Towards Optimal Algorithms for Prediction with Expert Advice

Nick Gravin, Yuval Peres, Balasubramanian Sivan

Published 2014-09-10Version 1

We study the classic problem of prediction with expert advice in the adversarial setting. Focusing on settings with a constant number of experts, we develop optimal algorithms and obtain precisely optimal regret values for the case of 2 and 3 experts. Further, we develop regret lower bounds for the multiplicative weights algorithm that exactly match the known upper bounds for an arbitrary number of experts k. This establishes a constant factor separation between the regrets achieved by the optimal algorithm and the multiplicative weights algorithm for a constant number of experts. All prior work on this problem has been restricted to optimal algorithms for special cases of adversary, or, algorithms that are optimal only in the doubly asymptotic sense: when both the number of experts and the time horizon go to infinity. In contrast to these results, our algorithms are exactly optimal for the most general adversary. Our main tool is the minimax principle which lets us analyze the optimal adversary to compute optimal regret values. While analyzing the optimal adversary, we establish deep connections with non-trivial aspects of random walk. We further use this connection to develop an improved regret bound for the case of 4 experts.