Out-of-Distribution Generalization via Risk Extrapolation (REx) (2003.00688v5)

Abstract: Distributional shift is one of the major obstacles when transferring machine learning prediction systems from the lab to the real world. To tackle this problem, we assume that variation across training domains is representative of the variation we might encounter at test time, but also that shifts at test time may be more extreme in magnitude. In particular, we show that reducing differences in risk across training domains can reduce a model's sensitivity to a wide range of extreme distributional shifts, including the challenging setting where the input contains both causal and anti-causal elements. We motivate this approach, Risk Extrapolation (REx), as a form of robust optimization over a perturbation set of extrapolated domains (MM-REx), and propose a penalty on the variance of training risks (V-REx) as a simpler variant. We prove that variants of REx can recover the causal mechanisms of the targets, while also providing some robustness to changes in the input distribution ("covariate shift"). By appropriately trading-off robustness to causally induced distributional shifts and covariate shift, REx is able to outperform alternative methods such as Invariant Risk Minimization in situations where these types of shift co-occur.

• The paper introduces REx, a novel approach that leverages training domain shifts to tackle out-of-distribution generalization.
• It details MM-REx, optimizing worst-case extrapolated risks, and V-REx, penalizing risk variance to boost model robustness.
• Experimental results highlight REx's robust performance on benchmarks like Colored MNIST and reinforcement learning tasks.

Out-of-Distribution Generalization via Risk Extrapolation

The paper, "Out-of-Distribution Generalization via Risk Extrapolation," addresses the persistent challenge of distributional shift in machine learning models when applied to real-world settings. The authors propose Risk Extrapolation (REx) as a method to tackle the problem of out-of-distribution (OOD) generalization by focusing on the differences observed across multiple training domains.

Introduction

Neural networks, while achieving super-human performance on training distributions, struggle with OOD generalization. This performance drop is often due to reliance on spurious features unrelated to the core prediction task. REx aims to mitigate this by using the distribution shifts observed during training to inform predictions about potential shifts at test time.

Methodology

REx is motivated by robust optimization over a perturbation set of extrapolated domains, leading to the formulation of Minimax-REx (MM-REx) and Variance-REx (V-REx). The former aims to optimize worst-case performance over affine combinations of training risks, while the latter introduces a penalty on the variance of training risks.

Minimax-REx (MM-REx)

The MM-REx approach extends conventional robust optimization (DRO) to include extrapolated risks by considering affine combinations of training risks: $MM-REx(\theta) = \max_{\substack{ \Sigma_e \lambda_e = 1 \ \lambda_e \geq \lambda_{\min}} \sum_{e=1}^m \lambda_e R_e(\theta),$ where $\lambda_{\min}$ is a hyperparameter controlling the degree of extrapolation.

Variance-REx (V-REx)

V-REx simplifies the approach by penalizing the variance of the risks directly: $$R_{\mathrm{V-REx}}(\theta) = \beta \; \mathrm{Var}(\{R_1(\theta), ..., R_m(\theta)\}) + \sum^m_{e=1} R_e(\theta),$$ where $\beta$ balances between minimizing average risk and equalizing the risks.

Theoretical Foundations

The authors prove the theoretical soundness of REx, showing that it recovers causal mechanisms by enforcing equality of risks across training domains. In particular:

1. Theorem 1 demonstrates that MM-REx can identify causal mechanisms in linear Structural Equation Models (SEMs) given sufficiently diverse interventions.
2. Theorem 2 extends this result to general SCMs, illustrating that exact risk equality ensures the model learns the causal mechanisms of the target variable.

Experimental Results

REx and its variants were compared with existing methods (e.g., Invariant Risk Minimization (IRM)) across different datasets and settings:

1. Colored MNIST: On the canonical CMNIST dataset and its variants with added covariate shift, V-REx demonstrated superior performance, particularly in settings where covariate shift co-occurs with interventional distribution shifts.
2. Linear SEMs: The authors showed that REx outperformed IRM in scenarios with domain-homoskedastic noise but struggled with domain-heteroskedastic noise.
3. Domain Generalization: In the DomainBed suite of benchmarks, REx, IRM, and ERM performed comparably, indicating no general method outperformed others consistently for domain generalization.
4. Reinforcement Learning: Extending the framework to reinforcement learning tasks, REx outperformed both IRM and ERM, suggesting robust learning benefits from REx in high-dimensional state spaces typical of RL tasks.

Implications and Future Work

Practically, REx provides a robust method against covariate shifts and varying domain noise levels. Theoretically, it consolidates understanding of causal discovery in OOD scenarios. Future work could focus on a deeper exploration of domain-specific noise handling and refining hyperparameter selection heuristics to enhance REx’s applicability across diverse domains.

Conclusion

REx, both in its MM-REx and V-REx forms, offers a substantial advancement in handling OOD generalization, going beyond invariant prediction to providing robustness against various kinds of distributional shifts. The rigorous theoretical backing, combined with extensive experimental validation across different domains, underscores REx as a potent tool for modern machine learning applications facing real-world distributional uncertainties.