Efficient Discrepancy Testing for Learning with Distribution Shift

TL;DR

This paper introduces efficient algorithms for testing localized discrepancy distances between train and test distributions, enabling robust learning under distribution shifts with broad applicability and improved performance.

Contribution

It provides the first provably efficient algorithms for localized discrepancy testing, advancing the field of Testable Learning with Distribution Shift (TDS learning).

Findings

Universal learners succeed across large test distribution classes.

Achieve near-optimal error rates.

Exponential improvements for constant depth circuits.

Abstract

A fundamental notion of distance between train and test distributions from the field of domain adaptation is discrepancy distance. While in general hard to compute, here we provide the first set of provably efficient algorithms for testing localized discrepancy distance, where discrepancy is computed with respect to a fixed output classifier. These results imply a broad set of new, efficient learning algorithms in the recently introduced model of Testable Learning with Distribution Shift (TDS learning) due to Klivans et al. (2023). Our approach generalizes and improves all prior work on TDS learning: (1) we obtain universal learners that succeed simultaneously for large classes of test distributions, (2) achieve near-optimal error rates, and (3) give exponential improvements for constant depth circuits. Our methods further extend to semi-parametric settings and imply the first positive results for low-dimensional convex sets. Additionally, we separate learning and testing phases and obtain algorithms that run in fully polynomial time at test time.