Online control of the false discovery rate with decaying memory
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 pvalue 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.
This paper unifies and extends concepts related to the online false discovery rate (FDR) control. This is a recent trendy setting where null hypotheses are coming in a sequential manner over time and the user should make an (irrevocable) decision about rejecting (or not) the null coming at each time. The aim is to make a sequential inference so that the expected proportion of errors among the rejected nulls is bounded by a prescribed value $\alpha$ (at any stopping time).
My opinion is that this paper is of very high quality: the writing is good and easy to follow, the motivation is clear and interesting, the new procedures and concepts are clever and the proofs are really nicely written.
My only concern is that the strongest innovation (decaying memory procedure) is maybe not 
sufficiently well put forward in the main paper. It only comes at the last 1.5 pages and there is no illustration for Alpha-death recovery. This concept seems however more important than the double-weighting of Section 4 that can be put in appendix safely (more standard). I would also suggest to put Section D.3 of the supplement in the main paper (illustration of Piggybacking).
Here are some other comments and typos : 
 line 58 "will equal 1" (only with high probability);
 line 131 "mst";
 line 143 "keeping with past work";
 Notation "W(t)" I suggest to use W_t (if you agree);
 Lemma 1 f_i is not F_i measurable but just measurable (right?);
 Section D.1, line 469-470 :" mu_j follows 0" should read "mu_j=0";
 line 528 : equation (13) should read equation (10);
 line 529-530 : -phi_t (R_t-1) should read +phi_t (R_t-1).