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.

翻译：本文通过数值分析及偏微分方程的相关实验，研究在线机器学习中对抗性环境下的预测与专家建议问题。该问题是一个重复双人博弈，在对抗性环境中每步需根据 $n$ 位专家提供的建议进行决策。该博弈在大步数下的连续极限是一个退化椭圆方程，其解编码了双方玩家的最优策略。我们利用方程及其解的对称性大幅缩减计算域规模，针对该方程在较高维度（$n\leq 10$）下发展数值逼近方法。基于数值结果，我们提出了关于多种对抗性策略最优性的若干猜想，特别是关于COMB策略的非最优性。