Simple online learning with consistent oracle

TL;DR

This paper introduces a simple online learning algorithm using a consistent oracle, achieving mistake bounds exponential in Littlestone dimension, with a simpler proof and improved bounds over previous work.

Contribution

It presents a novel, simpler online learning algorithm with mistake bounds of O(256^d) in the consistent oracle model, improving upon prior bounds.

Findings

Algorithm makes at most O(256^d) mistakes.

Proof relies on basic properties of Littlestone dimension.

Lower bound of 3^d mistakes established.

Abstract

We consider online learning in the model where a learning algorithm can access the class only via the \emph{consistent oracle} -- an oracle, that, at any moment, can give a function from the class that agrees with all examples seen so far. This model was recently considered by Assos et al.~(COLT'23). It is motivated by the fact that standard methods of online learning rely on computing the Littlestone dimension of subclasses, a computationally intractable problem. Assos et al.~gave an online learning algorithm in this model that makes at most $C^d$ mistakes on classes of Littlestone dimension $d$, for some absolute unspecified constant $C > 0$. We give a novel algorithm that makes at most $O(256^d)$ mistakes. Our proof is significantly simpler and uses only very basic properties of the Littlestone dimension. We also show that there exists no algorithm in this model that makes less than $3^d$ mistakes.