发表机构
University of Minnesota; Louisiana State University（明尼苏达大学; 路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推导了五专家预测PDE的显式解，证明COMB策略仅在低维子集上最优，从而反驳了Gravin等人的COMB最优性猜想。
AI 中文摘要
在本文中，我们推导了五专家情形下带专家建议的平稳预测PDE的显式解。该公式在三个区域中给出。在前两个区域中，解为四专家解加上一个具有初等正密度的单重积分。在第三个区域中，解为双曲函数乘积的有限和，其系数由一个标量求积确定。我们的公式确立了方向$(1,0,1,0,0)$在整个有序扇区上是最优的，而COMB策略$(1,0,1,0,1)$仅在扇区的低维子集上（其中$x_1=x_2$且$x_3=x_4$）是最优的。这反驳了Gravin、Peres和Sivan（2016）的COMB最优性猜想。哈密顿不等式的验证是一项繁琐的任务，其中部分通过计算机辅助证明完成。验证归结为21个标量不等式，我们使用147个精确的有理Bernstein多项式证书证明了这些不等式。精确证书及其独立的算术检查包含在本文的补充材料中。
英文摘要
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. 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.
