Sparsity and Out-of-Distribution Generalization

TL;DR

This paper presents a theoretical framework explaining out-of-distribution generalization through the lens of sparsity and feature relevance, supported by formal proofs and extensions to subspace classifiers.

Contribution

It introduces a formal account of OOD generalization based on sparse hypotheses and feature overlap, extending classic sample complexity bounds.

Findings

Sparse hypotheses can generalize across distributions with overlapping features.

A formal theorem generalizes classic sample complexity bounds to OOD scenarios.

Extension to subspace juntas shows ground truth depends on low-dimensional feature subspaces.

Abstract

Explaining out-of-distribution generalization has been a central problem in epistemology since Goodman's "grue" puzzle in 1946. Today it's a central problem in machine learning, including AI alignment. Here we propose a principled account of OOD generalization with three main ingredients. First, the world is always presented to experience not as an amorphous mass, but via distinguished features (for example, visual and auditory channels). Second, Occam's Razor favors hypotheses that are "sparse," meaning that they depend on as few features as possible. Third, sparse hypotheses will generalize from a training to a test distribution, provided the two distributions sufficiently overlap on their restrictions to the features that are either actually relevant or hypothesized to be. The two distributions could diverge arbitrarily on other features. We prove a simple theorem that formalizes the above intuitions, generalizing the classic sample complexity bound of Blumer et al. to an OOD context. We then generalize sparse classifiers to subspace juntas, where the ground truth classifier depends solely on a low-dimensional linear subspace of the features.