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 is less than 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.

本文研究通过在线机器学习的技术，建立了一个 “股票预测问题” 的模型，探讨了投资者和市场策略，并使用最优控制、图论和偏微分方程的方法确定了使用两个与历史相关的专家的股票行情预测的性能上下界。