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Gaussian Approximation and Multiplier Bootstrap for Federated Linear Stochastic Approximation

arXiv 2605.19629 ↗First posted 19 May 2026stat.ML cs.LG math.OCcited by 0

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Abstract (arXiv)

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.

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12 of 61 references.

References · 12

  1. 2018Local SGD Converges Fast and Communicates Littlecited by 605
  2. 2024Statistical Inference for Policy Evaluation with Temporal Difference Learningcited by 15
  3. 2023Online Bootstrap Inference with Nonconvex Stochastic Gradient Descent Estimatorcited by 13
  4. 2025Statistical inference for Linear Stochastic Approximation with Markovian Noisecited by 13
  5. 2025Uncertainty quantification for Markov chain induced martingales with application to temporal difference learningcited by 11
  6. 2025Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descentcited by 10
  7. 2022Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descentcited by 8
  8. 2020Local Stochastic Approximation: A Unified View of Federated Learning and Distributed Multi-Task Reinforcement Learning Algorithmscited by 4
  9. 2025Revisiting LocalSGD and SCAFFOLD: Improved Rates and Missing Analysiscited by 4
  10. 2025Central Limit Theorems for Asynchronous Averaged Q-Learningcited by 4
  11. 2025Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximationcited by 4
  12. 2025Achieving Tighter Finite-Time Rates for Heterogeneous Federated Stochastic Approximation under Markovian Samplingcited by 2

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