Equivalence of Coarse and Fine-Grained Models for Learning with Distribution Shift

TL;DR

This paper proves an equivalence between coarse and fine-grained models for learning under distribution shift, revealing new hardness results and proposing methods for learning halfspaces with membership queries.

Contribution

It establishes a surprising equivalence between PQ and TDS learning models in a distribution-free setting, and introduces a boosting method via branching programs.

Findings

Equivalence between PQ and TDS learning models in distribution-free setting.

First hardness results for distribution-free TDS learning of halfspaces.

Membership queries enable efficient distribution-free learning of halfspaces.

Abstract

Recent work on provably efficient algorithms for learning with distribution shift has focused on two models: PQ learning (Goldwasser et al. (2020)) and TDS learning (Klivans et al. (2024)). Algorithms for TDS learning are allowed to reject a test set entirely if distribution shift is detected. In contrast, PQ learners may only reject points that are deemed out-of-distribution on an individual basis. Our main result is a surprising equivalence between these two models in the distribution-free setting. In particular, we give an efficient black-box reduction from PQ learning to TDS learning for any Boolean concept class. This equivalence implies the first hardness results for distribution-free TDS learning of basic classes such as halfspaces. The main technical contribution underlying our equivalence is a method for boosting, via branching programs, the weak distinguishing power of TDS learners that have rejected the target domain. We also show that giving a learner access to membership queries sidesteps these hardness results and allows for efficient, distribution-free PQ learnability of halfspaces. Our algorithm iteratively recovers large-margin separators obtained by applying successive Forster transforms on the training data.