Online Learning with an Almost Perfect Expert

TL;DR

This paper analyzes the multiclass online learning problem, demonstrating that when the best expert makes very few mistakes, the forecaster's expected mistakes are tightly bounded, especially in binary prediction scenarios.

Contribution

It provides a tight bound on the expected mistakes of the optimal forecaster when the best expert makes few errors, extending understanding of online learning with expert advice.

Findings

Expected mistakes are at most log_4 n + o(log_4 n) when the best expert makes o(log_4 n) mistakes.

The bound is tight, with adversary strategies showing worst-case scenarios in binary prediction.

The analysis applies to the regime where the best expert's mistakes are very small relative to the number of experts.

Abstract

We study the multiclass online learning problem where a forecaster makes a sequence of predictions using the advice of $n$ experts. Our main contribution is to analyze the regime where the best expert makes at most $b$ mistakes and to show that when $b = o(\log_4{n})$, the expected number of mistakes made by the optimal forecaster is at most $\log_4{n} + o(\log_4{n})$. We also describe an adversary strategy showing that this bound is tight and that the worst case is attained for binary prediction.