Tight Convergence Rates for Online Distributed Linear Estimation with Adversarial Measurements

TL;DR

This paper derives tight finite-time convergence rates for distributed linear estimation under adversarial measurements and asynchrony, extending previous asymptotic results to practical finite-sample scenarios.

Contribution

It establishes non-asymptotic convergence rates under null-space conditions and identifies relaxed conditions affecting recovery of the expected signal.

Findings

Established tight non-asymptotic convergence rates.

Identified conditions where only projected recovery is possible.

Unified finite-time analysis of robustness and efficiency.

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.