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Random Aggregate Beamforming for Over-the-Air Federated Learning in Large-Scale Networks

Chunmei Xu, Shengheng Liu, Yongming Huang, Bjorn Ottersten, Dusit Niyato

TL;DR

The paper tackles joint device selection and over-the-air aggregation for large-scale FL by proposing a random aggregate beamforming scheme that samples a unit-sphere vector for $\mathbf{m}$ and selects devices based on effective channel gains or MSE constraints, avoiding channel estimation. It provides asymptotic analyses showing the random scheme can approach optimal MSE as the number of devices $K$ grows and derives bounds on the number of selected devices under an MSE constraint. A refined method introduces multiple randomizations $N_m$ to boost finite-$K$ performance, with theoretical guarantees that performance gaps diminish as $N_m$ increases. Simulation results corroborate the theory, demonstrating favorable MSE behavior, higher device participation, and improved learning performance on non-i.i.d. MNIST and CIFAR-10 datasets, highlighting practical impact for edge FL with AirComp in massive networks.

Abstract

At present, there is a trend to deploy ubiquitous artificial intelligence (AI) applications at the edge of the network. As a promising framework that enables secure edge intelligence, federated learning (FL) has received widespread attention, and over-the-air computing (AirComp) has been integrated to further improve the communication efficiency. In this paper, we consider a joint device selection and aggregate beamforming design with the objectives of minimizing the aggregate error and maximizing the number of selected devices. This yields a combinatorial problem, which is difficult to solve especially in large-scale networks. To tackle the problems in a cost-effective manner, we propose a random aggregate beamforming-based scheme, which generates the aggregator beamforming vector via random sampling rather than optimization. The implementation of the proposed scheme does not require the channel estimation. We additionally use asymptotic analysis to study the obtained aggregate error and the number of the selected devices when the number of devices becomes large. Furthermore, a refined method that runs with multiple randomizations is also proposed for performance improvement. Extensive simulation results are presented to demonstrate the effectiveness of the proposed random aggregate beamforming-based scheme as well as the refined method.

Random Aggregate Beamforming for Over-the-Air Federated Learning in Large-Scale Networks

TL;DR

The paper tackles joint device selection and over-the-air aggregation for large-scale FL by proposing a random aggregate beamforming scheme that samples a unit-sphere vector for and selects devices based on effective channel gains or MSE constraints, avoiding channel estimation. It provides asymptotic analyses showing the random scheme can approach optimal MSE as the number of devices grows and derives bounds on the number of selected devices under an MSE constraint. A refined method introduces multiple randomizations to boost finite- performance, with theoretical guarantees that performance gaps diminish as increases. Simulation results corroborate the theory, demonstrating favorable MSE behavior, higher device participation, and improved learning performance on non-i.i.d. MNIST and CIFAR-10 datasets, highlighting practical impact for edge FL with AirComp in massive networks.

Abstract

At present, there is a trend to deploy ubiquitous artificial intelligence (AI) applications at the edge of the network. As a promising framework that enables secure edge intelligence, federated learning (FL) has received widespread attention, and over-the-air computing (AirComp) has been integrated to further improve the communication efficiency. In this paper, we consider a joint device selection and aggregate beamforming design with the objectives of minimizing the aggregate error and maximizing the number of selected devices. This yields a combinatorial problem, which is difficult to solve especially in large-scale networks. To tackle the problems in a cost-effective manner, we propose a random aggregate beamforming-based scheme, which generates the aggregator beamforming vector via random sampling rather than optimization. The implementation of the proposed scheme does not require the channel estimation. We additionally use asymptotic analysis to study the obtained aggregate error and the number of the selected devices when the number of devices becomes large. Furthermore, a refined method that runs with multiple randomizations is also proposed for performance improvement. Extensive simulation results are presented to demonstrate the effectiveness of the proposed random aggregate beamforming-based scheme as well as the refined method.
Paper Structure (20 sections, 6 theorems, 40 equations, 9 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 6 theorems, 40 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Problem Formulation
  4. MSE Minimization
  5. Maximizing The Number of Selected Devices
  6. Random Aggregate Beamforming-Based Scheme
  7. Random Aggregate Beamforming-Based Methods
  8. The Distribution of $\left\Vert \mathbf{m}^{\mathrm{H}}\mathbf{h}_{k}\right\Vert$
  9. MSE Minimization
  10. Maximum Number of Selected Devices
  11. Refined Method
  12. MSE Minimization
  13. Maximizing The Number of Selected Devices
  14. Observations
  15. Simulation
  16. ...and 5 more sections

Key Result

Lemma 1

For any given feasible $\mathbf{m}$ and arbitrary $\theta\in\mathbb{R}$, the objectives of the considered problems (eq:SIMO1_1) and (eq:SIMO3) are identical under $\mathbf{m}$and $\mathbf{m}e^{j\theta}$.

Figures (9)

  • Figure 1: System hierarchy model under investigation.
  • Figure 2: Implementation of the proposed scheme for (a) problem (\ref{['eq:SIMO1_1']}) and (b) problem (\ref{['eq:SIMO3']}).
  • Figure 3:
  • Figure 5:
  • Figure 7: Numerical and theoretical PMFs. (a) $K=10^2$. (b) $K=10^3$. (c) $K=10^4$.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 3 more