PaperlayerView exact revision on arXiv

Source-linked AI summary

Federated Learning via Intelligent Reflecting Surface

Zhibin Wang, Jiahang Qiu, Yong Zhou, Yuanming Shi, Liqun Fu, Wei Chen, Khaled B. Lataief

arXiv:2011.05051v2cs.ITcs.LGeess.SP

TL;DR

AirComp-based federated learning suffers aggregation distortion from unfavorable wireless channels, limiting reliable participation. This paper uses an IRS-assisted two-step alternating low-rank optimization framework, achieving faster convergence, lower training loss, and higher test accuracy than baseline schemes.

  • Problem

    Unfavorable uplink channel conditions cause magnitude reduction and misalignment, enlarging AirComp aggregation error and complicating reliable device selection.

  • Method

    The paper jointly optimizes device selection, BS aggregation beamforming, and IRS phase shifts through a two-step alternating low-rank DC optimization framework.

  • Results

    The proposed IRS-assisted system schedules more devices under MSE requirements and achieves faster convergence, lower training loss, and higher test accuracy than baseline schemes.

  • Takeaways & Limitations

    IRS-assisted AirComp can support larger concurrent participation while improving the reported federated-learning training and prediction outcomes.

Abstract

from arXiv · show

Over-the-air computation (AirComp) based federated learning (FL) is capable of achieving fast model aggregation by exploiting the waveform superposition property of multiple access channels. However, the model aggregation performance is severely limited by the unfavorable wireless propagation channels. In this paper, we propose to leverage intelligent reflecting surface (IRS) to achieve fast yet reliable model aggregation for AirComp-based FL. To optimize the learning performance, we formulate an optimization problem that jointly optimizes the device selection, the aggregation beamformer at the base station (BS), and the phase shifts at the IRS to maximize the number of devices participating in the model aggregation of each communication round under certain mean-squared-error (MSE) requirements. To tackle the formulated highly-intractable problem, we propose a two-step optimization framework. Specifically, we induce the sparsity of device selection in the first step, followed by solving a series of MSE minimization problems to find the maximum feasible device set in the second step. We then propose an alternating optimization framework, supported by the difference-of-convex-functions programming algorithm for low-rank optimization, to efficiently design the aggregation beamformers at the BS and phase shifts at the IRS. Simulation results will demonstrate that our proposed algorithm and the deployment of an IRS can achieve a lower training loss and higher FL prediction accuracy than the baseline algorithms.

I. INTRODUCTION … A. FL Model

The paper develops an IRS-assisted AirComp-based federated learning system for fast, reliable model aggregation, addressing wireless distortion and the trade-off between device participation and aggregation error. It formulates and solves a joint device-selection, beamforming, and IRS phase-shift design problem, then evaluates its impact on federated learning.

  • I. INTRODUCTION: Federated learning coordinates edge devices to collaboratively train a global model through iterative model aggregation and model dissemination.The edge server averages received local model updates and broadcasts the updated global model back to selected devices.
  • I. INTRODUCTION: Wireless channel fading and additive noise distort model parameters received by the edge server, motivating reliable aggregation methods.Digital FL schemes use orthogonal resource blocks and assume correct decoding of local model parameters.
  • I. INTRODUCTION: AirComp-based analog FL exploits waveform superposition to merge concurrent device transmissions and compute aggregated models under bandwidth and latency constraints.The edge server is interested in the aggregated model rather than individual local updates.
  • I. INTRODUCTION: An IRS reconfigures wireless propagation through low-cost passive reflecting elements that adjust incident-signal phase shifts.This capability can mitigate channel-fading effects in wireless networks.
  • A. Contributions: The proposed IRS-assisted AirComp-based FL system mitigates magnitude misalignment so more devices can participate under each aggregated model’s MSE requirement.The stated objectives are lower training loss, higher test accuracy, and fewer communication rounds.
  • A. Contributions: The optimization jointly selects devices, designs the BS aggregation beamformer, and configures IRS phase shifts despite sparse objectives and coupled biquadratic constraints.The coupling makes the formulated problem highly intractable.
  • A. Contributions: A two-step framework induces device-selection sparsity with ℓ1-relaxation, then solves MSE minimization problems to identify the maximum feasible selection set.Alternating optimization subsequently decouples BS beamformer and IRS phase-shift design.
  • A. FL Model: The system comprises one M-antenna BS, K single-antenna edge devices, and an IRS with N passive reflecting elements, while FedAvg collaboratively updates the global model.Each device owns a local dataset, performs local updates, and the BS averages selected local model updates.

B. Communication Model for IRS-Assisted AirComp

The section models AirComp-based aggregation as a nomographic function and defines the transmitted model symbols, IRS-assisted wireless channels, BS estimation, and MSE distortion. It also specifies perfect CSI, IRS phase shifts, device power limits, and optimal transmit scalars for minimizing aggregation error.

  • Aggregation function: AirComp is used to efficiently compute the average model update because the aggregation function is nomographic.Devices use identity pre-processing, while the BS applies a post-processing function scaled by the selected-device count.
  • Signal model: Selected devices transmit normalized, independent model-update vectors whose dimensions permit transmission within one interval.The transmit symbols have zero mean and unit covariance, and a typical scalar entry is used for the communication model.
  • IRS-assisted channel: An IRS is deployed to mitigate AirComp distortion caused by channel-induced magnitude reduction and signal misalignment.The model includes device-to-BS, device-to-IRS, and IRS-to-BS channels, assumes invariant channel gains within an interval, and uses perfect CSI.
  • BS estimation: The BS estimates the aggregated target function using an aggregation beamformer and denoising factor, then applies post-processing to obtain the global model.The received signal includes device transmissions through the direct and IRS-reflected channels, together with additive white Gaussian noise.
  • MSE and power constraint: The aggregation distortion is quantified by the MSE between the BS estimate and the target value, with device transmit power constrained by P0.For fixed BS beamforming and IRS phase shifts, Proposition 1 gives transmit scalars that minimize the MSE under the power constraint.

C. Problem Formulation

The problem maximizes the number of selected devices under an MSE requirement γ > 0, then reformulates selection through sparse optimization, yielding a computationally difficult problem addressed by a two-step alternating low-rank framework.

  • C. Problem Formulation: The optimization maximizes selected devices while satisfying the model-aggregation MSE requirement γ > 0.This aims to speed training convergence and avoid notable prediction-accuracy reduction.
  • C. Problem Formulation: The MSE constraint is equivalently rewritten as nonconvex quadratic and biquadratic constraints.Proposition 2 provides the equivalent constraint representation used for algorithm design.
  • C. Problem Formulation: Maximizing feasible MSE constraints is transformed into minimizing the number of nonzero x_i’s through an auxiliary variable x.The objective |S| counts feasible MSE constraints under the regularity condition ∥m∥2 ≥ 1.
  • C. Problem Formulation: Device selection is encoded by x’s sparsity: x_i = 0 indicates device i can satisfy the MSE requirement and be selected.The resulting sparse objective and biquadratic and quadratic constraints make problem P computationally difficult.

III. ALTERNATING LOW-RANK OPTIMIZATION · A. Proposed Two-Step Framework for Solving Problem P · 1) Sparsity Inducing:

The paper introduces a two-step framework for IRS-assisted AirComp-based federated learning: first inducing device-selection sparsity, then identifying the largest feasible device set under MSE constraints. Alternating optimization is used to solve the problems in both steps.

  • III. ALTERNATING LOW-RANK OPTIMIZATION: The framework jointly addresses device selection for IRS-assisted AirComp-based federated learning through two sequential optimization steps.The authors then apply alternating optimization to solve the problem in each step.
  • A. Proposed Two-Step Framework for Solving Problem P: The first step induces sparsity in x to determine each device’s selection priority.The second step uses these priorities to identify the maximum feasible device set while satisfying the MSE requirement.
  • A. Proposed Two-Step Framework for Solving Problem P: The ℓ0-norm sparse objective is replaced with the convex ℓ1-norm surrogate when formulating the first-step problem.This surrogate is adopted to handle the nonconvex sparse objective.
  • 1) Sparsity Inducing:: The first-step problem is solved subject to constraints (16b), (16c), and (16d).These constraints define the feasibility conditions accompanying the sparsity-inducing formulation.
  • 1) Sparsity Inducing:: After solving problem P1, the method checks selected-device feasibility and seeks the maximum number of edge devices under the MSE constraint.This constitutes the second step of the proposed framework.
  • 1) Sparsity Inducing:: Each xi measures the disparity between device i’s MSE requirement and achievable MSE, so smaller xi gives device i higher selection priority.The resulting {xi}K values are sorted for second-step device selection.

2) Feasibility Detection: · B. Alternating Low-Rank Optimization · 1) Sparsity Inducing:

The method detects the largest feasible device set by ordering devices, using bisection, and testing minimum maximal MSE against the requirement. It then alternates optimization variables and applies matrix lifting with rank constraints to handle coupled nonconvex beamforming and IRS phase-shift design.

  • 2) Feasibility Detection:: Devices are ordered by ascending x_i to establish their selection priority.The ordering is x_π(1) ≤ · · · ≤ x_π(K), where x_π(i) is the i-th smallest element among {x_i}^K_i=1.
  • 2) Feasibility Detection:: Bisection finds the maximum k for which the priority set S[k] can be feasibly selected.The candidate set is S[k] = {π(1), π(2), ..., π(k)}.
  • 2) Feasibility Detection:: A candidate set is feasible when its optimal minimum maximal MSE is below the required MSE.Feasibility is checked by comparing the MSE requirement with the minimum maximal MSE obtained for the selected devices.
  • 2) Feasibility Detection:: The IRS phase-shift variables satisfy unit-modulus constraints |Θ_n,n| = 1 for every IRS element.This constraint is imposed for n ∈ {1, ..., N}.
  • B. Alternating Low-Rank Optimization: Alternating optimization addresses the nonconvexity caused by coupled variables in constraint (16b) and objective function (18a).The variables are optimized in alternating blocks rather than jointly.
  • B. Alternating Low-Rank Optimization: With Θ fixed, the auxiliary variables x and m are optimized alternately, while the combined channel vector h_i is fixed.The combined channel is h_i = GΘh^r_i.
  • 1) Sparsity Inducing:: Matrix lifting converts the fixed-Θ subproblem’s biquadratic and quadratic constraints into an SDP-based low-rank optimization problem.With M = mm^H, the lifted formulation represents the beamforming variables through a matrix constrained by positive semidefiniteness and rank structure.
  • 1) Sparsity Inducing:: With x and m fixed, phase-shift feasibility is represented by v = [e^jθ_1, ..., e^jθ_N]^T and lifted as V = v̄v̄^H with rank(V) = 1.The formulation also imposes |v_n| = 1 and V ⪰ 0, with V_n,n = 1 for n ∈ {1, ..., N + 1}.

2) Feasibility Detection:

The feasibility-detection step reformulates problem P2 and alternately optimizes the aggregation beamforming vector and IRS phase-shift matrix. The resulting low-rank problems are handled through an alternating DC algorithm with semidefinite relaxation (SDR).

  • Problem P2 is first reformulated for the second step of feasibility detection.
  • The aggregation beamforming vector m and phase-shift matrix Θ are optimized alternately to decouple the optimization variables.
  • Each alternating subproblem is further represented as a low-rank matrix optimization problem.
  • The alternating low-rank optimization retains fixed rank-one constraints, making problems P1,1, P1,2, P2,1, and P2,2 nonconvex.
  • The algorithm drops the nonconvex rank constraints using SDR to solve relaxed SDP problems.

IV. ALTERNATING DC APPROACH FOR LOW-RANK OPTIMIZATION · A. DC Formulation for Rank-One Constrained Problems

The paper formulates rank-one constraints in difference-of-convex form and embeds them in a two-step alternating DC algorithm for IRS-assisted AirComp-based federated learning. The method induces device-selection sparsity, detects feasibility, and recovers aggregation beamformers and IRS phase shifts from rank-one solutions.

  • IV. ALTERNATING DC APPROACH FOR LOW-RANK OPTIMIZATION: The proposed framework combines a DC formulation for rank-one-constrained semidefinite programs with a two-step alternating DC algorithm for problem P.The algorithm targets IRS-assisted AirComp-based federated learning with device selection.
  • A. DC Formulation for Rank-One Constrained Problems: Accurate rank-one detection is critical for assessing nonconvex quadratic-constraint feasibility in the two-step device-selection framework.The paper therefore provides DC representations for the rank-one constraints introduced earlier.
  • A. DC Formulation for Rank-One Constrained Problems: For a positive semidefinite matrix M, the rank-one constraint is rewritten through its singular values, trace norm, and spectral norm.This reformulation enables a difference-of-convex representation of the rank-one condition.
  • A. DC Formulation for Rank-One Constrained Problems: Step 1 induces sparsity in device selection by iteratively solving problem P1,2 from an initial point Θ0 until the objective decrease falls below ǫ.The algorithm outputs x⋆ after the stopping condition is met.
  • A. DC Formulation for Rank-One Constrained Problems: Step 2 detects feasibility by ordering x⋆, constructing candidate sets S[k], and iteratively narrowing the device count between Nlow and Nup.The procedure uses a predefined threshold ǫ and updates the candidate set through feasibility checks.
  • A. DC Formulation for Rank-One Constrained Problems: A DC penalty is applied to problem P1,1 so that enforcing the penalty term to zero yields a rank-one matrix under the stated trace and positive-semidefinite constraints.The formulation includes tr(M) − γ · tr(MH_i) ≤ x_i, tr(M) ≥ 1, and M ⪰ 0, with penalty parameter ρ > 0.
  • A. DC Formulation for Rank-One Constrained Problems: When the DC objective reaches zero, Cholesky decomposition recovers the aggregation beamformer from M⋆ = mmH and the IRS phase-shift matrix from V⋆ = v̄v̄H.For the IRS variables, the method extracts v̄ = [v0, t0]T, computes v = v0/t0, and sets Θ = diag(v); analogous DC formulations handle P2,1 and P2,2.

B. DC Algorithm

The DC programs are handled by successively linearizing their concave components, yielding convex subproblems for both optimization steps. These subproblems can be efficiently solved with CVX, and the procedure converges to critical points from any feasible initialization.

  • DC linearization: The algorithm exploits the difference-of-convex structure by successively linearizing each program’s concave objective component.This treatment is applied to the programs in both optimization steps.
  • First step: In the first step, the objectives of P′1,1 and P′1,2 are represented as g1 − h1 and g2 − h2, respectively.For P′1,1, g1 = ∥x∥1 + ρ · tr(M) and h1 = ρ · ∥M∥2.
  • Second step: In the second step, P′2,1 and P′2,2 are likewise converted into successive subproblems with objectives g3 − h3 and g4 − h4.For P′2,1, g3 = (1 + ρ) · tr(M) and h3 = ρ · ∥M∥2.
  • Subproblem solution: The resulting subproblems are convex and can therefore be efficiently solved using CVX.The required subgradients involve the largest-eigenvalue eigenspace through ∂∥X∥2.
  • Convergence: The DC solving procedure converges to critical points of the DC programs from any feasible initial point.This convergence property applies to the iterative solution procedure described for the DC programs.

C. Computation Complexity Analysis

Algorithm 1 solves multiple SDP problems across two steps, with complexity determined by the dimensions of M, N, K, and the DC iteration count T. The DC approach is more computationally complex than SDR, but this tradeoff improves system performance.

  • SDP Complexity: Each SDP solved by the second-order interior-point method has worst-case complexity O((M^2+K)^3.5) or O((N^2+K)^3.5), depending on the problem.Problems (39) and (43) use M, while problems (40) and (44) use N.
  • SDR Comparison: SDR requires solving each SDP only once after dropping rank-one constraints, corresponding to T = 1.This applies to problems P1,1, P1,2, P2,1, and P2,2.
  • Complexity–Performance Tradeoff: The proposed DC algorithm has higher computational complexity than SDR, while providing significant system-performance improvement.The performance improvement is identified as the benefit of the added computational cost and is demonstrated in the paper’s subsequent results.

V. SIMULATION RESULTS

Simulations evaluate the proposed two-step alternating DC algorithm for IRS-assisted AirComp-based federated learning, including its convergence, channel assumptions, and comparisons with alternative schemes. The results assess both device selection performance and the effectiveness of deploying an IRS.

  • Simulation setup: Simulations use Rician-fading channels with uniform linear BS antennas and a uniform rectangular IRS array in a three-dimensional coordinate system.The setup includes BS–IRS, IRS–device, and BS–device links with specified path-loss and Rician-factor assumptions.
  • Simulation setup: The default configuration sets C0 = −30 dB, P0 = 20 dBm, σ2 = −90 dBm, ǫ = 10−3, K = 20, M = 20, and N = 64.Path-loss exponents are 3.6 for BS–device, 2.2 for BS–IRS, and 2.8 for IRS–device channels.
  • Convergence: The sparse inducing and feasibility detection steps converge to stationary points while accurately finding rank-one solutions through DC programming.Convergence behavior is shown for problems P1 and P2 in Fig. 4 and Fig. 5, respectively.
  • Baseline comparisons: Algorithm 1 is compared with alternating SDR with IRS, random phase shifts, and a without-IRS scheme under the proposed two-step framework.Random phase shifts are independently uniform on [0, 2π), while the without-IRS scheme sets Θ = 0.

A. Device Selection · B. Performance Comparison for Federated Learning · VI. CONCLUSIONS

The IRS-assisted AirComp framework selects more devices under MSE constraints, while increasing IRS elements or BS antennas further improves selection capacity. Across FL experiments, the proposed two-step alternating DC method achieves lower training loss and higher test accuracy, supporting faster and more reliable aggregation than the no-IRS baseline.

  • A. Device Selection: Increasing the MSE threshold γ increases the average number of selected devices, with IRS deployment supporting substantially more participants than systems without IRS.Relaxing the aggregation-error requirement permits more devices to join each FL round.
  • A. Device Selection: More reflecting elements at the IRS increase selected devices by producing more accurate passive beamforming and reducing aggregation error at the BS.This trend is evaluated at γ = −20 dB.
  • A. Device Selection: Increasing BS antennas enables more device selection by enhancing channel gain, mitigating additive noise, and reducing aggregation error.This trend is evaluated at γ = −22 dB.
  • A. Device Selection: Even doubling BS antennas does not allow the no-IRS system to achieve similar performance to jointly optimized IRS-assisted aggregation, indicating reduced BS hardware complexity with IRS.The comparison concerns joint optimization of the BS aggregation beamformer and IRS phase shifts.
  • B. Performance Comparison for Federated Learning: On MNIST, the proposed two-step alternating DC algorithm selects more devices and richer datasets per round, yielding lower training loss and higher test accuracy than other schemes.The simulations use a deep CNN and average results over 10 realizations at γ = −17 dB.
  • B. Performance Comparison for Federated Learning: The proposed method collaboratively trains a global model to reach lower training loss and higher test accuracy in fewer communication rounds.The benchmark selects all devices without aggregation error.
  • VI. CONCLUSIONS: The paper proposes an IRS-assisted AirComp framework using two-step alternating low-rank optimization to maximize selected devices under model-aggregation MSE requirements.The alternating procedure uses a difference-of-convex formulation and DC algorithm for rank-one constrained problems, outperforming the no-IRS baseline.

APPENDIX A PROOF OF PROPOSITION 1 · APPENDIX B PROOF OF PROPOSITION 2

Appendix A derives the zero-forcing transmitter scalar in Proposition 1 by establishing an MSE lower bound and identifying the minimizing form. Appendix B equivalently reformulates constraint (14b) through normalization and the substitution ˜m = m/√τ.

  • APPENDIX A PROOF OF PROPOSITION 1: The transmitter scalar {wi} in Eq. (10) has a zero-forcing structure that enforces the required condition.This structure is used in the proof of Proposition 1.
  • APPENDIX A PROOF OF PROPOSITION 1: MSE(ˆg, g) ≥σ2 ∥m∥2 /η, and the resulting zero-forcing transmitter scalar minimizes the MSE.The inequality follows from Eq. (10) and yields the form stated in Proposition 1.
  • APPENDIX B PROOF OF PROPOSITION 2: Using ∥m∥2 ≥τ and τ > 0, the constraint is rewritten as Fi(m/√τ) = Fi(m)/τ ≤0 for i ∈S.The normalization preserves the inequality while introducing the positive scalar τ.
  • APPENDIX B PROOF OF PROPOSITION 2: Introducing ˜m = m/√τ transforms constraint (14b) into an expression involving ∥˜m∥2 and |˜mH(GΘhr i )|2.The substitution is the key variable change used in the equivalent formulation.
  • APPENDIX B PROOF OF PROPOSITION 2: The equivalent constraint requires ∥˜m∥2 ≥1 and satisfies ∥˜m∥2 −γ|˜mH(GΘhr i )|2 ≤0 for i ∈S, yielding Eq. (15).This completes the equivalence claimed for constraint (14b).
Loading 2011.05051v2…