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Optimized Power Control Design for Over-the-Air Federated Edge Learning

Xiaowen Cao, Guangxu Zhu, Jie Xu, Zhiqin Wang, Shuguang Cui

arXiv:2106.09316v3cs.ITeess.SP

TL;DR

Air-FEEL improves communication efficiency through over-the-air gradient aggregation, but channel and noise errors complicate convergence. This paper analyzes their round-dependent effect, designs power controls that minimize the resulting optimality gap, and reports faster convergence than fixed-power and conventional MSE-minimization schemes.

  • Problem

    Air-FEEL lacks an analytic learning-performance characterization and corresponding power-control design because aggregation errors can affect communication rounds differently.

  • Method

    The paper derives optimality-gap convergence results and solves convex power-control problems under biased and unbiased aggregation, average-power, and maximum-power constraints.

  • Results

    The proposed power-control policies achieve significantly faster convergence than fixed-power transmission and conventional MSE-minimization schemes.

  • Takeaways & Limitations

    Unbiased aggregation can converge exactly to the optimum, while biased aggregation produces an error floor determined by accumulated bias.

  • Takeaways & Limitations

    The paper leaves additional issues unaddressed and identifies optimizing learning performance through transmission power control as difficult.

Abstract

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This paper investigates the transmission power control in over-the-air federated edge learning (Air-FEEL) system. Different from conventional power control designs (e.g., to minimize the individual mean squared error (MSE) of the over-the-air aggregation at each round), we consider a new power control design aiming at directly maximizing the convergence speed. Towards this end, we first analyze the convergence behavior of Air-FEEL (in terms of the optimality gap) subject to aggregation errors at different communication rounds. It is revealed that if the aggregation estimates are unbiased, then the training algorithm would converge exactly to the optimal point with mild conditions; while if they are biased, then the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Next, building upon the convergence results, we optimize the power control to directly minimize the derived optimality gaps under both biased and unbiased aggregations, subject to a set of average and maximum power constraints at individual edge devices. We transform both problems into convex forms, and obtain their structured optimal solutions, both appearing in a form of regularized channel inversion, by using the Lagrangian duality method. Finally, numerical results show that the proposed power control policies achieve significantly faster convergence for Air-FEEL, as compared with benchmark policies with fixed power transmission or conventional MSE minimization.

I. INTRODUCTION

Air-FEEL uses over-the-air aggregation for communication-efficient federated edge learning, but channel fading and noise create aggregation errors that affect learning differently across rounds. This paper therefore analyzes convergence and designs power control to minimize the resulting optimality gap.

  • Motivation and system: Air-FEEL applies over-the-air computation to aggregate distributed gradient updates at each communication round.The framework trains a shared model across edge devices while avoiding centralized data upload.
  • Motivation and system: Aggregation errors create a tradeoff between Air-FEEL’s communication efficiency and degraded learning performance.The errors arise from channel fading and noise perturbation.
  • Research gap: Existing power controls based on channel inversion or per-round MSE minimization may be suboptimal because aggregation errors have different effects across communication rounds.Deep fading can amplify receiver noise under channel inversion.
  • Contributions: The paper links aggregation errors across rounds to the loss-function optimality gap under smooth models satisfying the Polyak-Łojasiewicz inequality.The analysis distinguishes aggregation bias and MSE as determinants of convergence performance.
  • Contributions: The proposed power-control problems minimize the optimality gap under individual average and maximum power constraints.Both formulations are transformed into convex problems and solved using Lagrangian duality, yielding regularized channel inversion.
  • Evaluation: Simulations show significantly faster convergence for the proposed policies than for fixed-power transmission and conventional MSE minimization.The evaluation includes ridge regression with synthetic data and MNIST digit recognition using a CNN.

B. Basic Assumptions on Learning Model

The Air-FEEL model uses distributed gradient estimation and simultaneous analog transmission over a multiple-access channel. Its analysis assumes regularity of the loss and gradient estimates while power control obeys per-device average and maximum constraints.

  • Learning assumptions: The convergence analysis assumes smoothness, the Polyak-Łojasiewicz inequality, and bounded-variance local gradient estimates.These assumptions support convergence analysis for the distributed learning updates.
  • Learning assumptions: The Polyak-Łojasiewicz condition is more general than strong convexity and applies to losses including logistic regression, linear regression, and least squares.It implies that every stationary point is a global minimum.
  • Over-the-air aggregation: Devices transmit gradient estimates simultaneously over the same time-frequency block, using AirComp to exploit the multiple-access channel’s superposition property.Devices are assumed to know their own channel state information for phase compensation.
  • Over-the-air aggregation: The received aggregation error contains signal-misalignment and noise-induced components.The resulting global gradient estimate is affected by both channel-related misalignment and additive white Gaussian noise.
  • Power constraints: Each device adaptively controls transmit power subject to both per-round maximum and whole-training-period average power budgets.The constraints apply individually across devices and communication rounds.
  • Power-control objective: The paper uses the derived optimality gap to formulate power-control optimization under individual power constraints.This connects the aggregation model directly to the subsequent learning-oriented design.

III. CONVERGENCE ANALYSIS

The convergence analysis expresses Air-FEEL’s optimality gap through aggregation-error bias and MSE across communication rounds. Unbiased aggregation can reach the optimum, whereas biased aggregation produces an error floor, with later-round errors receiving greater weight.

  • Error model: The analysis defines aggregation error bias and MSE for the global gradient estimate at each communication round.The expectation includes stochastic mini-batch sampling and receiver noise.
  • Convergence behavior: With biased aggregation, the algorithm may converge only to a neighborhood of the optimum because accumulated bias determines the error floor.The gap to that floor can vanish as the number of rounds increases.
  • Round-dependent sensitivity: Enforcing unbiased aggregation can increase MSE and enlarge the finite-round gap to the error floor, potentially slowing convergence.Thus unbiased aggregation improves the asymptotic limit but can impose a finite-round tradeoff.
  • Convergence behavior: With unbiased aggregation and sufficiently many rounds, Air-FEEL can converge exactly to the optimal point with zero error floor.This case imposes E[ε(n)] = 0 at every communication round.
  • Round-dependent sensitivity: Aggregation errors introduced in later communication rounds contribute more to the finite-round optimality gap than errors introduced early.Early effects are discounted by C^(N−n).
  • Learning-rate tradeoff: With diminishing learning rates, a smaller learning rate increases the error floor while reducing the gap to that floor.The learning-rate choice therefore creates an additional convergence tradeoff.

B. Optimality Gap versus Transmission Power Control

The paper converts the convergence-derived optimality gap into a transmission-power-control design objective. The formulation is built using a bounded sample-wise gradient assumption.

  • Optimization setup: The transmission-power-control formulation minimizes the optimality gap for cases with and without unbiased aggregation constraints.It uses the convergence results from the aggregation-error analysis.
  • Optimization setup: The power-control analysis assumes that each sample-wise gradient is bounded by a specified constant.This assumption supports bounding the aggregation-error terms in the objective.

1) Convergence Analysis for Air-FEEL in Case I:

The convergence analysis characterizes Air-FEEL's expected optimality gap under aggregation errors and distinguishes unbiased from biased aggregation settings. The subsequent treatment covers both diminishing and fixed learning rates, while presenting the power-control optimization using diminishing rates for brevity.

  • Convergence analysis: The analysis derives Air-FEEL's bias and mean-squared error at each communication round from the over-the-air gradient aggregation.These error quantities are then substituted into the convergence analysis to obtain optimality-gap bounds.
  • Case I: Without unbiased aggregation constraints: The expected optimality gap without unbiased aggregation constraints is upper bounded under the stated learning-rate conditions.The proposition applies to the case where aggregation estimates are not required to be unbiased.
  • Case II: With unbiased aggregation constraints: The expected optimality gap with unbiased aggregation constraints is likewise upper bounded under the corresponding learning-rate conditions.The unbiased-aggregation result is stated as Proposition 2.
  • Learning-rate cases: The subsequent power-control optimization is presented for diminishing learning rates because both learning-rate cases have similarly structured convergence results.The paper states that the resulting insights also hold for fixed learning rates.

IV. POWER CONTROL OPTIMIZATION

For the case without unbiased aggregation constraints, the paper reformulates the power-control problem into a convex optimization and derives an optimal policy through Lagrangian duality. The resulting policy has a regularized channel-inversion structure and reduces to conventional channel inversion under sufficiently large average-power budgets.

  • Problem formulation: The power-control design targets faster convergence by minimizing the effective optimality gap subject to individual-device power constraints.The reformulation uses the stated assumptions and inequalities to avoid requiring non-causal gradient information.
  • Convex reformulation: Introducing auxiliary variables yields an equivalent convex formulation that can be solved by standard convex optimization or Lagrangian duality.The paper uses Lagrangian duality to expose the structure of the optimal solution.
  • Optimal solution: The optimal power scaling factors follow a regularized channel-inversion structure.The regularization incorporates power-budget effects through the optimization variables and associated dual terms.
  • Special case: With sufficiently large average-power budgets at all devices, the dual variables become zero and the optimal policy reduces to channel inversion.In this special case, the policy is equivalent to minimizing MSE independently at each communication round.

B. Power Control Optimization fo Case II

For the case with unbiased aggregation constraints, the paper checks whether the power budget can support the required unbiased estimation level, then solves the feasible convex reformulation using Lagrangian duality. Its optimal policy again has regularized channel inversion, with regularization tied to each device's own average-power budget.

  • Problem formulation: The unbiased-aggregation power-control problem is formulated to minimize the optimality gap while enforcing the additional unbiasedness-related constraint.The original formulation is non-convex but is transformed into an equivalent convex form.
  • Feasibility: Feasibility is checked by comparing the maximum unbiased estimation level supported by the power budget with the required level.The feasibility test is itself posed as a convex problem and the optimization proceeds when the requirement is supportable.
  • Optimal solution: The feasible convex problem is solved using Lagrangian duality to obtain the optimal power scaling solution.The corresponding dual variables are associated with the relevant average- and maximum-power constraints.
  • Structure and power budgets: The optimal solution has a regularized channel-inversion structure whose regularization depends only on the device's own average-power budget through its dual variable.When that dual variable is positive, the device's average-power constraint is tight; when it is zero, channel inversion does not exhaust the budget.

V. SIMULATION RESULTS

Simulations evaluate Air-FEEL power-control policies on ridge regression and handwritten digit recognition under varying communication rounds, learning rates, and device counts. The proposed policies improve convergence and learning performance relative to fixed-power transmission and conventional MSE minimization, with Case II becoming advantageous in longer runs.

  • Experimental setup: Simulations cover ridge regression and handwritten digit recognition, using optimality gap, prediction error, loss value, and test accuracy as performance metrics.The experiments use Matlab and PyTorch, respectively, for the two tasks.
  • Compared policies: The experiments compare fixed-power transmission, conventional per-round MSE minimization, and the proposed power-control policies.The proposed policies include Cases I and II, corresponding to biased and unbiased aggregation constraints.
  • Ridge regression: Both proposed policies and conventional MSE minimization converge faster and achieve lower optimality gaps than fixed-power transmission.This result is reported for the ridge-regression experiments over communication rounds.
  • Ridge regression: The proposed policies significantly outperform conventional MSE minimization in reducing the optimality gap because aggregation errors affect different communication rounds differently.Case II achieves a lower optimality gap than Case I when N > 150 with fixed learning rates and N > 80 with diminishing learning rates.
  • Device-count sensitivity: As the number of devices K increases, all schemes achieve lower optimality gaps, while the proposed policies' performance advantage over benchmarks decreases and saturates in the large-K regime.The comparison uses diminishing and fixed learning rates across the optimality-gap and prediction-error plots.

C. Air-FEEL for Handwritten Digit Recognition

On MNIST, the proposed Air-FEEL power-control policies improve learning performance over fixed-power and conventional MSE-minimization schemes. The paper also identifies several extensions, including broader network settings, FedAvg, non-i.i.d. data, and fair comparison with digital FEEL.

  • C. Air-FEEL for Handwritten Digit Recognition: The MNIST experiment uses a 6-layer CNN classifier for handwritten digit recognition.The model includes convolutional, pooling, fully connected, and softmax layers.
  • C. Air-FEEL for Handwritten Digit Recognition: The proposed power-control policies achieve lower loss and higher test accuracy than fixed-power transmission and conventional MSE minimization.This pattern is observed across the tested communication-round settings and learning-rate choices.
  • C. Air-FEEL for Handwritten Digit Recognition: Case II outperforms Case I when N > 200 with fixed learning rates and N > 150 with diminishing learning rates.These observations are reported as consistent with the ridge-regression results.
  • Future work: Future work includes multi-cell Air-FEEL, FedAvg, non-i.i.d. data, and fair quantitative comparison with digital FEEL.These directions involve interference management, new error analysis, and training-latency or accuracy comparisons.

APPENDIX

The appendix derives convergence inequalities by relating gradient norms to expected per-round improvement and then applying algebraic and standard inequality steps. These manipulations yield the stated bound under the paper’s assumptions.

  • APPENDIX: The proof begins by relating the gradient norm to the expected improvement at each communication round.The update relation for w(n+1) is used in this argument.
  • APPENDIX: The derivation applies Assumption 1 and the bound L ≜ ∥L∥∞ before taking expectations.The expectation is taken on both sides of the resulting inequality.
  • APPENDIX: The proof uses the arithmetic–geometric-mean inequality and Cauchy–Schwarz inequality to bound intermediate terms.These steps produce successive inequalities used in the convergence result.
  • APPENDIX: Assumption 2 is then applied, and further algebra using m_b = N and C = 1 − δη yields equation (17).The appendix states that this completes the proof.

B. Proof of Proposition 3

The proof solves the first power-control problem through Lagrangian duality, exploiting convexity and decomposing the dual evaluation across communication rounds.

  • Problem (P1.1) is solved by forming its Lagrange dual because convexity establishes strong duality.
  • For fixed feasible dual variables, the dual function is evaluated by solving one subproblem for each communication round.
  • The resulting per-round optimization yields the primal solution through the lemma’s closed-form characterization.
  • Because the dual function is generally concave but nondifferentiable, an ellipsoid method can obtain the optimal dual variables.
  • Substituting the optimal dual variables into the lemma gives the optimal solution to problem (P1.1).

C. Proof of Proposition 4

The proof solves the second power-control problem using strong duality, per-round subproblems, and optimization of the associated dual variables.

  • Problem (P2.1) is convex and satisfies Slater’s condition, so strong duality permits solution through its Lagrange dual.
  • For fixed feasible dual variables, evaluating the dual function decomposes into subproblems for each device’s power-scaling factor in each communication round.
  • The first-order condition for each per-device, per-round subproblem produces the lemma characterizing its optimal solution.
  • The dual function is generally concave but nondifferentiable, so the ellipsoid method can obtain the optimal dual variables.
  • Using the optimal dual variables in Lemma 2 yields the optimal solution to problem (P2.1).
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