Gaussian Approximation and Multiplier Bootstrap for Federated Linear Stochastic Approximation

TL;DR

This paper develops Gaussian approximation bounds and a bootstrap inference method for federated linear stochastic approximation, explicitly accounting for communication, computation, and heterogeneity effects.

Contribution

It introduces the first federated Gaussian approximation for LSA that captures heterogeneity and communication trade-offs, along with a bootstrap inference procedure.

Findings

Established Berry-Esseen bounds for federated LSA.

Provided non-asymptotic validity guarantees for the bootstrap method.

Recovered recent convergence rates as special cases.

Abstract

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.