Numerical solution of a PDE arising from prediction with expert advice
CoRR(2024)
Abstract
This work investigates the online machine learning problem of prediction with
expert advice in an adversarial setting through numerical analysis of, and
experiments with, a related partial differential equation. The problem is a
repeated two-person game involving decision-making at each step informed by n
experts in an adversarial environment. The continuum limit of this game over a
large number of steps is a degenerate elliptic equation whose solution encodes
the optimal strategies for both players. We develop numerical methods for
approximating the solution of this equation in relatively high dimensions
(n≤ 10) by exploiting symmetries in the equation and the solution to
drastically reduce the size of the computational domain. Based on our numerical
results we make a number of conjectures about the optimality of various
adversarial strategies, in particular about the non-optimality of the COMB
strategy.
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