与历史相关的专家在线预测：一般情况
Online Prediction With History-Dependent Experts: The General Case
摘要：
我们研究的问题预测的二进制序列与专家的意见，在在线设置，这是一个经典的例子在线机器学习。我们将二元序列解释为股票的价格历史，并将预测者视为投资者，这将问题转化为股票预测问题。在这个框架中，预测股票每日走势的投资者和控制股票的敌对市场在$N$回合中相互博弈。投资者结合$n\geq 2$专家的预测，以决定在每轮投资多少，并旨在最大限度地减少他们对游戏结束时表现最好的专家的遗憾。我们考虑的问题与历史依赖的专家，其中每个专家使用前$d$天的历史市场在作出他们的预测。我们证明了这个博弈的值函数，适当地重新标度，以$O(N^{-1/6})$的速度收敛为$N\to \infty$到非线性退化椭圆型偏微分方程的黏性解，这可以理解为两人博弈的Hamilton-Jacobi-Issacs方程。因此，我们能够推导出投资者的渐近最优策略。我们的结果扩展了第一作者和R. V. Kohn [13]为$n=2$专家和$d\leq 4$历史日建立的结果。出现在纯数学和应用数学通讯。
英文摘要：
We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over $N$ turns. The investor combines the predictions of $n\geq 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous $d$ days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \infty$ at a rate of $O(N^{-1/6})$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V.Kohn [13] for $n=2$ experts and $d\leq 4$ days of history. To appear in Communications on Pure and Applied Mathematics.
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