# An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB

arXiv: https://arxiv.org/abs/2609.14892 · HTML: https://mlsanity.com/paper/2609.14892/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [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)
- [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)
- [Towards Minimax Online Learning with Unknown Time Horizon](https://mlsanity.com/paper/1307.8187/) (2013)
- [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)
- [Asymptotically optimal strategies for online prediction with history-dependent experts](https://mlsanity.com/paper/2008.13703/) (2020)
- [Numerical solution of a PDE arising from prediction with expert advice](https://mlsanity.com/paper/2406.05754/) (2024)

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

- [Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant](https://mlsanity.com/paper/2609.27206/) (2026)
- [The finite-horizon five-expert prediction problem](https://mlsanity.com/paper/2609.38035/) (2026)
- [On prediction from expert advice with more than five experts](https://mlsanity.com/paper/2610.05186/) (2026)

## Abstract (arXiv)

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction (1,0,1,0,0) is optimal throughout the ordered sector, and that the COMB strategy (1,0,1,0,1) is optimal only on a lower dimensional subset of the sector (where x1=x2 and x3=x4). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan. The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.