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Improving Wireless Federated Learning via Joint Downlink-Uplink Beamforming over Analog Transmission

Chong Zhang, Min Dong, Ben Liang, Ali Afana, Yahia Ahmed

TL;DR

This work tackles the degradation of federated learning caused by noisy wireless channels in a setting with analog downlink transmission and over-the-air uplink aggregation using a multi-antenna BS. It derives a round-trip global model update equation and upper bounds on the expected training loss after $T$ rounds, tying convergence to the joint downlink-uplink beamforming design. To optimize learning performance efficiently, it introduces JDUBF, with two variants JDUBF-R (receive-only uplink beamforming) and JDUBF-TR (joint transmit-receive uplink beamforming), solved online per round via a greedy BCD-PGD framework and aided by a closed-form initialization. Experiments on MNIST with realistic 5G-like channels show JDUBF-TR substantially improves convergence and can nearly match ideal FL under favorable conditions, underscoring the practical value of joint transceiver design for wireless FL.

Abstract

Federated learning (FL) over wireless networks using analog transmission can efficiently utilize the communication resource but is susceptible to errors caused by noisy wireless links. In this paper, assuming a multi-antenna base station, we jointly design downlink-uplink beamforming to maximize FL training convergence over time-varying wireless channels. We derive the round-trip model updating equation and use it to analyze the FL training convergence to capture the effects of downlink and uplink beamforming and the local model training on the global model update. Aiming to maximize the FL training convergence rate, we propose a low-complexity joint downlink-uplink beamforming (JDUBF) algorithm, which adopts a greedy approach to decompose the multi-round joint optimization and convert it into per-round online joint optimization problems. The per-round problem is further decomposed into three subproblems over a block coordinate descent framework, where we show that each subproblem can be efficiently solved by projected gradient descent with fast closed-form updates. An efficient initialization method that leads to a closed-form initial point is also proposed to accelerate the convergence of JDUBF. Simulation demonstrates that JDUBF substantially outperforms the conventional separate-link beamforming design.

Improving Wireless Federated Learning via Joint Downlink-Uplink Beamforming over Analog Transmission

TL;DR

This work tackles the degradation of federated learning caused by noisy wireless channels in a setting with analog downlink transmission and over-the-air uplink aggregation using a multi-antenna BS. It derives a round-trip global model update equation and upper bounds on the expected training loss after rounds, tying convergence to the joint downlink-uplink beamforming design. To optimize learning performance efficiently, it introduces JDUBF, with two variants JDUBF-R (receive-only uplink beamforming) and JDUBF-TR (joint transmit-receive uplink beamforming), solved online per round via a greedy BCD-PGD framework and aided by a closed-form initialization. Experiments on MNIST with realistic 5G-like channels show JDUBF-TR substantially improves convergence and can nearly match ideal FL under favorable conditions, underscoring the practical value of joint transceiver design for wireless FL.

Abstract

Federated learning (FL) over wireless networks using analog transmission can efficiently utilize the communication resource but is susceptible to errors caused by noisy wireless links. In this paper, assuming a multi-antenna base station, we jointly design downlink-uplink beamforming to maximize FL training convergence over time-varying wireless channels. We derive the round-trip model updating equation and use it to analyze the FL training convergence to capture the effects of downlink and uplink beamforming and the local model training on the global model update. Aiming to maximize the FL training convergence rate, we propose a low-complexity joint downlink-uplink beamforming (JDUBF) algorithm, which adopts a greedy approach to decompose the multi-round joint optimization and convert it into per-round online joint optimization problems. The per-round problem is further decomposed into three subproblems over a block coordinate descent framework, where we show that each subproblem can be efficiently solved by projected gradient descent with fast closed-form updates. An efficient initialization method that leads to a closed-form initial point is also proposed to accelerate the convergence of JDUBF. Simulation demonstrates that JDUBF substantially outperforms the conventional separate-link beamforming design.

Paper Structure

This paper contains 33 sections, 6 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization and Notation
  4. System Model
  5. FL System
  6. Wireless Communication Model
  7. Downlink-Uplink Analog Transmission for FL
  8. Downlink Broadcast of Global Model Update
  9. Local Model Update
  10. Uplink Aggregation of Local Model Updates
  11. Receive Beamforming Only
  12. Joint Transmit-Receive Beamforming
  13. Joint Downlink-Uplink Beamforming for Wireless FL
  14. Joint Downlink-Uplink Formulation
  15. Convergence Analysis on Global Training Loss
  16. ...and 18 more sections

Key Result

Lemma 1

Consider the FL system described in Section sec:FL_alg. Let $\beta^{\text{re}}_{ t} \triangleq \sum_{k=1}^{K}\!|\mathfrak{R}\mathfrak{e}\{\rho_{k,t}\}|$ and $\beta^{\text{im}}_{ t} \triangleq \sum_{k=1}^{K}\!|\mathfrak{I}\mathfrak{m}\{\rho_{k,t}\}|$. Let $Q_t\triangleq1 - 4\eta_t^2J^2L^2$, and assum

Figures (3)

  • Figure 1: Test accuracy vs. communication round $T$ with equal-distance devices for $(N,K) = (64,10)$. Left: $G_{k} = -145~\text{dB}$, $\forall k$, $(\text{SNR}_{\text{dl}},\text{SNR}_{\text{ul}}) = (-2,-10)$ dB. Middle: $G_{k} = -140~\text{dB}$, $\forall k$, $(\text{SNR}_{\text{dl}},\text{SNR}_{\text{ul}}) = (3,-5)$ dB. Right: $G_{k} = -133~\text{dB}$, $\forall k$, $(\text{SNR}_{\text{dl}},\text{SNR}_{\text{ul}}) = (10,2)$ dB.
  • Figure 2: The CDF of downlink and uplink SNRs with randomly located devices.
  • Figure 3: Test accuracy vs. communication round $T$ with randomly located devices. Left: $N = 16$, $K = 20$. Middle: $N = 64$, $K = 20$. Right: $N = 64$, $K = 10$.

Theorems & Definitions (8)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • Remark 2