.7
65.4
80.2
ReW
76.3
88.0 +0.2
70.7 #0.6
93.6
64.7 #0.2
#0.0
65.2
80.2
URM
76.9 11.9
87.3 10.7
63.6 11.2
93.4 10.3
66.2 10.55
80.1 10..4
67.7
.0.7
81.8 10.1
URM (DFR)
89.7 #0.2
91.7 #0.1
67.4 +0.8
93.5 #0.1
61.3 +1.0
81.5 =00.0
70.7 =0.9
80.8 =0.1
3.5
Uniformly Distributed Feature Representations Improve Domain Generalization
URM improves the generalization performance of models in novel domains, URM outperformed competing methods in the RotatedMNIST and VL.CS and PACS benchmarks (Table Moreover, when combined with
8

---

Published in Transactions on Machine Learning Research (12/2024)
Table 3: Domain generalization accuracies using leave-one-domain-out validation. Results were averaged
three random sceds. Results for other methods are from Gulrajani & Lopez-Par Paz 2021
Algorithm
ColoredMNIST
RotatedMINIST
VLCS
PACS
OfficeHome
Terralncognita
ERM
36.7 = 0.1
97.7 + 0.0
77.2 = 0.4
83.0 = 0.7
65.7 + 0.5
41.4 + 1.4
IRM
40.3 + 4.2
97.0 0.2
76.3 0.6
81.5 - 0.8
64.3
41.2 3.6
GroupDRO
36.8 = 0.1
97.6 4 0.1
77.9 = 0.5
83.5 = 0.2
65.2 = 0.2
44.9 : 1.4
Mixup
33.4 4.7
97.8 + 0.0
77.7 0.6
83.2 - 0.4
67.0 0.2
48.7 0.4
MLDG
36.7 = 0.2
97.6
77.2
82.9 1.7
66.1
46.2
CORAL
39.7 = 2.8
97.8
78.7
82.6 1.6 0.5
6
46.3 # 1.7
MMD
36.8 0.1
97.8
77.3
83.2 + 0.2
6.2
46.5 16.5 1.5
DANN
40.7 +
97.6
76.9
$10
6.2
444
CDANN
39.1 + 4.4
97.5 + 0.2
+.73
39.9 + 3.2
3.2
MTL
35.0 + 1.7
97.8 0.1
76.6
83.7 0.4
0.5
44.9 1.2
SagNet
36.5 = 0.1
94.0 3.0
77.3
82.3 0.1
0.3
47.2
ARM
36.8 + 0.0
98.1 + 0.1
76.6 0.5
81.7 1 0.2
0.2
42.6 + 2.7
VREx
36.9 = 0.3
93.6 4 3.4
7.0
81.3 - 0.9
1.3
37.3 3.0
RSC
36.5 + 0.2
97.6 + 0.1
77.5 0.5
82.6 + 0.7
65.8
40.0 + 0.8
URM
36.9 + 0.2
98.1 0.2
84.3 + 4.8
84.0 0.2
67.5 + 0.5
48.3 1.4
URM (Mixup)
32.4 + 3.6
96.7 + 0.4
77.1 + 0.2
87.2 3.4
68.9 + 0.6
49.3 + 0.9
Inter-domain Mixup, URM achieved the best accuracy on the PACS dataset achieving 3.5% better accuracy than the best competing method, as well as the OfficeHome and Terraincognita benchmarks (Table 3 This demonstrates that URM, robust representation learning method, can be combined with an orthogonal method such as Mixup, a data augmentation method. URM does not use any knowledge of the domain generalization task and yet achieves competitive accuracy with methods that do. As URM is performed by learning a feature representation that is uniformly distributed over a bounded range e.g. [-1, where Z is the dimensionality of the feature space, feature representations of test domain samples fall within the familiar feature space of the URM model, reducing the likelihood of unexpected outputs. Hence, the URM model the entire range of feature representations equally during training, improving domain generalization.
3.6 Visualizing Uniformity of Learned Feature Representations
Data distributions often have multiple peaks and troughs, and models trained on such distributions tend to be biased towards high-density regions (majority groups) since they contribute more to gradient updates during training. Consequently, the learned parameters are more optimized for these high-density regions, resulting in poorer performance on low-density regions (the troughs). URM addresses this by learning a feature space that more uniformly distributed, effectively flattening the training feature distribution and roducing bias. This approach encourages the model to learn features that more evenly distributed across both high- and low-density regions. To demonstrate this, we visualized the feature representations learned by URM compared to those learned by ERM using UMAP (Mclnnes et al. 2018 (Figure 2). We extracted feature vectors for test samples in the Waterbirds datasct using ERM and URM trained models. While UMAP does not provide a complete picture of the data due to significant dimensionality reduction, we observed that URM's features were significantly more uniformly distributed than ERM's. This suggests that URM achieves the training objective for the feature representations to be more uniformly distributed and contributes to the robustness of the model.
4 Related Work
Group robustness: The topic of robustness to sub-population shifts has been increasingly studied in recent years as machine learning models are being deployed in real-world applications where we wish avoid downstream performance biases. Many such methods involve training on group balanced dataset with improved performance. As shown above, URM in fact supports this approach of training on uniform

---

Published in Transactions on Machine Learning Research (12/2024)
Empirical Risk Minimization
Uniform Risk Minimization
Figure 2: UMAP McInnes et al. 2018 visualization of feature vectors generated by ERM and URM on Waterbirds dataset. URM generates more uniformly distributed feature representation space compared to ERM.
group distributions. For example, one of the existing best. methods, DFR, retrains the linear classifier head on group balanced dataset. However, group-attributes may not available if many practical scenarios. Hence, method such as URM that does not require group annotations would be useful. Morcover, as shown above, URM can also be combined with met such as DFR for improved performance (Section 3.4..
Uniform priors: We note that our work closely related to the ideas presented uniform priors for data-cfficient learning (Sinha et al. 2022). Compared (Sinha al. 2022), we proposed novel lcarning principle and framework to motivate the use of uniform priors for robustness. In fact, our results may explain the empirical results of (Sinha et al. 2022). In other work, uniform priors have also been employed in self-supervised representation learning, both explicitly and implicitly (Assran et al. 2023
Robust Representation Learning: URM is also related to the idea of state reification (Lamb et al, 2019), which aims to project out-of-distributiio hidden states back on to the manifold of familiar hidden states from the training distribution. Rather than training additional networks to map out-of-distribution hidden states back on the training data manifold, URM learns bounded feature spaces that are uniformly distributed. Hence, even -nt-of-distribuutin samples will fall in the familiar feature space of the model and is less likely provide unexpected outputs. Other work has introduced input-space transformations that miligate the effect of irrelevant input features to improve representation learning (Taghanaki el al. 2021
5 Conclusion
We presented novel learning principle called Uniform Risk Minimization (URM) and showed theoretically that uniform training data distributions an optimal choice lower wniform risk, a novel risk measure for ood robustness. Our results also provide unified perspective various existing algorithms that perform group-balanced training from the perspective of distribution shift. Finally, empirical method to Iearn uniformly distributed fcature representations in deep neural networks using adversarial distribution matching. We empirically demonstrated the elfectiveness of our method to improve robustness to sub- nab-population shifts as well as domain generalization, without knowledge of any privileged group or domain attribute information during training. Our work sheds light on the importance of the dis- tribution of learned feature representations for model robustness and fairness. We hope our work encourages further research into uniform data distributions and priors for robust and fair representation learning.
proposed
10

---

Published in Transactions on Machine Learning Research (12/2024)
References
Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant Risk Minimization, March 2020. URL
00w.19977799993393 arXiv:1907.0288 [os, stat].
Mido Assran, Randall Balestriero, Quentin Duval, Florian Bordes, Ishan Misra, Piotr Bojanowski, Pascal Vincent, Michael Rabbat, and Nicolas Ballas. The hidden uniform cluster prior in self-supervised learning. In The Eleventh International Conference on Learning Representations, 2023. URL httpts//oppenreeiie. met/Eforunn/ddoo4KPMMMk....
Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in Terra Incognita. In Vittorio Ferrari, Martial Hebert, Cristian Sminchisescu, and Yair Weiss (eds.), Computer Vision ECCV 2018, pp. 472 489, Cham, 2018. Springer International Publishing. ISBN 978-3-080-01120-00
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Percira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine Learning, 79(1):151 175, May 2010. ISSN 1573-0563. doi: 10.1007 0tt/://1199999----5111
URL http://6......000..0000000000000
Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from Several Related Classifica- tion Tasks to a New Unlabeled Sample. In J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems, volume 24. Cur- ran Associates, Inc., 2011. URL htttp::/ppooeediingg www.neeuuu.eeeeee wwt.c/ape..llll/pppppeeeeeeeeeee wfiles/paper/2011/Ei.. 3577 Paper pdff
Gilles Blanchard, Aniket Anand Deshmukh, Urun Dogan, Gyemin Lee, and Clayton Scott. Domain gen- eralization by marginal transfer learning. J. Mach. Learn. Res., 22(1), January 2021. ISSN 1532-4435. Publisher: JMLR.org.
Daniel Borkan, Lucas Dixon, Jeffrey Sorensen, Nithum Thain, and Lucy Vasserman. Nuanced Metrics for Mcasuring Unintended Bias with Real Data for Text Classification. In Companson Proceedings of The 2019 World Wide Web Conference, WWW '19, pp. 491-500, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 978-1-4503-6665555 doi: 10.1145/335 /1/5//3355500331553333 URL. URT htttps://dii.orrg 10. 0ee111143388550 33113888865633313333333333333333 event-place: San Francisco, USA.
Mateusz Buda, Atsuto Maki, and Maciej A. Mazurowski. A systematic study of the class imbalance problem in convolutional neural networks. Neural Networks, 106:249 259, October 2018. ISSN 08936080. doi: arXiv: 1710.05381.
10.1016//..euuee.2011 (10p://..eeuee.2211.00..00 8.0.011 URL http://arxiv.org wttp:/o/////a/////// 05381
Kaidi Cao, Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning Imbalanced Datasets with Jabel-Distrbuttion-Avare Margin Loss. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d' Alché-Bue, E. Fox, and R. Garnett (eds.), Advances in Newral Information Processing Systems, volume 32, Cur- ran Associates, Inc., 2019. URL https::/proceel//
neurips. ccsscccpppprr haarr//ss.c//pppre_..eeeee/ppppe ftt.e///paper//01///11.
pdf.
Y. Cui, M. Jia, T. Lin, Y. Song, and S. Belongie. Balance Loss Based on Effective Number of Samples. In 2019 IEEE/CVE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 9260-9269, Los Alamitos, CA, USA, June 2019. IEEE Computer Society. doi: 10. (1110//CVP.2001.00000 URL
https://doo http://do....ecccppteeeeeee www.cceeptteeesocctttt...... org/ 11w.//199//.00000 2019..0099..0099999900
Jacob Devlin, Ming-Wa Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of Deep Bidi- rectional Transformers for Language Understanding, In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Tong and Short Papers), pp. 4171 4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.1865//11NN19-1122 URL https://acclanthology/ www.oe/yyoooo////133333
Chen Fang, Ye Xu, and Daniel N. Rockmore. Unbiased Metric Learning: On the Utilization of Multiple Datasets and Web Images for Softening Bias. In 2013 IEEE International Conference on Computer Vision pp. 1667-1664, 2013. doi: 10.1109/CCCV22013.000
11

---

Published in Transactions on Machine Learning Research (12/2024)
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain-Adversarial Training of Neural Networks. Journal of Machine Learning Research, 17(59):1-30) 35, 2016. ISSN 1533-7928. URL http://jmlr. httpp://jiirrrrr/papppeee wwt.ooog///////... wttp/ppperr/v17/15-55555
html
M. Ghifary, W. Kleijn, M. Zhang, and D. Balduzzi. Domain Generalization for Object Recognition with Multi-task Autoencoders. In 2015 IEEE International Conference on Computer Vision (ICCV), pp. 2551- 2559, Los Alamitos, CA, USA, December 2015. TEEE Computer Society. doi: 10.1109/ 1010/ICCVV ICCV.2015.298. 22015.293. ISSN: 2380-7504.
URL http:ttt/dbbiiieeeecccee http://biiieeccoppppeeeerrreeeie org/10...1111. 1109/ICCV 2015
2015. 293
lan Goodfellow, Jean Pouget-Abadie, Mchdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. In Adoances in Newral Information Processing Systems, volume 27. Curran Associates, Inc., 2014. URL httpps://proceedinggs cwturips.ccpaapa..
Sindhu C. M. Gowda, Shalmali Joshi, Haoran Zhang, and Marzyeh Ghassemi. Pulling Up by the Causal Bootstraps: Causal Data Augmentation for Pre-Trainin Training Debiasing, In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, CIKM '21, pp. 606-616, New York, NY, USA, 2021. Association for Computing Machinery. ISBN 978-1-4503-8446-4 doi: 10.1145/3459637 /3459637. 3482380. URL https://doi org/ 1145/10..144///496333. 33/ 3482380 Australia.
event-place: Virtual Event, Queensland,
Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. A
Kernel Two-Sample Test. Journal of Machine Learning Research, 13(25):723 773, 2012. URL http:
//jmlr.org/pan wttp//pppers//////rtttoooooooooo html
Ishaan Guirajani and David Lopez-Paz. In Search of Lost Domain Generalization. In International Confer- ence on Learning Representations, 2021. URL
Tatsunori Hashimoto, Megha Srivastava, Hongseok Namkoong, and Percy Liang. Fairness Without De- mographics in Repeated Loss Minimization. In Proceedings of the 35th International Conference on Machine Learning, pp. 1929-1938. PMT.R, July 2018. URL htttps://proceedings.. .ml.. mml..pres//v80/ hashimotol8a.1 html ISSN: 2640-3498.
Kaiming Hc, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image Recognition. In 2016 IEEE Conference on Compuler Vision and Pallern Recognition (CVPR), pp. 770 778, June 2016. doi: 10.1109//VV /CVVCC22222....0 ISSN: 1063-6919.
Zeyi Huang, Haohan Wang, Fric Xing, and Dong Huang. Self-challenging Improves Cross-Domain Gen- eralization. In Andrea Vedaldi, Horst Bischof, Thomas Brox, and Jan-Michael Frahm (eds.), Com- puter Vision ECCV 2020, volume 12347, pp. 124-140. Springer International Publishing, Cham, 2020. ISBN 978-3-030-58535-8 978-3-030-58536-3 doi: 10.1007 (L1007/978-33 1977-99030-5885555 8,
8. URL https: Series Title: Lecture Notes in Computer Sci-
//1ink.springee. springer ence.
Badr Youbi Idrissi, Martin Arjovsky, Mohammad Pezeshki, and David Lopez-Paz. Simple data balancing achieves competitive worst-group-accuracyy In Proceedings of the First Conference on Causal Learning and Reasoning, pp. 336 351. PMLR, June 2022. URL httpp:://pocceddai wwrpangsmmmr..omm mlr, wttpp///se///srssssseee
html ISSN: 2640-3498.
Pavel Izmailov, Polina Kirichenko, Nate Gruver, and Andrew Gordon Wilson. On Feature Learning in the Presence of Spurious Correlations. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (ods.), Advances in Neural Information Processing Systems, 2022. URL https://openreviie net/
forum? wiwuuumd-=knhUUuuaaaaa
Nathalic Japkowicz. The Class Imbalance Problem: Significance and Strategies. Proceedings of the 2000
International Conference on Artificial Intelligence ICAI, June 2000.
12

---

Published in Transactions on Machine Learning Research (12/2024)
John C. Duchi and Hongscok Namkoong. Learning models with uniform performance via distributionally robust optimization. The Annals of Stalistics, 49(3):1378-1406, June 2021. doi: 10.1214/20-A052004 URL https://doi.org//
20044/20-A02200
Bingyi Kang, Saining Xie, Marcus Rohrbach, Zhicheng Yan, Albert Gordo, Jiashi Feng, and Yannis Kalan- tidis. Decoupling Representation and Classifier for Long-Tailed Tailed Recognition. In International Conference
on Learning Representations, 2020. URL. httpps///ppearreeeeserrr/roorooo
David Krueger, Ethan Caballero, Joern-Henrik Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-Distribution Generalization via Risk Extrapolation (REx). Proceedings of the 38th International Conference on Machine Learning, pp- 5815-5826. PMLR, July
2021. URL http:///ppopepeeo....oorpr////// pptess/v139/rruuer222222
mlr.
ISSN: 2640-3498.
Alex Lamb, Jonathan Binas, Anirudh Goyal, Sandeep Subramanian, Toannis Mitliagkas, Yoshua Bengio, and Michael Mozer.
Networks: Improving Generalization by Modeling the Distribution of Hidden Reprosentations. In Proceedings of the 36th International Conference on Machine Learning, pp. 3622 3631. PMLR, May 2019. URL https://proceeddingg mlr. wwwp:::sww...ap....eee.eeeeeeeee htt..eeee///9///ma1aaa... html ISSN: 2640-3498.
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M. Hospedales. Deeper, Broader and Artier Domain Generalization. In 2017 IEEE International Conference on Compuler Vision (ICCV), pp. 5543 5551, Venice, October 2017. IEEE. ISBN 278-1-5356-1032- doi: 10.1109/ICCV 20.109//CCCV2211..599. URL http:// ieeexplore leebe org/ document/82378553
Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M. Hospedales. Learning to generalize: meta-learning for domain generalization. In Proceedings of the Thirly-Secowe AAAT Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence, AAAII8//AAAT1//AAATT AAAI Press, 2018a. ISBN
978-1-37735-8000 Place: New Orleans, Louisiana, USA.
Haoliang Li, Sinno Jialin Pan, Shiqi Wang, and Alex C. Kot. Domain Generalization with Adversarial Feature Learning. In 2018 IEEE/CVE Conference on Computer Vision and Pattern Recognition, pp. 5400 5409, 2018b. doi: 10.1109 10.1109/CVPR. Cttp:/CCVR.....0.000
Tsung-Yi Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár. Focal Loss for Dense Object Detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(2):318 327, 2020. doi:
(..1109/PPAMI2000 2858886..
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 Infor- mation. In Proceedings of the 38th International Conference on Machine Learning, pp. 6781-6792. PMLR, July 2021. URL htttps://proceedinggs mlr wwwpss/v19//eleerfrrrrr 12t .ttml
ISSN: 2640-3498.
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep Learning Face Attributes in the Wild. In 2015 IEEE International Conference on Computer Vision (ICCV), pp. 3730 3738, 2015. doi: 10.1109/ ICCV 425..
Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, and Ian Goodfellow. Adversarial Autoencoders. In International Conference on Learning Representations, 2016. URL http:::/aaxw..orr/abbb//////////
L. McInnes, J. Healy, and J. Melville. UMAP: Uniform Manifold Approximation and Projection for Dimen- sion Reduction. ArXiv e-prints, February 2018. eprint: 1802.03426.
Aditya Krishna. Menon, Ankit Singh Rawat, and Sanjiv Kumar. Overparamcterisation and worst-case generalisation: friend or foe? In International Conference on Learning Representations, 2021. URL http:://pppaeeeiiee.eee//r/aaaap
13

---

Published in Transactions on Machine Learning Research (12/2024)
Krikamol Muandet, David Balduzzi, and Bernhard Schölkopf. Domain Generalization via Invariant Feature Representation. In Sanjoy Dasgupta and David McAllester (eds.), Proceedings of the 30th International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pp. 10 18, Atlanta, Georgia, USA, June 2013. PMLR. URL https://proceeddie .ml..ng.s.................. html
Issue:
H. Nam, H. Lec, J. Park, W. Yoon, and D. Yoo. Reducing Domain Gap by Reducing Style Bias. In 2021 IEEE/CVE /CFF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 8686 8695, Los Alamitos, CA, USA, June 2021. IEEE Computer Society. doi: 10.1109/ 10.1109/CVPRAAS /CWVC//437..000.000505 URL https://doi-
2021 00858
Junhyun Nam, Hyuntak Cha, Sungsoo Ahm, Jacho Leo, and Jinwoo Shin. Learning from Failure: De-biasing Classifier from Biased Classifier. In Advances in Neural Information Processing Systems, volume 33, pp. 20673 20684. Curran Associates, Inc., 2020. URL hatl://ccc4427777770002211111133 Abtseeeeeees--tttttttt.... html
Junhyun Nam, Jachyung Kim, Jacho Lee, and Jinwoo Shin. Spread Spurious Attribute: Improving Worst- group Accuracy with Spurious Attribule Estimation. In International Conference on Leurning Represen- tations, 2022. URL http::///ppprrreeeeeee///oooaaa 2
A. Tuan Nguyen, Toan Tran, Yarin Gal, Philip Torr, and Atilim Gunes Baydin. KL Guided Domain Adaptation. In International Conference on Learning Representations, 2022. URL https://openrevier net/forum??d=OIzqql.. IVWDD
Jiawci Ren, Cunjun Yu, shunan sheng, Xiao Ma, Haiyu Zhao, Shuai Yi, and hongsheng Li. Balanced Meta- Softmax for Long- Tailed Visual Recognition. In Advances in Neural Information Processing Systems, volume 33, pp. 4175 4186. Curran Associates, Inc., 2020. URL https: http:///poccce....e/// neurips.cel html
paper/2020/hash. httpa///ccccccaa4444111111222330
Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjecv Sathoesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision, 111(3)22 December 2015. ISSN 1573-1405. doi: 10.1007/511263-015-08111 611263-015-0816->
URL https://aci.oor//00
Shiori Sagawa, Pang Wei Koh, Tatsunori B. Hashimoto, and Percy Liang. Distributionally Robust Neu- ral Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization. arXiv: 1911.08731.
arXiv:1911.0879) [cs, stalf, April 2020. URL http::tttt::::: org/abs//911.000 08731
Victor Sanh, Thomas Wolf, Yonatan Belinkov, and Alexander M. Rush. Learning from others' mistakes: Avoiding dataset biases without modeling them. In International Conference Learning Representations, 2021. URL https::/ net Hf3gXoNNkR
Samarth Sinha, Karsten Roth, Anirudh Goyal, Marzyeh Ghassemi, Zeynep Akala, Hugo Larochelle, and Animesh Garg. Uniform Priors for Data-Efficient Learning. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp. 4016 4027, New Orleans, LA, USA, June 2022. URL https://ieeexplors
IEEE ISBN 978-1-665548-3399 leee. atg//occument/8553377/
doi: 10.1109 D0.108/CVPRN56647.20000
Baochen Sun and Kate Sacnko. Deep CORAL: Correlation Alignment for Deep Domain Adaptation. Gang Hua and Hervé Jégou (eds.), Compuler Vision ECCV 2016 Workshops, 2016.
Saeid A. Taghanaki, Kristy Choi, Amir Hosein Khasahmadi, and Anirudh Goyal. Robust Representa- tion Learning via Perceptual Similarity Metrics. Proceedings of the 38th International Conference on Machine Learning, pp. 10043-10053. PMLR, July 2021. URL TSSN: 2640-3498.
mlr. press/v139/ /ttp://ceddaaaa.....rrsee//////
taghanaki 21a. html
14

---

Published in Transactions on Machine Learning Research (12/2024)
Prasetya Ajie Utama, Nafise Sadat Moosavi, and Iryna Gurevych. Towards Debiasing NLU Models from Unknown Biases. In Bonnie Webber, Trevor Cohn, Yulan He, and Yang Liu (eds.), Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 7597 7610, Online, November 2020. Association for Computational Linguistics. doi: 11118653/1/2220...//mm https://acllantholog.. wwwpe/ooooooooo////////// wwgg e01200mmmmpp-mi..... 613 Vladimir Vapnik. Slalistical Learning Theory. Wiley, 1998. ISBN 978-0-471-03003-4..
URL
Hemanth Venkatcswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep Hashing Network for Unsupervised Domain Adaptation. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5385-5394, 2017. doi: N0.1108/CVPR.2207.5522
C. Wah, S. Branson, P. Welinder, P. Perona, and S. Belongie. Caltech-UCSD. Technical Report CNS-TR- 2011-001, California Institute of Technology, 2011.
Yufci Wang, Haoliang Li, and Alex C. Kot. Heterogeneous Domain Genoralization Via Domain Mixup. In ICASSP 2020 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3626, 2020. doi:
Adina Williams, Nikita Nangia, and Samuel Bowman A Broad-Covenr Cover-Ce Challenge Corpus for Sentence Understanding through Inference. In Marilyn Walker, Heng Ji, and Amanda Stent. (eds.), Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), pp. 1112-1122 New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/NV 1101, URL https://aclanthollgg org/N18-1100
Yadollah Yaghoobzadeh, Soroush Mehri, Remi Tachet des Combes, T. J. Hazen, and Alessandro Sor- doni. Increasing Robustness to Spurious Correlations using Forgettable Examples. In Paola Merlo, Jorg Tiedemann, and Reut Tsarfaty (eds.), Proceedings of the 16th Conference of the European Chap- ter of the Association for Computational Linguistics: Main Volume, pp. 5319-3333 Online, April URL https:
2021. Association for Computational Linguistics. doi: 22261////2222leaclcl 7/aclanthology.oo crc.laatoooooy..rg/22ll...oooo eacllalllll-- main. main.291
Yuzhe Yang, Haoran Zhang, Dina Katabi, and Marzych Ghassemi. Change is Hard: A Closer Look at Subpopulation Shift. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), Proceedings of the 40th International Conference on Machine Learn- ing, volume 202 of Proceedings of Machine Learning Research, pp. 39584-39622. PMLR, July 2023. URL https://proceediings mlr, wrwpppeesseeeeee vrress/v202/yaagg333 html
Huaxiu Yao, Yu Wang, Sai Li, Linjun Zhang, Weixin Liang, James Zou, and Chelsea Finn. Improving Out-of- Distribution Robustness via Selective Augmentation. In Proceedings of the 39th International Conference on Machine Learning, pp. 25407-25437. PMLR, June 2022. URL https://pproceedings. mml aalr.pres//
v162/yao22b. html ISSN: 2640-3498.
Haoran Zhang, Amy X. Lu, Mohamed Abdalla, Matthew McDermott, and Marzyeh Ghassemi. Hurt- ful Words: Quantifying Biases in Clinical Contextual Word Embeddings. In Proceedings of the ACM Conference on Health, Inference, and Learning, CHIL 220, pp. 110-120 120, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 978-1-4503--046-2 doi: 11145/3368555.3366 https://ddo..org//// httrg//0.114//200 1145/3355555 3384448 event-place: Toronto, Ontario, Canada.
URL
Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz. mixup: Beyond Empirical Risk Minimization. In International Conference on Learning Representations, 2018. URL https://
Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain Adaptation under Target and Conditional Shift. In Proceedings of the 30th International Conference on International Conference on Machine Learning Volume 28, ICML'13, pp. III-819 III-827. JMLR.org, 2013. Place: Atlanta, GA,
USA.
15

---

Published in Transactions on Machine Learning Research (12/2024)
Marvin Zhang, Henrik Marklund, Nikita Dhawan, Abhishek Gupta, Sergey Levine, and Chelsea Finn. Adap- tive Risk Mininization: Learning to Adapt to Domain Shift. In Advances in Newral Information Process- ing Systems, volume 34, pp. 23664 23678. Curran Associates, Inc., 2021. URL http://awwe.000..aaaaaaccc121aaa otst://ac...ee.s...a.a..aaaaa...
Abstract.htar
Zhilu Zhang and Mert Sabuneu. Generalized Cross Entropy Loss for Training Deep Neural Net- works with Noisy Labels. In Advances in Newral Information Processing Systems, volume 31. Cur- ran Associates, Inc., 2018. URL https::/proceeddngg www.npe..c..ppp.peee.ee http:/pppppppeeeee/ppppeeeeeeeee /ftles/paper2228/haaaa 22552277700222255222552222222221
Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 Million Image Database for Scene Recognition. JEEE Transactions on Pattern Analysis and Machine Intelligence, 40(6): 1452-1464 2018. doi: 1AA1009/TPAML2211227220000
16

---

Published in Transactions on Machine Learning Research (12/2024)
A Proof of Proposition 2.1
Proof.
M
Lest S train
KL Pio (x, (2,
(using Proposition of Nguyen et al.
2022
V2
M
S ltrain
KL (x, y) Per y)1
111
(square the square roo
2
M
M KL KIlpte( y) Ip y)l
-traiin
(7)
2
4
M
M
= -erain
Pp (a.v)
(lgg) + log Pte (y(az) log Per (x)
Po Per (b[z)] +
(8)
V2
4
M
-train
P... (x.1))
(log) log per (1)
V2
M
+ (a
(oopPeewlll (vlp))
log per (y)==]] + 4v2
(9)
M
lerain
Pie (z) loglloggee (tg (x) - lor Per (x)
2
M
M
+
E
(x))(x)
- log per (1/xx)]]
(10)
2
4v2
M
M
-erain
(L.ppt (.)
(L +
(11)
2
4
M
= lerain
(covariate-shil))
(12)
22
4v2
Using the covariate-shift assumption, the second KL divergence term in Eq. I1 involving the conditional probabilities plylz) equals 0 and can be dropped. This is a reasonable assumption as generalization to
distributions different from the training distribution is challenging if the labeling mechanism changes (Ben-
David et al. 2010). Hence, we can make the common assumption that KLJpte (y/=))ppr (y|æ)]] = O. We can then focus on the divergence between the marginal distributions i.. (=)pl. (.)).
B Proof of Proposition of 2.2 Proof.
arg min Epin( pi-(x)
(x)
(13)
N
Pie (xx) Der (xx²) )]
= arg min 1Dir(a=1) (x)(
Pte (n
(14)
arg min Epir(a=1)
Pie (=.))n(pte((xs) (e((x))n(pp
(15)
(x)
arg min Epi((a-1)
Pte (c.))n(pte(xx))
Σ Pie (te(
(16)
N Pie (x.)
arg min Epi((a=1) (w)
Pie (n:) Epirto
(17)
Note that diverreeeee is non-negative. is added as a buffer in case the KT=divergence is less than 1, as Vo >=
1
17

---

Published in Transactions on Machine Learning Research (12/2024)
N
N
= arg min ((r(x)
Σ
Dir (eee (x))
(18)
arg min Per(x)
(remove terms independent of por (z:))
arg min (a)la)
[n(er (Dr)
(expectation of Dirichlet dist. is ae)
arg min Piv(a)
(p (2:))
O3
(a, = 1, 0g = N)
ax
N
arg min (t)(a)
In(per (x+)
(19)
N
arg min E DD (LLp
arg min (-(
Σ (x(p))(x)))
(20)
Per (x)
N Per (2.) =
subject to
Σ
i=1
The Lagrangian for this constrained optimization problem is:
C(po ... Pp.(.)))
(x:)) + 2 Σ
Ptr (2,) 1)
(21)
1=1
solve (n. ( =0
1
= (ir( =
NX ViE fl....,N}
= >>= 1, Σ
Ptr (x:.) =1 (constraint)
(22)
= (z;) = N
Vi E N) (i.e. uniform distribution)
(23)
arg min Epaa Dir(a-) [L (x)) (tr)) = (x) (x)
(24)
1
where (I;) N Vi (I.
C Proof of Proposition of E.1
Proof.
M
hest S ltrain
/
(using Proposition of Nguyen et al. 2022
2
M
1 KL KL (x, y) Ip y)]
< Itrain
1
(square the square root
22
M
M
-Itrain
KT. KT.(ptc(a v) u)
(25)
22
4
M.
M
= -erain
Ep (r.w) [lo (1) + log Pte (ælyy) log (or - Por Per (xly)] +
(26)
22
4v2
18

---

Published in Transactions on Machine Learning Research (12/2024)
M 1 Pu (r, log Ptr (v)
terain
v2
M
+ Buellan Pog - por +
(27)
4v2
M
train
(v) (w)
Ep
Poeg (lg log
V2
M
M
(ep plgg Per (xrl))l
(28)
V2
4v2
M
M
= -erain
KL (pll((ee)) + Epic (v) (elllee
(29)
V2
4
M
M
train +
(target-shift]
(30)
V2
4
In Eq. 29 we assume that the conditional distribution of p(zly) does not change and that only the label distribution ply) has shifted. This assumes that the generative process p(a] y) has not changed at test time. This is referred to target shift in the domain adaptation literature (Zhang et al. 2013). Hence, we can drop the second term involving the conditional misalignment is zero and focus on the target shift in Eq. 30 Based on Eq. [30] we can similarly show that the expected test loss under target shift is minimized by the uniform training dins-distribution.
D Extending Propositions 2.1 and 2.2 to Deep Feature Layers
Here extend Propositions 2.1 and 2.2 to feature spaces such those learned by deep neural network. This will be useful below when we wish to apply the above results to method that modifies the feature space to be uniformly distributed in order to improve robustness. Let G be model parameterized by deep neural network that outputs a latent variable i.e. = G(wx), which can then be used for downstream lask such as classification. To extend the propositions to the latent space, 2, we make additional assumptions about the input data x, and the latent variable, 2:
Assumption D.1. (z, = ler(x,,), v), where ((5)) is mutual information, defined as:
(=, = Bpi((a.) fog (2, Pir (z)
] (x, = (e. [og (er v) Pir (x)
(31)
This a reasonable assumption that the latent variable & sufficient to solve the task and contains the same information about the output variable y as does the input variable x. As the latent representation z is learned while optimizing the task objective, must learn to retain information about the label.
Assumption D.2. (tlz)) = (Pt (vlz)), Vz, E x.Y
This assumption simply allows us re-write the true predictive distribution Py-(yl=) as a conditional ex- pectation over the latent variable z. When this assumption holds, the predictive distribution, p(yl=) Ep(stu; [[r]=)], will approximate (er(yla) when p(yll) ptr(y]z), which is a goal of training.
Assumption D.3. (a.y))
(ur((a)) <0,, N
This assumption is satisfied when train and test distributions have the same support set X, This is a reasonable assumption for sub-population shift scenarios when the lest distribution assumned to comprise the same groups as the training distribution but in different proportions.
Assumption D.4.
0
This the covariate-shift assumption.
Note that diverreeee is non-negative. is added as a buffer in case the KL=ddvvegeence is less than 1, as Vo >= 0
19

---

Published in Transactions on Machine Learning Research (12/2024)
Using these four assumptions, Proposition 1 of Nguuyn et al. 2022
can be re-written follows (see Nguyen
et al.
2022 for proofs):
M KL [KL (z)l
M
lest Arainn +
(32)
v2
4v2
We have thus replaced x (input space) with a (latent space). The rest of our claims Propositions 2.1 and
2.2 can then straightforwardly applied to the latent space rather than the input space.
E
Uniform Risk for Label Shifts and Sub-population Shifts
In the context of sub-population and label shifts, uniform risk can be shown to be the balanced risk across groups or classes, respectively.
Let Rg be the group-conditional risk computed over each group-specific distribution Po = p(a,, vlg) i.e. Rg = (z, y)]. In the case label-shift, the group can be treated as a class instead. Assume that the distribution shift is strictly over the group or class distribution.
Rv == Ep., (DraaDir(aa1)) [[sr]]
Ep Dir
Ep, [gyl
E (inaaair(a1)))
Bg Rgl, where A gl
95g
Pic Dir(a R
15
gEg
R
(balanced risk)
gEg
Thus, uniform risk is the balanced risk of model over groups or classes (Eq. bolanced risk) in the context of sub-population or label shifts. However, in the sub-population shift literature, authors generally focus on the worst-group performance. The limitations of over-emphasis on the worsl-group accuracy metric has been reported in prior work (Section 5.5 of Yang al. (2023)). In contrast, uniform risk provides balanced risk measure that equally weighs all groups or classes.
E.1 Class Balancing Lowers Uniform Risk
Here we discuss the common practice of class-balancing datasets from the perspective of uniform risk and robustness to label shifts at test time. In Eq. 7 of the Proof of Proposition 2.1 we can alternativoly decompose the KL-divergence between the joint training and test distributions (xx using the marginal misalignment in p(y) and the conditional misalignment in p(zly). We can then similarly show that training on data with uniform class or label distribution minimizes the upper bound on uniform risk, as pertaining the laiel-shift problem.
Proposition E. If the loss - log (y)c))) is bounded by M and the generalive process, p(æly), is the same between train and test distributions A we have:
M KL
M
hest S Itrain +
+
(33)
22
4
Proof. Proof in Appendix
4This is known the lahel or target slift assumption the distribution-shift literature.
20

---

Published in Transactions on Machine Learning Research (12/2024)
The expectation of Eq.
33
with respect to uniform prior distribution over test class-distributions is:
M
M
Bpn - Dir(a beett S lerain + B (pma()) -1 (ot +
(34)
v2
4
We can now aim to minimize the upper bound on the expectation of test-loss under label-shift i.e. uniform risk for label-shifts, by minimizing the expected KL-divergence between train and test class distributions. Let C be the number of classes in the classification task. The proof follows the same steps as the proof of Proposition 2.2 but for the dass-distribution p(y) instead of p(x).
Proposition E.2. The expected KL- divereneeeecce between train and test class distributions in Eq. is mini- mized when the training class distribution (t) is uniform:
arg min Dir(aa))) [KL[pte (v) (tr (1) Pe-(1)
(35)
1 C Vi (I,...,cC.
where uf
Proof- Same as proof of Proposition 2.2
Therefore our results support. the common practice of class-balancing from the perspective of robustness under label shifts. Indeed, prior work has demonstrated the efficacy of balancing classes during training [uda et al. 2018). We further experimentally demonstrate this below (Section 3.3
F
Datasets
We describe each dataset used in our group robustness benchmarks below.
Waterbirds (Wah al.) 2011). Waterbirds binary classification image dataset containing spurious correlations, constructed by placing images from the Caltech-UCSD Birds-200-2011 (CUB) dataset (aah et al. 2011] over backgrounds from the Places dataset (Zhou et al. 2018 The task is to classify landbirds from waterbirds and the spurious attribute is the background (water or land). As most images of waterbirds have a water background and most images of landbirds have a land background, models may latch on to the spurious correlation between the background and the type of bird.
CelebA (Liu et al., 2015). CelebA is a binary classification image dataset. The task is to predict hair color from images of celebrity faces (blond us. non-blond), where the spurious correlation is gender. As most blond haired people this dataset are female and most non-blond haired people are male, this creates a spurious correlation between gender and hair color.
CivilComments (Borkan et al., 2019). CivilComments is binary classification text dataset where models must predict whether an internet comment contains toxic language. The spurious autrbbuee is the presence of references to eight demographic identities (male, female, LGBTQ, Chris- tian, Muslim, other religions, Black, and White).
MultiNLI (Williams et al. 2018). MulGNLI is a text classification dataset with 3 classes and the target is the natural language inference relationship between the premise and the hypothesis (noutral, contradiction, or entailment). The spurious attribute is the presence of negation in the lext, negation is highly correlated with the contradiction label.
Domain Generalization Datasets: We benchmarked on multiple challenging DG datasets. ColoredM- NIST (Arovssy et al. 2020) a variant of the MNIST digit recognition dataset where each domain contains a disjoint set of digits colored either red or blue. The binary label noisy function of digit and color, such that color is correlated with the label to varying degree in each domain and the digit bears correlation 0.75 with the label. MotATedMNIST (Ghifary al. 2015) also variant of MNIST where each domain contains
21

---

Published in Transactions on Machine Learning Research (12/2024)
digits rotated to different degrees (0, 15, 30, 45, 60, 75 and contains 10 classes. PACS Li et al. 2017 contains
four diverse domains (art, cartoons, photos, sketches) and 7 classes. VI.CS (Fang et al. 2013) contains four photographic domains (Caltech101, LabelMe, SUN09, VOC2007) and 5 classes. OfficeHome /Venkateswara
ct al.
2017) also includes four domains (art, clipart, product, real) and 65 classes. Terralncogniaa (Beery
et al. 2018 comprises photographs of animals taken by camera traps at four different locations.
G
Baselines
Empirical Risk Minimization (ERM, (Vapnik) 1998)), minimizes the errors across all training samples. Subgroup Robust Methods: Group distributionally robust optimization (GroupDRO) (Sagawa et al. 2020) performs ERM while emphasising sub- sub-populations or domains with high losses during training. CVaRDRO (John C. Duchi & Hongseok Namkoong) 2021 is variant of GroupDRO that up- weights training samples with the highest losses. LfF' (Nam et al. 2020 trains two models where the first model is biased and the second one is debiased using a re-weighted objective. Just train twice (JTT)
Vanilla Training:
(Liu et al.) 2021) first trains a standard ERM model to identify minority sub-populations in the dataset and then trains another ERM model while up-weighting minority samples. LISA (Yao et al. 2022 trains
invariant predictors using data interpolation within and across attributes. Deep feature re-weighting (DFR)
[Imaillv et al. 2022) first trains an ERM model and then retrains only the last layer of the model using a
dataset balanced among different sub-populations. Data Augmentation Method: Mixup (Zhang et al.
2018) trains linear interpolations of randomly sampled training data points and their labels. Domain- Invariant Representation Learning Methods: Invariant risk minimization (IRM) (Arjovsky et al. 2020) learns a feature space such that the optimal linear classifier is the same across domains. Deep correlation alignment (CORAL) (Sun & Saenko 2016) matches the sccond order moments of the feature distributions.
Maximum mean discrepancy (MMD) {Li et al. 2018b) matches the MMD (Gretton el al. 2012) of feature distributions across domains. Domaín Adversarial Neural Networks (DANN, (Ganin et al. 2016)) employs adversarial training to match feature distributions across domains. Class-condiiional DANN (C-DANN, (Li
et al.
2018b:) is similar to DANN it matches the class-conditional feature distribution across domains.
Imbalanced Learning Methods: ReSample (Japkowicz) 2000) and Re ReWeight. (Japkowicz) 2000 simply
re-sample or re-weight the inputs according to the number of samples per class. Focal loss (Focal) Lin
et al.] 2020 reduces the relative loss for well-classified samples and focuses on difficult samples. Class-
balanced loss (CBLoss) (Cui et al. 2019 proposes re- re-weighting by the inverse effective number of samples.
The LDAM loss (L.DAM) (Cao et al. 2019) employs a modified marginal loss that favors minority samples
more. Balanced-Softtmax (BSofumax) (Ren et al. 2020) extends Softmax to an unbiased estimation that
considers the number of samples in each class. Classifier re-training (CRT) (Kang et al., 2020) decomposes the representation and classifier learning into two stages, where it fine-tunes the classifier using class- balanced sampling with representation fixed in the second stage. ReWeightCRT (Kang et al. 2020 is a re-weighting variant of CRT.
Domain Generalization Baselines: Inter-domain Mixup (Mixup, (Zhang et al. 2018 Wang et al. 2020)) trains on linear iterpolations between samples from different domains. Meta-Lnarning for Domain Generalization (MLDG, (Li et al. 2018a)) uses MAML to meta-learn generalizing across domains. Marginal Transfer Learning (MTI., (Blanchard et al.) 2011: 2021)) ) estimates mean embedding for each domain that is passed as second argument to the model. Adaptive Risk Minimization (ARM, (Zhang et al. 2021h) extends MTL using a separate embedding model. Style-Agnostic Networks (SagNet, (Nam et al, 2021 trains models by keeping image content and randomizing style. Representation Self Challenging (RSC, (Huang et al.] 2020)) learns robust models by iteratively pruning the most activated features. Risk Extrapolation (VREx, |Kuueger el al. 2021 approximates IRM using a variance penalty.
Training Hyper-parameters
22

---

Published in Transactions on Machine Learning Research (12/2024)
Table 4: Search ranges for URM-specific hyper-porameters. Other training parameters ranges are based on
Yang et al. 2023
Method
Parameter
Random Distribution
lambda À
Uniform(0,, 0.2)
generator output
RandomChoice [T am H, ReLUJ)
URM
number of layers discriminator
RandomChoice(11.2, 10Uniorm(-55,-3
learning rate (for image or tabular datasets) optimizer (for image or tabular datasets)
Handom(Choicc((SGU)) 10
learning rate (text datasets) optimizer (text datasets)
RandonChoice((AAmm)))
23


Reviews:
[{"review_id": "3HUIzIjjrG", "reviewer": "Reviewer_2jfY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nThe empirical result shows the strength of the approach.\n\nThe proposed algorithm is presented clearly. \n\n\nWeaknesses:\n\nThe assumption I(z,y)=I(x,y) is somehow problematic. This assumption indicates that the latent variable is sufficient to solve the “task”, which is a fixed distribution in the training environment. Then in the proof, the training distribution changes somehow, which leads to a circular argument. \n\nThough the theoretical result is intuitively correct, it is probably not quite informative as it only shows how to minimize the KL divergence between the training and test distribution without considering the utility of the model. In other words, the fair representation will be achieved if the model outputs uniformly distributed noise for any input, but that representation is not useful at all.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper first proves that training on uniformly distributed training samples mitigates the distribution shifts problem, then generalizes the result on samples to features to show that the features should also be uniformly distributed. To learn a uniform feature distribution, adversarial learning is used and the experiment shows that the proposed URM method is effective in several datasets with distribution shifts.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address the weaknesses in the above section.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9SHinq0sj3", "reviewer": "Reviewer_ysik", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The introduction of Uniform Risk Minimization (URM) provides a somewhat novel perspective on handling distribution shifts.\n- The authors offer rigorous theoretical support for their method, demonstrating that uniformly distributed training data or feature representations minimize uniform risk. \n- The submission includes thorough experiments across multiple benchmarks (e.g., sub-population shifts, domain generalization), demonstrating the effectiveness of URM and its competitive performance with previous methods.\n\nWeaknesses:\n- The use of adversarial training to enforce uniformity in feature representations may introduce additional computational complexity and training instability. \n- It appears that the improvement achieved by using URM alone over other methods is not highly significant. Additionally, the comparisons in Table 2 (URM + DFR vs. others) and Table 3 (URM + Mixup vs. others) may not be entirely fair, as they compare combination methods against standalone approaches. If the authors choose to include combination methods, they should consider incorporating other combinations of the baseline methods for a more balanced comparison.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors introduce a learning principle called Uniform Risk Minimization (URM), which addresses robustness to distribution shifts in machine learning, especially when test distributions differ from training distributions. URM trains DNNs to produce uniformly distributed feature representations in the final activation layer using adversarial training to address sub-population shifts and domain generalization without requiring prior knowledge of groups or domains during training. Theoretical analysis and experimental results over different datasets are performed to validate the effectiveness of the proposed URM learning strategy.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Please try to discuss the weaknesses.\n- Although there are many different baselines, the main characteristics of these methods are not clearly presented in Tables 2 and 3. It would be helpful for the authors to provide concise information about each baseline and reorganize the tables for better clarity.\n- Minor issue: try to fix the layout of Figure 2 for page 5 and page 6", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "VFqomhpkmr", "reviewer": "Reviewer_VVDg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- URM concept is interesting and results look promising\n- Paper is rigorous and proofs are clear and easy to follow\n- Theoretical study provides a strong mathematical grounding for URM\n- Training procedure (section 2.6) is elegant and offers a good pipeline to reuse for similar tasks\n\nWeaknesses:\n- Clarity: some assumptions / reasonings seem unclear, e.g. why choose the Dirichlet distribution in section 2.3\n- Lack of qualitative insights: some insights are provided (Figure 2), but I would have wanted additional comments and interpretations on how URM compares to ERM in different settings. Likewise, the paper did not spend much time analyzing specifically robustness and fairness (especially the former). A good example would have been analyzing a classification task with adversarial noise, either from a theoretical or empirical perspective", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a new learning principle called Uniform Risk Minimization as an alternative to Empirical Risk Minimization to ensure better fairness and robustness. The context is a classification task that can be subject to distribution shifts (covariate shifts, target shifts). In a first part, the paper provides a theoretical study on the Uniform Risk (UR) where the main result is proving that under some common assumptions, a uniform distribution in the training data minimizes the UR. This result is also proven for a uniform distribution for the learned feature representations of a neural network. Results are then corroborated by multiple experiments on various datasets, where a performance comparison with some known baseline models as well as some insights are provided.\nOverall, the URM concept is novel, and proofs and results look promising.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weaknesses section for more details. Some additional points:\n- Add the code for reproducibility\n- Section 2.6: shouldn't it be $\\lambda [E_{p_{tr}}(log(1 - D(G(x))) + E_{p_u}(log D(\\bar{z}))) ] $ (i.e. missing parentheses after $\\lambda$)? Also given that it is explained \"The same $\\lambda$ is applied to both update the discriminator and encoder so that one does not overpower the other\"", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: please see above for required changes in terms of clarification in the writeup in the theory section.

please address all the points that were not clear to the reviewers.

as promised, please upload the code to gitub for reproducibility.

Another point that would be nice to address is in terms of discussions on stability of the method when running experiments.
Decision: Accept (poster)