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Learning to Optimize: Training Deep Neural Networks for Wireless Resource Management

Haoran Sun, Xiangyi Chen, Qingjiang Shi, Mingyi Hong, Xiao Fu, Nicholas D. Sidiropoulos

arXiv:1705.09412v2cs.ITeess.SP

TL;DR

Wireless resource-management algorithms can be computationally too costly for real-time use. The paper approximates their input/output mappings with fully connected DNNs, establishes theoretical learnability conditions, and evaluates the approach experimentally. Its analysis links approximation accuracy to network depth and width, while experiments indicate potential for real-time wireless-resource management.

  • Problem

    High computational costs in iterative wireless resource-management algorithms create obstacles for real-time implementation, motivating questions about DNN learnability, required size, and practical robustness.

  • Method

    The paper treats a resource-allocation algorithm as a black-box input/output mapping and approximates it with a generic fully connected DNN.

  • Results

    Theoretical analysis identifies conditions for learnability and relates approximation error to DNN depth and width; experiments indicate potential for real-time wireless-resource management.

  • Takeaways & Limitations

    DNN-based algorithm approximation provides theoretical understanding and design guidance for wireless resource allocation and related tasks beyond it.

Abstract

from arXiv · show

For the past couple of decades, numerical optimization has played a central role in addressing wireless resource management problems such as power control and beamformer design. However, optimization algorithms often entail considerable complexity, which creates a serious gap between theoretical design/analysis and real-time processing. To address this challenge, we propose a new learning-based approach. The key idea is to treat the input and output of a resource allocation algorithm as an unknown non-linear mapping and use a deep neural network (DNN) to approximate it. If the non-linear mapping can be learned accurately by a DNN of moderate size, then resource allocation can be done in almost real time -- since passing the input through a DNN only requires a small number of simple operations. In this work, we address both the thereotical and practical aspects of DNN-based algorithm approximation with applications to wireless resource management. We first pin down a class of optimization algorithms that are `learnable' in theory by a fully connected DNN. Then, we focus on DNN-based approximation to a popular power allocation algorithm named WMMSE (Shi {\it et al} 2011). We show that using a DNN to approximate WMMSE can be fairly accurate -- the approximation error $ε$ depends mildly [in the order of $\log(1/ε)$] on the numbers of neurons and layers of the DNN. On the implementation side, we use extensive numerical simulations to demonstrate that DNNs can achieve orders of magnitude speedup in computational time compared to state-of-the-art power allocation algorithms based on optimization.

I. INTRODUCTION

The paper proposes approximating computationally intensive wireless resource-allocation algorithms with generic fully connected DNNs, targeting real-time processing. It develops theory for when algorithms are learnable and evaluates the approach through experiments.

  • Motivation: Wireless resource-management algorithms process real-time network parameters through potentially costly iterations to produce optimized allocation strategies.Examples include power control, beamformer design, and admission control.
  • Motivation: High computational costs, including matrix inversion, bisection, singular value decomposition, and repeated linear programs, hinder millisecond-scale implementation.These operations occur within iterative algorithms used for wireless resource management.
  • Proposed approach: The proposed method treats a resource-optimization algorithm as a black box and learns its input/output relation with a DNN.The network maps algorithm inputs to allocation outputs using simple operations such as matrix-vector multiplications.
  • Proposed approach: Unlike unfolding methods, the approach uses a generic fully connected DNN because complex wireless iterations are not amenable to representation by a single structured layer.The paper contrasts this design with methods relying on simple iteration structure or known gradient information.
  • Research questions: The paper asks which optimization algorithms are learnable, how network depth and width affect accuracy, and how robust the method is in practice.These questions address the theoretical and practical viability of DNN-based algorithm approximation.
  • Contributions: Theoretical analysis identifies conditions for DNN learnability and relates approximation error to network depth and width.The results are presented as understanding and design guidance for learn-to-optimize methods beyond wireless resource allocation.
  • Contributions: Extensive experiments on simulated and real DSL data demonstrate the proposed approach across varied scenarios and indicate potential for real-time wireless-resource management.The experiments are described as a proof of concept.

II. PRELIMINARIES

The paper formulates weighted sum-rate power allocation for an interference channel and reviews WMMSE as a stationary-point method for this nonconvex problem.

  • System model: The interference channel contains K single-antenna transmitter–receiver pairs with constant channel realizations within each resource allocation slot.Each transmitter sends an independent zero-mean Gaussian symbol whose variance is its transmission power.
  • Power-control problem: Power allocation maximizes weighted system throughput subject to per-transmitter power budgets, forming a nonconvex optimization problem.Pmax is the power budget, and αk > 0 are throughput weights.
  • Power-control problem: The power-control problem is known to be NP-hard, motivating efficient alternatives to conventional optimization algorithms.The paper identifies WMMSE as a popular existing power-control algorithm.
  • WMMSE algorithm: WMMSE converts weighted sum-rate maximization into an equivalent weighted MSE minimization problem using the MMSE-SINR equality.The transformed problem has identical stationary solutions to the original formulation.
  • WMMSE algorithm: WMMSE uses block coordinate descent, optimizing one variable set at a time while fixing the others.The cited result states that WMMSE can reach a stationary solution of the original problem.
  • WMMSE algorithm: The paper modifies WMMSE to operate in the real domain and applies it to both interference channels and interfering multiple-access channels.The real-domain modification exploits that replacing hkj by |hkj| leaves the rate function unchanged.

III. THE PROPOSED APPROACH

The proposed approach approximates iterative optimization algorithms with DNNs, supported by a universal-approximation result and an initialization-aware treatment of nonconvexity.

  • Proposed approach: The proposed approach uses a DNN to approximate the behavior of WMMSE, with theoretical results followed by network design and training procedures.The paper explicitly addresses training, validation, and testing stages.
  • Universal approximation: A gradient-descent example shows that learning z → xT can fail because different initializations converge to positive or negative stationary solutions.The network may consequently learn only the middle point rather than the correct behavior.
  • Universal approximation: Learning (x0, z) → xT instead accurately approximates the gradient-descent behavior in the example.The paper presents this result in Figure 3(b).
  • Universal approximation: A finite iterative algorithm is represented by continuous mappings xt+1 = ft(xt, z), where z is the problem parameter and x lies in the feasible region.The algorithm runs for a finite number of iterations indexed by t through T.
  • Universal approximation: For compact parameter and feasible sets, a sigmoid network can approximate the mapping from initialization x0 and problem parameter z to final output xT.The approximation can achieve any prescribed error with sufficiently many hidden units.
  • Universal approximation: Including initialization as an input feature enables a well-trained neural network to learn deterministic algorithms whose iterations are continuous mappings.For nonconvex problems, initialization is necessary because the mapping from parameters alone may be undefined when multiple isolated solutions exist.
  • Universal approximation: Because WMMSE iterations are continuous and the optimization variable lies in a compact set, WMMSE can be approximated arbitrarily well by a single-hidden-layer feedforward network when channel realizations lie in a compact set.This conclusion follows by applying the iterative universal approximation theorem.

B. Approximation Rate of WMMSE via DNN

The paper establishes that WMMSE can be approximated by a fully connected DNN under stated channel and initialization conditions, with approximation complexity growing mildly with target accuracy. The analysis composes neural approximations of basic operations, controls propagated error, and extends the framework to other resource-allocation algorithms.

  • Generalization: The framework applies to other resource-allocation algorithms expressible through multiplication, division, binary search, or thresholding operations.The paper presents WMMSE as an example and states that similar analysis steps can handle broader algorithm classes.
  • Approximation rate: O(log(1/ϵ)) is the approximation-error dependence of network size under the theorem’s bounds.The authors characterize this dependence as minor, while noting that the theorem gives upper bounds and practical networks may be smaller.
  • Theoretical construction: The analysis approximates WMMSE by composing neural networks for multiplication and division across its iterations.The construction controls error propagation from inner operations through the complete algorithm.
  • Theoretical guarantee: WMMSE can be approximated by a finite-size multi-layer network under admissible channel and initialization conditions.The theorem defines admissible channel realizations and assumes bounded channel strengths and positive iteration variables.
  • Scope and assumptions: The theoretical scope is limited by assumptions including lower-bounded channel realizations and WMMSE sequences satisfying prescribed conditions.These assumptions help ensure compact admissible channel sets and positive iteration variables.
  • Approximation rate: Multiple layers are essential for the stated rate bounds, whereas the approximation error has only a minor effect on network size.The result concerns the number of layers and total network units required to achieve a given approximation error.

IV. SYSTEM SETUP

The system setup trains a fully connected DNN to map channel magnitudes to optimized power allocations generated by WMMSE. It evaluates the learned allocation through sum-rate comparison and tests robustness on channel distributions differing from training.

  • Network Structure: The DNN takes channel-coefficient magnitudes as input and outputs a power allocation vector.The network uses ReLU hidden-layer activations and a constrained output activation to enforce the power limit.
  • Data Generation: Training samples pair generated channel realizations with optimized power vectors obtained by running WMMSE.WMMSE uses a fixed initialization and terminates when the objective change is below 10^-5 or iterations exceed 500.
  • Data Generation: The validation set supports cross-validation, model selection, and early stopping during training.It is typically smaller than the training set.
  • Training Stage: The network is trained with mean squared error and mini-batch RMSprop stochastic gradient descent.The learning rate and batch size are selected by cross-validation.
  • Testing Stage: Testing compares the DNN-generated power allocation’s sum-rate with WMMSE and evaluates robustness under different channel distributions.Test channels are first generated from the training distribution and then from distributions differing from training.

A. Simulation Setup

The simulations evaluate DNN-based power allocation across Gaussian IC and IMAC channel models, network sizes, and training settings, comparing performance with WMMSE and heuristic baselines.

  • Performance evaluation: The evaluation compares WMMSE with random and maximum-power allocation baselines using averaged sum-rate curves over 10,000 test points.WMMSE is implemented in both Python and C to compare same-platform performance and best computational efficiency.
  • Channel and network models: The experiments cover Gaussian IC and multi-cell interfering MAC (IMAC) channel models.Gaussian IC coefficients follow a standard normal distribution, while IMAC channels use Rayleigh fading with distance-dependent variance and shadow fading.
  • Channel and network models: Three network sizes are tested for Gaussian IC, while IMAC experiments use five (N, K) configurations.The Gaussian IC cases use K ∈ {10, 20, 30}; IMAC cases use (3,12), (3,18), (3,24), (7,28), and (20,80).
  • Data and evaluation: Each network scenario uses one million training realizations and ten thousand validation and testing realizations with distinct random seeds.The validation and testing sets are generated separately from the training data.
  • DNN configuration: The DNN uses three hidden layers with 200 neurons per layer, mapping channel coefficients to K power allocations.The input size depends on the channel model: K^2 for Gaussian IC and N × K for IMAC.
  • Performance evaluation: The DNN achieves 98.33% and 96.82% average accuracy relative to WMMSE for Gaussian IC and IMAC, respectively.WMMSE allocations are binary in most cases, motivating output discretization at the 0.5 threshold.

D. Scalability Performance

Scalability experiments show that the DNN maintains strong prediction and computational performance as network configurations change, although performance degrades under substantial mismatch.

  • Overall scalability: The proposed method achieves good scalability in prediction accuracy and computational efficiency across IC and IMAC scenarios.Tables I and II report sum-rate and computational performance for Gaussian IC and IMAC.
  • Overall scalability: 90.36% sum-rate is achieved against IMAC WMMSE with N = 20 and K = 80, alongside more than one hundred times speedup.The comparison uses the C implementation of IMAC WMMSE.
  • IC versus IMAC: The DNN approximates the full WMMSE sum-rate profile well across the test dataset.Figure 8 presents the distributions of sum-rates for the DNN and WMMSE in Gaussian IC scenarios.
  • IC versus IMAC: The DNN performs better in IMAC than in IC, with lower computational time and higher relative sum-rate.For equal K, IMAC provides K × N input coefficients while IC provides K^2, and WMMSE retains the same per-user iteration structure.
  • Model mismatch: The DNN generalizes across some network configurations, so one trained model is not necessary for every configuration.This finding is reported for tests with changed input dimensions, including the half-user Gaussian IC case.

F. Real Data Performance

The paper validates the DNN approach on measured VDSL channels, where it achieves relatively high sum-rate performance despite limited real-world data, while identifying further research needs.

  • Measured VDSL setup: The real-data experiment models measured VDSL channels as a 28-user interference channel and uses WMMSE to generate power-allocation targets.The dataset contains 6955 channel samples, with channel magnitudes used for power allocation.
  • Training procedure: The training data are synthetically generated from means and variances learned from the validation set, producing 50,000 training samples.The procedure separately models direct and interfering channels before generating samples.
  • Training procedure: 0.001 noise variance is used so that WMMSE produces meaningful outputs for the relatively small measured channel coefficients.Training, validation, and testing follow the procedures used in the earlier experiments.
  • Results and implications: Despite limited available data, the DNN achieves relatively high sum-rate performance on the measured testing set.The authors describe this result as further validation of the learning-to-optimize approach.
  • Results and implications: The theoretical and empirical results support using finite DNNs to approximate WMMSE and other expensive optimization algorithms for quasi-optimal, real-time allocation.The authors characterize the work as a preliminary step and identify complexity, architecture, and broader beamforming generalization as open questions.

APPENDIX A PROOFS

The appendix constructs finite neural-network approximations for arithmetic operations and propagates their errors through repeated WMMSE iterations to relate approximation accuracy to network size.

  • Arithmetic approximation: Multiplication is approximated by representing one input with an n-bit binary expansion and bounding the resulting truncation error.For bounded y, the error is controlled by the binary approximation of x.
  • Arithmetic approximation: Binary digits and quotient approximations are implemented with ReLU and binary units arranged across multiple layers.The construction uses two layers per binary component and yields an explicit finite-network representation.
  • Arithmetic approximation: The quotient construction uses recurrent intermediate values interpreted as x mod 2^(i+1).These intermediate values support the binary expansion of x/y.
  • Error bounds: The multiplication error under perturbed inputs is bounded by 3 max(Xmax, Ymax) max(ε1, ε2).The bound assumes bounded nonnegative inputs and bounded perturbations.
  • WMMSE approximation: Each WMMSE iteration is approximated by a DNN, and repeating the network T times approximates T WMMSE iterations.The proof rewrites the WMMSE updates using rational functions and approximates multiplication and division recursively.
  • WMMSE approximation: The approximation error is amplified by at most G through one iteration and accumulates across T consecutive iterations.The network-size trade-off is obtained by counting the required binary units, ReLUs, and layers.
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