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Reconfigurable Intelligent Surface Enabled Federated Learning: A Unified Communication-Learning Design Approach

Hang Liu, Xiaojun Yuan, Ying-Jun Angela Zhang

arXiv:2011.10282v4cs.ITcs.LGcs.NIeess.SP

TL;DR

Over-the-air FL faces a straggler bottleneck and a tradeoff between data exploitation and aggregation error. The paper develops a unified learning-aware RIS design that jointly optimizes device selection, transceivers, and RIS configuration. Experiments report improved accuracy and convergence over existing approaches under heterogeneous channels.

  • Problem

    Over-the-air FL must balance data exploitation against communication error because weak-channel devices can bottleneck aggregation.

  • Method

    The paper derives a learning-loss bound and formulates a unified optimization of device selection, over-the-air transceivers, and RIS phase shifts.

  • Results

    The proposed algorithm improves accuracy and convergence over existing approaches, including when channel conditions vary substantially across devices.

  • Takeaways & Limitations

    RIS-assisted over-the-air FL can address communication-learning tradeoffs through joint learning and wireless-system design.

Abstract

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To exploit massive amounts of data generated at mobile edge networks, federated learning (FL) has been proposed as an attractive substitute for centralized machine learning (ML). By collaboratively training a shared learning model at edge devices, FL avoids direct data transmission and thus overcomes high communication latency and privacy issues as compared to centralized ML. To improve the communication efficiency in FL model aggregation, over-the-air computation has been introduced to support a large number of simultaneous local model uploading by exploiting the inherent superposition property of wireless channels. However, due to the heterogeneity of communication capacities among edge devices, over-the-air FL suffers from the straggler issue in which the device with the weakest channel acts as a bottleneck of the model aggregation performance. This issue can be alleviated by device selection to some extent, but the latter still suffers from a tradeoff between data exploitation and model communication. In this paper, we leverage the reconfigurable intelligent surface (RIS) technology to relieve the straggler issue in over-the-air FL. Specifically, we develop a learning analysis framework to quantitatively characterize the impact of device selection and model aggregation error on the convergence of over-the-air FL. Then, we formulate a unified communication-learning optimization problem to jointly optimize device selection, over-the-air transceiver design, and RIS configuration. Numerical experiments show that the proposed design achieves substantial learning accuracy improvement compared with the state-of-the-art approaches, especially when channel conditions vary dramatically across edge devices.

I. INTRODUCTION

The paper addresses communication bottlenecks and stragglers in over-the-air federated learning by jointly modeling learning loss, device selection, aggregation error, and RIS-assisted wireless design. It formulates a unified communication-learning approach and reports improved accuracy under heterogeneous channels.

  • Motivation: Over-the-air aggregation lets devices concurrently upload model updates using signal superposition, without communication latency increasing with device count.This improves communication scalability compared with orthogonal multiple access.
  • Problem: Weak-channel devices become stragglers because stronger devices reduce transmit power to align local models at the parameter server.Excluding stragglers reduces communication error but also decreases the training data size.
  • Problem: Existing heuristic designs use exploited-data fractions, participant counts, scheduling, or manually tuned error constraints rather than fully characterizing learning performance.The paper identifies these ad hoc choices as potentially producing sub-optimal system designs.
  • Approach: RIS elements manipulate propagation channels through phase shifts, creating an opportunity to improve RIS-assisted over-the-air FL beyond communication-only optimization.Prior RIS-assisted work improves aggregation quality but does not fully incorporate the communication-learning tradeoff.
  • Contributions: The proposed framework tracks device-selection and communication-error effects on FL training loss, then jointly optimizes selection, receiver beamforming, and RIS phase shifts.The analysis yields a loss bound and motivates a unified optimization problem; Gibbs sampling is used for active-device selection.
  • Results: The proposed algorithm achieves accuracy close to the error-free ideal benchmark under strongly heterogeneous channel conditions and requires a smaller RIS for the same learning accuracy.The paper reports substantial improvement over existing FL solutions and state-of-the-art methods.

II. SYSTEM MODEL

The system consists of a parameter server and edge devices that iteratively select participants, broadcast a global model, compute local gradients, and aggregate updates over wireless channels. Channel fading and noise distort the aggregated gradient, while the model supports analysis of device selection and communication effects on learning.

  • Learning model: Local datasets contain labeled samples, and each device computes its loss gradient from its own data before participating in global-model updates.The adopted setup uses batch gradient descent for local model updates.
  • FL procedure: Each FL round selects active devices, broadcasts the current global model, computes local gradients, and uploads them for aggregation.The parameter server uses the received signals to compute a weighted sum for updating the global model.
  • Aggregation: The parameter server estimates the weighted global gradient from received signals, but fading and communication noise make the estimate differ from the true gradient.The resulting distortion affects the subsequent global-model update.
  • Assumptions: The analysis assumes channel invariance during learning and fixes device selection across iterations, while noting time-varying-channel studies separately.The transceiver design accounts for channel heterogeneity and data heterogeneity.
  • Extensions: Numerical results indicate that the proposed approach also improves learning performance for multiple-batch and multiple-epoch FL algorithms.The main analysis adopts a batch-gradient model, with broader algorithmic applicability evaluated numerically.

B. RIS-Assisted Communication System

The RIS-assisted system uses configurable phase shifts to support over-the-air aggregation between single-antenna devices and a multi-antenna parameter server. Local model updates are normalized, equalized, and superposed over direct and RIS-cascaded channels, while fading and noise create aggregation error affecting FL convergence.

  • System model: The system connects single-antenna edge devices to an N-antenna parameter server through an RIS with L phase-shift elements.The channel is block-fading, with coefficients assumed invariant throughout FL training.
  • RIS configuration: The RIS applies independent, invariant phase shifts during aggregation, with unit-amplitude continuous coefficients satisfying |θ_l|^2 = 1.The RIS phase-shift vector is denoted θ ∈ C^L×1.
  • Over-the-air communication: The received parameter-server signal is the superposition of transmissions through direct device-server and device-RIS-server cascaded channels, plus AWGN.The effective cascaded channel uses the RIS-server matrix and device-RIS channel vector.
  • Model aggregation: Devices upload local model updates as D-slot transmit sequences after computing and transmitting local gradient statistics for normalization.The transmit sequence is formed from normalized gradient entries, while transmit power is controlled by the equalization factor p_m under a maximum-power constraint P_0.
  • Learning impact: Fading and communication noise cause estimation error in the aggregated update, making the global model update inaccurate and affecting FL convergence.The analysis characterizes this communication error with respect to device selection, beamforming, RIS phases, normalization, and transmit equalization.
  • Normalization: Local-statistics normalization ensures the unit-variance condition even when active devices have substantially different gradients.This differs from prior normalization using uniform statistics, which may violate the required second-moment condition when local gradients vary significantly.

III. FL PERFORMANCE ANALYSIS AND PROBLEM FORMULATION

The paper analyzes FL under over-the-air aggregation by separating device-selection and communication errors, deriving a loss bound, and formulating a unified design problem over selection, beamforming, and RIS phases.

  • III. FL PERFORMANCE ANALYSIS AND PROBLEM FORMULATION: Under the stated assumptions, the paper derives an upper bound on the learning loss by analyzing device-selection and communication-noise errors.The bound is established for the model-update recursion with an appropriate learning rate.
  • III. FL PERFORMANCE ANALYSIS AND PROBLEM FORMULATION: For fixed device selection, receiver beamforming, and RIS phases, the analysis gives a closed-form choice of η and p_m that minimizes aggregation-error power.The optimization is subject to the transmit-power constraint.
  • III. FL PERFORMANCE ANALYSIS AND PROBLEM FORMULATION: The resulting system design minimizes learning loss jointly over device selection, receiver beamforming, and RIS phase configuration.The formulation treats these communication and participation variables as a unified learning-loss optimization problem.
  • III. FL PERFORMANCE ANALYSIS AND PROBLEM FORMULATION: The model-update recursion and error terms are defined relative to the communication-noise expectation, with the resulting convergence quantities depending on the selected devices and communication design.The section introduces the loss-bound framework before presenting the subsequent convergence characterization.
  • A. Assumptions and Preliminaries: The global model recursion includes a gradient error vector caused by device selection and model aggregation.The communication component includes fading and communication noise.
  • A. Assumptions and Preliminaries: Selecting fewer than all devices introduces selection error because it reduces the amount of exploited data.This selection error and aggregation error jointly affect FL performance.
  • A. Assumptions and Preliminaries: The analysis assumes a strongly convex, twice-continuously differentiable loss with Lipschitz-continuous gradients.These assumptions provide the regularity conditions used for the convergence analysis.
  • A. Assumptions and Preliminaries: A bounded-gradient assumption limits each local gradient norm using constants α1 and α2 and the global gradient norm.The assumptions are described as standard in stochastic optimization, and A1 ensures existence of a global optimum.

B. Learning Performance Analysis

The learning analysis separates selection and communication errors, optimizes transceiver parameters for fixed participation and RIS settings, and derives convergence bounds dependent on the joint communication-learning design.

  • Error decomposition: The total gradient error is decomposed into device-selection error and aggregation error, with communication error depending on M, f, θ, η, and p_m.Selection error depends on the chosen device set, while aggregation error also depends on the communication design.
  • Aggregation-error optimization: The optimal η and p_m minimize E[∥e_2,t∥^2] under the transmit-power constraint for fixed M, θ, and f.The resulting solution is given in closed form as a function of the selected devices, RIS phases, and receive beamforming.
  • Convergence bound: The loss bound depends on d(M, f, θ) and Ψ(M, f, θ), which summarize the effects of device selection and communication design.The parameters of the loss assumptions and the transmit-power limit also enter the bound.
  • Convergence behavior: The bound converges with speed Ψ(M, f, θ) when Ψ(M, f, θ) < 1.This condition determines whether the recursive bound contracts over training rounds.
  • Convergence behavior: If d(M, f, θ) ≤ 1/(2α2), the asymptotic excess loss is bounded by α1d(M, f, θ)/(ω − µ) + 2µα2d(M, f, θ).Under this condition, the transient term vanishes as the number of training rounds tends to infinity.

C. Problem Formulation

The formulation uses d(M, f, θ) to connect device selection and communication error with FL convergence and asymptotic learning performance. It also exposes the data-versus-communication tradeoff and the role of RIS phase shifts in mitigating weak channels.

  • C. Problem Formulation: The converged loss generally remains above the optimal loss because of device selection loss and communication noise.
  • C. Problem Formulation: d(M, f, θ) represents the impact of device selection and communication error on convergence rate and asymptotic learning performance.
  • C. Problem Formulation: A smaller d(·) leads to faster convergence and a smaller asymptotic gap from the optimal loss.
  • C. Problem Formulation: The design optimizes d(M, f, θ) over device selection, receiver beamforming, and RIS phase shifts.
  • C. Problem Formulation: The theoretical results require assumptions A1–A4, and they do not apply to widely used non-strongly-convex cross-entropy loss.
  • D. Further Discussions: The formulation directly unifies device selection loss and communication error without requiring an MSE threshold to be tuned.
  • D. Further Discussions: It accounts for heterogeneous dataset sizes, so selecting a device with large K_m can be unfavorable and selection must balance data exploitation against communication error.
  • D. Further Discussions: RIS phase shifts can enhance weak channels and break the tension between selecting many devices and reducing communication MSE.

IV. COMMUNICATION-LEARNING CO-DESIGN

The co-design problem jointly optimizes device selection, receiver beamforming, and RIS phase shifts, but is mixed-integer and non-convex. The proposed solution combines successive convex approximation with dual optimization to avoid combinatorial search.

  • IV. COMMUNICATION-LEARNING CO-DESIGN: Device selection, receiver beamforming, and RIS phase shifts should be jointly optimized to maximize learning performance.
  • IV. COMMUNICATION-LEARNING CO-DESIGN: The resulting problem is mixed-integer and non-convex because it combines combinatorial search over x with non-convex objectives and constraints.
  • IV. COMMUNICATION-LEARNING CO-DESIGN: An iterative Gibbs-sampling-based algorithm optimizes device selection, receiver beamforming, and RIS phase shifts while avoiding exhaustive combinatorial search.
  • IV. COMMUNICATION-LEARNING CO-DESIGN: For fixed device selection, successive convex approximation updates f and θ by solving a sequence of convex surrogate problems.
  • IV. COMMUNICATION-LEARNING CO-DESIGN: The f-and-θ subproblem remains non-convex because the feasible set is non-convex, so the method approximately solves its Lagrange dual problem.
  • IV. COMMUNICATION-LEARNING CO-DESIGN: The resulting solution is generally sub-optimal because strong duality is not guaranteed for the non-convex primal problem.

B. Optimizing Device Selection

Device selection is optimized iteratively with Gibbs sampling, using the objective obtained from the beamforming and RIS optimization as the selection score. Warm starts and a cooling schedule reduce repeated optimization cost.

  • B. Optimizing Device Selection: The device-selection algorithm uses Gibbs sampling to optimize x based on the objective J(x)=min_f,θ d(x, f, θ).
  • B. Optimizing Device Selection: At each iteration, candidate selection vectors are evaluated, a new x is sampled, and the corresponding f and θ solution is returned.
  • B. Optimizing Device Selection: Algorithm 1 is repeatedly invoked to compute J(x), while warm starts reduce the computational complexity of these repeated solves.
  • B. Optimizing Device Selection: The temperature parameter β_j decreases geometrically through β_j=ρβ_j−1 with 0<ρ<1 to accelerate convergence.
  • B. Optimizing Device Selection: A warm-start technique reuses the previous beamforming and RIS solution because successive selection candidates usually differ in at most two entries.

C. Computational Complexity

The proposed optimization has lower stated complexity than the compared RIS and non-RIS alternatives, while simulations evaluate its learning performance across device and channel settings.

  • The interior-point solution of (29) has worst-case complexity O(|M|^3), giving Algorithm 1 an upper bound of O(ImaxM^3).
  • Algorithm 2 invokes Algorithm 1 JmaxM times, so its overall complexity grows with the repeated inner optimization.
  • The FL optimization algorithm in [8] has complexity O(M(N^6 + (N^2 + M)^3)), also stated to have larger order than Algorithm 2.
  • The experiments evaluate test accuracy on Fashion-MNIST under concentrated equal-data devices and two-cluster unequal-data devices, with the latter producing a severe straggler issue.
  • Under Setting 1, the proposed method has similar accuracy to and RIS-enabled methods avoid the large aggregation errors of non-RIS baselines.

C. Simulations on Mini-Batch Gradient Descent

Mini-batch gradient descent experiments compare the proposed method with baselines under balanced and heterogeneous device settings. Device selection is especially valuable when channel and data heterogeneity create stragglers.

  • Mini-batch updates replace the gradient computation with sequential batch updates, and the local model change replaces the gradient in global aggregation.
  • Under Setting 1, all simulated algorithms converge with relatively high prediction accuracy because strong straggler effects are absent.
  • Under Setting 2, device selection balances data exploitation and communication-error mitigation, and the proposed algorithm outperforms all baselines.
  • The proposed method reaches near-optimal accuracy with smaller RIS sizes than: L ≥10 and L ≥30 versus L ≥30 and L ≥60 in the two settings.

D. Simulations on Discrete RIS Phase Shifts

The paper extends the RIS-assisted FL evaluation to discrete phase shifts and time-varying Rician fading. Accuracy remains close to the error-free benchmark when phase resolution is sufficient and channels vary over training.

  • Discrete RIS phase shifts: Discrete RIS phase shifts use b bits per element, with b →∞ recovering the continuous feasible set.
  • Discrete RIS phase shifts: The discrete-phase algorithm first optimizes continuous phase shifts, then projects the solution onto the discrete feasible set.
  • Discrete RIS phase shifts: For b = 1 or 2, accuracy degrades by ≈0.1 relative to continuous shifts, whereas b = 3 makes the loss negligible.
  • Time-varying Rician fading: Under time-varying Rician fading with channel updates every 100 rounds, the proposed algorithm remains close to the error-free benchmark.
  • The paper formulates a unified communication-learning optimization over device selection, receiver beamforming, and RIS phase shifts, solved by a Gibbs-sampling and SCA-based algorithm.
  • The authors identify non-i.i.d. data distributions and gradient compression or quantization as future directions.

APPENDIX A

The appendix proves bounds for aggregation error and learning performance by combining channel-error expressions, inequalities, and recursive application of the resulting bound.

  • The derivation substitutes the optimal power-control expression into the aggregation-error bound and completes the associated proof.
  • The learning-performance proof bounds the first error term, applies Lemma 2 and the triangle inequality, and substitutes the resulting inequality.
  • Applying the recursive inequality t + 1 times yields the stated performance result.
  • The appendix also establishes feasibility-related behavior by minimizing the Lagrangian expression under the stated condition and substitution.
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