This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite horizon and random stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor’s and the opponent’s minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin et al. (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, SODA ’16, (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2016). Our techniques are similar to those of Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.

中文翻译：

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借助专家建议进行预测：PDE观点

这项工作通过专家建议解决了在线预测的经典问题。我们假设一个对抗性对手，我们同时考虑了这个零和两人游戏的有限视野和随机停止版本。着眼于适当的连续极限，并使用最佳控制方法，我们将博弈的值表征为某个非线性偏微分方程的粘性解。分析还揭示了预测者和对手的最小最大最优策略。我们的工作尤其提供了Gravin等人最近工作的连续性观点。（在：第27届ACM-SIAM年度离散算法研讨会论文集，SODA '16，（美国宾夕法尼亚州费城），工业和应用数学学会，2016年）。