# Prediction with Expert Advice: a PDE Perspective

## TL;DR

This paper models online prediction with expert advice as a zero-sum game and characterizes its value through a nonlinear PDE, providing a continuum perspective and revealing optimal strategies for both predictor and adversary.

## Contribution

It introduces a PDE-based framework for analyzing online prediction with expert advice, connecting game theory with optimal control and continuum limits.

## Key findings

- Game value characterized as viscosity solution of a nonlinear PDE
- Optimal strategies for predictor and adversary derived from PDE analysis
- Provides a continuum perspective linking discrete prediction games to PDEs

## Abstract

This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor's and the opponent's minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin, Peres, and Sivan (Proc SODA 2016). Our techniques are similar to those of Kohn and Serfaty (Comm Pure Appl Math 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.