An Online Learning Approach to Interpolation and Extrapolation in Domain Generalization

TL;DR

This paper models domain generalization as an online game, demonstrating that extrapolation is computationally harder than interpolation, but ERM remains minimax-optimal for both tasks, providing a new theoretical framework.

Contribution

It introduces a formal online game framework for domain generalization, analyzing the computational and statistical aspects of interpolation and extrapolation.

Findings

Extrapolation is computationally more difficult than interpolation.

ERM is minimax-optimal for both interpolation and extrapolation.

Statistical complexity of both tasks is similar.

Abstract

A popular assumption for out-of-distribution generalization is that the training data comprises sub-datasets, each drawn from a distinct distribution; the goal is then to "interpolate" these distributions and "extrapolate" beyond them -- this objective is broadly known as domain generalization. A common belief is that ERM can interpolate but not extrapolate and that the latter is considerably more difficult, but these claims are vague and lack formal justification. In this work, we recast generalization over sub-groups as an online game between a player minimizing risk and an adversary presenting new test distributions. Under an existing notion of inter- and extrapolation based on reweighting of sub-group likelihoods, we rigorously demonstrate that extrapolation is computationally much harder than interpolation, though their statistical complexity is not significantly different. Furthermore, we show that ERM -- or a noisy variant -- is provably minimax-optimal for both tasks. Our framework presents a new avenue for the formal analysis of domain generalization algorithms which may be of independent interest.