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# Efficient and Optimal Fixed-Time Regret with Two Experts

*Proceedings of The 33rd International Conference on Algorithmic Learning Theory*, PMLR 167:436-464, 2022.

#### Abstract

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 \(\sqrt{(T/2)\ln n}\) 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 \(\sqrt{T/2\pi} + O(1)\) regret with \(O(T^2)\) pre-processing time. In this work, we propose an optimal algorithm for prediction with two expertsâ€™ 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.