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Deep Learning for Distributed Optimization: Applications to Wireless Resource Management

Hoon Lee, Sang Hyun Lee, Tony Q. S. Quek

arXiv:1905.13378v1cs.ITcs.LGeess.SPmath.IT

TL;DR

Distributed wireless optimization must handle non-convex constraints and limited information exchange, while existing DL training approaches do not straightforwardly address these conditions. The paper develops constrained DNN training for centralized and quantized distributed settings; in the IFC, its distributed method with B=1 attains WMMSE performance, and with B=0 it outperforms a naive distributed DNN.

Abstract

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This paper studies a deep learning (DL) framework to solve distributed non-convex constrained optimizations in wireless networks where multiple computing nodes, interconnected via backhaul links, desire to determine an efficient assignment of their states based on local observations. Two different configurations are considered: First, an infinite-capacity backhaul enables nodes to communicate in a lossless way, thereby obtaining the solution by centralized computations. Second, a practical finite-capacity backhaul leads to the deployment of distributed solvers equipped along with quantizers for communication through capacity-limited backhaul. The distributed nature and the nonconvexity of the optimizations render the identification of the solution unwieldy. To handle them, deep neural networks (DNNs) are introduced to approximate an unknown computation for the solution accurately. In consequence, the original problems are transformed to training tasks of the DNNs subject to non-convex constraints where existing DL libraries fail to extend straightforwardly. A constrained training strategy is developed based on the primal-dual method. For distributed implementation, a novel binarization technique at the output layer is developed for quantization at each node. Our proposed distributed DL framework is examined in various network configurations of wireless resource management. Numerical results verify the effectiveness of our proposed approach over existing optimization techniques.

I. INTRODUCTION

The paper targets non-convex wireless optimization where distributed computation and limited cooperation complicate solution finding. It proposes learning-based centralized and distributed approaches for constrained problems, including quantized information exchange over finite-capacity backhaul links.

I. INTRODUCTION

  • Existing methods rely on restrictive mathematical structure, continuous-message exchange, or ideal cooperation, limiting their fit to practical distributed wireless networks.Prior DL approaches also face constraints in DNN training and often require global network information.Source: §I · p. 1, A. Related Works · p. 2

A. Related Works

  • Prior wireless DL work includes supervised and unsupervised approaches, but distributed designs lack optimized quantization and constrained optimization capabilities.Supervised learning requires solution labels, while penalty-based constrained training does not analytically ensure feasibility.Source: A. Related Works · pp. 1–2

B. Contributions and Organization

  • The framework covers infinite-capacity centralized computation and finite-capacity distributed computation, jointly training node-specific quantizers and optimizers with a constrained primal-dual strategy.The finite-capacity approach uses stochastic binarization to produce bipolar messages and address the vanishing-gradient issue.Source: B. Contributions and Organization · pp. 2–3

II. GENERAL NETWORK SETUP AND FORMULATION

  • The network model assigns each node a local observation and state, with the overall cost minimized subject to expected constraints and feasible per-node assignments.The objective and constraints may be nonconvex, and the setup includes both long-term and instantaneous regulations.Source: §II · p. 3

A. Formulation

The formulation distinguishes a centralized solution using global observations from a distributed solution using quantized local measurements. In the finite-capacity case, each node combines its local observation with received quantized messages to compute its state.

A. Formulation

  • With infinite-capacity links, nodes share local observations losslessly, enabling a centralized rule to compute the network solution from global information.The formulation masks the global solution to identify the state associated with each node.Source: A. Formulation · pp. 3–4
  • With finite-capacity links, each node quantizes its local measurement into a bipolar message and computes its state from local information and messages received from neighbors.The quantization length is determined by the link capacity in bits/sec/Hz.Source: A. Formulation · p. 4
  • The binary quantization constraint makes searching for an efficient message computationally demanding, so traditional optimization is not straightforward for this formulation.Source: A. Formulation · p. 4

B. Learning to Optimize

The paper replaces unknown optimization mappings with DNNs and develops constrained training for the resulting non-convex problem. In the centralized approach, convex projection enforces per-node feasibility, while time-sharing and strong duality support primal-dual training of the long-term constraints.

B. Learning to Optimize

  • A feedforward DNN maps an input through successive weighted, biased, activated layers, with training selecting parameters to minimize its cost.The paper notes that the nonlinear activations make closed-form optimal parameters unavailable, motivating gradient-based training.Source: B. Learning to Optimize · p. 4
  • The paper identifies gaps in prior wireless DL optimization: long-term constraints are not properly handled, centralized solutions may be impractical, and distributed heuristics lack optimized quantization.The proposed constrained primal-dual training and output binarization target these limitations.Source: B. Learning to Optimize · p. 4

III. CENTRALIZED APPROACH

  • The centralized DNN replaces the unknown solution mapping, and a convex projection activation enforces each node’s deterministic feasible-set constraint.The projection is differentiable for backpropagation, allowing the constrained assignment to be represented in the DNN output layer.Source: §III · pp. 4–5
  • Universal approximation results support representing continuous solution mappings with sufficiently large DNNs and, under the stated integrability condition, approximating discrete mappings as well.The latter result supports learning mappings such as the quantization operation.Source: §III · pp. 4–5
  • Existing DL training does not straightforwardly handle the long-term constraints; penalty methods do not analytically ensure feasibility and require trial-and-error tuning.The proposed approach incorporates the constraints through Lagrange duality rather than relying on an arbitrary regularization hyperparameter.Source: §III · p. 5
  • Time-sharing of feasible DNN parameter sets satisfies the time-sharing condition, and Slater’s condition supports strong duality for the constrained training problem.These properties justify using Lagrange duality despite the non-convex dependence of costs and constraints on the DNN parameters.Source: §III · pp. 5–6
  • Primal-dual training jointly updates DNN parameters and dual variables, using an unsupervised objective to learn efficient solutions without prior optimization labels.Upon convergence, the method guarantees primal feasibility; trained parameters are then configured at nodes for solution computation.Source: §III · p. 6

IV. DISTRIBUTED APPROACH

The distributed approach jointly optimizes local quantizers and distributed solvers for finite-capacity communication. Each node uses local observations to generate quantized messages and combines them with received messages to compute its assignment.

IV. DISTRIBUTED APPROACH

  • The distributed framework jointly optimizes quantization and distributed optimization, unlike the centralized approach that relies on perfect observations from other nodes.The centralized approach presumes infinite-capacity backhaul links; the distributed approach addresses the constrained-backhaul setting.Source: §IV · p. 6
  • Each node uses a quantizer DNN to produce messages from its local observation and an optimizer DNN to compute its solution from local and received information.The approach jointly trains 2N DNNs across N nodes.Source: §IV · p. 7
  • A convex projection activation enforces each optimizer’s deterministic feasible-set constraint, while a direct signum quantizer is unsuitable because its null gradient prevents SGD training.The binary constraint is non-convex, motivating the separate stochastic binarization design.Source: §IV · p. 7

A. Stochastic binarization layer

The stochastic binarization layer produces bipolar messages while preserving a usable gradient for training. It uses tanh outputs and stochastic rounding in the forward pass, with a deterministic expected-gradient approximation in backpropagation.

A. Stochastic binarization layer

  • The quantizer DNN’s tanh output lies in [-1, 1], and a stochastic operation maps it to a bipolar vector in {-1, +1}.The paper uses β(x) = (1+x)/2, yielding zero-mean quantization noise and an unbiased bipolar estimate of the unquantized vector.Source: A. Stochastic binarization layer · p. 7
  • The stochastic layer addresses the binary constraint while retaining an SGD-compatible training path, unlike deterministic sign activation with a null gradient.The random quantization noise is defined elementwise, with probabilities determined by the continuous tanh output.Source: A. Stochastic binarization layer · p. 7
  • The forward pass sends the quantized bipolar vector to the optimizer, while backpropagation uses the unquantized tanh output and its non-vanishing gradient.The gradient of the stochastic output is approximated by the gradient of its expectation over quantization noise.Source: A. Stochastic binarization layer · pp. 7–8

B. Centralized Training and Distributed Implementation

The optimizer and quantizer DNNs are jointly trained offline in a centralized manner, then run locally at each node. Pretraining DNNs for candidate backhaul capacities lets nodes select a suitable stored model during operation.

B. Centralized Training and Distributed Implementation

  • The joint strategy trains 2N DNNs for the distributed problem, using stochastic binarization to handle its binary constraint without loss of optimality.The primal-dual method is applied because the time-sharing condition still holds for the distributed formulation.Source: B. Centralized Training and… · p. 8
  • Training is performed offline and centrally, while each node uses its stored optimizer and quantizer parameters for real-time distributed computation.For a given backhaul capacity, nodes select among DNNs trained in advance for the service provider’s candidate capacities.Source: B. Centralized Training and… · p. 8
  • Decentralized training would require DNNs to exchange computation results, training samples, costs, and gradients across layers, creating a huge communication burden per iteration.The paper therefore presents centralized joint training followed by distributed deployment as practical for capacity-limited backhaul links.Source: B. Centralized Training and… · p. 8

V. APPLICATIONS TO WIRELESS RESOURCE ALLOCATION

The section evaluates the proposed DL approaches for wireless power control in C-MAC and interference-channel systems. It examines optimality and performance across several optimization tasks, including a non-smooth objective and constraints.

  • The evaluation covers C-MAC and IFC wireless systems, with power control managed by the proposed DL strategies.Source: §V · p. 8
  • For C-MAC, numerical investigations verify the proposed methods’ analytical optimality against the solution in in both primal and dual domains.Source: §V · p. 8
  • For IFC, the section tests trained DNNs on sum-rate maximization against WMMSE and applies the framework to minimum-rate maximization.Source: §V · p. 8
  • The results support using DL for optimization with non-smooth objectives and constraints without additional reformulation.Source: §V · p. 8

1) Cognitive multiple access channels:

The C-MAC application considers secondary users sharing resources with a licensed primary user and seeking to maximize average sum capacity. The proposed implementation maps local channel information to transmit powers, using centralized or distributed DNNs.

1) Cognitive multiple access channels:

2) Interference channels:

The IFC section considers sum-rate and minimum-rate power control with average and peak power constraints. It describes limitations of existing approaches and applies the proposed DL framework to both problems, including the non-smooth minimum-rate objective without traditional reformulation.

2) Interference channels:

  • The IFC setup has N transmitter-receiver pairs sharing a frequency band, with channel state information denoted by hij and power-control strategies based on local CSI.Source: 2) Interference channels · p. 9
  • The sum-rate problem includes average and peak transmit-power constraints, and is described as non-convex and NP hard.Source: 2) Interference channels · p. 9
  • For sum-rate maximization, WMMSE provides a local optimum when PA,i = PP,i, while prior DL methods rely on global CSI and centralized computation.Source: 2) Interference channels · p. 9
  • For minimum-rate maximization, the problem is generally non-convex and non-smooth, and no efficient solution is available in the general case PA,i ≤ PP,i.Source: 2) Interference channels · p. 9
  • The proposed DL approaches address minimum-rate maximization without epigraph or Perron-Frobenius reformulations, which can add auxiliary variables and increase solution-space dimension.Source: 2) Interference channels · p. 9

B. Numerical Results

The proposed DL framework is evaluated on wireless power-control problems, including centralized and distributed approaches. Results show centralized performance matching the optimum and distributed performance improving as backhaul capacity increases.

B. Numerical Results

  • The evaluation trains on 2.5 × 10^9 samples using Adam and tests performance on held-out samples; similar trends are reported for practical distance-based channels and Rayleigh fading.The stated setup uses mini-batch size S = 5000 and learning rate η = 5 × 10^-5.Source: B. Numerical Results · pp. 9–10

1) Cognitive multiple access channels:

  • The centralized DNN matches optimal average sum capacity, while the distributed method has a slight gap that shrinks with backhaul capacity and is nearly identical at B = 3.The distributed method also outperforms the other baseline schemes.Source: 1) Cognitive multiple access channels · p. 10
  • The centralized training converges to optimal primal and dual performance across tested SNR regimes, with complementary slackness supporting zero duality gap.The IT constraint is initially violated, prompting λ3 to rise before settling; λ1 and λ2 rise when the power constraints reach their bounds.Source: 1) Cognitive multiple access channels · p. 10, Figure 4 · p. 11
  • At SNR = 5 dB, mini-batches of 100, 500, and 1000 degrade performance; S = 5000 balances convergence behavior with per-iteration computation cost.Small batches approximate the average cost and constraints inaccurately, yielding incorrect dual-variable estimates.Source: 1) Cognitive multiple access channels · p. 10, Figure 5 · p. 11
  • At SNR = 5 dB, the distributed method converges to optimal performance with B = 5 bits, but at B = 0 it fails to identify the optimal IT-constraint dual variable.The failure causes performance loss, indicating that quantization and data sharing matter for distributed implementation.Source: 1) Cognitive multiple access channels · pp. 10–11, Figure 6 · p. 12
  • At SNR = 0 dB, the centralized DNN yields the optimum transmit-power CDF, while the distributed CDF approaches it as backhaul capacity increases.With B = 0, separate SUs tend toward higher transmit power because they lack shared information.Source: 1) Cognitive multiple access channels · p. 11, Figure 7 · p. 12
  • Average sum capacity increases with hidden-layer count and saturates at four layers; empirical results led to a four-layer DNN with dimension 20 per layer.The paper notes that optimizing DNN structure for these applications is out of scope.Source: B. Numerical Results · p. 9, 1) Cognitive multiple access channels · p. 12, Figure 8 · p. 12

2) Interference channels:

In interference-channel resource management, the proposed DL methods perform strongly against optimization baselines, while distributed performance improves with backhaul capacity. Peak-power limits and the task affect how much capacity is needed to approach centralized performance.

2) Interference channels:

  • The centralized DNN exceeds locally optimal WMMSE across SNR regimes, while distributed DL with B=1 attains WMMSE performance.The B=0 proposed distributed method also outperforms the naive distributed DNN.Source: Figure 9 · p. 13, 2) Interference channels · p. 13
  • At B=0, the proposed distributed DNN beats the naive DNN; centralized and distributed DNNs use 4% and 6.7% of WMMSE CPU time, respectively.DNN inference uses matrix multiplications, whereas WMMSE uses iterative calculations.Source: 2) Interference channels · p. 13
  • With both average and peak power constraints, PP = 2.5PA gives almost the unbounded sum-rate performance, and distributed performance improves monotonically with B.Fig. 10(b) reports the backhaul-capacity comparison at SNR = 20 dB.Source: 2) Interference channels · p. 13, Figure 10 · p. 14
  • For max-min rate, centralized DL is near-optimal across SNR regimes; matching it with distributed DL requires more backhaul than sum-rate maximization, especially at moderate-to-high SNR.Even with B = 0, distributed DL significantly outperforms the baseline power-allocation schemes.Source: Figure 11 · p. 14, 2) Interference channels · p. 14
  • For max-min rate, PP = 2PA is sufficient to reach the unbounded-peak-power performance, and all DNNs match the PA = PP optimal solution at high SNR.At low SNR the peak-power budget is crucial; at SNR = 5 dB, distributed performance improves as B increases toward the centralized DNN.Source: 2) Interference channels · p. 14

VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

The paper presents a DL framework for non-convex constrained optimization in wireless networks, using primal-dual constrained training and stochastic binarization for limited-capacity backhaul. It reports efficacy in C-MAC and IFC examples and identifies several extensions for future work.

VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

  • The framework approximates unknown optimal-solution computations with DNNs, using primal-dual constrained training and stochastic binarization for finite-capacity backhaul.The centralized configuration assumes infinite-capacity, lossless information sharing.Source: §VI · pp. 14–15
  • Numerical examples in C-MAC and IFC are reported to demonstrate the framework's efficacy for wireless-communication optimization.Source: §VI · p. 15
  • Future directions include multi-antenna and realistic-channel applications, DNN architecture optimization, mixed-integer tasks, and joint quantization and resource allocation in C-RAN.The C-RAN extension could use centralized control with distributed devices linked by limited-capacity fronthaul and backhaul.Source: §VI · p. 15

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