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Unfolding WMMSE using Graph Neural Networks for Efficient Power Allocation

Arindam Chowdhury, Gunjan Verma, Chirag Rao, Ananthram Swami, Santiago Segarra

arXiv:2009.10812v4eess.SP

TL;DR

The paper addresses efficient, distributed, and effective power allocation in single-hop ad hoc wireless networks without relying solely on model-based optimization or labeled solutions. It proposes UWMMSE, which unfolds WMMSE into a GNN-parameterized architecture trained unsupervised on channel-state samples. The resulting method is reported as effective, distributed, efficient, and generalizable across network sizes and densities.

  • Problem

    Power allocation requires effective, distributed, and efficient optimization of generally non-convex wireless utility functions, while supervised neural methods may generalize poorly and require costly labels.

  • Method

    UWMMSE unfolds WMMSE into a layered neural architecture with learnable modules parameterized by GNNs and trains them through stochastic gradient descent using channel-state samples without optimal-allocation labels.

  • Results

    UWMMSE achieves state-of-the-art utility-maximization performance and computational efficiency, with experiments covering hyper-parameter effects and generalization to unseen network sizes and densities.

  • Takeaways & Limitations

    The approach combines WMMSE’s model-based structure with trainable GNN components, yielding solutions described as effective, distributed, and faster to execute after training than model-based methods.

Abstract

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We study the problem of optimal power allocation in a single-hop ad hoc wireless network. In solving this problem, we depart from classical purely model-based approaches and propose a hybrid method that retains key modeling elements in conjunction with data-driven components. More precisely, we put forth a neural network architecture inspired by the algorithmic unfolding of the iterative weighted minimum mean squared error (WMMSE) method, that we denote by unfolded WMMSE (UWMMSE). The learnable weights within UWMMSE are parameterized using graph neural networks (GNNs), where the time-varying underlying graphs are given by the fading interference coefficients in the wireless network. These GNNs are trained through a gradient descent approach based on multiple instances of the power allocation problem. We show that the proposed architecture is permutation equivariant, thus facilitating generalizability across network topologies. Comprehensive numerical experiments illustrate the performance attained by UWMMSE along with its robustness to hyper-parameter selection and generalizability to unseen scenarios such as different network densities and network sizes.

I. INTRODUCTION

Power allocation in wireless ad hoc networks is important but difficult because utility maximization is generally non-convex and NP-hard. The paper proposes UWMMSE, an algorithm-unfolded, GNN-parameterized approach that combines model-based structure with data-driven learning.

  • Motivation: Power allocation supports efficient wireless operation, fairness, and quality of service, but must adapt to randomly varying channel characteristics and user demands.The paper also notes its importance for mitigating interference and prolonging mobile-user battery life.
  • Problem formulation: Power allocation optimizes network utilities such as sum-rate, min-rate, or harmonic-rate under resource constraints, yet these formulations are often non-convex and NP-hard.The formulation includes increasing utility functions of achievable data rates and per-transmitter maximum-power constraints.
  • Related approaches: Supervised deep-learning approaches can accelerate inference but generalize poorly beyond the channel characteristics, user counts, and mobility represented in training data.Obtaining sufficiently broad ground-truth datasets would require repeatedly running computationally heavy iterative algorithms.
  • Proposed approach: UWMMSE addresses these trade-offs by unfolding WMMSE iterations into a neural architecture whose learnable modules are parameterized by GNNs.The approach is unsupervised, using channel-state samples without requiring optimal power allocations as labels.
  • Analysis and evaluation: The architecture is theoretically permutation equivariant and is evaluated for performance, hyper-parameter sensitivity, distributed implementation, computational efficiency, and generalization to unseen network sizes and densities.The paper studies a single-hop network of M single-antenna transmitter-receiver pairs with time-varying channel coefficients represented by a channel-state matrix.

III. UNFOLDED WEIGHTED MINIMUM MEAN SQUARED ERROR (UWMMSE)

UWMMSE unfolds the classical WMMSE iterations into a trainable layered architecture whose weights are parameterized by differentiable functions, ultimately implemented with GNNs. It uses unsupervised optimization over channel samples to learn updates intended to accelerate convergence while preserving good power-allocation performance.

  • Architecture: UWMMSE unfolds WMMSE iterations into a layered neural architecture with trainable parameters and output power p = Φ(H; Θ).The output is obtained by elementwise squaring the final layer vector, while saturation enforces the power constraint.
  • Architecture: The unfolded formulation is based on an equivalent WMMSE problem whose block-coordinate updates have closed-form solutions and recover the original power allocation.The transformed variables satisfy √p* = v*, linking the WMMSE formulation to the power-allocation solution.
  • Learned updates: Learned variables a^(k) and b^(k) modify the WMMSE updates to accelerate convergence while maintaining good performance with fewer iterations.The paper uses a learned affine transformation in the w^(k) update as a simple implementation choice.
  • Training: UWMMSE is trained by stochastic gradient descent on a loss based on expected utility over channel-state samples, without optimal power-allocation labels.The optimization remains non-convex, but only channel matrices H are required for unsupervised training.
  • General utilities: The architecture can accommodate utility functions beyond sum-rate by changing the loss and adapting the unfolded update corresponding to the modified WMMSE solution.The remaining equations stay unchanged, and the modified iteration has the optimal allocation as a fixed point under the stated concavity condition.

A. Graph neural networks: Permutation equivariance

The paper uses permutation-equivariant functions to ensure that UWMMSE does not depend on arbitrary node indexing. A GNN-based parameterization satisfies this property, and the equivariance propagates through the complete unfolded architecture.

  • Permutation equivariance: Permutation equivariance means permuting network labels permutes the outputs identically, leaving the underlying allocation rule independent of node indexing.Formally, f(ΠHΠ^T) = Πf(H).
  • Architecture-wide property: If the parameterizing function Ψ is permutation equivariant, Proposition 1 guarantees that the full UWMMSE mapping Φ is also permutation equivariant.The result follows by propagating the permutation through each unfolded layer.
  • GNN parameterization: The proposed GNN function Ψ is permutation equivariant, as established in Proposition 2.Its construction uses graph interactions together with ReLU and sigmoid nonlinearities.
  • GNN parameterization: Because equivariance is modular, practitioners can choose another permutation-equivariant GNN while retaining the property for UWMMSE.The paper explicitly separates the architecture-level guarantee from the particular choice of Ψ.

B. UWMMSE convergence: A necessary condition

The convergence analysis derives a necessary asymptotic condition on the learned b^(k) weights when UWMMSE approaches the optimal allocation with increasing depth. The condition is architecture-level and does not depend on the specific parameterization function Ψ.

  • Proof intuition: The proof analyzes perturbations around the optimal fixed point and shows that the layer deviation term g_k must vanish as k increases.This follows from uniform convergence and the layer updates around v* + ε.
  • Necessary condition: If an infinite-depth UWMMSE with a^(k) = 1 converges uniformly to the optimal allocation, its learned b^(k) weights must asymptotically satisfy the condition in (13).The theorem applies to nodes whose optimal powers lie strictly between 0 and pmax.
  • Interpretation: The expected practical regime is b^(k) → 0, which reduces the unfolded layer toward the classical WMMSE iteration.Other limits can satisfy the necessary condition, but they require Ψ to represent the optimal receiver-side variable sufficiently well.
  • Scope: The convergence condition is independent of the specific function Ψ used to parameterize b^(k).Thus, the theorem concerns the unfolded architecture in equations (5)–(8), rather than one particular GNN choice.

C. Distributed implementation, scalability, and complexity analysis

UWMMSE can be deployed for distributed inference, but training remains centralized and communication overhead limits responsiveness. Its per-layer complexity scales quadratically with network size, matching WMMSE per-iteration scaling.

  • Distributed implementation: Distributed inference requires each node to access relevant indexed channel information and the trainable weights of both GCN modules across unrolled layers.Each receiver computes u_i^(k), which is passed to transmitters for computing v_i^(k).
  • Distributed implementation: UWMMSE cannot be trained in a distributed fashion because the GCN formulation requires the entire CSI matrix.After centralized training, the model can compute an individual node’s output from its indexed row and column.
  • Distributed implementation: Distributed deployment incurs communication overhead, and allocation speed is limited by the feedback link between each transceiver pair.
  • Complexity analysis: UWMMSE has O(M^2) complexity per unrolled layer, matching the per-iteration scaling of WMMSE for an M-user single-antenna interference network.Each GCN adds O(M^2F) complexity when the hidden dimension is F.
  • Complexity analysis: Limiting the number of unrolled layers provides a processing-time reduction that is validated empirically.

IV. NUMERICAL EXPERIMENTS

The experiments use randomly generated Rayleigh-fading geometric networks with time-varying fading and mostly fixed topology. The baseline UWMMSE configuration uses four unrolled layers, two-layer GCN modules, four hidden features, and ADAM training.

  • Experimental scope: The study evaluates UWMMSE against competing methods, varies architectural parameters, and examines node features and other settings.
  • Experimental setup: The experiments construct two-dimensional random geometric graphs with M transmitter–receiver pairs under a Rayleigh fading channel model.Transmitters and paired receivers are randomly placed, while channel gains combine distance-based path gain and randomly drawn fading coefficients.
  • Experimental setup: Topology is fixed in most experiments, while fading conditions change at each discrete-time instant and determine the instantaneous channel information used for allocation.
  • Reproducibility: Code for reproducing the numerical experiments is publicly available in the authors’ GitHub repository.
  • Baseline configuration: The baseline model uses 4 unrolled WMMSE layers, two 2-layer GCNs per layer, hidden dimension 4, and ADAM training for 10000 steps.Training uses batches of 64 randomly sampled fading states, learning rate 1 × 10^-3, and at most 20 epochs.

A. Performance comparison

UWMMSE matches or exceeds WMMSE utility in the low-noise regime while using few unrolled iterations, and it offers a strong performance–latency trade-off. Its advantage is clearest against truncated WMMSE and other learning-based baselines under comparable runtime.

  • Low-noise comparison: 83.21 average sum-rate: UWMMSE exceeds WMMSE’s 82.94 and Tr-WMMSE’s 76.49 in the low-noise regime.The corresponding standard deviations are 6.10, 5.76, and 6.28, respectively.
  • Low-noise comparison: UWMMSE matches WMMSE for most low-noise channel realizations and surpasses it in some cases while using as many iterations as truncated WMMSE.
  • Learning-based baselines: 78.17 average sum-rate: MLP beats Tr-WMMSE but remains substantially below WMMSE in the low-noise regime.REGNN and IGCNet, designed for high-noise settings, also fail to match WMMSE in the more challenging low-noise setting.
  • High-noise comparison: In the high-noise regime, all algorithms tend toward binary transmission decisions because sum-capacity depends less on precise power allocation.Consequently, performance values are generally low and vary little across methods.
  • Runtime comparison: Close to 2 ms per sample: UWMMSE is substantially faster than WMMSE at around 16 ms while matching the runtime of other learning-based methods.Those other learning-based methods do not achieve UWMMSE’s performance within that time, while UWMMSE training takes close to 15 minutes.

B. Impact of depth, width, and GNN functional form

UWMMSE performance is evaluated across unrolling depth, hidden-feature width, and alternative GNN modules. The GCN and REGNN implementations produce substantially overlapping sum-rate distributions, supporting flexibility in the learning module choice.

  • Depth and width: UWMMSE performance is evaluated for unrolling depths from 2 to 7 and GCN hidden-feature widths from 2 to 15.
  • GNN functional form: The study compares the canonical GCN learning module with REGNN as an alternative for computing the affine parameters a^(k) and b^(k).REGNN is selected because of its simple and lightweight architecture.
  • GNN functional form: The GCN-based and REGNN-based UWMMSE models have substantially overlapping empirical sum-rate distributions.The authors conclude that the unfolded architecture is not restricted to a specific GNN and allows choices based on computation and memory constraints.

C. Incorporating node features and different utility functions

The paper extends UWMMSE to alternative utility functions and node features while showing that learned parameters can compensate for model mismatch.

  • Different utility functions: UWMMSE supports other utility functions β(z) when γ(z) = −β(−log(z)) is strictly concave for z > 0.The paper validates this condition using β(z) = z^2, for which γ(z) = −(log(z))^2.
  • Model mismatch: UWMMSE maintains high performance without the modified WMMSE update by learning parameters that enforce an appropriate update trajectory.With and without the modified update, the reported means are 1141.38 and 1131.86, respectively.
  • Node features: The architecture can incorporate node attributes such as transmitter–receiver distance and shortest distance to other transmitters.These features are introduced to encode information specific to individual nodes.

D. Empirical validation of Theorem 1

Layerwise histograms show that UWMMSE changes its learned parameters across depth, using flexible updates early and approaching the classical WMMSE rule in later layers.

  • Experimental setup: The experiment evaluates a fully trained 5-layer UWMMSE model using a(k) and b(k) values from layers 1, 3, and 5.The histograms contain 12,800 values from 6,400 test vectors of size M = 20.
  • Learned variables: a(k) remains concentrated near the lower range in early layers but shifts toward values close to 1 in the final layer.The major shift occurs at layer 5.
  • Learned variables: b(k) decreases across layers, with most b(5) values converging near 0 at the fifth layer.The lower-end peak becomes higher in deeper layers.
  • Interpretation: The learned a(k) and b(k) values accelerate convergence in early layers before approximating the WMMSE update rule in final layers.The final-layer behavior corresponds to learning a(k) = 1 and b(k) = 0.

E. Generalization to variations in network density and size

UWMMSE is evaluated under changing spatial density and network size, while Ro-UWMMSE uses topology-diverse training to improve robustness across these variations.

  • Spatial density: The density experiment varies transmitter spacing through a factor d from 0.5 to 5.0, covering sparser and denser topologies.The ranges 0.5 ≤ d ≤ 1.0 and 1.0 ≤ d ≤ 5.0 are plotted separately.
  • Spatial density: Ro-UWMMSE is trained on batches of independently generated topologies with randomly sampled d ∈ [0.5, 5.0].This regularized training differs from baseline UWMMSE, which uses a constant topology across training cycles.
  • Spatial density: In sparse settings, Ro-UWMMSE follows WMMSE closely, whereas UWMMSE has a constant moderate performance gap.The comparison includes WMMSE and Tr-WMMSE baselines.
  • Spatial density: Ro-UWMMSE continues to follow WMMSE closely in denser settings, illustrating generalization under regularized training.Figure 6 compares mean sum-rate across density factor ranges.
  • Network size: For networks of size M ∈ [10, 30], UWMMSE performs best at the training size and declines relative to WMMSE when nodes are added or removed.The model is trained at N = 20; despite the decline, performance remains close to WMMSE under regularized training.

V. CONCLUSIONS

The paper presents UWMMSE as a GNN-parameterized unfolding of WMMSE that combines model structure with learned components for wireless power allocation. It reports comparable performance, distributed implementability, and lower forward-pass cost, while identifying extensions to complex-valued channels and imperfect channel information.

  • Contributions: UWMMSE unfolds classical WMMSE into a layered neural architecture whose trainable components are parameterized by GNNs.The design uses the graph representation of communication networks.
  • Conclusions: The reported method is effective, distributed, and efficient relative to established model-based benchmarks.The claims refer to comparable performance, decentralized implementation, and reduced computation after training.
  • Future directions: Future work includes efficient power allocation for complex-valued signals and channels, alternative unfolding designs, and settings with missing, noisy, or adversarial channel information.The paper also proposes studying computation–performance trade-offs and other wireless resource-allocation problems.
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