Numerical solution of a PDE arising from prediction with expert advice

TL;DR

This paper explores the connection between prediction with expert advice in adversarial settings and a related PDE, developing numerical methods to analyze optimal strategies in high-dimensional cases.

Contribution

It introduces numerical techniques for solving a degenerate elliptic PDE arising from an online learning problem, enabling analysis of strategies in up to 10 dimensions.

Findings

Numerical methods effectively approximate PDE solutions in high dimensions.

Conjectures on the optimality of adversarial strategies, including the non-optimality of the COMB strategy.

Insights into the continuum limit of the game and strategy optimality.

Abstract

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.