  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
($n\leq 10$) 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.
