[Paper Review] Out-of-Distribution Generalization via Risk Extrapolation (REx)

David Krueger, Ethan Caballero+6 more|arXiv (Cornell University)|Mar 2, 2020

Domain Adaptation and Few-Shot LearningComputer Science68 references260 citations

TL;DR

REx introduces Risk Extrapolation (MM-REx and V-REx) to improve OOD generalization by equalizing training risks across domains, outperforming IRM under covariate shift in several tasks.

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.

Motivation & Objective

• Address distributional shift in multi-domain learning and enable robust generalization to unseen domains.
• Introduce Risk Extrapolation (REx) as a principled way to achieve invariant prediction through risk equality across training domains.
• Propose MM-REx (minimax extrapolation) and V-REx (variance penalty) as scalable, practical variants.
• Provide theoretical connections to causal mechanisms and compare with Invariant Risk Minimization (IRM) and DRO approaches.
• Demonstrate empirical benefits of REx on CMNIST variants, domain generalization benchmarks, and reinforcement learning tasks.

Proposed method

• Define OOD risk as the maximum risk over extrapolated domain distributions (MM-REx).
• Allow affine combinations of training risks with extrapolation via a tunable minimum weight (lambda_min).
• Introduce a simpler, stabilizing variant using the variance of domain risks (V-REx) with a beta regularizer.
• Show that equalizing risks leads to invariant prediction and can recover causal mechanisms under stated assumptions.
• Theoretically relate equalizing risks to learning the causal mechanism Y via Homoskedastic SCM assumptions; provide Theorems 1 and 2.
• Compare REx with IRM and other domain generalization methods, highlighting robustness to covariate shift as a key advantage.

Experimental results

Research questions

• RQ1 Can REx recover the causal mechanism of the target variable under multisource domains with interventions?
• RQ2 Does forcing equality (or low variance) of training risks yield invariant predictors and improved OOD robustness, especially under covariate shift and interventional shifts?
• RQ3 How does REx compare to IRM and standard ERM across tasks involving covariate shift, interventional shift, and mixed causal structures?
• RQ4 What are the theoretical guarantees linking risk equality to invariant prediction and causal discovery under specified assumptions?

Key findings

• REx outperforms IRM in settings with covariate shift and invariant-prediction requirements (e.g., CMNIST variants).
• Risk extrapolation flattens the training-risk plane, improving worst-case (OOD) performance.
• V-REx and MM-REx can achieve robust OOD generalization even when covariate shift and interventional shifts co-occur.
• REx can uncover invariant relationships and, under certain assumptions, identify the causal mechanism; however, performance depends on data noise and heteroskedasticity.
• In domain generalization benchmarks and reinforcement learning tasks, REx provides robustness and competitive performance relative to IRM and ERM.