Gaussian Approximation and Multiplier Bootstrap for Federated Linear Stochastic Approximation

2026-05-19

AI Summary

This work addresses the lack of non-asymptotic Gaussian approximation and valid statistical inference methods in federated linear stochastic approximation, particularly under data heterogeneity and communication-computation trade-offs where theoretical guarantees are scarce. The authors establish a Berry–Esseen-type non-asymptotic bound that explicitly quantifies the impact of local stepsize, number of local updates, and data heterogeneity on convergence rates. Building on this, they develop a Gaussian approximation framework applicable to both constant and diminishing stepsizes and propose an online multiplier bootstrap procedure that avoids explicit covariance estimation, yielding confidence intervals for the final iterate with non-asymptotic validity. This approach recovers and extends existing optimal convergence rates under both stepsize regimes.

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.

Problem

• Federated Learning
• Linear Stochastic Approximation
• Gaussian Approximation
• Heterogeneity
• Statistical Inference

Innovation

• Federated Learning
• Linear Stochastic Approximation
• Gaussian Approximation
• Multiplier Bootstrap
• Heterogeneity-aware Analysis