Towards Optimal Algorithms for Prediction with Expert Advice
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.
Towards Optimal Algorithms For Prediction With Expert Advice
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. 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 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. 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…