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arxiv: 2605.19629 · v1 · pith:5FCX7CCBnew · submitted 2026-05-19 · 📊 stat.ML · cs.LG· math.OC

Gaussian Approximation and Multiplier Bootstrap for Federated Linear Stochastic Approximation

Ilya Levin , Maksim Shuklin , Eric Moulines , Paul Mangold , Sergey Samsonov This is my paper

Pith reviewed 2026-05-20 02:05 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.OC
keywords federated learningstochastic approximationBerry-Esseen boundsGaussian approximationmultiplier bootstraplast-iterate inferenceheterogeneous data
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The pith

Federated linear stochastic approximation admits explicit Berry-Esseen bounds that track local updates, communication rounds and client heterogeneity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes Berry-Esseen-type bounds for the last iterate of federated linear stochastic approximation. These bounds make the approach to a Gaussian limit explicit in the local step size, the number of local updates per communication round, and the level of heterogeneity across clients. The results hold for both constant step-size schedules and decreasing step sizes paired with increasing local iterations, recovering earlier centralized rates when heterogeneity vanishes. A direct application is an online multiplier bootstrap that supplies non-asymptotic validity guarantees for inference without estimating the asymptotic covariance matrix. Practitioners in distributed learning settings need such guarantees because communication is limited and data distributions differ across nodes.

Core claim

We establish Berry-Esseen-type bounds for federated linear stochastic approximation that 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. The bounds are derived for both constant step-size regimes and decreasing step-size regimes with an increasing number of local iterations. As a primary application we develop an online multiplier bootstrap procedure for inference on the last iterate that avoids explicit estimation of the asymptotic covariance matrix and obtain non-asymptotic validity guarantees for this procedure.

What carries the argument

Federated Berry-Esseen bounds that bound the Kolmogorov distance of the scaled last-iterate estimator to a Gaussian limit through additive terms for local step size, local iteration count per round, and heterogeneity.

If this is right

  • Increasing the number of local updates per round reduces the contribution of heterogeneity to the overall error.
  • The multiplier bootstrap yields valid confidence intervals for the last iterate under the same communication and heterogeneity conditions used for the Gaussian approximation.
  • Centralized rates are recovered as a special case when the number of clients is one or heterogeneity is zero.
  • The same bounding technique applies to both constant and diminishing step-size policies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of federated systems could use the explicit trade-off terms to choose how many local updates to perform before each communication round.
  • The linear structure exploited here suggests the same bounding approach may extend to certain non-linear stochastic approximation problems in distributed settings.
  • Large-scale numerical checks on real heterogeneous datasets would reveal how tight the derived bounds are in practice.

Load-bearing premise

The noise terms satisfy uniform moment bounds across all heterogeneous local distributions and the step-size conditions permit a single global recursion whose error terms remain controllable.

What would settle it

A simulation in which the empirical distribution of the normalized last-iterate estimator in a federated linear regression task is compared against the Gaussian limit and the observed deviation exceeds the paper's explicit Berry-Esseen bound for moderate numbers of communication rounds.

Figures

Figures reproduced from arXiv: 2605.19629 by Eric Moulines, Ilya Levin, Maksim Shuklin, Paul Mangold, Sergey Samsonov.

Figure 1
Figure 1. Figure 1: Illustration of the multiplier bootstrap bounds: [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). It provides the first federated Gaussian approximations that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying effects of local step size, number of local updates, and heterogeneity. Results are given for both constant step-size and decreasing step-size regimes (recovering Bonnerjee et al. [2025] rates as a special case). As an application, an online multiplier bootstrap for inference on the last iterate is developed with non-asymptotic validity guarantees.

Significance. If the derivations hold, the work is significant for extending non-asymptotic Gaussian approximation and bootstrap theory to the federated LSA setting while explicitly incorporating practical factors such as communication rounds and client heterogeneity. The recovery of prior centralized rates as a special case and the covariance-free bootstrap procedure are clear strengths. Non-asymptotic bounds and explicit trade-off quantification add practical value for distributed inference.

major comments (2)
  1. [Main results on constant step-size regime] The central global recursion that absorbs local heterogeneity must be shown to keep all additive error terms controllable under the stated uniform bounded-moment assumptions; without an explicit bound on the heterogeneity-induced term in the Berry-Esseen error (e.g., in the constant-step-size case), the claimed communication-computation trade-off quantification is not yet load-bearing.
  2. [Bootstrap validity theorem] The non-asymptotic validity of the online multiplier bootstrap is derived from the Gaussian approximation; the proof must verify that the bootstrap multiplier distribution matches the last-iterate fluctuation up to the same o(1) rate as the Gaussian approximation itself, particularly when the number of local updates grows.
minor comments (3)
  1. [Introduction] The introduction should include a short table or paragraph contrasting the new federated rates with the centralized rates of Bonnerjee et al. [2025] to make the improvement explicit.
  2. [Preliminaries] Notation for the heterogeneity measure (e.g., variance of local means) should be introduced once and used consistently in all error-term statements.
  3. [Numerical experiments] Figure captions for any convergence plots should state the precise values of local step size, communication rounds, and heterogeneity level used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and positive recommendation for minor revision. We address the major comments below, providing clarifications and indicating planned revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Main results on constant step-size regime] The central global recursion that absorbs local heterogeneity must be shown to keep all additive error terms controllable under the stated uniform bounded-moment assumptions; without an explicit bound on the heterogeneity-induced term in the Berry-Esseen error (e.g., in the constant-step-size case), the claimed communication-computation trade-off quantification is not yet load-bearing.

    Authors: We agree with the referee that an explicit bound on the heterogeneity-induced term would make the argument more transparent. In the manuscript, the global recursion is given in (4.3), and under Assumption 2.1-2.3, the heterogeneity term is bounded by C * delta * eta * K in the error analysis for the constant step-size case (see the derivation leading to (4.12)). This term is then absorbed into the overall Berry-Esseen bound as an additive O(delta * sqrt(eta)) term, which is controllable and vanishes under the appropriate scaling of communication rounds and local steps. We will revise the manuscript to include a new display equation isolating this term and a short paragraph explaining its controllability, confirming the load-bearing nature of the trade-off quantification. revision: yes

  2. Referee: [Bootstrap validity theorem] The non-asymptotic validity of the online multiplier bootstrap is derived from the Gaussian approximation; the proof must verify that the bootstrap multiplier distribution matches the last-iterate fluctuation up to the same o(1) rate as the Gaussian approximation itself, particularly when the number of local updates grows.

    Authors: We appreciate this comment. The non-asymptotic validity in Theorem 5.1 is proven by coupling the bootstrap process to the Gaussian approximation via the multiplier central limit theorem, with the error rate matching that of the GA up to o(1). When the number of local updates increases, the additional variance from local steps is accounted for in the last-iterate analysis, and the proof shows the bootstrap distribution converges at the same rate provided the total iterations satisfy the conditions for the GA to hold. To make this explicit, we will add a remark in the proof of Theorem 5.1 detailing the rate matching for growing local updates. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from standard assumptions

full rationale

The paper derives Berry-Esseen bounds and bootstrap validity for federated LSA from a global recursion under uniform standard linear SA assumptions (bounded moments, step-size conditions) across heterogeneous clients. It recovers Bonnerjee et al. [2025] rates only as a special case without reducing the central claims to fitted parameters or self-citation chains. No load-bearing step equates outputs to inputs by construction; the communication-computation trade-offs and heterogeneity terms are explicitly quantified from the recursion rather than smuggled in via ansatz or renaming. This is the expected non-circular outcome for a non-asymptotic analysis paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit list of free parameters, axioms, or invented entities; standard stochastic approximation assumptions (moment bounds, step-size conditions) are implicitly invoked but not detailed.

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