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

2026-09-13

AI Summary

本文解决了五专家预测PDE的显式解问题，通过分区域解析方法，并证明了COMB策略仅在特定条件下最优。

Abstract

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 $x_1=x_2$ and $x_3=x_4$). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan (2016). 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.

Problem

• five-expert prediction PDE
• exact optimality set
• COMB strategy

Innovation

• explicit solution
• prediction PDE
• five experts
• COMB strategy
• computer assisted proof