Computable universal online learning

TL;DR

This paper investigates the conditions under which universal online learning can be implemented as a computable algorithm, revealing limitations and characterizing learnable classes in the computability context.

Contribution

It provides the first characterization of computable universal online learning and explores its variants, bridging the gap between theoretical learnability and practical computability.

Findings

Universal online learning does not imply computable universal online learning.

Exact characterization of classes learnable in the computable agnostic setting.

Conditions under which proper universal online learning is possible.

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.