e-GAI: e-value-based Generalized $\alpha$-Investing for Online False Discovery Rate Control

TL;DR

This paper introduces e-GAI, a new e-value-based framework for online FDR control that improves power and generalizes to dependent data, with practical algorithms e-LORD and e-SAFFRON.

Contribution

The paper proposes the e-GAI framework, enabling online FDR control under broader dependencies and incorporating dynamic, risk-aware level allocation, along with two novel procedures e-LORD and e-SAFFRON.

Findings

e-GAI ensures provable FDR control under general dependence.

e-LORD and e-SAFFRON outperform existing methods in power and FDR control.

Experimental results validate the effectiveness of the proposed methods.

Abstract

Online multiple hypothesis testing has attracted a lot of attention in many applications, e.g., anomaly status detection and stock market price monitoring. The state-of-the-art generalized $\alpha$-investing (GAI) algorithms can control online false discovery rate (FDR) on p-values only under specific dependence structures, a situation that rarely occurs in practice. The e-LOND algorithm (Xu & Ramdas, 2024) utilizes e-values to achieve online FDR control under arbitrary dependence but suffers from a significant loss in power as testing levels are derived from pre-specified descent sequences. To address these limitations, we propose a novel framework on valid e-values named e-GAI. The proposed e-GAI can ensure provable online FDR control under more general dependency conditions while improving the power by dynamically allocating the testing levels. These testing levels are updated not only by relying on both the number of previous rejections and the prior costs, but also, differing from the GAI framework, by assigning less $\alpha$-wealth for each rejection from a risk aversion perspective. Within the e-GAI framework, we introduce two new online FDR procedures, e-LORD and e-SAFFRON, and provide strategies for the long-term performance to address the issue of $\alpha$-death, a common phenomenon within the GAI framework. Furthermore, we demonstrate that e-GAI can be generalized to conditionally super-uniform p-values. Both simulated and real data experiments demonstrate the advantages of both e-LORD and e-SAFFRON in FDR control and power.