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Power-Efficient Over-the-Air Aggregation with Receive Beamforming for Federated Learning

Faeze Moradi Kalarde, Min Dong, Ben Liang, Yahia A. Eldemerdash Ahmed, Ho Ting Cheng

TL;DR

This paper tackles power-efficient uplink design for federated learning using over-the-air analog aggregation with a multi-antenna server. It derives sufficient aggregation-error conditions that guarantee convergence and reformulates the problem into a bi-convex structure, enabling a monotonic, alternating optimization algorithm for joint transmit weights and receive beamforming. The method is extended to imperfect CSI via a CSI-error-aware design, preserving convex subproblems and achieving additional power savings in practice. Across MNIST and CIFAR-10 experiments, the proposed PoMFL framework attains the same convergence rate as benchmarks but with significantly reduced uplink power; a CSI-aware variant further improves performance under channel uncertainties, highlighting practical relevance for energy-constrained wireless FL systems.

Abstract

This paper studies power-efficient uplink transmission design for federated learning (FL) that employs over-the-air analog aggregation and multi-antenna beamforming at the server. We jointly optimize device transmit weights and receive beamforming at each FL communication round to minimize the total device transmit power while ensuring convergence in FL training. Through our convergence analysis, we establish sufficient conditions on the aggregation error to guarantee FL training convergence. Utilizing these conditions, we reformulate the power minimization problem into a unique bi-convex structure that contains a transmit beamforming optimization subproblem and a receive beamforming feasibility subproblem. Despite this unconventional structure, we propose a novel alternating optimization approach that guarantees monotonic decrease of the objective value, to allow convergence to a partial optimum. We further consider imperfect channel state information (CSI), which requires accounting for the channel estimation errors in the power minimization problem and FL convergence analysis. We propose a CSI-error-aware joint beamforming algorithm, which can substantially outperform one that does not account for channel estimation errors. Simulation with canonical classification datasets demonstrates that our proposed methods achieve significant power reduction compared to existing benchmarks across a wide range of parameter settings, while attaining the same target accuracy under the same convergence rate.

Power-Efficient Over-the-Air Aggregation with Receive Beamforming for Federated Learning

TL;DR

This paper tackles power-efficient uplink design for federated learning using over-the-air analog aggregation with a multi-antenna server. It derives sufficient aggregation-error conditions that guarantee convergence and reformulates the problem into a bi-convex structure, enabling a monotonic, alternating optimization algorithm for joint transmit weights and receive beamforming. The method is extended to imperfect CSI via a CSI-error-aware design, preserving convex subproblems and achieving additional power savings in practice. Across MNIST and CIFAR-10 experiments, the proposed PoMFL framework attains the same convergence rate as benchmarks but with significantly reduced uplink power; a CSI-aware variant further improves performance under channel uncertainties, highlighting practical relevance for energy-constrained wireless FL systems.

Abstract

This paper studies power-efficient uplink transmission design for federated learning (FL) that employs over-the-air analog aggregation and multi-antenna beamforming at the server. We jointly optimize device transmit weights and receive beamforming at each FL communication round to minimize the total device transmit power while ensuring convergence in FL training. Through our convergence analysis, we establish sufficient conditions on the aggregation error to guarantee FL training convergence. Utilizing these conditions, we reformulate the power minimization problem into a unique bi-convex structure that contains a transmit beamforming optimization subproblem and a receive beamforming feasibility subproblem. Despite this unconventional structure, we propose a novel alternating optimization approach that guarantees monotonic decrease of the objective value, to allow convergence to a partial optimum. We further consider imperfect channel state information (CSI), which requires accounting for the channel estimation errors in the power minimization problem and FL convergence analysis. We propose a CSI-error-aware joint beamforming algorithm, which can substantially outperform one that does not account for channel estimation errors. Simulation with canonical classification datasets demonstrates that our proposed methods achieve significant power reduction compared to existing benchmarks across a wide range of parameter settings, while attaining the same target accuracy under the same convergence rate.

Paper Structure

This paper contains 35 sections, 9 theorems, 43 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contribution
  4. Organization
  5. System Model and Problem Formulation
  6. FL System
  7. FL with Over-the-Air Aggregation
  8. Problem Formulation
  9. Joint Transmit and Receive Beamforming Design
  10. Training Convergence under Communication Error
  11. Problem Reformulation
  12. Proposed Joint Beamforming Algorithm
  13. Transmit Weights Optimization
  14. Receive Beamforming Optimization
  15. Final Scaling of Receive Beamforming and Transmit Weights
  16. ...and 20 more sections

Key Result

Theorem 1

Under A1-A3, assume the following conditions on the error vector $\mathbf{e}_t$ and the learning rate $\gamma_t$ hold: where the expectation $\mathbb{E}[\cdot]$ in C1 and C2 is w.r.t. the receiver noise. Then, for any initial point $\mathbf{w}_0$, the expected optimality gap between the global loss function $F(\mathbf{w}_t)$ and its minimum value $F(\mathbf{w^\star})$ converges to zero, i.e., $

Figures (4)

  • Figure 1: An illustration of federated learning with over-the-air aggregation.
  • Figure 2: Average number of epochs vs. power limit $P_0$ (MNIST).
  • Figure 4: Average number of epochs vs. power limit $P_0$ (CIFAR-10).
  • Figure 6: Average number of epochs (CIFAR-10 with imperfect CSI).

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Remark 3
  • Remark 4
  • Proposition 3
  • Proposition 4
  • ...and 5 more