A Theory of Optimistically Universal Online Learnability for General Concept Classes

TL;DR

This paper characterizes which concept classes are optimistically universally online learnable with binary labels, providing minimal assumptions on data processes and designing algorithms that succeed universally, extending results to the agnostic case.

Contribution

It fully characterizes optimistically universal online learnability for all concept classes and designs general algorithms that work under minimal assumptions, including the agnostic setting.

Findings

Complete characterization of optimistically universal online learnability.

Design of general learning algorithms for minimal assumptions.

Extension of results to the agnostic case showing equivalences.

Abstract

We provide a full characterization of the concept classes that are optimistically universally online learnable with $\{0, 1\}$ labels. The notion of optimistically universal online learning was defined in [Hanneke, 2021] in order to understand learnability under minimal assumptions. In this paper, following the philosophy behind that work, we investigate two questions, namely, for every concept class: (1) What are the minimal assumptions on the data process admitting online learnability? (2) Is there a learning algorithm which succeeds under every data process satisfying the minimal assumptions? Such an algorithm is said to be optimistically universal for the given concept class. We resolve both of these questions for all concept classes, and moreover, as part of our solution, we design general learning algorithms for each case. Finally, we extend these algorithms and results to the agnostic case, showing an equivalence between the minimal assumptions on the data process for learnability in the agnostic and realizable cases, for every concept class, as well as the equivalence of optimistically universal learnability.