Online Prediction With History-Dependent Experts: The General Case

TL;DR

This paper analyzes online binary sequence prediction with history-dependent experts, modeling it as a game between an investor and an adversarial market, and derives asymptotically optimal strategies using PDE methods.

Contribution

It extends previous work to multiple experts and longer history lengths, providing a PDE-based framework for optimal decision strategies in online prediction.

Findings

Convergence of the value function to a viscosity solution of a nonlinear PDE.

Asymptotically optimal strategies for the investor.

Extension of prior results to more experts and longer history.

Abstract

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over $N$ turns. The investor combines the predictions of $n\geq 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous $d$ days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \infty$ at a rate of $O(N^{-1/6})$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V.Kohn [13] for $n=2$ experts and $d\leq 4$ days of history. To appear in Communications on Pure and Applied Mathematics.