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A Deep Learning Framework for Optimization of MISO Downlink Beamforming

Wenchao Xia, Gan Zheng, Yongxu Zhu, Jun Zhang, Jiangzhou Wang, Athina P. Petropulu

arXiv:1901.00354v2cs.IT

TL;DR

The paper addresses the latency and computational burden of iterative optimization for multiuser MISO downlink beamforming. It develops CNN-based BNNs that predict compact solution features and recover beamforming vectors using expert knowledge, with supervised or hybrid learning across three optimization problems. Simulations show near-optimal performance for SINR balancing and power minimization and performance close to WMMSE for sum rate maximization, with reduced computational complexity.

  • Problem

    Iterative beamforming optimization has high computational delay, while sum rate maximization lacks a practically useful algorithm for finding its global optimum.

  • Method

    The framework combines CNN-based neural networks with expert-guided beamforming recovery and uses supervised BNNs for two problems and hybrid supervised-unsupervised learning for sum rate maximization.

  • Results

    Simulations show near-optimal SINR balancing and power minimization, performance close to WMMSE for sum rate maximization, and significantly reduced computational complexity.

  • Takeaways & Limitations

    Predicting compact key features instead of the full beamforming matrix supports fast beamforming optimization while retaining the framework’s problem-specific structure.

Abstract

from arXiv · show

Beamforming is an effective means to improve the quality of the received signals in multiuser multiple-input-single-output (MISO) systems. Traditionally, finding the optimal beamforming solution relies on iterative algorithms, which introduces high computational delay and is thus not suitable for real-time implementation. In this paper, we propose a deep learning framework for the optimization of downlink beamforming. In particular, the solution is obtained based on convolutional neural networks and exploitation of expert knowledge, such as the uplink-downlink duality and the known structure of optimal solutions. Using this framework, we construct three beamforming neural networks (BNNs) for three typical optimization problems, i.e., the signal-to-interference-plus-noise ratio (SINR) balancing problem, the power minimization problem, and the sum rate maximization problem. For the former two problems the BNNs adopt the supervised learning approach, while for the sum rate maximization problem a hybrid method of supervised and unsupervised learning is employed. Simulation results show that the BNNs can achieve near-optimal solutions to the SINR balancing and power minimization problems, and a performance close to that of the weighted minimum mean squared error algorithm for the sum rate maximization problem, while in all cases enjoy significantly reduced computational complexity. In summary, this work paves the way for fast realization of optimal beamforming in multiuser MISO systems.

I. INTRODUCTION

The paper targets low-latency downlink beamforming in multiuser MISO systems, where conventional iterative optimization is computationally burdensome. It proposes CNN-based BNNs that exploit expert knowledge for three optimization problems and use supervised or hybrid learning as appropriate.

  • Motivation: Iterative beamforming algorithms introduce high computational complexity and latency, making solutions unsuitable for real-time and rapidly varying wireless applications.The paper notes that even millisecond-scale channel fading can make iterative solutions outdated.
  • Proposed framework: The framework combines a CNN-based neural network with signal-processing recovery based on expert knowledge, including uplink-downlink duality and optimal-solution structure.This design predicts problem-specific parameters rather than the full beamforming matrix.
  • Target problems: Three BNNs address SINR balancing, power minimization, and sum rate maximization in the MISO downlink.The problems use total-power or QoS constraints as specified in the system formulation.
  • Expert-guided prediction: For supervised problems, the networks predict a K-element virtual uplink power vector instead of NK beamforming variables, reducing prediction, training, and implementation costs.For sum rate maximization, the network predicts two power allocation vectors totaling 2K elements.
  • Learning methods: The SINR balancing and power minimization BNNs use supervised learning, while the sum rate BNN uses supervised pretraining followed by unsupervised optimization.The hybrid sum rate method first learns from WMMSE outputs and then updates parameters using a sum-rate loss.

III. A DL-BASED FRAMEWORK FOR BEAMFORMING OPTIMIZATION

The proposed framework is a gray-box CNN architecture that predicts compact beamforming features and reconstructs the beamforming matrix using problem-specific expert knowledge. This reduces the burden of directly learning complex beamforming variables, while complex channel inputs require real-valued transformations.

  • Framework rationale: Directly predicting the complex NK-element beamforming matrix can create high network complexity and may discard problem-specific structure.The paper also notes that excessively wide or deep networks can increase implementation complexity and cause over-fitting.
  • Framework rationale: The framework uses CNNs to extract features and predicts key features of the beamforming solution rather than the beamforming matrix itself.CNN weight sharing reduces the number of learned parameters.
  • Module structure: The architecture has neural-network and beamforming-recovery modules, with the latter converting predicted key features into the beamforming matrix.Recovery functions and selected features are specified by expert knowledge for each optimization problem.
  • Input representation: Complex channel coefficients are transformed into real-valued representations before entering the neural network.The paper describes I/Q separation and P/M phase-magnitude transformations.

2) Convolutional Layer:

The convolutional module extracts channel features using shared convolution kernels, while batch normalization stabilizes intermediate representations through normalization followed by trainable scaling and shifting.

  • Convolutional Layer: Each convolutional layer applies shared kernels and biases across feature-map elements to extract features.The layer creates c_l kernels of size a_l×a_l.
  • Convolutional Layer: The convolution operation produces feature maps whose dimensions are determined by the layer configuration.The supplied description identifies the convolution operator and output feature-map representation.
  • Batch Normalization: Batch normalization normalizes convolutional outputs by subtracting batch means and dividing by batch standard deviations.A small variance offset prevents division by zero.
  • Batch Normalization: Trainable scaling and shifting parameters restore representational flexibility after normalization without changing all network parameters.The paper states that batch normalization can also reduce over-fitting probability and enable higher learning rates.

4) Activation Layer:

The activation, flatten, fully connected, and output layers transform extracted features into predicted continuous variables, while the loss function guides parameter updates. The framework’s recovery module then maps compact predictions to feasible beamforming solutions using expert knowledge.

  • Activation Layer: ReLU and sigmoid are recommended for the final activation because the predicted variables are continuous positive real numbers.The paper advises against tanh and linear functions in the last activation layer because they can produce negative values.
  • Activation Layer: Intermediate activation functions operate element-wise and preserve the shapes of their inputs.ReLU is identified as the common choice for intermediate activation layers.
  • Flatten, Fully Connected, and Output Layers: The flatten layer reshapes its input into a vector for the fully connected layer, whose weights and biases generate learned outputs.The output layer produces predicted results after training.
  • Loss Function: The loss function and learning rate guide network parameter updates for continuous predictions.MAE and MSE are suggested metrics; this work chooses MSE because simulated training data lack problematic outliers and its gradient is easier to calculate.
  • Beamforming Recovery Module: The beamforming recovery module converts predicted key features into a beamforming matrix using problem-dependent expert knowledge.This reduces the number of variables that the neural network must predict compared with the full beamforming matrix.

B. Computational Complexity

The framework shifts complexity from online beamforming prediction to offline training and reduces prediction complexity by estimating uplink power allocation rather than the full beamforming matrix.

  • Complexity analysis: Offline training complexity is assumed affordable, so the analysis focuses on online prediction complexity.Training occurs at a much longer time scale than online prediction.
  • Complexity analysis: The neural-network module predicts its output with approximate complexity O(NK).The prediction requires arithmetic operations including multiplications, divisions, and exponentiations.
  • Complexity reduction: Predicting the power allocation vector reduces complexity compared with directly predicting the beamforming matrix.The beamforming matrix is recovered afterward using uplink-downlink duality.
  • Expert-knowledge integration: Uplink-downlink duality enables solving an easier uplink problem before recovering the downlink beamforming solution.The uplink and downlink scenarios have the same achievable SINR region.
  • SINR balancing: For SINR balancing, the optimal objective value is the reciprocal of the largest eigenvalue of the specified nonnegative matrix.The iterative training-sample algorithm produces a strictly monotonically increasing target sequence.

B. BNN Structure

The SINR-balancing BNN uses the proposed framework’s standard layers, with ReLU in intermediate activations and sigmoid in the final activation.

  • BNN Structure: The SINR-balancing BNN uses input, convolutional, batch-normalization, and output layers from the general BNN framework.Its intermediate activation layers use ReLU, while the last activation layer uses sigmoid.

1) Scaling Layer:

The BNNs use scaling and conversion layers to turn predicted uplink power allocations into downlink beamforming matrices, with problem-specific recovery procedures.

  • 1) Scaling Layer:: The SINR-balancing BNN scales its predicted uplink power vector to satisfy the total power constraint.The scaling is needed because prediction error can otherwise violate the constraint.
  • 1) Scaling Layer:: The conversion layer recovers the final downlink beamforming matrix from the scaled predicted power allocation.Recovery includes normalized beamforming construction, eigenvalue/eigenvector calculation, and output formation.
  • 1) Scaling Layer:: The power-minimization BNN predicts uplink power allocation, infers normalized beamforming, and then recovers downlink beamforming using uplink-downlink duality.Its conversion method differs from SINR balancing because the power budget is unknown.
  • 1) Scaling Layer:: For power minimization, the downlink beamforming vectors are obtained by scaling normalized beamforming vectors with the optimal downlink power allocation.The scaling factors are derived from the normalized beamforming matrix through the applicable conversion lemma.
  • 1) Scaling Layer:: The power-minimization formulation assumes that a solution always exists, excluding infeasible QoS constraints from consideration.User selection is suggested as a possible additional neural-network stage when only some users can be satisfied.

B. BNN Structure

The power-minimization BNN recovers downlink beamforming from predicted uplink powers, but prediction errors can cause inaccurate or infeasible power allocations.

  • BNN Structure: Prediction error in uplink powers propagates to the inferred downlink power allocation and beamforming matrix.The predicted vector must be sufficiently close to the target vector for the stated suboptimal-solution guarantee.
  • BNN Structure: A significant uplink-power prediction error can make the inferred downlink solution infeasible because some downlink powers become negative.The paper reports a failure probability below 1% in most settings.

VI. BNN FOR SUM RATE MAXIMIZATION PROBLEM

For sum rate maximization, the paper exploits the connection between power minimization and sum rate to derive expert features, then uses WMMSE-derived supervision because no practical globally optimal algorithm is available.

  • Solution Structure: Problem P3 lacks a practically useful algorithm for finding its optimal solution, so uplink-downlink duality cannot be applied directly.
  • Solution Structure: The optimal power-minimization solution can provide SINR values for constructing targets in the sum rate maximization problem when its power equals Pmax.
  • Solution Structure: The solution structure identifies the virtual power allocation vector λ and power vector p as key features for designing the BNN for problem P3.
  • Solution Structure: WMMSE supplies power allocation vectors p and λ for supervised training, although its solution for P3 is only locally optimal.

B. Hybrid BNN Structure

The hybrid BNN for sum rate maximization uses two training stages and a structured output-and-recovery pipeline. Its evaluation uses a defined simulation setup with fixed network and channel-processing choices.

  • Hybrid BNN Structure: The BNN for problem P3 first uses supervised MSE pre-training and then unsupervised objective-based enhanced training.The second stage replaces the MSE metric with the sum-rate objective and is intended to improve performance and accelerate convergence.
  • Hybrid BNN Structure: The output layer generates 2K values containing the predicted power allocation vectors p̂ and λ̂.
  • Hybrid BNN Structure: A scaling layer adjusts the output to satisfy the power constraint, and a construction layer forms the downlink beamforming vectors.
  • Hybrid BNN Structure: The beamforming recovery module for problem P3 has time complexity O(KN^2 + N^3).
  • Simulation Setup: Simulations use N = 6 antennas, K = 4 single-antenna users, a 500 m coverage radius, and benchmarks including optimal beamforming, ZF, RZF, and WMMSE.
  • Simulation Setup: The experiments use 20000 training samples and 5000 testing samples with a 0.2 validation split and normalized channel coefficients.

A. BNN for the SINR Balancing Problem

For SINR balancing, the supervised BNN closely matches the optimal solution across fading settings while avoiding the iterative process required for optimal beamforming. It also outperforms ZF and RZF in the reported comparisons, with substantially lower execution complexity than iterative optimization.

  • The supervised BNN achieves SINR performance very close to the optimal solution with and without large-scale fading.The comparison averages performance over 5000 samples for K = 4 and N = 6.
  • At low transmit power, optimal beamforming clearly outperforms ZF, although the gap decreases as transmit power increases.
  • With K = N and Pmax = 20 dBm, the BNN has lower SINR than the optimal solution but outperforms both ZF and RZF beamforming.
  • All four beamforming solutions lose SINR as the number of transmit antennas and users increases, with ZF suffering the greatest performance loss.
  • The BNN's computational complexity and time are similar to ZF and RZF but much lower than the optimal iterative solution.
  • A single BNN trained with K = 4 and N = 10 produces SINR results close to optimal while being used with transmit-antenna settings from 4 to 10.Redundant input and output entries are filled with zeros when the antenna number varies.

C. BNN for the Sum Rate Maximization Problem

The sum rate maximization BNN uses hybrid learning and is evaluated against iterative and heuristic beamforming baselines. The proposed hybrid solution achieves performance close to WMMSE while targeting lower computational complexity.

  • Evaluation setup: Under K = 4 and N = 4, the evaluation compares hybrid-learning and supervised-learning BNNs with ZF and RZF baselines.The baselines use equal power allocation, with pk = Pmax/K for ZF and pk = λk = Pmax/K for RZF.
  • Evaluation setup: WMMSE performance depends strongly on initialization, so RZF and random initializations are evaluated separately.WMMSE with RZF initialization generates the supervised-learning samples used in the first stage.
  • Performance: The hybrid-learning BNN consistently performs close to WMMSE with RZF initialization under both small-scale-only and combined small- and large-scale fading.The supervised-learning BNN is less satisfactory, while the hybrid method further maximizes sum rate during its second stage.
  • Performance and complexity: Figure 11 compares sum rate and execution time per sample for WMMSE, two BNN variants, RZF, and ZF as antenna and user numbers increase together.The comparison averages execution time over 5000 samples at Pmax = 30 dBm, with WMMSE limited to at most 10 iterations.
  • Method: The hybrid BNN combines supervised pre-training with unsupervised updates because no practically useful algorithm is available for the nonconvex sum rate problem.The framework also extracts key features using expert knowledge rather than directly predicting the beamforming matrix.
  • Future work: Open extensions include input representation, joint user selection and beamforming, mobility, imperfect CSI, feasibility detection, and multi-cell scenarios.These topics are identified as directions for further study.
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