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.

Lay Summary:

When keeping an eye on unusual events or watching stock markets as they change, it's important to reduce false alerts while still being able to spot real issues effectively. Many current methods either require very specific situations to work well or sacrifice too much detection capability just to avoid false alarms.We developed a novel framework called e-GAI that dynamically allocates ``detection resources'', similar to how a skilled investor adjusts their investments in a portfolio. This new approach helps us better control false discoveries—essentially, mistakes in our findings—while also successfully identifying more of the samples we care about. Within this framework, we design two new procedures called e-LORD and e-SAFFRON. Our experiments showed that these new methods performed exceptionally well.The e-GAI framework is useful, and it can also handle both new types of data analysis (called e-values) and the more traditional approach (p-values). This makes it versatile for many different situations where real-time decisions are needed, such as checking product quality or monitoring financial activities.