Computable universal online learning

2025-10-21

AI Summary

This paper investigates the feasibility of universal online learning under computability constraints, specifically whether learning processes for binary classification can be realized as effective computer programs—a dimension overlooked in classical online learning theory. Method: Integrating computability theory with online learning frameworks, the authors employ adversarial analysis and logical techniques to formalize and characterize computable universal online learning. Contribution/Results: They establish the first rigorous model of computable universal online learning and reveal a fundamental gap between abstract learnability and algorithmic realizability. Crucially, they prove that even when a hypothesis class is itself computable, universal online learnability does not guarantee the existence of a computable learner. By analyzing both agnostic and proper learning settings, they derive necessary and sufficient conditions for computable learnability in multiple scenarios, precisely linking structural properties of hypothesis classes—such as finite Littlestone dimension or computable empirical risk minimization—to the existence of computable online learners.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Understanding when learning is possible is a fundamental task in the theory of machine learning. However, many characterizations known from the literature deal with abstract learning as a mathematical object and ignore the crucial question: when can learning be implemented as a computer program? We address this question for universal online learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC'21). In this model, there is no hypothesis fixed in advance; instead, Adversary -- playing the role of Nature -- can change their mind as long as local consistency with the given class of hypotheses is maintained. We require Learner to achieve a finite number of mistakes while using a strategy that can be implemented as a computer program. We show that universal online learning does not imply computable universal online learning, even if the class of hypotheses is relatively easy from a computability-theoretic perspective. We then study the agnostic variant of computable universal online learning and provide an exact characterization of classes that are learnable in this sense. We also consider a variant of proper universal online learning and show exactly when it is possible. Together, our results give a more realistic perspective on the existing theory of online binary classification and the related problem of inductive inference.

Problem

• Characterizing when universal online learning can be implemented computationally
• Establishing computable learnability conditions for agnostic online classification
• Determining feasibility of proper universal online learning with programs

Innovation

• Computable universal online learning implementation
• Characterizing agnostic computable online learning
• Proper universal online learning feasibility conditions