论文标题
与历史依赖专家的在线预测：一般情况
Online Prediction With History-Dependent Experts: The General Case
论文作者
论文摘要
我们研究了在线环境中使用专家建议预测二进制序列的问题，这是在线机器学习的一个典型示例。我们将二进制序列解释为股票的价格历史记录，并将预测变量视为投资者，将问题转换为股票预测问题。在此框架内，一个预测股票日常移动的投资者和控制股票的对抗市场，在$ n $ n $ thew中相互对抗。投资者结合了$ n \ geq 2 $专家的预测，以便决定每回合要投资多少，并旨在最大程度地减少对游戏结束时表现最好的专家的遗憾。我们考虑了与历史有关的专家的问题，在这些专家中，每个专家都使用前面的$ d $市场历史来做出预测。我们证明，适当重新缩放的该游戏的价值功能以$ n \ to \ infty $的收敛，以$ o（n^{ - 1/6}）$的速率汇总到非线性退化椭圆形PDE的粘度解决方案，可以理解为hamilton-Jacobi-issacsSacsSacsSequation for Twe-equariation for Two-for Tast-eqeration for Two-for Taster-for Taster-for Taster-for Taster-for Taster-for Taster。结果，我们能够为投资者推断出渐近最佳的策略。我们的结果扩展了第一作者建立的结果和R.V.Kohn [13]以$ n = 2 $专家和$ d \ leq 4 $天的历史。出现在有关纯数学和应用数学的通信中。
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.
