The finite-horizon five-expert prediction problem

2026-09-29

AI Summary

This study addresses the finite-horizon prediction problem with five experts by determining optimal strategies through solving an associated partial differential equation (PDE). Methodologically, it integrates Laplace transforms, Gaussian kernel series expansions, and the Poisson summation formula to derive analytical solutions, while employing interval arithmetic and the Lean proof assistant for rigorous verification. The primary contributions include obtaining the first explicit solution to this PDE, establishing the global optimality of the adversary's rank-based strategy alongside the optimality of the COMB strategy over specific sets, and successfully certifying forty-one Gaussian series identities with a formalized proof of the main theorem. Collectively, these results provide a solid mathematical foundation for online learning theory.

Abstract

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.

Problem

• prediction with expert advice
• finite-horizon
• five-expert problem
• partial differential equation
• adversarial strategy

Innovation

• expert prediction
• partial differential equation
• Laplace transform
• computer-assisted proof
• formal verification