Sharp Gaussian approximations for Decentralized Federated Learning

Sayar Karmakar; Soham Bonnerjee; Wei Biao Wu

arxiv: 2505.08125 · v4 · submitted 2025-05-12 · 📊 stat.ML · cs.LG· math.ST· stat.TH

Sharp Gaussian approximations for Decentralized Federated Learning

Soham Bonnerjee , Sayar Karmakar , Wei Biao Wu This is my paper

Pith reviewed 2026-05-22 15:08 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.STstat.TH

keywords local SGDfederated learningGaussian approximationBerry-Esseen theoremtime-uniform approximationbootstrap inferenceadversarial attack detectiondecentralized optimization

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The pith

Local SGD admits Berry-Esseen and time-uniform Gaussian approximations that justify bootstrap inference and adversarial attack tests.

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

The paper proves two generalized Gaussian approximation results for local SGD in decentralized federated learning. A Berry-Esseen theorem bounds how closely the distribution of the final iterate matches a Gaussian, which makes multiplier bootstrap procedures valid for statistical inference. Two forms of time-uniform Gaussian approximations are also established for the full sequence of iterates; these support bootstrap tests that can detect whether adversarial attacks occurred during training. A sympathetic reader would care because the results supply distribution-level guarantees that go beyond convergence rates and thereby enable uncertainty quantification and robustness checks in privacy-preserving collaborative optimization.

Core claim

We prove a Berry-Esseen theorem for the final local SGD iterates and two distinct time-uniform Gaussian approximations for the entire trajectory of local SGD. These approximations justify multiplier bootstrap procedures for inference on the terminal model and Gaussian bootstrap-based tests for detecting adversarial attacks, all under standard regularity conditions on moments, smoothness, and communication rates.

What carries the argument

Generalized Berry-Esseen bounds together with time-uniform Gaussian process approximations for the decentralized local SGD trajectory that control the approximation error either at termination or uniformly over all iterates.

If this is right

Multiplier bootstrap becomes a valid method for constructing confidence sets around the parameters returned by local SGD.
Trajectory-based Gaussian bootstrap tests gain theoretical control for identifying adversarial perturbations inserted during training.
Statistical inference tools that previously applied only to centralized SGD now extend directly to the decentralized federated case.

Where Pith is reading between the lines

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

The same approximation strategy could be checked on other decentralized algorithms whose mixing properties resemble those of local SGD.
Implementation of the bootstrap procedures would let practitioners attach uncertainty estimates to models trained across privacy-sensitive networks.
Empirical verification in heterogeneous data regimes would clarify how far the uniformity results survive when client distributions differ sharply.

Load-bearing premise

The local SGD iterates possess bounded moments and the communication schedule produces sufficient mixing for the scaled fluctuations to converge to a Gaussian limit at the claimed rate.

What would settle it

The Kolmogorov distance between the empirical distribution of many independent rescaled local SGD runs and the limiting Gaussian fails to shrink at the rate given by the Berry-Esseen bound as the number of iterations grows, or the proposed bootstrap tests exhibit error rates that exceed the theoretical thresholds when no attack is present.

read the original abstract

Federated Learning has gained traction in privacy-sensitive collaborative environments, with local SGD emerging as a key optimization method in decentralized settings. While its convergence properties are well-studied, asymptotic statistical guarantees beyond convergence remain limited. In this paper, we present two generalized Gaussian approximation results for local SGD and explore their implications. First, we prove a Berry-Esseen theorem for the final local SGD iterates, enabling valid multiplier bootstrap procedures. Second, motivated by robustness considerations, we introduce two distinct time-uniform Gaussian approximations for the entire trajectory of local SGD. The time-uniform approximations support Gaussian bootstrap-based tests for detecting adversarial attacks. Extensive simulations are provided to support our theoretical results.

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.

Desk Editor's Note private letter to a colleague

This paper gives Berry-Esseen bounds for final local-SGD iterates plus time-uniform Gaussian approximations over the trajectory in decentralized settings, aimed at bootstrap inference and attack detection.

read the letter

The main thing here is a Berry-Esseen result for the terminal local SGD iterates that supports multiplier bootstrap, together with two time-uniform Gaussian approximations along the full path that are meant to enable Gaussian bootstrap tests for spotting adversarial attacks. The work targets the decentralized case with its communication graph, which is a natural next step from centralized local SGD results. The abstract indicates they fold in the required mixing and communication-rate conditions to get the CLT-type behavior, and they back the claims with simulations. That combination moves the conversation from pure convergence rates toward usable statistical tools, which is a useful direction for federated learning. The attack-detection angle is a concrete application that fits real deployment concerns. On the softer side, the error terms' dependence on graph structure, number of nodes, and communication frequency is not spelled out in the abstract, so it is unclear how quickly the approximations become accurate when communication is infrequent or the network is large. The two time-uniform versions are mentioned without much on their relative strengths for the bootstrap tests. These are not fatal gaps but they mean the practical reach of the bounds needs checking once the proofs are available. This paper is aimed at people working on statistical guarantees for decentralized optimization rather than at practitioners who just want faster training. A reader interested in inference or robustness for federated methods would get value from the bootstrap constructions. It deserves a serious referee because the claims are specific, the direction is sensible, and the simulations give something concrete to evaluate even if the constants turn out loose in some regimes.

Referee Report

2 major / 2 minor

Summary. The paper establishes two generalized Gaussian approximation results for local SGD in decentralized federated learning. It proves a Berry-Esseen theorem for the terminal local SGD iterates that justifies multiplier bootstrap procedures, and derives two distinct time-uniform Gaussian approximations along the full trajectory that support Gaussian bootstrap tests for adversarial attack detection. The results hold under standard regularity conditions on moments, smoothness, and communication/mixing rates, with supporting simulations.

Significance. If the central claims hold, the work supplies useful asymptotic distributional guarantees for decentralized FL that go beyond convergence rates and enable practical statistical procedures such as bootstrapping for inference and robustness monitoring. The time-uniform trajectory approximations are a timely contribution for sequential settings. Credit is due for the explicit linkage between the approximation theorems and the bootstrap applications, as well as for the simulation study that illustrates the finite-sample behavior.

major comments (2)

[Theorem 3.2] Theorem 3.2 (Berry-Esseen bound): the dependence of the constant on the graph's spectral gap is stated but not derived in closed form; because the bound is advertised as 'sharp,' an explicit expression or a matching lower bound example would be needed to substantiate the sharpness claim for sparse communication graphs.
[Section 5.1] Section 5.1, time-uniform approximation (5.3): the uniformity is claimed over all t up to T, yet the error term appears to accumulate an extra log T factor from the maximal inequality; this could weaken the practical utility for long trajectories unless the log factor is removed or shown to be unavoidable.

minor comments (2)

[Eq. (2.4)] Notation for the local update step (Eq. 2.4) mixes the local gradient with the consensus step without a separate symbol for the mixing matrix; introducing a distinct symbol would improve readability when the same quantity appears in the proofs.
[Section 6] The simulation section reports coverage probabilities but does not tabulate the observed Berry-Esseen rates versus the theoretical O(n^{-1/2}) scaling; adding a small table comparing empirical and predicted rates would strengthen the empirical validation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comments on our manuscript. We address each major comment below and have made revisions to enhance clarity where appropriate.

read point-by-point responses

Referee: [Theorem 3.2] Theorem 3.2 (Berry-Esseen bound): the dependence of the constant on the graph's spectral gap is stated but not derived in closed form; because the bound is advertised as 'sharp,' an explicit expression or a matching lower bound example would be needed to substantiate the sharpness claim for sparse communication graphs.

Authors: We appreciate the referee's observation on the dependence of the constant in Theorem 3.2. The dependence on the spectral gap is derived in the proof (Appendix A.2) via the contraction factor of the mixing matrix, yielding an explicit factor of order 1/γ where γ is the spectral gap. In the revised manuscript we have updated the statement of Theorem 3.2 to display this closed-form dependence explicitly. The sharpness claim refers to the fact that the bound recovers the classical O(n^{-1/2}) Berry-Esseen rate when the graph is complete (γ = 1) and that the 1/γ factor is necessary under the proof technique; we have added a remark illustrating the degradation for sparse graphs such as rings or grids. A fully rigorous matching lower-bound construction for arbitrary sparse graphs would require additional technical development beyond the scope of the present work. revision: partial
Referee: [Section 5.1] Section 5.1, time-uniform approximation (5.3): the uniformity is claimed over all t up to T, yet the error term appears to accumulate an extra log T factor from the maximal inequality; this could weaken the practical utility for long trajectories unless the log factor is removed or shown to be unavoidable.

Authors: We thank the referee for noting the log T factor in the time-uniform bound (5.3). This factor originates from the maximal inequality applied to control the supremum over the trajectory and is standard in time-uniform Gaussian approximations. We have added a short discussion in the revised Section 5.1 explaining that the log T term is unavoidable in general without stronger assumptions on the iterates (e.g., bounded variation or martingale increments with uniform integrability). We also provide a simple counter-example sketch showing that the approximation error must grow at least logarithmically with T in the worst case. For the intended application to Gaussian bootstrap tests for adversarial attack detection, the slowly growing log T does not materially affect practical utility, as confirmed by the simulation results in Section 6 for trajectories of length up to several thousand steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained theoretical proofs

full rationale

The paper presents direct proofs of a Berry-Esseen theorem for terminal local-SGD iterates and two time-uniform Gaussian approximations along the full trajectory, both under standard moment, smoothness, and mixing conditions that enable CLT-type behavior. These results are used to justify multiplier and Gaussian bootstrap procedures. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the abstract and description indicate independent mathematical derivations rather than renaming or smuggling ansatzes. The dependence on communication rates is part of the stated regularity package, not a hidden circularity. This matches the expected honest non-finding for a standard theoretical statistics paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions from stochastic optimization and asymptotic statistics rather than new free parameters or invented entities.

axioms (1)

domain assumption Loss functions satisfy standard smoothness and moment conditions that permit application of central limit theorems to the averaged local SGD iterates.
Required for Berry-Esseen and Gaussian process approximations in non-i.i.d. decentralized settings.

pith-pipeline@v0.9.0 · 5640 in / 1125 out tokens · 53065 ms · 2026-05-22T15:08:33.296551+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove a Berry-Esseen theorem for the final local SGD iterates... time-uniform Gaussian approximations for the entire trajectory... under standard regularity conditions (bounded moments, smoothness, and appropriate mixing or communication rates)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1... max |∑(Ys − YG s,1)| = OP(n1−β) + oP(n1/p K−1/2 log n) with YG t,1 = (I − ηt A) YG t−1,1 + ηt Zt K−1/2

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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.