# Numerical solution of a PDE arising from prediction with expert advice

arXiv: https://arxiv.org/abs/2406.05754 · HTML: https://mlsanity.com/paper/2406.05754/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 9 Jun 2024.

## Builds on (most cited first, 18 of 18)

- [The game theoretic p-Laplacian and semi-supervised learning with few labels](https://mlsanity.com/paper/1711.10144/) (2017)
- [Prediction with Expert Advice: a PDE Perspective](https://mlsanity.com/paper/1904.11401/) (2019)
- [New Potential-Based Bounds for Prediction with Expert Advice](https://mlsanity.com/paper/1911.01641/) (2019)
- [Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data](https://mlsanity.com/paper/1710.10364/) (2017)
- [Analysis and algorithms for ℓp-based semi-supervised learning on graphs](https://mlsanity.com/paper/1901.05031/) (2019)
- [Finite-Time 4-Expert Prediction Problem](https://mlsanity.com/paper/1911.10936/) (2019)
- [Towards Optimal Algorithms for Prediction with Expert Advice](https://mlsanity.com/paper/1409.3040/) (2014)
- [On the asymptotic optimality of the comb strategy for prediction with expert advice](https://mlsanity.com/paper/1902.02368/) (2019)
- [Tight Lower Bounds for Multiplicative Weights Algorithmic Families](https://mlsanity.com/paper/1607.02834/) (2016)
- [A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts](https://mlsanity.com/paper/2007.12732/) (2020)
- [The limit shape of convex hull peeling](https://mlsanity.com/paper/1805.08278/) (2018)
- [Experimental Evidence for Asymptotic Non-Optimality of Comb Adversary Strategy](https://mlsanity.com/paper/1912.01548/) (2019)
- [New Potential-Based Bounds for the Geometric-Stopping Version of Prediction with Expert Advice](https://mlsanity.com/paper/1912.03132/) (2019)
- [Malicious Experts versus the multiplicative weights algorithm in online prediction](https://mlsanity.com/paper/2003.08457/) (2020)
- [Online Prediction With History-Dependent Experts: The General Case](https://mlsanity.com/paper/2008.00052/) (2020)
- [PDE approach to the problem of online prediction with expert advice: a construction of potential-based strategies](https://mlsanity.com/paper/1705.01091/) (2017)
- [Asymptotically optimal strategies for online prediction with history-dependent experts](https://mlsanity.com/paper/2008.13703/) (2020)
- [Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian](https://mlsanity.com/paper/2401.07463/) (2024)

## Built on by (most cited first, 3 of 3)

- [An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB](https://mlsanity.com/paper/2609.14892/) (2026)
- [Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant](https://mlsanity.com/paper/2609.27206/) (2026)
- [On prediction from expert advice with more than five experts](https://mlsanity.com/paper/2610.05186/) (2026)

## Abstract (arXiv)

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≤ 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.