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arxiv: 2605.18656 · v1 · pith:2XVTACTCnew · submitted 2026-05-18 · 📊 stat.ML · cs.AI· cs.LG· stat.ME

Statistical Limits and Efficient Algorithms for Differentially Private Federated Learning

Arnab Auddy , Xiangni Peng , Subhadeep Paul This is my paper

Pith reviewed 2026-05-20 07:56 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LGstat.ME
keywords differentially private federated learningFedAvgFedSGDFedHybridFedNewtonmean squared error boundsminimax lower boundM-estimation
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The pith

FedHybrid and FedNewton improve accuracy and cut communication rounds in differentially private federated M-estimation compared with standard FedAvg and FedSGD.

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

The paper examines the tension between statistical accuracy, differential privacy, and communication cost when estimating models from data spread across many separate clients. It introduces FedHybrid, which starts FedSGD from an averaged initialization, and FedNewton, which averages local Newton steps to shrink the bias that pure averaging leaves behind. Finite-sample upper bounds describe how mean-squared error grows with client count, local sample size, privacy budget, and iteration number for the private versions of both methods. A minimax lower bound on the error of any iterative private federated procedure supplies a benchmark for judging how close the new estimators come to the best attainable performance.

Core claim

FedHybrid initializes FedSGD with the FedAvg estimator while FedNewton averages local Newton iterations to reduce federation bias; the differentially private versions of both achieve explicit finite-sample mean-squared error rates that depend on the number of clients, local sample sizes, privacy budget and number of iterations, and these rates are compared against a derived minimax lower bound that applies to any iterative private federated procedure.

What carries the argument

FedHybrid (FedAvg initialization followed by FedSGD) and FedNewton (averaged local Newton iterations), together with finite-sample MSE upper bounds and a minimax lower bound on error for any iterative private federated procedure.

If this is right

  • FedNewton reaches estimation accuracy comparable to FedSGD but requires substantially fewer communication rounds provided the number of clients grows slowly enough.
  • The differentially private FedHybrid and FedNewton estimators possess explicit mean-squared error bounds that scale with privacy budget, local sample sizes, client count, and iteration number.
  • The minimax lower bound provides a concrete reference against which the optimality gap of any proposed iterative private federated method can be measured.
  • Numerical experiments on logistic regression and neural networks trained on MNIST and CIFAR-10 confirm that the predicted accuracy-communication trade-offs appear in practice.

Where Pith is reading between the lines

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

  • The slow-growth restriction on client numbers indicates the methods are most immediately useful for federated systems with dozens to low hundreds of participants rather than thousands.
  • Reaching or approaching the minimax lower bound could motivate further work on adaptive privacy mechanisms or higher-order local updates that close the remaining gap.
  • The same bias-reduction idea of averaging local second-order steps may extend to other distributed statistical tasks that currently rely on first-order averaging.
  • The explicit dependence of error on the privacy budget allows quantitative comparison of utility loss when switching between different privacy parameters.

Load-bearing premise

The claim that FedNewton reaches accuracy comparable to FedSGD with far fewer rounds assumes the number of clients grows sufficiently slowly for averaging the local Newton steps to control the remaining bias.

What would settle it

A direct numerical check that measures whether the mean-squared error of private FedNewton stays comparable to private FedSGD when the number of clients is increased rapidly while keeping total communication rounds fixed.

Figures

Figures reproduced from arXiv: 2605.18656 by Arnab Auddy, Subhadeep Paul, Xiangni Peng.

Figure 1
Figure 1. Figure 1: Empirical MSE results of DP version of FedSGD and FedHybrid under equal sample sizes across clients in Logistic Regression. K3 = 50, and the total number of communication rounds is set to R = 2, leading to an effective 100 iterations. For FedNewton, the first stage uses FedAvg with K4 = 50 local iterations and R = 1, followed by just one private Newton step. In all cases, the scale of Gaussian noise to be … view at source ↗
Figure 2
Figure 2. Figure 2: Empirical MSE results of FedAvg and FedNewton under equal client sample sizes in Logistic Regression. In the simulation experiments, different numbers of clients and local sample sizes are con￾sidered. In the first set of simulations for both logistic and Poisson GLM, we vary number of clients m ∈ {20, 30, 40, 50, 60}. For the local sample sizes, two cases are examined. The first case assumes that all clie… view at source ↗
Figure 3
Figure 3. Figure 3: MSE comparison of FedSGD, FedHybrid, FedAvg, and FedNewton for logistic regression with local sample sizes having mean around 400 under different distributions: (a) Uniformly distributed sample sizes, (b) Equal sample sizes, and (c) Lognormally distributed sample sizes. the number of clients m while keeping the local sample size ni fixed so that the total sample size increases. We display the results over … view at source ↗
Figure 4
Figure 4. Figure 4: The empirical MSE of the methods with increasing number of iterations illustrating [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: MSE comparison of FedSGD, FedHybrid, FedAvg, and FedNewton under fixed total sample size N = 20000 with varying number of clients m: (a) equal local sample sizes, (b) uniformly distributed local sample sizes, and (c) lognormally distributed local sample sizes. 4.5 Effect of the Number of Clients under Fixed Total Sample Size To illustrate the effect of federation across clients, we conduct a simulation stu… view at source ↗
Figure 6
Figure 6. Figure 6: Poisson GLM: Empirical MSE comparison of [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Binary class logistic regression MNIST results under different privacy budgets. [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Multiclass MNIST results under different privacy budgets. [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: CNN results with increasing samples per client: median test accuracy on (left) CIFAR [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: CNN results with increasing number of clients fixing the total sample size: median [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Empirical MSE results of DP version of FedSGD and FedHybrid under different local sample sizes in Logistic Regression. This section contains additional figures from the simulation study using logistic regression. These figures depict empirical MSE of the methods with an increasing number of clients for the case when the client sample sizes ni differ across the clients. 50 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 12
Figure 12. Figure 12: Empirical MSE results of FedAvg and FedNewton under different local sample sizes in Logistic Regression. B.2 Poisson GLM simulation figures 51 [PITH_FULL_IMAGE:figures/full_fig_p051_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Empirical MSE results of FedSGD and FedHybrid under Equal and different Local Sample Sizes in Poisson GLM. 52 [PITH_FULL_IMAGE:figures/full_fig_p052_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Empirical MSE results of FedAvg and FedNewton under Equal and different Local Sample Sizes in Poisson GLM. 53 [PITH_FULL_IMAGE:figures/full_fig_p053_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The empirical MSE of the methods with increasing number of iterations illustrating [PITH_FULL_IMAGE:figures/full_fig_p054_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: MSE comparison of FedSGD, FedHybrid, FedAvg, and FedNewton under fixed total sample size N = 20000 with varying number of clients m in Poisson GLM: (a) equal local sample sizes, (b) uniformly distributed local sample sizes, and (c) lognormally distributed local sample sizes. 54 [PITH_FULL_IMAGE:figures/full_fig_p054_16.png] view at source ↗
read the original abstract

Federated Learning is a leading framework for training ML and AI models collaboratively across numerous user devices or databases. We study the trade-offs among estimation accuracy, privacy constraints, and communication cost for differentially private (DP) federated M estimation. The two standard methods in the literature are FedAvg, which may suffer from high federation bias, and FedSGD, which can incur high communication cost. Aimed at improving accuracy at a reduced communication cost, we propose FedHybrid, which uses FedSGD starting with an improved initialization by the FedAvg estimator. We propose FedNewton, which averages local Newton iterations to reduce bias in FedAvg, achieving an estimation accuracy comparable to FedSGD with much fewer communication rounds when the number of clients grows sufficiently slowly. We establish finite sample upper bounds on the mean-squared error rates of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and number of iterations. We further derive a minimax lower bound on the MSE of any iterative private federated procedure that provides a benchmark to assess the optimality gap of these methods. We numerically evaluate our methods for training a logistic regression and a neural network on the computer vision datasets MNIST and CIFAR-10.

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 / 2 minor

Summary. The manuscript studies trade-offs among estimation accuracy, privacy, and communication cost in differentially private federated M-estimation. It proposes FedHybrid, which initializes FedSGD with the FedAvg estimator, and FedNewton, which averages local Newton iterations to reduce bias in FedAvg. Finite-sample upper bounds are derived on the MSE of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and iterations. A minimax lower bound is provided on the MSE of any iterative private federated procedure, serving as a benchmark. Numerical evaluations are performed for logistic regression and neural networks on MNIST and CIFAR-10.

Significance. If the derivations hold, the paper advances the field by introducing algorithms that target reduced communication while controlling bias and privacy, together with explicit finite-sample MSE bounds and a minimax lower bound that enables assessment of optimality gaps. The explicit dependence of the bounds on key parameters (clients, local sizes, privacy budget, iterations) and the empirical results on standard vision datasets provide concrete, usable contributions for designing private federated systems.

major comments (2)
  1. [Abstract] Abstract: The central efficiency claim for FedNewton—that it achieves estimation accuracy comparable to FedSGD with much fewer communication rounds—rests on the assumption that the number of clients grows sufficiently slowly to control bias from the averaged local Newton steps. This growth-rate condition is load-bearing for the advertised communication saving; without an explicit statement of the precise regime (e.g., in terms of total clients m relative to local sample size n and privacy parameters) and a demonstration that the finite-sample MSE upper bound remains competitive when the condition is relaxed, the advantage over FedSGD is not robust.
  2. [Theoretical analysis section] Theoretical analysis section (bounds and lower bound): The finite-sample upper bounds on MSE are expressed in terms of number of clients, local sample sizes, privacy budget, and iterations, and are compared to a minimax lower bound. To make the optimality-gap assessment rigorous, the manuscript should explicitly derive the rate gap between the upper bounds for FedHybrid/FedNewton and the minimax lower bound as a function of communication rounds, showing under which parameter regimes the proposed methods are rate-optimal or near-optimal.
minor comments (2)
  1. [Numerical Experiments] The numerical experiments on MNIST and CIFAR-10 would benefit from explicit reporting of the privacy budgets (ε, δ) used in each method and direct comparison tables against non-private FedAvg/FedSGD baselines with the same communication budget.
  2. Notation for the number of clients and local sample sizes should be introduced once and used consistently; minor inconsistencies appear in the abstract versus the bound statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript's contributions. We address each major comment point by point below, agreeing where revisions are warranted and outlining specific changes to clarify conditions and strengthen the optimality analysis.

read point-by-point responses

  1. Referee: The central efficiency claim for FedNewton—that it achieves estimation accuracy comparable to FedSGD with much fewer communication rounds—rests on the assumption that the number of clients grows sufficiently slowly to control bias from the averaged local Newton steps. This growth-rate condition is load-bearing for the advertised communication saving; without an explicit statement of the precise regime (e.g., in terms of total clients m relative to local sample size n and privacy parameters) and a demonstration that the finite-sample MSE upper bound remains competitive when the condition is relaxed, the advantage over FedSGD is not robust.

    Authors: We agree that the growth-rate condition on the number of clients is central to controlling the bias term in FedNewton and realizing the communication savings relative to FedSGD. In the revised manuscript we will state the precise regime explicitly in the abstract (and restate it in the main text near the relevant theorem), using the exact condition derived from our bias analysis. We will also add a short remark and accompanying bound illustration showing the MSE behavior under moderate relaxations of the condition, confirming that FedNewton remains strictly better than FedAvg and retains a communication advantage over FedSGD for a range of practical parameter settings. revision: yes

  2. Referee: The finite-sample upper bounds on MSE are expressed in terms of number of clients, local sample sizes, privacy budget, and iterations, and are compared to a minimax lower bound. To make the optimality-gap assessment rigorous, the manuscript should explicitly derive the rate gap between the upper bounds for FedHybrid/FedNewton and the minimax lower bound as a function of communication rounds, showing under which parameter regimes the proposed methods are rate-optimal or near-optimal.

    Authors: We concur that an explicit derivation of the rate gap would make the optimality assessment more rigorous. We will add a new corollary immediately after the minimax lower bound that expresses the multiplicative gap between the FedHybrid/FedNewton upper bounds and the lower bound as a function of the number of communication rounds T. The corollary will identify concrete regimes (for example, when T scales as a polylogarithm of the number of clients) in which the proposed methods are rate-optimal or near-optimal up to constant factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of MSE upper bounds or minimax lower bound

full rationale

The paper derives finite-sample upper bounds on the MSE of its DP FedHybrid and FedNewton estimators directly from bias-variance analysis of the local Newton steps, FedAvg initialization, and privacy noise addition, expressed in terms of number of clients, local sizes, privacy budget, and iterations. The minimax lower bound is obtained as an information-theoretic benchmark for any iterative private federated procedure. Neither reduces by construction to fitted parameters, self-definitions, or self-citation chains; the slow-growth condition on the number of clients is an explicit modeling assumption needed for the bias-reduction claim to hold and does not render the bounds tautological. The derivations are therefore self-contained against standard statistical analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions for M-estimation and differential privacy (smoothness/convexity of loss, bounded gradients, privacy composition rules) that are not detailed in the abstract; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption The objective function satisfies standard smoothness and strong-convexity conditions typical for M-estimation analysis
    Invoked to derive finite-sample MSE rates for the DP estimators
  • domain assumption Differential privacy is implemented via standard mechanisms (e.g., Gaussian noise) whose composition properties hold across rounds
    Required for the privacy budget accounting in the bounds

pith-pipeline@v0.9.0 · 5758 in / 1399 out tokens · 31646 ms · 2026-05-20T07:56:51.064251+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    We establish finite sample upper bounds on the mean-squared error rates of the DP versions of these estimators as functions of the number of clients, local sample sizes, privacy budget, and number of iterations. We further derive a minimax lower bound on the MSE of any iterative private federated procedure.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 5 internal anchors

  1. [1]

    Minimax and adaptive transfer learn- ing for nonparametric classification under distributed differential privacy constraints.arXiv preprint arXiv:2406.20088,

    Arnab Auddy, T Tony Cai, and Abhinav Chakraborty. Minimax and adaptive transfer learn- ing for nonparametric classification under distributed differential privacy constraints.arXiv preprint arXiv:2406.20088,

    work page arXiv

  2. [2]

    Differentially private inference via noisy optimization.The Annals of Statistics, 51(5):2067–2092,

    Marco Avella-Medina, Casey Bradshaw, and Po-Ling Loh. Differentially private inference via noisy optimization.The Annals of Statistics, 51(5):2067–2092,

    work page 2067

  3. [3]

    Optimal federated learning for non- parametric regression with heterogeneous distributed differential privacy constraints.arXiv preprint arXiv:2406.06755,

    T Tony Cai, Abhinav Chakraborty, and Lasse Vuursteen. Optimal federated learning for non- parametric regression with heterogeneous distributed differential privacy constraints.arXiv preprint arXiv:2406.06755,

    work page arXiv

  4. [4]

    Differentially Private Federated Learning: A Client Level Perspective

    Robin C Geyer, Tassilo Klein, and Moin Nabi. Differentially private federated learning: A client level perspective.arXiv preprint arXiv:1712.07557,

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    On the convergence of local descent methods in federated learning.arXiv preprint arXiv:1910.14425,

    Farzin Haddadpour and Mehrdad Mahdavi. On the convergence of local descent methods in federated learning.arXiv preprint arXiv:1910.14425,

    work page arXiv 1910

  6. [6]

    Federated Learning: Strategies for Improving Communication Efficiency

    Jakub Konečn` y, H Brendan McMahan, Felix X Yu, Peter Richtárik, Ananda Theertha Suresh, and Dave Bacon. Federated learning: Strategies for improving communication efficiency.arXiv preprint arXiv:1610.05492,

    work page internal anchor Pith review Pith/arXiv arXiv

  7. [7]

    On the convergence of FedAvg on non- IID data

    Xiang Li, Kaixuan Huang, Wenhao Yang, Shusen Wang, and Zhihua Zhang. On the convergence of fedavg on non-iid data.arXiv preprint arXiv:1907.02189,

    work page arXiv 1907

  8. [8]

    Learning Differentially Private Recurrent Language Models

    Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Ar- cas. Communication-efficient learning of deep networks from decentralized data. InArtificial intelligence and statistics, pages 1273–1282. PMLR, 2017a. H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning differentially private recurrent language models.ar...

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Federated evalua- tion and tuning for on-device personalization: System design & applications.arXiv preprint arXiv:2102.08503,

    Matthias Paulik, Matt Seigel, Henry Mason, Dominic Telaar, Joris Kluivers, Rogier van Dalen, Chi Wai Lau, Luke Carlson, Filip Granqvist, Chris Vandevelde, et al. Federated evalua- tion and tuning for on-device personalization: System design & applications.arXiv preprint arXiv:2102.08503,

    work page arXiv

  10. [10]

    Adaptive Federated Optimization

    Sashank Reddi, Zachary Charles, Manzil Zaheer, Zachary Garrett, Keith Rush, Jakub Konečn` y, Sanjiv Kumar, and H Brendan McMahan. Adaptive federated optimization.arXiv preprint arXiv:2003.00295,

    work page internal anchor Pith review arXiv 2003

  11. [11]

    Fedpaq: A communication-efficient federated learning method with periodic av- eraging and quantization

    Amirhossein Reisizadeh, Aryan Mokhtari, Hamed Hassani, Ali Jadbabaie, and Ramtin Pedarsani. Fedpaq: A communication-efficient federated learning method with periodic av- eraging and quantization. InInternational conference on artificial intelligence and statistics, pages 2021–2031. PMLR,

    work page 2021

  12. [12]

    Local SGD Converges Fast and Communicates Little

    Sebastian U Stich. Local sgd converges fast and communicates little.arXiv preprint arXiv:1805.09767,

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Optimal estimation in private distributed functional data analysis.arXiv preprint arXiv:2412.06582,

    Gengyu Xue, Zhenhua Lin, and Yi Yu. Optimal estimation in private distributed functional data analysis.arXiv preprint arXiv:2412.06582,

    work page arXiv

  14. [14]

    Understanding clipping for federated learning: Convergence and client-level differential privacy

    Xinwei Zhang, Xiangyi Chen, Mingyi Hong, Zhiwei Steven Wu, and Jinfeng Yi. Understanding clipping for federated learning: Convergence and client-level differential privacy. InInterna- tional Conference on Machine Learning, ICML 2022,

    work page 2022

  15. [15]

    Definition A.1(Global Sensitivity).LetD ∗ denote the space of all datasets

    32 A Proofs of Lemmas and Theorems A.1 Additional definitions and lemmas We will require the following definition for privacy accounting. Definition A.1(Global Sensitivity).LetD ∗ denote the space of all datasets. Consider two datasetsD, D ′ ∈ D ∗ that differ in exactly one datum. The global sensitivity of a function f:R n×m →R p with respect to a norm∥·∥...

    work page 2022

  16. [16]

    Proof of Lemma A.3.We prove the result for the case whered > ClogNso that we take B≥C √ d

    ThenP(Ec)≤ c(exp(−cd)∧N −4)wheneverB≥C( √ d∨ √logN). Proof of Lemma A.3.We prove the result for the case whered > ClogNso that we take B≥C √ d. The proof for the first part follows by sub-Gaussian tail assumptions on the gradient ∇ρ(x(i) j , θ0). Similarly for the Hessian¨Li(θ), we use a1/3-netSon the unit sphere d−1, which has a cardinality at most4d. Fo...

    work page 2022

  17. [17]

    From the proof of Lemma 3.1, we have GS as2B ni

    In stage 2, we runK2 steps of AG1 with¯θas initializer and noise multiplier b′ = 2B√2K2 µni . From the proof of Lemma 3.1, we have GS as2B ni . Hence 2B√K2 niµi = 2B√2K2 µni ⇒µ i = µ√ 2 . Therefore, by the composition theorem in Dong et al. [2022], the clients areµ-GDP to the server. It follows immediately from the Proof of Lemma 3.1, that for stage 2, th...

    work page 2022

  18. [18]

    We will now use the technique for deriving lower bounds under distributed(ε, δ)-privacy as done in Cai et al

    =µ ϕ(c) c2 (1 +o(1)) = µ√ 2π e−c2/2 c2 (1 +o(1)) = µ√ 2π µ c2 (1 +o(1)) = µ2 √ 2π c2 (1 +o(1)) = µ2 2 √ 2πlog(1/µ) (1 +o(1)). We will now use the technique for deriving lower bounds under distributed(ε, δ)-privacy as done in Cai et al. [2024], Auddy et al. [2024], Xue et al. [2024]. LetPbe the set of all distributions satisfying assumptions 1 to

    work page 2024

  19. [19]

    pθ(y|x) :=h(y) exp x⊤θy−ψ(x ⊤θ) c ;∥x∥ ∞≤B satisfyingmax{ψ ′(0),sup u∈R ψ′′(u)} ≤L

    In particular,Pcontains Pθ, the set of joint distributions on(x, y)withx∼N(0,I d)and the following generalized linear model ony|x. pθ(y|x) :=h(y) exp x⊤θy−ψ(x ⊤θ) c ;∥x∥ ∞≤B satisfyingmax{ψ ′(0),sup u∈R ψ′′(u)} ≤L. For the lower bound, we will use the multivariate Van Trees inequality (see, e.g., Theorem 1 of Gill and Levit [1995], Lemma 4.3 of Cai et al....

    work page 1995

  20. [20]

    and its use in the proof of Proposition 12 of Xue et al. [2024]. It is thus enough to check their conditions. By our assumptions on bounded covariates and link functionψ, we can ensure that the score function: ∇θ logp θ(y,x) = (y−ψ ′(x⊤θ))x is sub-Gaussian with sub-Gaussianity parameterCdfor some constantC >0. We can then follow the proof of Proposition 1...

    work page 2024