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On Large-Scale Multiple Testing Over Networks: An Asymptotic Approach

Mehrdad Pournaderi, Yu Xiang

TL;DR

This work proposes two methods, proportion-matching and greedy aggregation, tailored to distributed settings, that achieves the global BH performance yet only requires a one-shot communication of the (estimated) proportion of true null hypotheses as well as the number of p-values at each node.

Abstract

This work concerns developing communication- and computation-efficient methods for large-scale multiple testing over networks, which is of interest to many practical applications. We take an asymptotic approach and propose two methods, proportion-matching and greedy aggregation, tailored to distributed settings. The proportion-matching method achieves the global BH performance yet only requires a one-shot communication of the (estimated) proportion of true null hypotheses as well as the number of p-values at each node. By focusing on the asymptotic optimal power, we go beyond the BH procedure by providing an explicit characterization of the asymptotic optimal solution. This leads to the greedy aggregation method that effectively approximates the optimal rejection regions at each node, while computation efficiency comes from the greedy-type approach naturally. Moreover, for both methods, we provide the rate of convergence for both the FDR and power. Extensive numerical results over a variety of challenging settings are provided to support our theoretical findings.

On Large-Scale Multiple Testing Over Networks: An Asymptotic Approach

TL;DR

This work proposes two methods, proportion-matching and greedy aggregation, tailored to distributed settings, that achieves the global BH performance yet only requires a one-shot communication of the (estimated) proportion of true null hypotheses as well as the number of p-values at each node.

Abstract

This work concerns developing communication- and computation-efficient methods for large-scale multiple testing over networks, which is of interest to many practical applications. We take an asymptotic approach and propose two methods, proportion-matching and greedy aggregation, tailored to distributed settings. The proportion-matching method achieves the global BH performance yet only requires a one-shot communication of the (estimated) proportion of true null hypotheses as well as the number of p-values at each node. By focusing on the asymptotic optimal power, we go beyond the BH procedure by providing an explicit characterization of the asymptotic optimal solution. This leads to the greedy aggregation method that effectively approximates the optimal rejection regions at each node, while computation efficiency comes from the greedy-type approach naturally. Moreover, for both methods, we provide the rate of convergence for both the FDR and power. Extensive numerical results over a variety of challenging settings are provided to support our theoretical findings.
Paper Structure (20 sections, 19 theorems, 100 equations, 3 figures)

This paper contains 20 sections, 19 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Multiple Testing and False Discovery Rate Control
  4. Distributed False Discovery Control: Problem Setting
  5. Communication-Efficient Algorithms
  6. Local Multiple Testing (No Communication)
  7. Proportion-Matching Algorithm
  8. Estimation of $r_0^{(i)}$
  9. Asymptotic Equivalence to the Global BH
  10. Robustness Against Inconsistent Estimators and heterogeneous Alternatives
  11. Towards Optimality: Adaptive Segmentation
  12. Oracle Solution
  13. Greedy Aggregation Procedure
  14. Asymptotic Performance
  15. Simulations
  16. ...and 5 more sections

Key Result

Proposition 1

Let $R_m^{(i)}$, $V_m^{(i)}$, and $\mathsf{FDP}^{(i)}$ denote the number of rejections, false rejections, and FDP of an arbitrary procedure at note $i$, respectively. If $\underset{m\rightarrow\infty}{\overline{\lim}}\mathsf{FDP}^{(i)}\leq \alpha\ a.s.$, $1\leq i \leq N$, then $\underset{m\rightarr

Figures (3)

  • Figure 1: Experiment 1: The effect of the number of p-values on the performance of a heterogeneous network with Gaussian statistics ($m^{(i)}=n(1.2-0.2i)$).
  • Figure 2: From left to right, Experiment $2$ (a): Gaussian statistics ($\underline{\mu}_{\text{base}}=\eta\cdot[1,2,3,4,5]$); Experiment $2$ (b): Cauchy statistics ($\mu^{(i)}=\mu$); Experiment $2$ (c): nodes $2$ and $4$ own p-values generated according to Gaussian statistics ($\mu^{(i)}=\mu$).
  • Figure 3: Experiment 3: p-values generated according to Gaussian statistics with tapering correlation structure.

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 31 more