Achievable distributional robustness when the robust risk is only partially identified

TL;DR

This paper explores the limits of distributional robustness in machine learning when the robust risk cannot be fully identified, proposing a new measure that improves generalization in partially identifiable scenarios.

Contribution

It introduces the worst-case robust risk as a new robustness measure applicable under partial identifiability, and demonstrates its advantages over existing methods in theory and real data.

Findings

Existing robustness methods are suboptimal under partial identifiability.

The proposed measure improves test error in gene expression data.

Partial identifiability allows for better robustness than traditional approaches.

Abstract

In safety-critical applications, machine learning models should generalize well under worst-case distribution shifts, that is, have a small robust risk. Invariance-based algorithms can provably take advantage of structural assumptions on the shifts when the training distributions are heterogeneous enough to identify the robust risk. However, in practice, such identifiability conditions are rarely satisfied -- a scenario so far underexplored in the theoretical literature. In this paper, we aim to fill the gap and propose to study the more general setting when the robust risk is only partially identifiable. In particular, we introduce the worst-case robust risk as a new measure of robustness that is always well-defined regardless of identifiability. Its minimum corresponds to an algorithm-independent (population) minimax quantity that measures the best achievable robustness under partial identifiability. While these concepts can be defined more broadly, in this paper we introduce and derive them explicitly for a linear model for concreteness of the presentation. First, we show that existing robustness methods are provably suboptimal in the partially identifiable case. We then evaluate these methods and the minimizer of the (empirical) worst-case robust risk on real-world gene expression data and find a similar trend: the test error of existing robustness methods grows increasingly suboptimal as the fraction of data from unseen environments increases, whereas accounting for partial identifiability allows for better generalization.