# Prediction with Expert Advice: a PDE Perspective

arXiv: https://arxiv.org/abs/1904.11401 · HTML: https://mlsanity.com/paper/1904.11401/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2019.

## Built on by (most cited first, 17 of 17)

- [Online Minimax Multiobjective Optimization: Multicalibeating and Other Applications](https://mlsanity.com/paper/2108.03837/) (2021)
- [PDE-Based Optimal Strategy for Unconstrained Online Learning](https://mlsanity.com/paper/2201.07877/) (2022)
- [Discounted Adaptive Online Learning: Towards Better Regularization](https://mlsanity.com/paper/2402.02720/) (2024)
- [Unconstrained Dynamic Regret via Sparse Coding](https://mlsanity.com/paper/2301.13349/) (2023)
- [Optimal Comparator Adaptive Online Learning with Switching Cost](https://mlsanity.com/paper/2205.06846/) (2022)
- [Improving Adaptive Online Learning Using Refined Discretization](https://mlsanity.com/paper/2309.16044/) (2023)
- [Continuous Prediction with Experts' Advice](https://mlsanity.com/paper/2206.00236/) (2022)
- [A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit](https://mlsanity.com/paper/2202.05767/) (2022)
- [Numerical solution of a PDE arising from prediction with expert advice](https://mlsanity.com/paper/2406.05754/) (2024)
- [An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB](https://mlsanity.com/paper/2609.14892/) (2026)
- [A New Approach to Drifting Games, Based on Asymptotically Optimal Potentials](https://mlsanity.com/paper/2207.11405/) (2022)
- [Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant](https://mlsanity.com/paper/2609.27206/) (2026)
- [Optimal Tracking in Prediction with Expert Advice](https://mlsanity.com/paper/2208.03708/) (2022)
- [A PDE approach for regret bounds under partial monitoring](https://mlsanity.com/paper/2209.01256/) (2022)
- [Reasoning without Regret](https://mlsanity.com/paper/2504.09777/) (2025)
- [Operationalizing Stein's Method for Online Linear Optimization: CLT-Based Optimal Tradeoffs](https://mlsanity.com/paper/2602.06545/) (2026)
- [On prediction from expert advice with more than five experts](https://mlsanity.com/paper/2610.05186/) (2026)

## Abstract (arXiv)

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.