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Non-Coherent Over-the-Air Federated Learning: Protocol, Convergence, and Device Scheduling

Haifeng Wen, Nicolò Michelusi, Osvaldo Simeone, Yang Yang, Hong Xing

arXiv:2609.08312v1cs.ITcs.LGeess.SPmath.IT

TL;DR

AirFL addresses AirFL’s communication bottleneck through MAC superposition, but coherent aggregation imposes demanding CSI, synchronization, and calibration requirements. NCAirFL uses non-coherent aggregation with binary dithering and error feedback; the paper reports FedAvg-like learning performance and faster convergence with optimized scheduling.

  • Problem

    Coherent AirFL requires accurate CSI, tight synchronization, and RF-chain calibration, which incur substantial overhead in broadband MAC settings.

    Source: §I · p. 1

  • Method

    NCAirFL combines binary dithering, unbiased non-coherent detection, and long-term error feedback for broadband AirFL without instantaneous CSI.

    Source: §VII · p. 10

  • Results

    NCAirFL approximates ideal FedAvg, and optimized device scheduling substantially accelerates convergence.

    Source: §VII · p. 11

  • Takeaways & Limitations

    The proposed scheduling approach formulates joint device selection and power control as a surrogate optimization with a low-complexity optimal solution.

    Source: §VII · p. 10

Abstract

from arXiv · show

To mitigate the scalability bottleneck in the radio access network (RAN) in federated edge learning (FEEL), over-the-air federated learning (AirFL) exploits waveform superposition over multiple-access channels (MACs) for analog model aggregation. However, coherent AirFL typically relies on stringent PHY-layer conditions such as accurate channel state information (CSI), tight time/frequency synchronization, and frequent transceiver calibration for signal alignment. However, these requirements, if not impossible to be met, incur substantial communication and computation overhead. In this paper, we propose a non-coherent AirFL (NCAirFL) protocol over a broadband single-antenna MAC, leveraging binary dithering, unbiased non-coherent detection, and long-term error feedback to waive the need for instantaneous CSI. For NCAirFL with general smooth non-convex objectives and a constant learning rate, we establish a convergence bound achieving the convergence rate in the same order of $\mathcal{O}(1/\sqrt{T})$ as communication-ideal FedAvg, where $T$ is the total number of communication rounds. To further improve communication efficiency under data and wireless resource heterogeneity, we also derive a lower bound on the expected single-round objective decrease in the global loss conditioned on device scheduling, building upon which a surrogate objective function is obtained for jointly optimal device selection and power control. Experimental results on MNIST and CIFAR-10 corroborate that NCAirFL achieves learning performance close to FedAvg in practical settings, with the proposed device scheduling policy substantially accelerating convergence.

I. INTRODUCTION

AirFL reduces uplink bandwidth by aggregating updates through MAC superposition, but coherent designs require demanding CSI, synchronization, and calibration. The paper introduces NCAirFL for broadband AirFL and outlines its relationship to prior work, system model, and FedAvg protocol.

A. Related Works

  • AirFL reduces bandwidth from O(n) to O(1) through MAC superposition, while coupling wireless impairments more tightly to learning dynamics.Coherent aggregation relies on CSI, synchronization, and RF-chain calibration; related CSI-light approaches include massive-MIMO systems and decentralized methods with different models or objective classes.Source: §I · p. 1, A. Related Works · p. 2
  • Prior work explored coherent AirFL, CSI-light aggregation, and learning-aware scheduling, but the cited approaches differ from broadband NCAirFL in detection, system setting, or objective class.The closest non-coherent decentralized methods provide guarantees for strongly convex objectives, unlike the general smooth non-convex objectives considered here.Source: A. Related Works · p. 2
  • The related-work discussion identifies learning-aware device selection as important under data and wireless-resource heterogeneity, beyond selection by channel quality alone.Prior approaches include controlled aggregation bias, gradient-based rules, and dynamic or probabilistic AirFL scheduling.Source: A. Related Works · p. 2

B. Contributions

  • NCAirFL combines binary dithering, unbiased non-coherent detection, and long-term error feedback to approximate communication-ideal FedAvg without instantaneous CSI.For smooth non-convex objectives, its convergence rate has the same order as FedAvg, and its joint device-selection and power-control policy is designed to accelerate convergence.Source: B. Contributions · p. 2

II. SYSTEM MODEL

  • Devices communicate with an edge server over a broadband fading MAC using analog OFDM, with neither transmitters nor the server accessing instantaneous CSI.The considered system is single antenna, and its channel model is illustrated in Fig. 1.Source: §II · p. 2, Figure 1 · p. 3

A. Learning Protocol (FedAvg)

  • FedAvg has each selected device initialize from the global model, perform local SGD, and transmit its model innovation for server aggregation and broadcast.The procedure continues until the prescribed round count or stopping criterion is reached.Source: A. Learning Protocol (FedAvg) · p. 3
  • With ideal communication, the server updates the global model by averaging active devices’ model innovations; fading and noise make this average unavailable directly over wireless.This motivates the paper’s communication model and non-coherent aggregation design.Source: A. Learning Protocol (FedAvg) · p. 3

B. Communication Model

The communication model uses a broadband fading MAC without instantaneous CSI and compensates only for large-scale fading. Square-law detection produces an unbiased statistic, which the decoder uses to estimate the aggregate model innovation.

B. Communication Model

  • The channel includes device-specific small-scale fading and additive Gaussian noise; fading is independent across devices and rounds, but need not be independent across subcarriers.Each device’s fading remains constant within a communication round and may vary across rounds.Source: B. Communication Model · p. 4
  • A common power scaling coefficient is chosen each round so that every active device satisfies its per-round average transmit-power constraint.The scaling sequences used in the analysis are initialized independently of the learning trajectory.Source: B. Communication Model · p. 4
  • The design avoids small-scale fading compensation and coherent phase alignment, compensating only for large-scale fading.The server applies square-law detection to the received signal and decodes an estimate of the aggregate model innovation.Source: B. Communication Model · p. 4
  • The square-law detector’s statistic is an unbiased estimate of the sum of transmitted nonnegative vectors, and the decoder maps it to an aggregate innovation estimate.The preprocessing and decoding functions must make detection unbiased for the preprocessed sum while controlling estimation error.Source: B. Communication Model · p. 4

III. NON-COHERENT AIRFL (NCAIRFL)

NCAirFL addresses CSI-free non-coherent aggregation by preprocessing innovations with multiplicative binary dithering and maintaining device memory. The memory feeds back portions not conveyed in a round, and its residual diminishes in mean square near a stationary point under the stated contraction property.

III. NON-COHERENT AIRFL (NCAIRFL)

  • NCAirFL is designed to aggregate wirelessly without CSI while preserving learning performance close to ideal FedAvg.Its preprocessing and decoding functions are tailored to the non-coherent transmission model.Source: §III · p. 4
  • Each device maintains memory of past model-innovation portions not represented in earlier transmissions, and multiplicative binary dithering maps the dithered innovation plus memory to a nonnegative vector.The dither entries are independent signs with probabilities p and 1−p; the nonnegative representation is suitable for non-coherent transmission.Source: §III · p. 4
  • Memory acts as error feedback by carrying forward innovation portions not conveyed in the current round; near a stationary point, accumulated residual diminishes in mean square under the established contraction property.A selected device updates memory by subtracting the conveyed surrogate, while an unselected device retains its memory.Source: §III · p. 4

2 Initialize m(0)

The algorithm initializes device memories and the global model, then repeats its round-level operations until the communication-round limit.

2 Initialize m(0)

  • Initialize each device’s memory to zero and initialize the global model before setting the round index to zero.The loop runs while t<T.Source: §2 · p. 5
  • All devices perform their parallel round operations, while only active devices update memory.The global model is then updated for the next round.Source: §2 · p. 5

28 Output: θ(T )

The NCAirFL round ends with server-side square-law detection, dither removal, a global-model update, and broadcast. Its convergence analysis bounds detector error and establishes a stationary-point rate matching FedAvg’s order without instantaneous CSI.

28 Output: θ(T )

  • The round concludes with server-side square-law detection, shared-dither removal, a global-model update, and broadcast.Only active devices commit their memory update and transmit.Source: §28 · p. 5

IV. CONVERGENCE ANALYSIS

  • The analysis conditions on the initial-state information that determines the scheduling policy {I(t), ρ(t)}.The convergence metric is the expected squared gradient norm averaged over global rounds.Source: §IV · pp. 5–6
  • The memory-mechanism upper bound is minimized at dithering probability p = 1/2, yielding λ = 1/2.The contraction property is stated for each device and communication round.Source: §IV · p. 5
  • The non-coherent detector’s MSE bound includes fading fourth-moment and noise-product terms, while cross-device signal products scale as s(s−1).The bound applies under Assumption 2 when ρ(t) is the largest feasible common scaling coefficient.Source: §IV · p. 6
  • The convergence bound separates initialization error, non-coherent aggregation error, stochastic-gradient and local-update error, and accumulated memory error.Aggregation error grows with detector MSE and decreases with the worst device’s average received SNR; the memory term depends on the dithering parameter λ.Source: §IV · p. 6
  • NCAirFL reaches a stationary-point rate of O(1/sqrt T), matching FedAvg’s order for smooth non-convex objectives without instantaneous CSI.The result assumes a constant learning rate and averages over mini-batch sampling, dithering, device selection, fading, and channel noise.Source: §IV · p. 6

V. JOINT OPTIMIZATION OF DEVICE SELECTION AND POWER CONTROL

The section frames device selection as a trade-off between aggregation utility and common power scaling, then develops a joint selection and power-control policy using a lower bound on expected one-round loss reduction.

V. JOINT OPTIMIZATION OF DEVICE SELECTION AND POWER CONTROL

  • Partial participation can reduce the quadratic non-coherent cross-term and remove a power bottleneck, but discards useful innovations.Centralized selection coordinates this trade-off; independent local decisions may not identify the common bottleneck.Source: §V · p. 6
  • The policy formulation uses a lower bound on expected one-round forward reduction for each prescribed active-device set to jointly optimize selection and power control.The section targets improved convergence performance under data and wireless-resource heterogeneity.Source: §V · p. 6
  • Because the required moments are unavailable in closed form, the method substitutes causal estimates and optimally solves the resulting surrogate problem.The resulting policy is evaluated in Section VI.Source: §V · p. 6

A. Optimization Problem Formulation

The formulation uses a lower bound on expected one-round loss reduction to motivate joint device selection and power control. The resulting surrogate problem has a globally optimal solution under its stated condition, while convergence under state-dependent scheduling remains open.

A. Optimization Problem Formulation

  • The one-round reduction bound separates gradient-descent utility, SGD error and data heterogeneity, and effective noise from non-coherent aggregation.The effective-noise term depends on the fading moment and power-scaling coefficient.Source: A. Optimization Problem Formulation · p. 7
  • The bound motivates jointly choosing the active devices and power control in each communication round.The formulation includes a fixed active-device count and an average-power constraint.Source: A. Optimization Problem Formulation · p. 7

B. Proposed solution to (P1)

  • Causal exponential moving averages replace unavailable closed-form moments used in the selection statistics.Every device refreshes its statistics each round, so inactivity alone does not cause an unselected device’s EMA to decay.Source: B. Proposed solution to (P1) · p. 7
  • When bui > 0 for every device, Algorithm 2 yields the jointly optimal active set and power-control factor.For a fixed active set, the optimal power scaling is the largest feasible value; selection then evaluates candidate bottleneck devices.Source: B. Proposed solution to (P1) · p. 7
  • O(n^2+n log n) operations solve the surrogate selection problem, below the stated exponential-order exhaustive search complexity.Sorting takes O(n log n), followed by O(n) work for each candidate bottleneck.Source: B. Proposed solution to (P1) · p. 8
  • Jointly optimizing participation ratio, device selection, and power control is left as an extension; this work fixes the ratio.The remark states the joint formulation has the same O(n^2+n log n) complexity.Source: B. Proposed solution to (P1) · p. 8
  • Convergence under deterministic, state-dependent device selection is left for future work.The stated convergence result assumes uniform sampling without replacement; the optimized rule selects using state-dependent statistics and power constraints.Source: B. Proposed solution to (P1) · p. 8

VI. NUMERICAL RESULTS

The evaluation uses 20 devices on MNIST and CIFAR-10 to assess NCAirFL against communication-ideal FedAvg and quantify the gain from joint device selection and power control.

VI. NUMERICAL RESULTS

  • The evaluation uses 20 devices on MNIST and CIFAR-10.It examines how closely NCAirFL approximates communication-ideal FedAvg and quantifies the gain from joint device selection and power control.Source: §VI · p. 8
  • NCAirFL is evaluated for its approximation to communication-ideal FedAvg.The passage states this as an evaluation objective, not as a reported outcome.Source: §VI · p. 8
  • The evaluation also quantifies the gain from joint device selection and power control.No measured gain is reported in the supplied passage.Source: §VI · p. 8

A. Experimental Settings

The experiments use MNIST with a non-i.i.d. split and CIFAR-10 with an i.i.d. split, alongside specified channel, training, and power settings. Comparisons include FedAvg and coherent AirFL baselines.

A. Experimental Settings

  • MNIST uses a Dirichlet non-i.i.d. split with ϱ = 0.1 and a two-layer fully connected network; CIFAR-10 uses an i.i.d. split and ResNet-18.The MNIST network has 79,510 parameters, and the CIFAR-10 network has 11,181,642 parameters.Source: A. Experimental Settings · p. 8
  • The channel setup uses Rayleigh small-scale fading and devices distributed within a 1000 m-radius disk around the server.The path-loss model uses a 2.4 GHz carrier frequency.Source: A. Experimental Settings · p. 8
  • Training uses five local SGD steps, participation ratio 0.5, dithering parameter 1/2, and EMA forgetting factor 0.1.Mini-batches contain 64 samples; reported curves average 50 Monte Carlo trials.Source: A. Experimental Settings · p. 8
  • The comparisons include ideal-communication FedAvg, optimized-selection FedAvg, CAirFL, and AirFL-Mem.In the stated typical 5G NR configuration, coherent baselines incur approximately 14.3% CSI-estimation overhead, which NCAirFL eliminates.Source: A. Experimental Settings · pp. 8–9, Table I · p. 9

B. Training Performance

NCAirFL achieves test accuracy close to FedAvg, while optimized scheduling improves learning behavior and convergence. The results also show that participation ratio and data heterogeneity shape scheduling benefits, and the conclusion reports convergence guarantees for smooth non-convex objectives.

B. Training Performance

  • On both datasets, NCAirFL and NCAirFL(Opt.) closely track their ideal-communication counterparts, and NCAirFL avoids CAirFL's observed error floor.The figures report training loss versus communication round index T.Source: B. Training Performance · p. 9, Figure 2 · p. 9, Figure 3 · p. 9
  • On CIFAR-10, NCAirFL(Opt.) reaches near-zero training loss within roughly 50 rounds; optimized scheduling can also outperform AirFL-Mem in the considered settings.Vanilla NCAirFL still has non-negligible residual loss at that communication budget.Source: B. Training Performance · p. 9, Figure 3 · p. 9
  • NCAirFL attains test accuracy very close to FedAvg on MNIST and CIFAR-10, while NCAirFL(Opt.) improves early- and mid-stage accuracy over vanilla and coherent baselines.The optimized version's advantage is more visible on CIFAR-10.Source: B. Training Performance · p. 9, Figure 4 · p. 10, Figure 5 · p. 10

C. Effect of the Participation Ratio and Data Heterogeneity

  • Optimized selection performs best at an intermediate participation ratio rather than r = 1, and improves average accuracy across a broad range of ratios on both datasets.At full participation, optimized and non-optimized schemes necessarily coincide.Source: C. Effect of the Participation Ratio… · p. 10, Figure 6 · p. 10, Figure 7 · p. 11
  • As MNIST data heterogeneity grows, the benefit of optimized selection becomes more pronounced; at 1/ϱ = 10, it improves average test accuracy by about 10 percentage points.Increasing heterogeneity also raises average loss and lowers average accuracy for all methods.Source: C. Effect of the Participation Ratio… · p. 10

VII. CONCLUSION

  • For smooth non-convex objectives, NCAirFL retains the convergence rate, and its low-complexity joint scheduling solution substantially accelerates convergence.The conclusion says NCAirFL approximates ideal FedAvg and derives a lower bound on single-round global-loss reduction to formulate joint device selection and power control.Source: §VII · pp. 10–11

APPENDIX A. Proof Sketch of Lemma 4.1

The proof sketches establish a dithering-dependent contraction, decompose detector error into four contributions, and use a perturbed-iterate argument to derive the convergence bound. The appendix also identifies the quantities plotted against heterogeneity for MNIST.

APPENDIX A. Proof Sketch of Lemma 4.1

  • Setting the binary-dithering probability to 1/2 maximizes λ and minimizes the contraction factor in the Lemma 4.1 argument.For nonzero squared entries, the bound uses λ = min(p, 1 − p).Source: Appendix A · p. 11
  • Figure 8 plots average MNIST training loss and test accuracy against heterogeneity level 1/ϱ, averaging each metric over communication rounds.The supplied caption identifies the NCAirFL and FedAvg comparison but does not state its outcome.Source: Figure 8 · p. 11

B. Proof Sketch of Lemma 4.2

  • The Lemma 4.2 proof attributes the error bound’s four right-hand-side terms to fading, noise–noise, inter-device, and signal–noise contributions.It uses vanishing mixed conditional second moments and an expected memory bound.Source: B. Proof Sketch of Lemma 4.2 · pp. 11–12

C. Proof Sketch of Proposition 4.1

  • The Proposition 4.1 proof compares real and virtual iterates, then sums the one-step bounds and uses f(˜θ(T)) ≥ f* to obtain (24).Uniform device selection is independent of the entering iterate, making the active-device average unbiased under total expectation.Source: C. Proof Sketch of Proposition 4.1 · p. 12

D. Proof Sketch of Proposition 5.1

The Proposition 5.1 proof conditions on the scheduled devices and power, decomposes the first-order update term, and combines its bounds with a detector-noise bound to obtain (25).

D. Proof Sketch of Proposition 5.1

  • The proof decomposes the first-order term into local-model drift, data heterogeneity, stochastic-gradient error, and a memory-difference term.It applies L-smoothness with device selection and power fixed conditional on F0.Source: D. Proof Sketch of Proposition 5.1 · p. 12
  • The local-model drift and memory-difference terms receive separate bounds using the local-update bound, assumptions, and expected memory bound.The passage also bounds the memory-difference term using λ2.Source: D. Proof Sketch of Proposition 5.1 · p. 12
  • The conditional zero-mean detector property eliminates the second-order cross term before the proof applies the intermediate MSE bound.Substituting the first- and second-order bounds into (46) yields (25).Source: D. Proof Sketch of Proposition 5.1 · p. 13

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