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e-GAI: e-value-based Generalized $\alpha$-Investing for Online False Discovery Rate Control
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Proposes e-GAI, a generalized alpha-investing framework built on e-values that controls online FDR under conditional validity, with risk-aversion investing producing e-LORD and e-SAFFRON (plus memory-decay variants and p-value extensions).
Problem
Online FDR control for sequential hypothesis testing under arbitrary or unknown dependence, where GAI on p-values requires independence or PRDS and e-LOND wastes budget and loses power.
Approach
Define an oracle e-value-based FDP estimate using the lagged denominator R_{j-1}+1 and invest testing levels via a risk-aversion investing (RAI) rule that updates the allocation proportion omega_t from past rejections and acceptances, yielding e-LORD (LORD-type) and e-SAFFRON (adaptive, Storey-type) procedures.
Key results
- On the NYC taxi data at alpha=0.1, e-LORD made 47 discoveries and e-SAFFRON 46, versus 33 for e-LOND, with estimated FDP 0.085 and 0.087 (below target).
- In AR(1) simulation with T=500, mu_c=4, pi_1=0.4, alpha=0.05, e-LORD (omega_1=1/T) reached 70.0% power versus 30.9% for e-LOND.
- Simulations show SAFFRON inflates FDR heavily under correlated data while e-LORD and e-SAFFRON maintain FDR control.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use e-LORD instead of e-LOND | online anomaly detection · NYC Taxi anomalies | Num Discovery: 33 -> 47 | Helped |
| use e-SAFFRON instead of e-LOND | online anomaly detection · NYC Taxi anomalies | Num Discovery: 33 -> 46 | Helped |
| use e-SAFFRON instead of e-LORD | online anomaly detection · NYC Taxi anomalies | Num Discovery: 47 -> 46 | Hurt |
| set ω1=1/T instead of ω1=1/√T (ablation) | e-LORD · AR(1), T=500 | Power (%): 22.2 -> 70.0 | Helped |
| set ω1=1/T instead of ω1=1/T² (ablation) | e-LORD · AR(1), T=500 | Power (%): 8.6 -> 70.0 | Helped |
| set ω1=1/T instead of ω1=1/√T (ablation) | e-SAFFRON · AR(1), T=500 | Power (%): 38.7 -> 70.5 | Helped |
| set ω1=1/T instead of ω1=1/T² (ablation) | e-SAFFRON · AR(1), T=500 | Power (%): 8.0 -> 70.5 | Helped |
| set ω1=1/T instead of ω1=1/√T (ablation) | e-LORD · AR(1), T=1000 | Power (%): 16.2 -> 70.1 | Helped |
| set ω1=1/T instead of ω1=1/T² (ablation) | e-LORD · AR(1), T=1000 | Power (%): 4.5 -> 70.1 | Helped |
| set ω1=1/T instead of ω1=1/√T (ablation) | e-SAFFRON · AR(1), T=1000 | Power (%): 28.0 -> 70.9 | Helped |
| set ω1=1/T instead of ω1=1/T² (ablation) | e-SAFFRON · AR(1), T=1000 | Power (%): 4.2 -> 70.9 | Helped |
Surprising findings
- Standard SAFFRON inflates FDR heavily under the paper's correlated Gaussian simulation, while LORD++ controls FDR empirically despite lacking theory under that dependence.
Limitations
- Theory guarantees FDR control only under conditional validity of e-values; denominator approximation may be conservative under strong dependence structures.
- Performance depends on user parameters (omega_1, varphi, psi, lambda) whose choice is heuristic (e.g., omega_1=O(1/T), lambda=0.1).
Experimental setup
Datasets: NYC taxi passenger counts (Numenta Anomaly Benchmark) Nasdaq monthly stock prices
Benchmarks: simulated Gaussian mean testing NYC taxi anomaly detection Nasdaq bubble dating
Metrics: FDR power FDP estimate number of discoveries runtime
Baselines: e-LOND LORD++ SAFFRON SupLORD
Builds on: generalized alpha-investing LORD++ SAFFRON e-LOND SupLORD
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