Tight Convergence Rates for Online Distributed Linear Estimation with Adversarial Measurements

2026-04-07

AI Summary

This work addresses the problem of online mean estimation for a random vector under a distributed parameter server architecture with adversarial measurements and asynchronous communication, where only linearly projected samples are observable. The authors propose a two-timescale ℓ₁-minimization algorithm and establish tight non-asymptotic convergence rate bounds. Their main contributions include providing the first finite-time error bound under this setting, introducing a relaxed condition on the sensing matrix—based on a null-space-like property—that enables accurate estimation of specific projected components even when the full mean vector is not recoverable, and offering a unified characterization of the trade-offs among robustness, identifiability, and statistical efficiency, thereby laying a theoretical foundation for applications such as network tomography.

Abstract

We study mean estimation of a random vector $X$ in a distributed parameter-server-worker setup. Worker $i$ observes samples of $a_i^\top X$, where $a_i^\top$ is the $i$th row of a known sensing matrix $A$. The key challenges are adversarial measurements and asynchrony: a fixed subset of workers may transmit corrupted measurements, and workers are activated asynchronously--only one is active at any time. In our previous work, we proposed a two-timescale $\ell_1$-minimization algorithm and established asymptotic recovery under a null-space-property-like condition on $A$. In this work, we establish tight non-asymptotic convergence rates under the same null-space-property-like condition. We also identify relaxed conditions on $A$ under which exact recovery may fail but recovery of a projected component of $\mathbb{E}[X]$ remains possible. Overall, our results provide a unified finite-time characterization of robustness, identifiability, and statistical efficiency in distributed linear estimation with adversarial workers, with implications for network tomography and related distributed sensing problems.

Problem

• distributed estimation
• adversarial measurements
• asynchronous updates
• linear sensing
• robustness

Innovation

• tight convergence rates
• adversarial measurements
• distributed estimation
• non-asymptotic analysis
• null-space property