Efficient and Optimal Fixed-Time Regret with Two Experts
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Prediction with expert advice is a foundational problem in online learning. In instances with T rounds and n experts, the classical Multiplicative Weights Update method suffers at most √((T/2)ln n) regret when T is known beforehand. Moreover, this is asymptotically optimal when both T and n grow to infinity. However, when the number of experts n is small/fixed, algorithms with better regret guarantees exist. Cover showed in 1967 a dynamic programming algorithm for the two-experts problem restricted to {0,1} costs that suffers at most √(T/2π) + O(1) regret with O(T^2) pre-processing time. In this work, we propose an optimal algorithm for prediction with two experts' advice that works even for costs in [0,1] and with O(1) processing time per turn. Our algorithm builds up on recent work on the experts problem based on techniques and tools from stochastic calculus.
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