Towards Optimal Algorithms for Prediction with Expert Advice

TL;DR

This paper develops optimal algorithms and adversaries for the prediction with expert advice problem, extending known results from 2 experts to 3 and beyond, using probabilistic and asymptotic analysis.

Contribution

It introduces the first optimal algorithms for 3 experts, generalizes to more experts, and connects probability matching to minimax optimality in adversarial settings.

Findings

Optimal algorithms for 3 experts are probability matching against a specific adversary.

The probability matching algorithm is both adversarial and minimax optimal.

Extended analysis provides improved algorithms and bounds for 4 or more experts.

Abstract

We study the classical problem of prediction with expert advice in the adversarial setting with a geometric stopping time. In 1965, Cover gave the optimal algorithm for the case of 2 experts. In this paper, we design the optimal algorithm, adversary and regret for the case of 3 experts. Further, we show that the optimal algorithm for $2$ and $3$ experts is a probability matching algorithm (analogous to Thompson sampling) against a particular randomized adversary. Remarkably, our proof shows that the probability matching algorithm is not only optimal against this particular randomized adversary, but also minimax optimal. Our analysis develops upper and lower bounds simultaneously, analogous to the primal-dual method. Our analysis of the optimal adversary goes through delicate asymptotics of the random walk of a particle between multiple walls. We use the connection we develop to random walks to derive an improved algorithm and regret bound for the case of $4$ experts, and, provide a general framework for designing the optimal algorithm and adversary for an arbitrary number of experts.