SCORE: A Unified Framework for Overshoot Refund in Online FDR Control

We propose a unified framework to enhance the power of online multiple hypothesis testing procedures based on $e$-values. While $e$-value-based methods offer robust online False Discovery Rate (FDR) control under minimal assumptions, they often suffer from power loss by discarding evidence that exceeds the rejection threshold. We address this inefficiency via the \textbf{S}equential \textbf{C}ontrol with \textbf{O}vershoot \textbf{R}efund for \textbf{E}-values (SCORE) framework, which leverages the inequality $\mathbb{I}(y \ge 1) \le y - (y-1)_+$, valid for all $y\ge 0$, to reclaim this otherwise ``wasted'' evidence. This simple yet powerful insight yields a unified principle for improving a broad class of online testing algorithms. Building on this framework, we develop SCORE-enhanced versions of several state-of-the-art procedures, including SCORE-LOND, SCORE-LORD, and SCORE-SAFFRON, all of which strictly dominate their original counterparts while preserving valid finite-sample FDR control. Furthermore, under mild assumptions, SCORE permits retroactive updates of alpha‑wealth by using the latest decision twice: first to determine its reward or loss, and then to refresh past wealth. Such a mechanism enables more aggressive testing strategies while maintaining valid FDR control, thereby further improving statistical power. The effectiveness of the proposed methods is validated through extensive simulation and real-data experiments.

Online decision systems often test many hypotheses one after another, such as in large-scale experiments or scientific screening studies. To avoid too many false discoveries, these systems must carefully control their error budget over time. This paper shows that when a test result is much stronger than needed for a discovery, the extra evidence can be refunded back into the testing budget instead of being discarded. This allows future tests to be run more effectively while still maintaining rigorous error control. Experiments and a real-data example show that the proposed method can increase the number of discoveries.