# PDE approach to the problem of online prediction with expert advice: a construction of potential-based strategies

## TL;DR

This paper introduces a PDE-based framework for online prediction with expert advice, linking supersolutions of a nonlinear PDE to potential functions that guide regret-minimizing strategies.

## Contribution

It develops a novel PDE approach to construct potential-based strategies for online prediction, extending classical methods with a rigorous mathematical foundation.

## Key findings

- Supersolutions of a nonlinear PDE relate to potential functions in prediction.
- Potential-based strategies satisfy the Blackwell condition.
- A new upper bound for worst-case regret is established.

## Abstract

We consider a sequence of repeated prediction games and formally pass to the limit. The supersolutions of the resulting non-linear parabolic partial differential equation are closely related to the potential functions in the sense of N.\,Cesa-Bianci, G.\,Lugosi (2003). Any such supersolution gives an upper bound for forecaster's regret and suggests a potential-based prediction strategy, satisfying the Blackwell condition. A conventional upper bound for the worst-case regret is justified by a simple verification argument.