On Computable Online Learning

TL;DR

This paper explores the limits of computable online learning, introducing new optimality concepts, revealing that classical complexity measures no longer suffice, and establishing computational separations between different learning paradigms.

Contribution

It provides a necessary and sufficient condition for optimal computable online learning and introduces the concept of anytime optimal learning, expanding the theoretical understanding of online learning.

Findings

Littlestone dimension does not characterize optimal mistake bounds in c-online learning.

Existence of a computational separation between a-optimal and optimal online learning.

Finiteness of Littlestone dimension is not necessary for c-online learnability under weaker computability.

Abstract

We initiate a study of computable online (c-online) learning, which we analyze under varying requirements for "optimality" in terms of the mistake bound. Our main contribution is to give a necessary and sufficient condition for optimal c-online learning and show that the Littlestone dimension no longer characterizes the optimal mistake bound of c-online learning. Furthermore, we introduce anytime optimal (a-optimal) online learning, a more natural conceptualization of "optimality" and a generalization of Littlestone's Standard Optimal Algorithm. We show the existence of a computational separation between a-optimal and optimal online learning, proving that a-optimal online learning is computationally more difficult. Finally, we consider online learning with no requirements for optimality, and show, under a weaker notion of computability, that the finiteness of the Littlestone dimension no longer characterizes whether a class is c-online learnable with finite mistake bound. A potential avenue for strengthening this result is suggested by exploring the relationship between c-online and CPAC learning, where we show that c-online learning is as difficult as improper CPAC learning.