有限时域五专家预测问题

中文摘要 AI 辅助

我们给出了有限时间时域设置下五专家预测与专家建议偏微分方程（PDE）的显式解。该解公式确立了对手的秩策略 $(1,0,1,0,0)$ 是全局最优的，而 COMB 策略 $(1,0,1,0,1)$ 恰好仅在 $x_1=x_2$ 且 $x_3=x_4$ 的集合上是最优的。该公式源自我们姊妹论文中给出的几何停止问题的解，通过 Bayraktar、Ekren 和 Zhang 的变换原理，利用拉普拉斯变换将两个问题联系起来。逐项反演变换后，解通过一系列高斯函数和互补误差函数核的级数表示。$(1,0,1,0,0)$ 的最优性归结为 $41$ 个单变量高斯级数的符号判定，这些符号通过泊松求和、第一模态支配以及 $1616$ 个有理单元上的区间算术在计算机辅助下得到验证。我们主要定理的证明（包括证书）也在 Lean 证明助手中形式化。

英文摘要

We give an explicit solution to the five expert prediction with expert advice partial differential equation (PDE) in the finite-time horizon setting. The solution formula establishes that the adversary's rank strategy $(1,0,1,0,0)$ is globally optimal, and the COMB strategy $(1,0,1,0,1)$ is optimal exactly on the set where $x_1=x_2$ and $x_3=x_4$. The formula is derived from the solution of the geometric-stopping problem given in our companion paper through the transform principle of Bayraktar, Ekren and Zhang, which links the two problems by a Laplace transform. Inverting the transform term by term expresses the solution through a series of Gaussian and complementary error function kernels. The optimality of $(1,0,1,0,0)$ is reduced to the signs of $41$ one-variable Gaussian series, which are certified with computer assistance by Poisson summation, first-mode domination and interval arithmetic on $1616$ rational cells. The proofs of our main theorems, certificates included, are also formalized in the Lean proof assistant.

发表机构

• University of Minnesota（明尼苏达大学）
• Louisiana State University（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。