# Finite-Time 4-Expert Prediction Problem

arXiv: https://arxiv.org/abs/1911.10936 · HTML: https://mlsanity.com/paper/1911.10936/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2019.

## Built on by (most cited first, 14 of 14)

- [PDE-Based Optimal Strategy for Unconstrained Online Learning](https://mlsanity.com/paper/2201.07877/) (2022)
- [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)
- [Efficient and Optimal Fixed-Time Regret with Two Experts](https://mlsanity.com/paper/2203.07577/) (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)
- [A PDE approach for regret bounds under partial monitoring](https://mlsanity.com/paper/2209.01256/) (2022)
- [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)

We explicitly solve the nonlinear PDE that is the continuous limit of dynamic programming of expert prediction problem in finite horizon setting with N=4 experts. The expert prediction problem is formulated as a zero sum game between a player and an adversary. By showing that the solution is C2, we are able to show that the strategies conjectured in arXiv:1409.3040G form an asymptotic Nash equilibrium. We also prove the "Finite vs Geometric regret" conjecture proposed in arXiv:1409.3040G for N=4, and and show that this conjecture in fact follows from the conjecture that the comb strategies are optimal.