The prediction of a binary sequence is a classic example of online machine learning. We like to call it the “stock prediction problem,” viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more “experts,” and strives to minimize her final-time regret with respect to the best-performing expert. Probability plays no role; rather, the market is assumed to be adversarial. We consider the case when there are two history-dependent experts, whose predictions are determined by the d most recent stock moves. Focusing on an appropriate continuum limit and using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market, and we determine associated upper and lower bounds for the investor's final-time regret. When d ≤ 4 our upper and lower bounds coalesce, so the proposed strategies are asymptotically optimal. Compared to other recent applications of partial differential equations to prediction, ours has a new element: there are two timescales, since the recent history changes at every step whereas regret accumulates more slowly. © 2022 Wiley Periodicals LLC.

中文翻译：

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根据两位历史相关专家的建议预测二进制序列的 PDE 方法

二进制序列的预测是在线机器学习的一个经典例子。我们喜欢将其称为“股票预测问题”，将序列视为股票在每个时间步长上涨或下跌一个单位的价格历史。在这个问题中，投资者可以获得两个或更多“专家”的预测，并努力将她对表现最好的专家的最终遗憾降到最低。概率不起作用；相反，市场被假定为具有对抗性。我们考虑有两个依赖历史的专家的情况，他们的预测由d最近的股票走势。着眼于适当的连续极限并使用最优控制、图论和偏微分方程的方法，我们讨论了投资者和对抗市场的策略，并确定了投资者最终后悔的相关上限和下限。当d ≤ 4 时，我们的上限和下限合并，因此所提出的策略是渐近最优的。与最近将偏微分方程用于预测的其他应用相比，我们的应用有一个新元素：有两个时间尺度，因为最近的历史每一步都在变化，而遗憾的积累速度更慢。© 2022 Wiley Periodicals LLC。