In the online multiple testing problem, p-values corresponding to different null hypotheses are observed one by one, and the decision of whether or not to reject the current hypothesis must be made immediately, after which the next p-value is observed. Alpha-investing algorithms to control the false discovery rate (FDR), formulated by Foster and Stine, have been generalized and applied to many settings, including quality-preserving databases in science and multiple A/B or multi-armed bandit tests for internet commerce. This paper improves the class of generalized alpha-investing algorithms (GAI) in four ways: (a) we show how to uniformly improve the power of the entire class of monotone GAI procedures by awarding more alpha-wealth for each rejection, giving a win-win resolution to a recent dilemma raised by Javanmard and Montanari, (b) we demonstrate how to incorporate prior weights to indicate domain knowledge of which hypotheses are likely to be non-null, (c) we allow for differing penalties for false discoveries to indicate that some hypotheses may be more important than others, (d) we define a new quantity called the decaying memory false discovery rate (mem-FDR) that may be more meaningful for truly temporal applications, and which alleviates problems that we describe and refer to as "piggybacking" and "alpha-death". Our GAI++ algorithms incorporate all four generalizations simultaneously, and reduce to more powerful variants of earlier algorithms when the weights and decay are all set to unity. Finally, we also describe a simple method to derive new online FDR rules based on an estimated false discovery proportion.

对广义α投资算法（GAI）算法类进行了四个方面的改进：（a）通过为每个拒绝授予更多的α财富，统一提高整个单调GAI程序的功率；（b）演示如何将先前的权重纳入考虑；（c）允许为错误发现指定不同的处罚；（d）定义了一种称为mem-FDR的新数量，它可以在真正的时间应用中更有意义，并减轻我们描述并称之为“piggybacking”和“alpha-death”的问题。 最后，我们还描述了一种基于估计的假阳性发现比例推导新的在线FDR规则的简单方法。

带有衰减记忆的假阳性发现率在线控制