Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization

Shiori SagawaPang Wei KohTatsunori B. HashimotoPercy Liang

article2019arXiv1,781 citations

Demonstrates that pairing group DRO with strong regularization solves worst-case generalization failures in overparameterized neural networks, improving minority group test accuracy by 10 to 40 percentage points on language and vision benchmarks.

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Modern deep learning systems routinely achieve exceptional average accuracy on standard test sets but frequently fail when deployed in real-world settings where spurious correlations break down. These models learn misleading shortcuts—such as associating water backgrounds with waterbirds, gender with hair color, or negation words with sentence contradictions—leading to severe drops in performance on minority groups. This failure poses serious operational, safety, and fairness risks in critical domains such as clinical diagnosis and automated text analysis.

The article evaluates why standard robust training techniques fail to protect complex, overparameterized neural networks against these group shifts and demonstrates how pairing distributionally robust optimization with deliberate regularization enables models to maintain high accuracy across all subgroups.

The researchers conducted an empirical and theoretical investigation using standard deep architectures, specifically fine-tuning ResNet-50 image models on bird classification (Waterbirds) and facial attribute recognition (CelebA), as well as a BERT language model on natural language inference (MultiNLI). They compared standard empirical risk minimization against group-based distributionally robust optimization—an approach that minimizes the worst-case loss across predefined subgroups. Additionally, the authors developed a scalable online stochastic optimization algorithm that adaptively updates model weights alongside a probability distribution over the data groups.

The investigation produced three critical findings. First, when models are trained to near-zero training error without strong regularization, robust optimization fails completely: both standard and robustly trained models achieve near-perfect training scores but suffer catastrophic test errors on minority groups (e.g., yielding worst-group test accuracies of only 41.1% to 65.7%). Second, applying strong regularization—such as increasing weight decay penalties by multiple orders of magnitude or halting training early—enables robust optimization to dramatically outperform standard approaches, improving worst-group test accuracy by 10 to 40 percentage points while maintaining average accuracy. Third, introducing group adjustments that penalize smaller groups more heavily to account for their higher risk of overfitting provides further performance gains, outperforming common heuristic techniques like standard importance upweighting, which can destabilize non-convex neural network training.

These findings demonstrate that overparameterized neural networks suffer from uneven generalization gaps across different subgroups rather than an inability to learn minority patterns. Consequently, conventional wisdom—which holds that large neural networks do not need strong regularization to achieve high average accuracy—does not apply when worst-case robustness is required. For decision-makers, this highlights that high average performance benchmarks can conceal severe vulnerabilities in rare scenarios, and mitigating these risks requires explicitly constraining model capacity during training.

Organizations developing or deploying neural networks where minority group accuracy is critical should adopt group-based robust optimization paired with elevated regularization and size-based group adjustments. Standard sample upweighting should be avoided as a standalone fix in complex models due to its theoretical and empirical shortcomings. Before deployment, engineering teams must define known confounding variables to form subgroups and evaluate performance on a group-by-group basis rather than relying exclusively on aggregate test metrics.

These conclusions are supported by strong empirical results and theoretical convergence proofs, though important boundaries remain. The proposed approach currently relies on knowing the relevant confounding attributes in advance to specify subgroups, and hyperparameter tuning requires access to balanced validation data. While supplemental experiments suggest the method remains effective under noisy group definitions, further research is required to automatically identify unmodeled subgroup shifts where prior attribute labels are entirely unavailable.

Microsoft Stanford University inference natural languages optimization distributionally robust neural networks group shifts worst-case generalization

arXiv: 1911.08731 kohpangwei/group_DRO

Paper: Fairness Without Demographics in Repeated Loss Minimization, Tatsunori B. Hashimoto et al. (2018). This paper establishes the distributionally robust optimization (DRO) formulation for controlling worst-case group risk that the source directly adapts and modifies for overparameterized neural networks.
Paper: Understanding deep learning requires rethinking generalization, Chiyuan Zhang et al. (2017). This foundational work demonstrates how overparameterized networks can achieve vanishing training loss, explaining why naive group DRO achieves zero worst-case training error without achieving worst-group generalization.
Paper: Towards Deep Learning Models Resistant to Adversarial Attacks, A. Ma̧dry et al. (2017). This work introduces the standard min-max robust optimization paradigm in deep learning that provides the conceptual foundation for distributionally robust training objectives.

Paper: WILDS: A Benchmark of in-the-Wild Distribution Shifts, Pang Wei Koh et al. (2020). This benchmark paper builds directly upon group shift formulations and establishes large-scale real-world datasets to evaluate worst-group generalization methods like Group DRO.
Paper: Shortcut learning in deep neural networks, Robert Geirhos et al. (2020). This perspective synthesizes and extends the broader phenomenon of models relying on spurious features and failing on atypical subgroups explored in the source.
Paper: Generalizing to Unseen Domains: A Survey on Domain Generalization, Jindong Wang et al. (2021). This comprehensive survey contextualizes group-robust optimization techniques within the wider landscape of domain generalization and out-of-distribution robustness.

Cover for Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization

Abstract

Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-defined groups. However, we find that naively applying group DRO to overparameterized neural networks fails: these models can perfectly fit the training data, and any model with vanishing average training loss also already has vanishing worst-case training loss. Instead, the poor worst-case performance arises from poor generalization on some groups. By coupling group DRO models with increased regularization---a stronger-than-typical L2 penalty or early stopping---we achieve substantially higher worst-group accuracies, with 10-40 percentage point improvements on a natural language inference task and two image tasks, while maintaining high average accuracies. Our results suggest that regularization is important for worst-group generalization in the overparameterized regime, even if it is not needed for average generalization. Finally, we introduce a stochastic optimization algorithm, with convergence guarantees, to efficiently train group DRO models.

Topics

inference natural languages optimization optimization problems regularization stochastic stochastic optimization algorithm test samples probabilistic inference network architecture

Keywords

distributionally robust neural networks group shifts worst-case generalization worst-group generalization gap group DRO models robust optimization empirical risk minimization

1 Introduction
2 Setup
2.1 Applications
3 Comparison between group DRO and ERM
3.1 ERM and DRO have poor worst-group accuracy in the overparameterized regime
3.2 DRO improves worst-group accuracy under appropriate regularization
3.3 Accounting for generalization through group adjustments improves DRO
4 Comparison between DRO and importance weighting
5 Algorithm
6 Related work
7 Discussion
References
A Proofs
A.1 Equivalence of DRO and importance weighting in the convex setting
A.2 Convergence rate of Algorithm
B Supplementary experiments
C Experimental details
C.1 Datasets
C.2 Models

Knowls

Knowl 1 — Group Distributionally Robust Optimization Formulation for Group Shifts
model/method
To build models robust to subpopulation distribution shifts and spurious correlations, the data distribution is modeled as a mixture of $m$ predefined groups indexed by $\mathcal{G} = \{1, \dots, m\}$ , where each group has distribution $P_g$ . The uncertainty set $\mathcal{Q}$ is defined as the set of all probability mixtures over these groups:

$\mathcal{Q} := \left\{ \sum_{g=1}^m q_g P_g : q \in \Delta_m \right\}$

where $\Delta_m = \{q \in \mathbb{R}^m : \sum_{g=1}^m q_g = 1, q_g \ge 0\}$ is the $(m-1)$ -dimensional probability simplex.

For a parameter space $\Theta$ , inputs $x \in \mathcal{X}$ , labels $y \in \mathcal{Y}$ , and a non-negative loss function $\ell: \Theta \times (\mathcal{X} \times \mathcal{Y}) \to \mathbb{R}_+$ , the worst-case expected loss over $\mathcal{Q}$ reduces to the maximum expected loss across groups:

$R(\theta) := \sup_{Q \in \mathcal{Q}} \mathbb{E}_{(x,y)\sim Q}[\ell(\theta; (x, y))] = \max_{g \in \mathcal{G}} \mathbb{E}_{(x,y)\sim P_g}[\ell(\theta; (x, y))]$

Given training samples split into empirical group distributions $\hat{P}_g$ over $n_g$ samples from group $g$ , the Group Distributionally Robust Optimization (Group DRO) objective minimizes the empirical worst-group risk:

$\hat{\theta}_{\text{DRO}} := \arg\min_{\theta \in \Theta} \left\{ \hat{R}(\theta) := \max_{g \in \mathcal{G}} \mathbb{E}_{(x,y)\sim \hat{P}_g}[\ell(\theta; (x, y))] \right\}$
Knowl 2 — Online Stochastic Minimax Optimization Algorithm for Group DRO
algorithm
The Group DRO empirical objective $\min_{\theta \in \Theta} \max_{q \in \Delta_m} \sum_{g=1}^m q_g \mathbb{E}_{(x,y)\sim P_g}[\ell(\theta; (x,y))]$ can be solved efficiently via a stochastic online minimax procedure that alternates between gradient descent updates on model parameters $\theta$ and exponentiated gradient ascent updates on the group probability distribution $q \in \Delta_m$ .
Input: Step sizes $\eta_q > 0$ , $\eta_\theta > 0$ ; group data distributions $P_g$ for each $g \in \{1, \dots, m\}$ ; total iterations $T$
Initialize $\theta^{(0)} \in \Theta$ and $q^{(0)} = (1/m, \dots, 1/m) \in \Delta_m$
for $t = 1, \dots, T$ do
    Sample group index $g \sim \text{Uniform}(1, \dots, m)$
    Sample training point $(x, y) \sim P_g$
     $q' \leftarrow q^{(t-1)}$
     $q'_g \leftarrow q^{(t-1)}_g \exp(\eta_q \ell(\theta^{(t-1)}; (x, y)))$
     $q^{(t)} \leftarrow q' / \sum_{g'=1}^m q'_{g'}$
     $\theta^{(t)} \leftarrow \theta^{(t-1)} - \eta_\theta q^{(t)}_g \nabla_\theta \ell(\theta^{(t-1)}; (x, y))$
end
Output: Final parameters $\theta^{(T)}$ or iterate average $\bar{\theta}^{(1:T)} = \frac{1}{T}\sum_{t=1}^T \theta^{(t)}$
In practice, stochastic gradient updates on $\theta$ are computed over minibatches with momentum, and the per-epoch compute time is within 5% of standard Empirical Risk Minimization SGD.
Knowl 3 — Group-Adjusted DRO for Disparate Group Sample Sizes
model/method
Because groups in imbalanced datasets differ substantially in sample size $n_g$ , the generalization gap $\delta_g := \mathbb{E}_{P_g}[\ell(\theta; (x, y))] - \mathbb{E}_{\hat{P}_g}[\ell(\theta; (x, y))]$ varies widely across groups, causing smaller groups to overfit more severely. To optimize an upper bound on worst-group test loss, the Group-Adjusted DRO estimator adds a group-size-dependent surrogate penalty $\hat{\delta}_g = C / \sqrt{n_g}$ to the empirical training risk of each group:

$\hat{\theta}_{\text{adj}} := \arg\min_{\theta \in \Theta} \max_{g \in \mathcal{G}} \left\{ \mathbb{E}_{(x,y)\sim \hat{P}_g}[\ell(\theta; (x, y))] + \frac{C}{\sqrt{n_g}} \right\}$

where $C \ge 0$ is a capacity-dependent tuning constant treated as a hyperparameter and $n_g$ is the number of training points in group $g$ . This penalty forces the optimization to prioritize achieving lower training loss on smaller groups to compensate for their larger anticipated generalization gaps.
Knowl 4 — Failure of Unregularized Group DRO in Overparameterized Neural Networks
empirical result
In the overparameterized regime, neural networks trained to convergence with default regularization (e.g., standard small $\ell_2$ penalty $\lambda = 10^{-4}$ and batch normalization) achieve near 100% training accuracy and vanishing training loss across all training groups. Under vanishing training loss, the worst-group empirical risk $\hat{R}(\theta) = \max_g \mathbb{E}_{\hat{P}_g}[\ell(\theta; (x, y))]$ is zero, which renders the empirical DRO objective mathematically identical to the empirical risk minimization (ERM) objective.

As a consequence, unregularized Group DRO performs identically to ERM on test data:
- On Waterbirds, ERM achieves 60.0% worst-group test accuracy and DRO achieves 76.9% worst-group test accuracy (with identical 100% worst-group training accuracy).
- On CelebA, ERM achieves 41.1% worst-group test accuracy and DRO achieves 41.1% worst-group test accuracy (with 99.9% and 100% worst-group training accuracy, respectively).
- On MultiNLI, ERM achieves 65.7% worst-group test accuracy and DRO achieves 66.4% worst-group test accuracy (both reaching $\ge 99.0\%$ worst-group training accuracy).
The poor worst-case test performance in overparameterized models stems from large worst-group generalization gaps rather than optimization failure on training loss.
Knowl 5 — Necessity of Strong Regularization for Group DRO Generalization
empirical result
Applying strong explicit regularization (substantially larger $\ell_2$ penalty) or implicit regularization (early stopping) prevents overparameterized neural networks from reaching zero training loss, which allows Group DRO to trade off group training errors and generalize well to worst-case groups.
1. Strong $\ell_2$ Regularization: Increasing the $\ell_2$ penalty coefficient to $\lambda = 1.0$ for ResNet-50 on Waterbirds and $\lambda = 0.1$ on CelebA reduces group generalization gaps:
  - Waterbirds worst-group test accuracy: ERM drops to 21.3% (train worst-group 35.7%), whereas Group DRO achieves 84.6% (train worst-group 97.5%).
  - CelebA worst-group test accuracy: ERM drops to 37.8% (train worst-group 40.4%), whereas Group DRO achieves 86.7% (train worst-group 93.4%).
2. Early Stopping: Curtailing training to 1 epoch for ResNet-50 on Waterbirds/CelebA and 3 epochs for BERT on MultiNLI yields:
  - Waterbirds worst-group test accuracy: ERM 6.7% vs. Group DRO 86.0%.
  - CelebA worst-group test accuracy: ERM 25.0% vs. Group DRO 88.3%.
  - MultiNLI worst-group test accuracy: ERM 66.0% vs. Group DRO 77.7%.
While strong regularization is not necessary for high average test accuracy in overparameterized models, it is essential for worst-group generalization.
Knowl 6 — Convergence Rate of Stochastic Online Group DRO in Convex Settings
theoretical result
Let the parameter set $\Theta \subseteq \mathbb{R}^d$ be convex and compact with $\|\theta\|_2 \le B_\Theta$ for all $\theta \in \Theta$ . Assume the loss function $\ell(\cdot; (x, y))$ is non-negative, convex, bounded by $B_\ell$ for all $(x, y) \in \mathcal{X} \times \mathcal{Y}$ , and $B_\nabla$ -Lipschitz continuous (meaning $\|\nabla_\theta \ell(\theta; (x, y))\|_2 \le B_\nabla$ ).

For the saddle-point objective $L(\theta, q) := \sum_{g=1}^m q_g \mathbb{E}_{(x,y)\sim P_g}[\ell(\theta; (x, y))]$ , the error of the iterate average $\bar{\theta}^{(1:T)} = \frac{1}{T}\sum_{t=1}^T \theta^{(t)}$ produced by the stochastic online Group DRO algorithm after $T$ steps is defined as:

$\varepsilon_T := \max_{q \in \Delta_m} L(\bar{\theta}^{(1:T)}, q) - \min_{\theta \in \Theta} \max_{q \in \Delta_m} L(\theta, q)$

The expected error is bounded by:

$\mathbb{E}[\varepsilon_T] \le 2m \sqrt{\frac{10(B_\Theta^2 B_\nabla^2 + B_\ell^2 \log m)}{T}}$

where the expectation is taken over the stochastic sampling of groups and data points.
Knowl 7 — Convex Equivalence and Non-Convex Inequivalence of Group DRO and Importance Weighting
theoretical result
For an uncertainty set $\mathcal{Q}$ over distributions on $\mathcal{Z} = \mathcal{X} \times \mathcal{Y}$ and a parameter space $\Theta \subseteq \mathbb{R}^d$ :
1. Convex Equivalence: If the loss $\ell(\cdot; z)$ is continuous and convex for all $z \in \mathcal{Z}$ , and both $\mathcal{Q}$ and $\Theta$ are compact and convex, then for any worst-group minimizer $\theta^* \in \arg\min_{\theta \in \Theta} \sup_{Q \in \mathcal{Q}} \mathbb{E}_{z \sim Q}[\ell(\theta; z)]$ , there exists a fixed distribution $Q^* \in \mathcal{Q}$ such that:
$\theta^* \in \arg\min_{\theta \in \Theta} \mathbb{E}_{z \sim Q^*}[\ell(\theta; z)]$

Thus, in convex settings, DRO is equivalent to static importance weighting under some target distribution $Q^*$ .
1. Non-Convex Inequivalence: When $\ell(\cdot; z)$ is non-convex, this equivalence fails. There exist finite datasets where no fixed importance weights $w \in \Delta_m$ can recover the robust minimax parameter $\theta^*$ . Even when stationary points match, the global minimizers of the importance-weighted risk do not coincide with the minimax DRO optimal solution $\theta^*$ .

Knowl 8 — Benchmark Accuracy Comparison of ERM, Upweighting, and Group DRO

data/table

Performance across benchmark tasks under full hyperparameter grid search (optimizing over $\ell_2$ penalty strength, training epochs, and group adjustments $C \in \{0, 1, 2, 3, 4, 5\}$ on validation worst-group accuracy). Accuracies are reported as percentages with binomial standard deviations in parentheses:

	Average Accuracy (%)			Worst-Group Accuracy (%)
Dataset	ERM	UW	DRO	ERM	UW	DRO
Waterbirds	97.0 (0.2)	95.1 (0.3)	93.5 (0.3)	63.7 (1.9)	88.0 (1.3)	91.4 (1.1)
CelebA	94.9 (0.2)	92.9 (0.2)	92.9 (0.2)	47.8 (3.7)	83.3 (2.8)	88.9 (2.3)
MultiNLI	82.8 (0.1)	81.2 (0.1)	81.4 (0.1)	66.4 (1.6)	64.8 (1.6)	77.7 (1.4)

Group DRO consistently outperforms standard Empirical Risk Minimization (ERM) on worst-group accuracy across all domains by 11.3 to 27.7 percentage points. It also outperforms standard inverse-frequency importance upweighting (UW), which fails on MultiNLI because static reweighting over-prioritizes minority groups at the expense of others when group losses are non-convex and unequally difficult to optimize.

Knowl 9 — Spurious Correlation Benchmark Datasets: Waterbirds, CelebA, and MultiNLI
experimental setup
Three datasets evaluate robustness to group shifts where a spurious attribute $a \in \mathcal{A}$ correlates with label $y \in \mathcal{Y}$ during training ( $m = |\mathcal{A}| \times |\mathcal{Y}|$ groups):
1. Waterbirds ( $m = 4$ ): Combines Caltech-UCSD Birds-200-2011 (CUB) foreground bird photographs cropped and pasted onto Places image backgrounds. Target $y \in \{\text{waterbird}, \text{landbird}\}$ , spurious background $a \in \{\text{water}, \text{land}\}$ . In training ( $n = 4795$ ), 95% of waterbirds appear on water and 95% of landbirds on land. The smallest training group (waterbirds on land) has 56 examples. Validation and test sets are balanced across groups.
2. CelebA ( $m = 4$ ): Celebrity face image dataset with target label $y \in \{\text{blond}, \text{dark}\}$ hair and spurious attribute $a \in \{\text{male}, \text{female}\}$ . Training set contains $n = 162770$ examples, with the minority group (blond males) having 1387 examples.
3. MultiNLI ( $m = 6$ ): Natural language inference classification with $y \in \{\text{entailment}, \text{neutral}, \text{contradiction}\}$ and spurious attribute $a \in \{\text{negation word}, \text{no negation word}\}$ (annotated by presence of 'nobody', 'no', 'never', or 'nothing' in the hypothesis). Training set contains $n = 206175$ examples with a 50-20-30 train-val-test split; the smallest group (entailment with negation) contains 1521 examples.

Coverage note — None was omitted; all key theoretical definitions, optimization algorithms, bounds, propositions, counterexamples, experimental setups, and empirical benchmark findings have been extracted into self-contained knowls.

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Citation

MLA

Sagawa, S., et al. “Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization”. arXiv, 2019, http://arxiv.org/abs/1911.08731v2.

APA

Sagawa, S., Koh, P. W., Hashimoto, T. B., & Liang, P. (2019). Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization. arXiv. http://arxiv.org/abs/1911.08731v2

Chicago

Sagawa, S., P. W. Koh, T. B. Hashimoto, and P. Liang. 2019. “Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization”. arXiv. http://arxiv.org/abs/1911.08731v2.

Harvard

Sagawa, S. et al. (2019) “Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1911.08731v2.

Vancouver

1. Sagawa S, Koh PW, Hashimoto TB, Liang P (2019) Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization. arXiv

BibTeX

@article{sagawa2019distributionally,
  title = {Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization},
  author = {Sagawa, Shiori and Koh, Pang Wei and Hashimoto, Tatsunori B. and Liang, Percy},
  year = {2019},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1911.08731v2},
  eprint = {1911.08731}
}

Metadata:arXiv

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Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization

Abstract

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Table of Contents

Knowls

Knowl 1 — Group Distributionally Robust Optimization Formulation for Group Shifts

Knowl 2 — Online Stochastic Minimax Optimization Algorithm for Group DRO

Knowl 3 — Group-Adjusted DRO for Disparate Group Sample Sizes

Knowl 4 — Failure of Unregularized Group DRO in Overparameterized Neural Networks

Knowl 5 — Necessity of Strong Regularization for Group DRO Generalization

Knowl 6 — Convergence Rate of Stochastic Online Group DRO in Convex Settings

Knowl 7 — Convex Equivalence and Non-Convex Inequivalence of Group DRO and Importance Weighting