Rethinking Federated Learning Over the Air: The Blessing of Scaling Up

Bikramjit Das; Howard H. Yang; Jiaqi Zhu; Nikolaos Pappas; Yong Xie

REVIEW 3 major objections 5 minor 55 references

Rethinking Federated Learning Over the Air: The Blessing of Scaling Up

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read In over-the-air federated learning, scaling up the client count shrinks privacy leakage, fading, and convergence error.

desk verdict Solid convergence and channel-hardening results, but the privacy theorem relies on a false decorrelation claim and should not be trusted as stated. read the letter →

arxiv 2508.17697 v1 pith:KQWUPD7S submitted 2025-08-25 cs.LG

Jiaqi Zhu , Bikramjit Das , Yong Xie , Nikolaos Pappas , Howard H. Yang This is my paper · ORCID

classification cs.LG

keywords over-the-aircomputationfederatedlearningprivacyleakagemutualinformationchannelhardeningconvergenceratepowercontrollarge-scalesystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Federated learning over the air sends client updates as analog signals that add together on the wireless channel, so the server receives a noisy but automatically aggregated gradient. This paper argues that the standard worry about that noise is inverted: when many clients participate, the aggregation itself becomes a blessing. The mutual information between one client's local gradient and the server's aggregated gradient is bounded by a term of order $O(1/N)$, so each participant hides better in a larger crowd. A concentration inequality shows channel hardening, meaning small-scale fading is averaged away as $N$ grows, and the convergence error contributions from thermal noise and mini-batch sampling carry $N^2$ denominators and vanish at scale. If these results hold, over-the-air federated learning becomes attractive precisely in the large-scale regime where digital communication would be most costly.

What carries the argument

The mechanism is the averaging of $N$ independent, channel-faded client gradients at the server; $N$ independent fading coefficients make the effective channel concentrate at its mean. Formally, the argument rides on three pieces: a Berry–Esseen-style entropic central limit theorem that lower-bounds the differential entropy of the aggregated signal and yields the $O(1/N)$ mutual-information upper bound; a Bernstein/McDiarmid concentration inequality (Lemma 1) showing the difference between the faded and mean-faded gradients is exponentially small in $N$; and standard smoothness and SGD-drift bounds that place sampling noise into terms with $1/(N^2 B)$ denominators. Together these convert "more clients" from an added communication burden into a noise-reduction and privacy-enhancing resource.

What would settle it

Compute or measure the mutual information between one client's gradient and the server's aggregated gradient for $N = 10$, $50$, $100$, and $500$ on the same task, but skip the random-flipping decorrelation step; if the decay flattens or the bound breaks once gradient entries stay correlated, the $O(1/N)$ privacy result rests on preprocessing rather than on the number of clients. A smaller-scale check: at $N = 10$, compare convergence with and without instantaneous-CSI power control; the paper's own experiments already show power control helps there, so the "not essential" claim is limited to large $N$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the wireless impairments in over-the-air federated learning—fading, thermal noise, and sampling noise—are progressively neutralized as the number of participating clients $N$ grows, so scale improves privacy, robustness, and convergence simultaneously. Theorem 1 bounds the per-client mutual information between a local gradient and the aggregated gradient by $C_{\bar{g}} d^*/N$ plus a logarithmic term, decaying as $O(1/N)$ under an honest-but-curious server and average-case leakage. Lemma 1 establishes exponential concentration of the channel-corrupted aggregated gradient around the mean-fading aggregated gradient, a channel-hardening effect that removes small-scale fading from the global update. Theorem 3 bounds the non-convex convergence rate by a leading term plus $d\sigma_z^2/(\mu_c^2 N^2)$ and $\sum \sigma_{s,n}^2/(N^2 B)$, so the noisy analog update becomes equivalent to a noiseless global gradient as $N$ grows. The paper concludes that instantaneous channel estimation and power control are not essential in this regime, since the averaging already does the work that power control was designed to do.

Load-bearing premise

The argument assumes the system is large ($N$ big) and that privacy is measured as average mutual information after a preprocessing step that makes each client's gradient entries statistically independent; if $N$ is only in the tens or an adversary needs worst-case guarantees, the $O(1/N)$ privacy decay and the dismissal of power control do not follow.

Editorial extensions

If this is right

Per-client privacy leakage, measured by mutual information, falls as $O(1/N)$, so larger systems can achieve the same average privacy with less added artificial noise.
Small-scale fading disappears from the global gradient through channel hardening, so blind transmission without instantaneous CSI becomes nearly as accurate as ideal power control.
The non-convex convergence error from thermal noise and mini-batch sampling vanishes as $N$ grows, meaning over-the-air training with many clients behaves like noiseless training.
Instantaneous-CSI power control is not essential at scale: Theorem 4 shows it introduces client dropout and channel-estimation-error penalties, and the experiments show its benefit nearly vanishes for large $N$.
Larger client populations also make training more robust to label-flipping and noisy-label attacks and allow second-order updates such as L-BFGS to work, as the experiments show.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the mutual-information bound is average-case and assumes preprocessing decorrelates gradient entries, a natural next test is whether worst-case gradient-inversion attacks on a single client also decay with $N$; the paper's own framing suggests they are not covered by the bound.
The channel-hardening logic suggests that client population size can substitute for many-antenna hardware in suppressing fading, which would change deployment cost calculations for edge-learning infrastructure.
The $N^2$ denominators in the convergence bounds imply a design trade-off not explored in the paper: for a fixed communication budget, operators could trade client count against local epochs and batch size to reach a target error floor.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is worth a serious look, but you should know up front that its first advertised contribution, the O(1/N) privacy bound in Theorem 1, is not proven. The proof in Appendix A (Eqs. 32-35) asserts that multiplying each entry of a local gradient by an independent Rademacher variable makes the entries mutually independent. That is false for a general correlated vector. Even with zero covariance, the squares remain dependent, so the entropy decomposition and the Berry-Esseen lower bound do not follow. The paper explicitly cites "random flipping" as a decorrelation technique, but that only diagonalizes the covariance matrix; it does not make the joint distribution a product. This is a load-bearing flaw, since the privacy pillar is a major part of the paper's message.

What is actually new and good: the channel-hardening effect in the client-count dimension, rather than the antenna-count dimension, is a real insight, and Lemma 1's concentration inequality is a reasonable way to capture it. The convergence analysis (Theorems 2-3) follows standard templates and the noise terms shrinking as N grows are correctly identified. The experiments align with the qualitative predictions, and the power-control discussion is honest about its own conditions. The paper also does a fair job of positioning itself relative to prior work.

Soft spots beyond the privacy proof: the power-control claim is conditional on a specific truncated channel-inversion scheme and on large N, and their own N=10 experiment shows power control still helps. That is a minor overstatement, not a fatal flaw. There is no code or data shipped, which limits reproducibility, but the theory is mostly self-contained. The MINE-based MI estimates show trends but do not repair the proof gap, and the paper does not quantify the approximation error.

Who is this for? Researchers working on over-the-air federated learning, especially those interested in whether power control can be dropped in large-scale systems. The convergence and hardening results are likely to be useful even if the privacy theorem is fixed or weakened to a conjecture with an explicit independence assumption.

Recommendation: send it to peer review, because the core question is important and most of the analysis is sound. But the referee should demand a corrected or substantially weakened statement of Theorem 1. As it stands, the privacy result is unproven, and the authors should either add a genuine independence assumption or provide a multivariate entropic CLT argument.

Referee Report

3 major / 5 minor

Summary. This paper studies an over-the-air federated edge learning system in which N clients transmit accumulated local gradients as analog signals over i.i.d. fading channels and the server receives a noisy linear combination. The authors derive three sets of analytical results: a mutual-information upper bound on privacy leakage that is claimed to scale as O(1/N) (Theorem 1, Section III-A), a concentration inequality for the aggregated gradient that is used to claim a channel-hardening effect (Lemma 1, Section III-B), and convergence rates for strongly convex (Theorem 2, Corollary 1) and non-convex (Theorem 3, Corollary 2) objectives in which thermal noise and SGD noise terms vanish with N. Section III-D analyzes truncated channel-inversion power control under imperfect CSI (Theorem 4) and concludes that instantaneous power control is unnecessary in large-scale systems. The theoretical claims are supplemented with experiments on EMNIST and CIFAR-10 using CNN, ResNet-18, and logistic-regression/L-BFGS setups.

Significance. If the results held as stated, the paper would make a useful contribution to over-the-air federated learning by showing that a large number of clients improves privacy, suppresses fading, and accelerates convergence, rather than only exacerbating communication cost. The paper's strengths are the clarity of the system model, the use of a standard FedAvg proof template for the convergence analysis, the explicit treatment of power control with imperfect CSI, and the experimental section that directly checks the predicted privacy, channel-hardening, and convergence trends. The main reservations are proof gaps in Theorem 1 and in the convergence proofs, which are central to the paper's three headline claims rather than presentation issues.

major comments (3)

[Appendix A, Eqs. (32)-(35)] The proof of Theorem 1 asserts that after multiplying entries of s_{n,t} by independent Rademacher variables S, the entries of \hat s_{n,t} are mutually independent. This is false for a general correlated vector. For example, take X1=Z and X2=Z+W with independent standard Gaussians Z,W and independent Rademachers S1,S2; then S1X1 and S2X2 have zero covariance but are dependent because E[(S1X1)^2(S2X2)^2]=4 while E[(S1X1)^2]E[(S2X2)^2]=2. Consequently, the coordinate-wise entropy decomposition and the Berry-Esseen entropic CLT lower bound in Eq. (35) do not follow for correlated gradient entries, and the O(1/N) privacy bound in Eq. (10) with its consequence in Eq. (25) is not established as stated for general local gradients. The footnote about random flipping reducing correlations is insufficient; the proof needs either an explicit entrywise-independence assumption on the preprocessed gradient or a genuinely multivariate entropic CLT argument. The MINE experiments provide empirical evidence but do not close this gap.
[Appendix C, Eq. (44); Appendix E, Eq. (48); Theorems 2-3] The convergence proofs replace the random fading coefficients c_{n,t} in the global update (6) by their mean \mu_c without a residual term. In Appendix C, the line before Eq. (44) writes \eta_t/N \sum_n \mu_c(w_t-w_{n,t}^{(E)})/\eta_l, which equals the true update only when c_{n,t}=\mu_c; similarly, Eq. (48) in Appendix E computes E[\|\hat g_t\|^2] from an expression containing no c_{n,t}. As a result, the bounds in Theorems 2 and 3 (and Corollaries 1 and 2) are proved for a channel-mean system, not for the random-fading model of Section II-B; a term of order \sigma_c^2 G^2/N would enter the residual. Lemma 1 provides only a high-probability concentration statement and is not integrated into the expectation bounds in these proofs. The claim that communication impairments vanish as N grows may survive a repair, but the theorem statements as proved are not supported.
[Corollary 1 and Appendix D, Eqs. (45)-(46)] The displayed algebra in the proof of Corollary 1 is inconsistent with the stated \kappa. Under the stated condition \eta_0\le 1/(4\mu_c L), we have \lambda\mu_c\eta_0-1<0 because \lambda\le L, so the first argument in the max in Eq. (22) is negative; then \kappa collapses to \|w_0-w^*\|^2 and all noise and heterogeneity terms disappear from the bound. The induction step in Eq. (46) requires \kappa to be at least on the order of \mu_c^2\eta_0^2 Q1/(\lambda\mu_c\eta_0), not the expression with \lambda\mu_c\eta_0-1 in the denominator. Please correct the formula or the proof; as printed, Corollary 1 is not derived.

minor comments (5)

[Section III-A] The statement that 'after certain preprocessing steps, the local gradient entries can be statistically independent of each other' is an assumption in all but name; since it drives Theorem 1, it should be stated as an explicit assumption with a precise description of the preprocessing.
[Eq. (8) and Appendix A] The privacy metric conditions on the past gradients {g_p}, but the proof of Theorem 1 computes entropies without making the conditioning visible in every step; please make the conditioning explicit and explain why past public models do not change the bounds.
[Eq. (18)] The notation for the term involving \beta_\nu is typeset ambiguously; please define \beta_\nu^N (or the intended expression) explicitly and state the dependence of the final bound on \nu.
[Section IV, Figs. 2 and 4] The MI estimates in Fig. 2 are obtained with MINE, an approximate estimator with known bias, and no error bars or seeds are reported; the O(1/N) privacy law is therefore only qualitatively validated. In addition, increasing N in Fig. 4 also increases the total amount of training data, so part of the observed accuracy gain may be a data-volume effect rather than the noise-reduction effect claimed in the analysis.
[Section III-C.2] The statement that the gradient discrepancy 'decreases at the rate of O(exp(-N))' is stronger than what Lemma 1 proves; Lemma 1 is a high-probability large-deviation bound, so the conclusion should be phrased as an exponential concentration statement rather than a bound on the mean discrepancy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's analytical claims derive from standard concentration inequalities, entropic CLT bounds, and established FedAvg proof techniques, with no fitted parameter renamed as a prediction.

full rationale

The derivation chain in this paper is self-contained and does not reduce its conclusions to its inputs by construction. Theorem 1's mutual-information bound is derived from entropy calculus, a linear-transformation entropy identity, and the entropic central limit theorem [52], under the explicitly stated conditional-independence preprocessing assumption; it is not a restatement of the definition of the metric. Lemma 1 is a direct Bernstein/McDiarmid concentration bound on the channel-averaging term, with no fitted constants. Theorems 2 and 3 follow the standard proof templates of [37] and [55] for FedAvg-type updates, and the O(1/N^2) noise terms enter algebraically from the global averaging and AWGN variance, not from tuning parameters to match the desired conclusion. The power-control comparison in Theorem 4 is likewise an analytical bound: the extra dropout and CSI-error terms are shown to vanish as N grows, and the claim that instant power control is unnecessary is explicitly conditional on large N and on the specific truncated channel-inversion scheme, so it is a derived comparison, not a circular one. Self-citations such as [10], [11], [12], and [33] are contextual (system model, prior OTA results, DP definitions) and are not load-bearing for the new theorems. Two flagged items are correctness/completeness concerns rather than circularity: the independence assertion after Rademacher flipping in Eq. (32) is not generally valid for correlated gradient entries, and Theorem 4 is stated without a corresponding appendix proof (the appendix stops at Corollary 2). These do not constitute circular reductions, so the circularity score remains 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard optimization assumptions (smoothness, convexity, bounded variance, bounded gradients), a specific statistical model for fading and noise, and the honest-but-curious privacy model. The only entities introduced are analytical constructs (channel hardening, MI metric), not new physical or algorithmic objects, so no invented entities are recorded. The main unquantified ingredients are C_g, the channel statistics, and the noise variances.

free parameters (4)

mu_c and sigma_c^2 (channel fading mean and variance) = mu_c = 1 in experiments; Rayleigh fading used
The channel statistics are treated as known inputs, but the convergence bounds require the mean mu_c to be known to set the global learning rate eta_t = eta_l / mu_c. In practice mu_c is estimated from long-term averages. The experiments set mu_c = 1. This is not fitted to the target result, but it is a system parameter.
sigma_z^2 (thermal noise variance) = not specified numerically
Appears in the privacy and convergence bounds as a noise variance. It is a physical system parameter, not a fitted constant, but its value is not reported in the experiments, making quantitative verification difficult.
sigma_s^2 (SGD sampling variance) = not specified
Bounded variance assumption in Assumption 3. The bounds depend on it, but the experiments do not estimate or report it.
C_g in Theorem 1 = unspecified constant
The mutual information bound contains an unspecified constant C_g depending on the fourth moment of the preprocessed gradient. Its value is not estimated, so the bound is not numerically evaluable as a privacy guarantee.

assumptions (6)

domain assumption Local gradient entries can be made statistically independent via decorrelation preprocessing such as random flipping (Section III-A, footnote 2).
This is required for the diagonal covariance structure in Theorem 1 and is assumed rather than proven for the actual networks used in experiments.
domain assumption All clients' local gradient covariance matrices share the same rank d* (Section III-A).
The privacy bound requires d*_n = d* for all clients; this is a simplifying assumption that may fail for heterogeneous data and networks.
domain assumption Honest-but-curious server and no malicious attackers for the privacy analysis (Section III-A).
The privacy claim is restricted to a server that follows the protocol; the experiments with label-flipping and noisy-label attacks are about robustness, not privacy.
domain assumption Assumption 4: expected squared norm of stochastic gradients is bounded by G_n^2.
Used in Lemma 1 and Theorems 3, 4. Justified by the finite transmit power budget, but it is an assumption on the loss landscape and data.
standard math The entropic central limit theorem bound from Bobkov et al. [52] holds for the preprocessed gradient vectors.
This is the external mathematical result that drives the lower bound on entropy in the proof of Theorem 1. It is cited, not proved, and its finite-fourth-moment conditions are assumed.
domain assumption For the power control analysis, the CSI estimation error is bounded by |Delta| <= Delta_max with known distributional assumptions (Section III-D).
This is an adopted model for imperfect CSI, and the conclusion that power control is unnecessary is conditional on this model and on the truncated channel inversion scheme.

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Pith. "Pith review of Rethinking Federated Learning Over the Air: The Blessing of Scaling Up." pith.science (2026). https://pith.science/paper/KQWUPD7S

@misc{pith2026250817697,
  author       = {Pith},
  title        = {Pith review of: Rethinking Federated Learning Over the Air: The Blessing of Scaling Up},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQWUPD7S}},
  note         = {Machine review of arXiv:2508.17697}
}

read the original abstract

Federated learning facilitates collaborative model training across multiple clients while preserving data privacy. However, its performance is often constrained by limited communication resources, particularly in systems supporting a large number of clients. To address this challenge, integrating over-the-air computations into the training process has emerged as a promising solution to alleviate communication bottlenecks. The system significantly increases the number of clients it can support in each communication round by transmitting intermediate parameters via analog signals rather than digital ones. This improvement, however, comes at the cost of channel-induced distortions, such as fading and noise, which affect the aggregated global parameters. To elucidate these effects, this paper develops a theoretical framework to analyze the performance of over-the-air federated learning in large-scale client scenarios. Our analysis reveals three key advantages of scaling up the number of participating clients: (1) Enhanced Privacy: The mutual information between a client's local gradient and the server's aggregated gradient diminishes, effectively reducing privacy leakage. (2) Mitigation of Channel Fading: The channel hardening effect eliminates the impact of small-scale fading in the noisy global gradient. (3) Improved Convergence: Reduced thermal noise and gradient estimation errors benefit the convergence rate. These findings solidify over-the-air model training as a viable approach for federated learning in networks with a large number of clients. The theoretical insights are further substantiated through extensive experimental evaluations.

Figures

Figures reproduced from arXiv: 2508.17697 by the authors.

**Figure 1.** An overview of the edge learning system. The following steps are repeated until the model converges: (1) each client [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Impact of client number N on privacy leakage, evaluated from training a CNN on the EMNIST dataset. participants accordingly. We further consider the same learning rate η = 0.03 for both local and global training (since µc = 1), local epoch E = ⌊ M B ⌋, and local batch size B = 50. All experiments are implemented with Pytorch on NVIDIA RTX 3090 GPU. 3) MI Estimation: We use Mutual Information Neural Estimator (MINE… view at source ↗

**Figure 3.** Visualizing the channel hardening effect, exemplified [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: , we can see that an increase in data heterogeneity impedes the training process, whereby it not only slows down the convergence rate but also inflicts additional fluctuations. However, with more clients participating in the system, the impact of data heterogeneity is …

**Figure 6.** Figure 6: Impact of power control on training efficiency, exem [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 8.** Figure 8: Impact of the number of clients N on test accuracy, exemplified by training a logistic regression model on the EMNIST dataset. challenge. To evaluate this, we train a logistic regression model for multi-class classification tasks on the EMNIST dataset and assess its pe…

**Figure 7.** Figure 7: Performance in the presence of malicious attacks, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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