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 \(O(\varepsilon )\) optimal strategies for \(n=2\) experts and \(d\le 4\) days of history, while Drenska and Kohn (J Nonlinear Sci 30. 30(1), 137–173, 2020) established \(O(\varepsilon ^{1/3})\) optimal strategies for all \(n\ge 2\) and all \(d\ge 1\), where the game is played for N steps and \(\varepsilon =N^{-1/2}\). In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish \(O(\varepsilon )\) optimal strategies for all values of n and d.

对于历史依赖的专家，我们为在线预测问题建立了清晰的渐近最优策略。预测问题（部分地）在称为d维de Bruijn图的离散图上播放，其中d是专家使用的历史天数。Drenska和Kohn（arXiv：2007.12732，2020）的先前工作为\（n = 2 \）专家和\（d \ le 4 \）天的历史建立了\（O（\ varepsilon）\）最佳策略，而Drenska和Kohn （J Nonlinear Sci 30. 30（1），137–173，2020 ）为所有\（n \ ge 2 \）和所有\（O（\ varepsilon ^ {1/3}）\）建立了最优策略\（d \ ge 1 \），其中游戏进行了N步和\（\ varepsilon = N ^ {-1/2} \）的游戏。在本文中，我们证明了de Bruijn图上的最优条件与图泊松方程相对应，并且针对所有n和d值建立了\（O（\ varepsilon）\）最优策略。