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Gradient and Channel Aware Dynamic Scheduling for Over-the-Air Computation in Federated Edge Learning Systems

Jun Du, Bingqing Jiang, Chunxiao Jiang, Yuanming Shi, Zhu Han

arXiv:2212.00491v1eess.SYcs.SY

TL;DR

AirComp-enabled federated edge learning must account for data-update quality, channel conditions, and device energy when scheduling transmissions. This paper proposes gradient- and channel-aware scheduling with power control and retained updates from unselected devices; simulations report higher test accuracy, faster convergence, and robustness across channel conditions.

  • Problem

    AirComp-FEEL scheduling must balance device energy and channel requirements because stricter energy weighting or received-SNR thresholds can reduce participation and test accuracy.

    Source: 1) Influence of · pp. 10–11

  • Method

    The paper schedules devices using update quality, channel status, and energy, with channel-inversion power control, Lyapunov-based optimization, and accumulated residual feedback for unselected updates.

    Source: A. Contributions and Organization · p. 2

  • Results

    Simulations on MNIST and CIFAR-10 report improved test accuracy and convergence rate, and robustness across channel conditions in both i.i.d. and non-i.i.d. cases versus baselines.

    Source: §VI · p. 13

  • Takeaways & Limitations

    The simulations indicate that channel-aware scheduling can vary participant count and transmit power to support energy-efficient and accurate model training.

    Source: §VI · p. 13

Abstract

from arXiv · show

To satisfy the expected plethora of computation-heavy applications, federated edge learning (FEEL) is a new paradigm featuring distributed learning to carry the capacities of low-latency and privacy-preserving. To further improve the efficiency of wireless data aggregation and model learning, over-the-air computation (AirComp) is emerging as a promising solution by using the superposition characteristics of wireless channels. However, the fading and noise of wireless channels can cause aggregate distortions in AirComp enabled federated learning. In addition, the quality of collected data and energy consumption of edge devices may also impact the accuracy and efficiency of model aggregation as well as convergence. To solve these problems, this work proposes a dynamic device scheduling mechanism, which can select qualified edge devices to transmit their local models with a proper power control policy so as to participate the model training at the server in federated learning via AirComp. In this mechanism, the data importance is measured by the gradient of local model parameter, channel condition and energy consumption of the device jointly. In particular, to fully use distributed datasets and accelerate the convergence rate of federated learning, the local updates of unselected devices are also retained and accumulated for future potential transmission, instead of being discarded directly. Furthermore, the Lyapunov drift-plus-penalty optimization problem is formulated for searching the optimal device selection strategy. Simulation results validate that the proposed scheduling mechanism can achieve higher test accuracy and faster convergence rate, and is robust against different channel conditions.

I. INTRODUCTION … A. Federated Learning Model

The paper develops AirComp-enabled federated edge learning to address communication, computation, and privacy challenges, proposing device scheduling and accumulated residual feedback alongside its system and learning model.

I. INTRODUCTION

  • Centralized training burdens wireless IoT with computation, communication, latency, and privacy challenges; federated learning avoids sharing local data, while AirComp uses channel superposition for communication-efficient aggregation.AirComp-enabled FEEL is presented as suitable for high communication efficiency and low latency.Source: §I · pp. 1–2

A. Contributions and Organization

  • The proposed scheduling selects devices using gradient, channel, and energy information, combines channel-inversion power control with distributed online Lyapunov optimization.Device quality is represented by a defined indicator, and the optimization algorithm addresses the scheduling problem.Source: A. Contributions and Organization · p. 2
  • Accumulated residual feedback retains unselected devices’ local updates for future use rather than discarding them, and the paper characterizes the mechanism’s convergence bound.The paper states that this approach can improve global aggregation efficiency and accuracy as well as convergence.Source: A. Contributions and Organization · p. 2
  • Simulations on MNIST and CIFAR-10 report improvements in test accuracy, energy consumption, and adaptation and efficiency in non-i.i.d. scenarios versus state-of-the-art baselines.The contributions passage reports these findings but provides no quantitative values.Source: A. Contributions and Organization · p. 2

B. Related Work

  • Prior scheduling work often considers channel conditions and model updates but omits energy efficiency, while unselected devices’ updates are discarded; AirComp research also addresses channel distortion and learning performance.The cited review describes prior approaches involving power control, user selection, resource allocation, gradient statistics, and energy-aware scheduling.Source: B. Related Work · pp. 2–3

II. SYSTEM MODEL AND LEARNING MECHANISM

  • The system consists of a finite set of single-antenna edge devices and an edge server; devices upload locally trained model gradients for aggregation, avoiding raw-data sharing.The paper describes reduced communication overhead, server computation workload, and privacy disclosure relative to centralized aggregation and training.Source: §II · p. 3, A. Federated Learning Model · p. 3

A. Federated Learning Model

  • The federated learning model uses cross-entropy local losses on independently collected, disjoint datasets and seeks a model minimizing the global loss across their union.The passages define the global objective over the distributed datasets and state that it measures fit to the entire corpus on average.Source: A. Federated Learning Model · p. 3

B. Over-the-air Gradient Aggregation

The section describes AirComp-based gradient collection in which selected edge devices transmit concurrently over a shared radio resource, allowing the server to aggregate their updates through channel superposition. Device selection and channel-aware power control are designed to support reliable aggregation and minimize expected final training loss.

B. Over-the-air Gradient Aggregation

  • The method selects a subset of devices with suitable channel conditions and data quality, optimizing round-by-round selection to minimize expected final training loss.A binary variable indicates whether each device is selected in each communication round.Source: B. Over-the-air Gradient Aggregation · p. 4
  • Concurrent gradient transmissions over one shared radio resource let the server aggregate a channel- and power-weighted sum, integrating communication and computation for low-latency, low-complexity collection.The devices are assumed to transmit in strict synchronization.Source: B. Over-the-air Gradient Aggregation · p. 4
  • Channel gains are modeled as quasi-static within each radio block but varying across blocks and rounds, and transmit power follows channel inversion to resist fading,,.A power-scaling factor determines the received SNR at the server.Source: B. Over-the-air Gradient Aggregation · p. 4
  • Devices normalize their gradient vectors before transmission, while the server denormalizes the noisy aggregate to update the global model using the round’s learning rate.The model assumes the server can broadcast the global model to all devices without error.Source: B. Over-the-air Gradient Aggregation · p. 4

C. AirComp enabled Federated Learning Mechanism … III. PROBLEM FORMULATION OF ACCUMULATED RESIDUAL BASED FEDERATED LEARNING

The FEEL mechanism schedules devices using local-update importance, channel quality, and energy, while retaining unselected gradients as residuals for possible future aggregation. The paper then develops the residual-based aggregation and device-selection problem and analyzes convergence.

C. AirComp enabled Federated Learning Mechanism

1) Global Model Broadcast:

2) Local Model Update:

  • Local mini-batches are sampled uniformly and randomly from each device’s dataset to reduce training complexity.Only a subset of devices transmits gradients for aggregation in a communication round.Source: 2) Local Model Update · p. 5, 3) Edge Device Selection · p. 5
  • Device selection considers candidate count for robustness to channel noise and weighs channel gain, local-update significance, and transmit energy to support reliable, efficient transmissions.The selection is intended to reduce latency and resource overhead by choosing only part of the devices in each round.Source: 3) Edge Device Selection · p. 5

3) Edge Device Selection:

  • Devices compute data-state and channel-state indicators from gradient magnitude and channel gain, then report these indicators with energy consumption for server-side selection.The server applies a selection mechanism to pursue accurate and energy-efficient gradient aggregation.Source: 3) Edge Device Selection · p. 5

4) Local Models Aggregation and Global Model Update:

  • Unselected devices’ gradients are accumulated as residuals, and selected devices upload their current gradient plus accumulated residual through AirComp for global-model updating.The server aggregates the selected devices’ local update information and updates the global model for the next round.Source: 4) Local Models Aggregation and Global… · p. 5

III. PROBLEM FORMULATION OF ACCUMULATED RESIDUAL BASED FEDERATED LEARNING

  • The paper introduces accumulated-residual uploading and aggregation, formulates server-side device selection, and analyzes convergence to transform the selection problem into a tractable form.The residual mechanism is designed to use gradients from unselected devices and accelerate convergence.Source: §III · p. 5

A. Accumulated Residual Transmission Mechanism · B. Problem Formulation · C. Convergence Analysis and Problem Transformation

The sections describe retaining unselected devices’ updates and formulating device scheduling as expected-loss minimization. Convergence bounds motivate an online selection strategy that accounts for channel noise and device energy constraints.

A. Accumulated Residual Transmission Mechanism

  • Selected devices combine current gradients with accumulated residuals (ξ=1), while unselected devices retain updates for future transmission when data and channel quality suffice.The mechanism is designed to use updates withheld in prior rounds due to energy constraints, while meeting energy requirements and targeting the all-noiseless-local-gradient-average benchmark.Source: A. Accumulated Residual Transmission… · pp. 5–6

B. Problem Formulation

  • The device-selection objective minimizes expected final training loss to attain desired model accuracy, subject to each device’s average transmit-energy constraint over T rounds.Source: B. Problem Formulation · p. 6
  • Because neural-network parameters make the expected one-round loss change difficult to express explicitly, P1 uses an approximate closed-form upper bound from convergence analysis.Source: B. Problem Formulation · p. 6

C. Convergence Analysis and Problem Transformation

  • The convergence analysis assumes L-smooth, µ-strongly convex losses and unbiased stochastic gradients with bounded variance.Source: C. Convergence Analysis and Problem… · p. 6
  • Lemma 1 gives an upper bound on average accumulated residual per round in terms of the global model gradient, local-gradient variance, and average probability of device nonselection.Source: C. Convergence Analysis and Problem… · p. 6
  • Lemma 2 bounds the one-round expected loss change for selected devices and mini-batches, accounting jointly for transmit power, channel noise, residual bounds, and stochastic gradients.Source: C. Convergence Analysis and Problem… · p. 6
  • The selected-device count affects resistance to channel noise, so scheduling minimizes the convergence upper bound rather than maximizing device count, using an online round-by-round decision.The future system state is unavailable, and the bound guides selection according to each round’s channel-noise level.Source: C. Convergence Analysis and Problem… · p. 6
  • With the dataset and specified hyperparameters fixed, the first three terms of Lemma 2 are constants and are ignored when transforming P1.Source: C. Convergence Analysis and Problem… · p. 6

IV. GRADIENT AND CHANNEL AWARE DEVICE SCHEDULING FOR AIRCOMP · A. Indicator of Device Quality

Section IV proposes dynamic device scheduling for AirComp-enabled FEEL based on data quality, channel conditions, and energy consumption; subsection A defines the quality indicator used to support later Lyapunov-based selection.

IV. GRADIENT AND CHANNEL AWARE DEVICE SCHEDULING FOR AIRCOMP

  • The proposed mechanism uses device quality to select AirComp participants, then applies Lyapunov-drift optimization to dynamic selection and designs an algorithm for that problem.Source: §IV · p. 7

A. Indicator of Device Quality

  • Local-update significance is measured by its l2 norm, while channel quality is measured by |hn,t|; larger update norms and channel magnitudes indicate greater contribution and better channels [40],.Source: A. Indicator of Device Quality · p. 7
  • The method places update- and channel-quality indicators on unified values and combines them using weights ρ1 and ρ2, where ρ1 + ρ2 = 1.Source: A. Indicator of Device Quality · p. 7
  • Because communication consumes more energy than local training, the selection design also accounts for each device’s transmit energy under channel-inversion power control.Source: A. Indicator of Device Quality · p. 7
  • The comprehensive indicator balances the gain from update and channel qualities against energy-consumption loss using adjustable weights λE and λV, with λE + λV = 1.Source: A. Indicator of Device Quality · p. 7
  • Each device calculates its indicator using current and previously untransmitted local updates, then sends the indicator to the server for selection, avoiding disclosure of local datasets.Source: A. Indicator of Device Quality · p. 7

B. Lyapunov-Drift-Minimization Based Problem Formulation

The section formulates device scheduling as a joint tradeoff between limiting channel-noise impact through device quantity and maximizing selected-device quality, using Lyapunov drift-plus-penalty optimization.

B. Lyapunov-Drift-Minimization Based Problem Formulation

  • The scheduling objective jointly chooses how many devices transmit to reduce channel-noise impact and favors devices with higher training quality.The paper frames this as a linear tradeoff between maximizing device quality and minimizing the FEEL objective function.Source: B. Lyapunov-Drift-Minimization Based… · p. 7
  • The server sorts device importance scores, selects k* by minimizing p_t(k), then updates residuals, sets transmit power, and aggregates transmitted models.These steps describe the scheduling algorithm’s candidate-count selection and subsequent model transmission and aggregation.Source: B. Lyapunov-Drift-Minimization Based… · p. 8
  • Device quality is I_n,t = λV V_n,t − λE E_n,t, with α balancing it against loss; tuning λE, λV, and α also supports acceptable per-round sum energy.The mechanism includes individual energy consumption and can also serve sum-energy-constrained FEEL systems by distributing limited power to energy-efficient devices.Source: B. Lyapunov-Drift-Minimization Based… · p. 8

C. Dynamic Device Scheduling Algorithm … A. Experiment Settings

The paper presents Lyapunov-based scheduling that selects devices using a quality indicator and retains unselected updates, then analyzes convergence and evaluates the mechanism in FEEL experiments.

C. Dynamic Device Scheduling Algorithm

  • The server selects the k devices with the highest indicator values, choosing k to minimize drift-plus-penalty; the algorithm’s per-round complexity is reported as o(NlogN).The indicator ranks device quality, and the search evaluates candidate quantities from 1 to N.Source: C. Dynamic Device Scheduling Algorithm · p. 8

D. Performance Analysis

  • Theorem 1 bounds the T-round Lyapunov drift, and its three bound terms converge to zero as t increases, implying a vanishing expected gap between global training loss and its optimum.The supplied passage does not include the bound’s formula.Source: D. Performance Analysis · p. 8

C. Dynamic Device Scheduling Algorithm

  • The indicator combines local-update quality, energy consumption, and channel condition; accumulated residuals preserve unselected updates for future transmission and support fairness.Indicator transmission overhead is described as negligible relative to local updates, and the mechanism can be adapted by redefining the indicator.Source: D. Performance Analysis · pp. 8–9, C. Dynamic Device Scheduling Algorithm · p. 8

V. SIMULATION RESULTS

  • The experiments evaluate scheduling over fading noisy channels in a system with 100 devices, i.i.d. Rayleigh fading, and noise variance selected from {0.5, 1, 3}.The average energy budget is 1.5 and the received-SNR threshold is 0 dB.Source: §V · p. 9, A. Experiment Settings · p. 9

A. Experiment Settings

  • Experiments use a CNN for MNIST and CIFAR-10 under both i.i.d. and non-i.i.d. data assignments, with each non-i.i.d. device receiving two label-specific shards.The datasets contain 60,000 MNIST and 50,000 CIFAR-10 training samples; the CNN has two 5 × 5 convolution layers, a 512-unit fully connected layer, and softmax output.Source: A. Experiment Settings · p. 9
  • The study compares an ideal noiseless-gradient benchmark with random selection, channel-threshold selection, channel-then-gradient selection, and local energy-based selection.The specified settings include 30 randomly selected devices, channel threshold |h|th = 1, and kc = 50 followed by k = 20 for Baseline 3.Source: A. Experiment Settings · p. 9

B. Influence of Parameter Settings on Training Efficiency · 1) Influence of

The section examines how α, λE/λV, and γthr shape device selection, energy consumption, and training accuracy under different channel conditions. The results show tradeoffs among these outcomes and that test accuracy can converge across the channel conditions studied.

1) Influence of

B. Influence of Parameter Settings on Training Efficiency

  • Increasing α selects more devices and raises their transmit energy consumption, while lower α favors devices with higher local-update and channel quality.Source: 1) Influence of · p. 10

1) Influence of

  • As channel noise σ2 decreases, lower transmit energy can let the server select more devices; with poor channel conditions, it selects fewer devices with higher transmit energy.Source: 1) Influence of · p. 10
  • A proper setting of λE and λV balances energy consumption and training accuracy, supporting improved energy-efficiency in the FEEL system.Source: 1) Influence of · pp. 10–11
  • Increasing γthr and worsening channel conditions reduce device selection; higher γthr requires more transmit power and energy, but does not guarantee higher training accuracy.Source: Figure 6 · p. 11, 1) Influence of · p. 11

C. Training Performance Improvements · D. Proposed Mechanism in Non-i.i.d. Data Distribution · VI. CONCLUSION

The paper’s dynamic AirComp scheduling mechanism combines gradient, channel, and energy awareness with power control and accumulated residual feedback to improve learning accuracy and convergence across i.i.d. and non-i.i.d. data and varying channel conditions. The sections report its performance, explain its operation in non-i.i.d. settings, and summarize the mechanism and future directions.

C. Training Performance Improvements

  • Baseline 1’s random selection ignores device differences, baselines 2 and 3 do not jointly consider CSI and DSI, and baseline 4 uses fixed power that cannot adapt to fading.The passage also notes that all baselines select a quantity of devices independent of channel condition.Source: C. Training Performance Improvements · p. 11
  • The proposed mechanism achieves the highest test accuracy and fastest convergence on MNIST and CIFAR-10, while maintaining convergence as channel noise worsens and baselines deteriorate.At σ^2_0 = 3, the baselines deteriorate rapidly, whereas the proposed mechanism ensures convergence and is described as robust across channel conditions.Source: C. Training Performance Improvements · pp. 11–12, Figure 8 · p. 12
  • Dynamic residual feedback improves test accuracy and convergence under all three evaluated channel conditions by retaining unselected devices’ gradients for later use when data and channel qualities suffice.Without this feedback, unselected devices’ updates are underused, potentially causing training bias and reducing training efficiency and accuracy.Source: C. Training Performance Improvements · p. 12, Figure 9 · p. 12

D. Proposed Mechanism in Non-i.i.d. Data Distribution

  • In non-i.i.d. data, channel noise affects test accuracy more strongly than in i.i.d. data, yet the proposed mechanism still outperforms baselines as noise increases.Accumulated residuals help use more important non-i.i.d. data, while dynamic scheduling with power control adapts to different noise levels.Source: Figure 10 · p. 12, D. Proposed Mechanism in Non-i.i.d.… · p. 12
  • Baseline 4 outperforms the other baselines in non-i.i.d. test accuracy, indicating the value of jointly considering CSI and DSI in device scheduling.Source: D. Proposed Mechanism in Non-i.i.d.… · p. 12

VI. CONCLUSION

  • The proposed AirComp-enabled FEEL mechanism schedules devices using data-update quality, channel condition, and energy consumption, with channel-inversion power control and accumulated residual feedback for model updates.A Lyapunov-drift-minimization algorithm provides distributed online scheduling, and the paper characterizes convergence of the mechanism.Source: §VI · pp. 12–13
  • When channel conditions are poor, the mechanism selects fewer contributive devices with higher transmit power; otherwise, it schedules more devices at lower power for energy-efficient, accurate training.Simulations report improved test accuracy and convergence and robustness across channel conditions for both i.i.d. and non-i.i.d. cases.Source: §VI · p. 13
  • Future work may consider multi-frequency channels for large-scale AirComp-enabled FEEL, along with privacy protection, security, and more insecure circumstances.Source: §VI · p. 13

APPENDIX A PROOF OF LEMMA 1

The proof decomposes the average residual into accumulated residuals from selected and unselected devices, with the unselection probability determined by channel condition.

APPENDIX A PROOF OF LEMMA 1

  • The proof separates the average residual into two parts.The following passages identify the parts as residuals accumulated by selected and unselected devices.Source: Appendix A · p. 13
  • One part is the accumulated residual of selected devices at t+1.Source: Appendix A · p. 13
  • The other part is the accumulated residual of unselected devices at t+1, whose average unselection possibility in round t is determined by channel condition.The proof concludes that this completes Lemma 1.Source: Appendix A · p. 13

APPENDIX B PROOF OF LEMMA 2

The proof derives Lemma 2 from (7), (11), and the assumptions in Section III-C, taking expectations on both sides. It then uses Lemma 1 and the approximation Lζt+1≈Lζt.

APPENDIX B PROOF OF LEMMA 2

  • The proof starts from (7) and (11), applies the assumptions in Section III-C, and takes expectations on both sides.It also uses m as an estimate of the global gradient mean.Source: Appendix B · p. 13
  • The derivation obtains inequality (a) from Lemma 1 and assumes Lζt+1≈Lζt.The appendix states that this completes the proof of Lemma 2.Source: Appendix B · pp. 13–14

APPENDIX C PROOF OF THEOREM 1

The proof defines a Lyapunov function from the loss gap and derives its expected one-round drift. It then extends the drift analysis across T rounds to complete Theorem 1.

APPENDIX C PROOF OF THEOREM 1

  • The Lyapunov function is defined as L_t = f(w_t) − f*, the gap between the loss and its optimum.The one-round drift is the expected change in this Lyapunov function from round t−1 to t.Source: Appendix C · p. 14
  • The proof derives the T-round drift term and concludes the proof of Theorem 1.The supplied passage does not include the full expression for the T-round drift.Source: Appendix C · p. 14
  • Using the third assumption in Section III-C and the one-round convergence rate, the proof derives a recursive bound across rounds.The displayed inequality substitutes the preceding-round bound into the T-round expression.Source: Appendix C · p. 14

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