  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 [11] established $O(\varepsilon)$ optimal strategies for $n=2$ experts and
$d\leq 4$ days of history, while [10] established $O(\varepsilon^{1/3})$
optimal strategies for all $n\geq 2$ and all $d\geq 1$, 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 $d$.

Optimization and Control;Computer Science and Game Theory;Machine Learning;Analysis of PDEs