Stability and List-Replicability for Agnostic Learners

2025-01-09

AI Summary

This paper addresses the theoretical gap between stability and learnability in agnostic online binary classification, where classical stable learning frameworks fail when the Littlestone dimension is infinite or only a small number of errors is permitted. Method: The authors introduce and rigorously characterize two relaxed notions of stability; they combine learning-theoretic analysis, constructive design of stable algorithms, and adversarial arguments to establish precise learnability criteria. Results: They prove, for the first time, that stability defined via excess error is equivalent to finiteness of the Littlestone dimension—and disprove stable learnability for classes with infinite Littlestone dimension. This fully resolves two open problems posed by Chase et al. Specifically, one relaxation is shown to be equivalent to online learnability, while the other remains restricted to finite hypothesis classes. The work establishes novel, rigorous foundations for stability in agnostic online learning.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Two seminal papers--Alon, Livni, Malliaris, Moran (STOC 2019) and Bun, Livni, and Moran (FOCS 2020)--established the equivalence between online learnability and globally stable PAC learnability in binary classification. However, Chase, Chornomaz, Moran, and Yehudayoff (STOC 2024) recently showed that this equivalence does not hold in the agnostic setting. Specifically, they proved that in the agnostic setting, only finite hypothesis classes are globally stable learnable. Therefore, agnostic global stability is too restrictive to capture interesting hypothesis classes. To address this limitation, Chase emph{et al.} introduced two relaxations of agnostic global stability. In this paper, we characterize the classes that are learnable under their proposed relaxed conditions, resolving the two open problems raised in their work. First, we prove that in the setting where the stability parameter can depend on the excess error (the gap between the learner's error and the best achievable error by the hypothesis class), agnostic stability is fully characterized by the Littlestone dimension. Consequently, as in the realizable case, this form of learnability is equivalent to online learnability. As part of the proof of this theorem, we strengthen the celebrated result of Bun et al. by showing that classes with infinite Littlestone dimension are not stably PAC learnable, even if we allow the stability parameter to depend on the excess error. For the second relaxation proposed by Chase et al., we prove that only finite hypothesis classes are globally stable learnable even if we restrict the agnostic setting to distributions with small population loss.

Problem

• Online Learning
• Stability
• Binary Classification

Innovation

• Littlestone Dimension
• Online Learning Stability
• Finite Class Hypotheses