On Large-Scale Multiple Testing Over Networks: A Non-Asymptotic Approach

Distributed multiple testing asks $N$ sites to control a global false discovery rate (FDR) under a tight communication budget. The greedy interval-aggregation algorithm of Pournaderi and Xiang (2024) solves this asymptotically but can violate $\mathrm{F}\mathrm{D}\mathrm{R}\le \alpha$ at finite samples. We trace the violation to a winner’s-curse bias in the selected density statistics, of exact order $\mathrm{\Theta }(m^{-1\mathrm{/}4}\sqrt{\log m})$ at the standard bandwidth $\epsilon \asymp m^{-1\mathrm{/}2}$, with $m$ the total number of p-values in the network. Cross-Fit Greedy Aggregation (CFGA) eliminates the curse by selecting the nested rejection family on one half of each node’s data and scoring it on the other, achieving finite-sample $\mathrm{F}\mathrm{D}\mathrm{R}\le \alpha$ when per-node null rates are known; an inflated variant covers the plug-in setting at a vanishing $\eta =1\mathrm{/}m$ slack. BONuS-GA instead masks a bag of synthetic uniform nulls calibrated by counting knockoffs, so every p-value serves both selection and inference; a per-node budgeted variant removes all oracle input, controlling $\mathrm{F}\mathrm{D}\mathrm{R}\le \alpha$ for any data-independent bag. Aggregating the CFGA folds by e-values over random splits (e-CFGA) removes the Bonferroni factor and averages the split randomness. All variants keep the $O(\sqrt{m}\log m)$ communication budget, up to e-CFGA’s $O(\bar{R}\log m)$ reporting round; empirically, BONuS-GA dominates at moderate-to-large per-node samples and CFGA with adaptive bandwidth at small ones.