Towards Optimal Algorithms for Prediction with Expert Advice
This work provides incremental improvements in online learning theory by solving a specific case and extending it, benefiting researchers in algorithmic decision-making under uncertainty.
The paper tackles the problem of prediction with expert advice in an adversarial setting with a geometric stopping time, designing the optimal algorithm, adversary, and regret for 3 experts and extending it to 4 experts and a general framework for arbitrary numbers. It shows that the optimal algorithm is a probability matching method analogous to Thompson sampling, proving it is minimax optimal with specific regret bounds.
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.