Asymptotically Optimal Strategies for Online Prediction with History-Dependent Experts

We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the d dimensional de Bruijn graph, where d is the number of days of history used by the experts. Previous work Drenska and Kohn (arXiv:2007.12732, 2020) established O(ε) optimal strategies for n= 2 experts and d≤ 4 days of history, while Drenska and Kohn (J Nonlinear Sci 30. 30(1), 137–173, 2020) established O(ε^{1 / 3}) optimal strategies for all n≥ 2 and all d≥ 1 , where the game is played for N steps and ε= N^{- 1 / 2}. In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish O(ε) optimal strategies for all values of n and d.