Asymptotically optimal strategies for online prediction with history-dependent experts

TL;DR

This paper develops sharp, asymptotically optimal strategies for online prediction with history-dependent experts, using a graph Poisson equation approach to handle complex dependencies over de Bruijn graphs.

Contribution

It introduces a novel connection between optimal strategies and graph Poisson equations, extending optimality results to all expert and history lengths.

Findings

Established $O(rac{1}{ oot 2 })$ optimal strategies for all expert and history lengths.

Connected optimality conditions to a graph Poisson equation.

Extended previous results to a broader setting with general $n$ and $d$.

Abstract

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 $d$.