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Towards Optimal Power Control via Ensembling Deep Neural Networks

Fei Liang, Cong Shen, Wei Yu, Feng Wu

arXiv:1807.10025v2eess.SPcs.ITstat.ML

TL;DR

The paper addresses non-convex sum-rate maximization for multi-user interference-channel power control, where optimal allocations are unknown. It introduces PCNet with unsupervised sum-rate training and ePCNet, an ensemble of independently trained PCNets. Simulations report improved power control with lower computational-resource use, while ideal channel estimation remains an acknowledged limitation.

  • Problem

    Power control seeks to maximize the sum rate of a multi-user interference channel, but the optimization is non-convex and the optimal power allocation is unknown.

  • Method

    PCNet is trained by directly maximizing system sum rate, while ePCNet ensembles independently trained PCNets and selects the highest-sum-rate power profile.

  • Results

    ePCNet(+) achieves better power control while consuming less computational resources.

  • Takeaways & Limitations

    The proposed PCNet family combines deep learning, unsupervised training, and ensemble learning for power control across system settings.

  • Takeaways & Limitations

    The current approach assumes ideal channel estimation, which the paper identifies as unrealistic in practice.

Abstract

from arXiv · show

A deep neural network (DNN) based power control method is proposed, which aims at solving the non-convex optimization problem of maximizing the sum rate of a multi-user interference channel. Towards this end, we first present PCNet, which is a multi-layer fully connected neural network that is specifically designed for the power control problem. PCNet takes the channel coefficients as input and outputs the transmit power of all users. A key challenge in training a DNN for the power control problem is the lack of ground truth, i.e., the optimal power allocation is unknown. To address this issue, PCNet leverages the unsupervised learning strategy and directly maximizes the sum rate in the training phase. Observing that a single PCNet does not globally outperform the existing solutions, we further propose ePCNet, a network ensemble with multiple PCNets trained independently. Simulation results show that for the standard symmetric multi-user Gaussian interference channel, ePCNet can outperform all state-of-the-art power control methods by 1.2%-4.6% under a variety of system configurations. Furthermore, the performance improvement of ePCNet comes with a reduced computational complexity.

I. INTRODUCTION

Power control in multi-user interference channels is difficult because the optimization is generally non-convex and can be NP-hard, while existing methods face performance, convergence, and complexity challenges. The paper proposes PCNet and ePCNet, using unsupervised training and ensembling to pursue higher sum rates with lower computational complexity.

  • Problem: Power control for multi-user interference channels generally yields non-convex optimization problems, including formulations that are NP-hard.These properties make finding solutions non-trivial.
  • Existing methods: Existing approaches often rely on exhaustive search or iterative optimization of approximate sub-problems, creating practicality concerns.The paper specifically identifies performance, convergence, and complexity issues.
  • Proposed perspective: The paper investigates deep learning as a data-driven alternative for achieving better power-control performance with lower complexity.This reframes the problem rather than directly solving its non-convex formulation analytically.
  • PCNet: PCNet connects sum-rate maximization with DNN loss minimization and uses network training to approach globally optimal power control.The approach relies on efficient network training and an ensembling mechanism.
  • ePCNet: ePCNet independently trains multiple PCNets and selects the power profile producing the largest sum rate.A single PCNet may not universally outperform existing methods, motivating the ensemble.
  • Results: The proposed DNN ensemble outperforms state-of-the-art power-control methods across varied system configurations and is especially efficient with many users.The paper contrasts this with traditional methods that do not always provide satisfactory results in large-user settings.

B. Applications of deep learning in communications

Deep learning has been applied across communications tasks, including channel decoding, end-to-end system design, detection, feedback reduction, coding, classification, and channel estimation. Reported benefits include improved performance, lower complexity, reduced feedback overhead, and automatically constructed codebooks, while power-control work motivates unsupervised and ensemble learning.

  • Channel decoding: Deep learning has been explored for channel decoding using feed-forward and recurrent neural networks built on belief-propagation architectures.Different neural-network weights are assigned to edges in the Tanner graph.
  • Channel decoding: Neural-network decoders can achieve better performance than traditional belief propagation, while related designs target lower decoding complexity.Other approaches combine neural networks with iterative algorithms or divide decoding into separately trained sub-blocks.
  • End-to-end communications: Deep learning has enabled end-to-end communication-system designs with transmitters and receivers implemented entirely by neural networks.Neural networks have also been applied to MIMO detection with comparable performance and much lower complexity.
  • Feedback and coding: CsiNet learns transformations between channel state information and codewords to address excessive feedback overhead.Neural networks have also been used for SCMA encoder-decoder design and can automatically construct efficient codebooks.
  • Other applications: Applications also include radio-signal classification, traffic-type recognition, channel estimation, and optical-fiber communications.These examples extend beyond decoding and end-to-end communication design.
  • Power control: For power control, prior supervised learning used WMMSE outputs as benchmarks, limiting the learned sum rate by WMMSE rather than the global optimum.This limitation motivates unsupervised learning, ensemble learning, and batch normalization to improve sum rate while reducing online complexity.

III. SYSTEM MODEL

The paper models a K-user single-antenna interference channel with interference treated as noise and studies power allocation for sum-rate maximization. The resulting SRM problem is non-convex and NP-hard.

  • Each receiver decodes only its intended signal, while interference from other users is treated as additive noise.
  • Transmit powers satisfy 0 ≤ P_i ≤ P_max, and the joint power profile contains the powers of all K users.
  • The SRM objective maximizes the sum of users’ achievable rates subject to individual power limits.
  • The paper focuses on sum-rate maximization and sum-rate maximization under minimum-rate quality-of-service constraints.
  • The SRM problem is difficult because it is non-convex with respect to the power profile and is known to be NP-hard.

2) Sum Rate Maximization under QoS Constraints (SRM-QC):

SRM-QC maximizes total rate while requiring every receiver to meet a minimum rate. The section introduces PCNet, a fully connected DNN that maps channel realizations to transmit powers while explicitly handling power constraints and penalizing rate violations.

  • SRM-QC formulation: SRM-QC maximizes the sum rate subject to each receiver meeting its minimum required rate.
  • SRM-QC formulation: SRM-QC may be infeasible when target rates are too large, and feasibility can be assessed using a matrix-based criterion.
  • PCNet design: PCNet is a fully connected DNN that maps a channel realization h to a power profile P using an input layer, hidden layers, and an output layer.
  • PCNet design: The output layer uses a sigmoid activation followed by scaling by P_max so each predicted transmit power lies within the allowed range.
  • Constraint handling: Minimum-rate constraints are harder to encode directly, so the SRM-QC training loss adds penalties for violated receiver-rate requirements.

B. Training PCNet

PCNet is trained without optimal power-allocation labels because obtaining them is computationally difficult, especially for large user counts. Instead, training directly optimizes the sum-rate objective using sampled channel realizations and stochastic gradient updates.

  • Rather than using WMMSE outputs as ground truth, PCNet directly uses the original optimization objective as its training loss.
  • Optimal power allocations are difficult to obtain efficiently, especially when the number of users K is large, motivating unsupervised training.
  • For SRM, the loss is defined to maximize the sum rate over channel realizations and is differentiable with respect to the network parameters.
  • The network is trained with stochastic gradient descent because the loss is differentiable with respect to the trainable parameters.
  • Mini-batch gradient descent estimates the distribution-level loss by generating multiple channel realizations in each training iteration.

2) Training PCNet for SRM-QC:

For QoS-constrained training, PCNet penalizes minimum-rate violations while retaining the sum-rate objective, though feasibility is not guaranteed in every case. PCNet+ adds noise power as an input, and ePCNet selects the best output from independently trained PCNets to improve performance with linear online overhead.

  • Training PCNet for SRM-QC: The SRM-QC loss combines negative sum rate with ReLU penalties for receivers whose achieved rates fall below their minimum requirements.
  • Training PCNet for SRM-QC: The penalty weight λ trades off satisfying rate constraints against increasing system sum rate.
  • Training PCNet for SRM-QC: Even with the penalty loss, PCNet may produce an infeasible profile because the problem itself may be infeasible or the network may fail to find feasibility.
  • PCNet+: PCNet+ takes noise power σ^2 alongside channel coefficients, allowing one model to handle a range of noise levels with very little performance loss.
  • ePCNet: ePCNet ensembles independently trained PCNets with diverse initial parameters and training data, then selects the output with the highest sum rate.
  • ePCNet: Ensembling can improve performance by exploring multiple local optima, while online complexity grows linearly with the number of local learners without changing its order in K.

VI. NUMERICAL EVALUATIONS FOR SRM

The section evaluates PCNet and ePCNet against established power-control methods for the SRM problem using numerical simulations. It also describes the training and implementation setup, including binary rounding for PCNet outputs.

  • Evaluation design: The evaluations compare PCNet and ePCNet with RR, WMMSE, GBPC, and GP-based methods across SRM and SRM-QC settings.The section focuses first on SRM and uses the listed methods as comparison baselines.
  • Implementation and training: PCNet and ePCNet are implemented in TensorFlow, trained with stochastic gradient descent and ADAM, and monitored using a validation set.Training uses randomly generated channel data, periodic validation, model checkpointing, and 10^5 iterations.
  • Implementation and training: The training procedure generates 10^8 channel samples from mini-batches of size 10^3 over 10^5 iterations.The validation set contains 10^4 samples and is evaluated every 50 iterations.
  • Power-control output: For SRM, PCNet outputs are almost always close to binary power profiles, so the reported profiles are rounded to binary values.Binary power control is motivated by practical transmitter configuration and prior results for K = 2 and K > 2.
  • Power-control output: The observed binary-power behavior is consistent with theoretical studies showing binary control is optimal for K = 2 and performs well, though not optimally, for K > 2.The paper explicitly states that its result is consistent with these studies.

2) Round-Robin Power Control (RR):

This section compares ePCNet and PCNet with iterative and binary power-control baselines under symmetric interference-channel settings. Ensembles generally improve performance over a single network and can outperform baselines while avoiding some of their complexity.

  • Baseline methods: RR updates one user’s power at a time and stops when consecutive sum rates satisfy a convergence condition.Each one-user subproblem is solved through polynomial optimization, with complexity increasing for larger K.
  • Performance comparison: 0.14%, 0.77%, and 1.57% are the ePCNet-over-WMMSE gains with ten networks versus ten WMMSE random initializations in three cases.With only one WMMSE initialization, the reported ePCNet advantage becomes more pronounced in selected cases.
  • Performance comparison: A single PCNet does not universally outperform existing methods, but increasing the ePCNet ensemble size quickly improves performance against standard WMMSE, RR, and GBPC in most cases.The main exception is K = 20 and EsN0 = 0 dB; with K = 20 and EsN0 = 10 dB, M = 2 already outperforms GBPC.
  • Performance comparison: 3.5%, 4.6%, and 1.2% are the gains of ePCNet with M = 10 over GBPC in the reported configurations.Further increasing M continues to improve performance, but the additional sum-rate gain becomes marginal.
  • Performance comparison: ePCNet achieves near-optimal performance in the K = 10 comparison and improves the sum rate for approximately 40% of channel realizations when K = 20 and EsN0 = 10 dB.About 10% of channel realizations still do not show this improvement in that setting.

C. Analyzing the optimization landscape for ePCNet and WMMSE

The paper investigates why ensembling helps ePCNet and why WMMSE behaves differently across noise levels. The evidence links broadly similar PCNet local learners with a smooth landscape, while lower noise increases WMMSE local-optimum variation.

  • ePCNet landscape: Each ePCNet local learner is selected with approximately equal probability, indicating that the ensemble’s networks contribute similarly to overall performance.The experiment uses M = 10, K = 20, EsN0 = 10 dB, and network structure {400, 400, 200, 20}.
  • ePCNet landscape: The similar selection frequencies suggest that PCNet learners reach the same or very similar local optima, supporting a smooth optimization landscape.The paper presents this as an empirical interpretation rather than an exact analytical explanation.
  • WMMSE landscape: WMMSE local-optimum rates are evaluated by running the algorithm 100 times with i.i.d. random initializations for each channel realization.The study estimates mean, variance, and coefficient of variation across the resulting local solutions.
  • WMMSE landscape: Decreasing background noise power increases variation among WMMSE local optima and may lead to bad local optima.The empirical CDFs use 10^4 channel samples, with 100 WMMSE runs per realization.
  • Interpretation: The increased WMMSE variation at lower noise is offered as an explanation for why ePCNet can outperform WMMSE at high EsN0.The paper explicitly notes that a precise neural-network analysis is difficult.

D. Complexity analysis

The complexity analysis compares measured running times under a common single-core environment. It finds that implementation libraries affect PCNet timing, while iterative RR and GBPC require substantially more running time.

  • Methodology: Running-time comparisons are approximate because execution depends on library implementation, architecture, parallelism, and iteration structure, not only FLOP counts.The experiments use the same processor and restrict all methods to one CPU core.
  • Scope: A comprehensive complexity comparison on real-world platforms remains open because neural-network libraries and hardware optimization can materially affect execution.The authors identify this as a topic for further investigation.
  • Results: PCNet running time depends strongly on the library: NumPy is faster for K = 10, whereas TensorFlow is more efficient for K = 20.The paper therefore cautions against judging PCNet complexity from FLOPs alone.
  • Results: PCNet-TF runs in less time than WMMSE, while PCNet-NP is slightly slower; overall, PCNet and WMMSE have comparable running times.RR and GBPC consume much more running time, consistent with the earlier complexity discussion.

E. Performance comparison in Rician fading and different geometries

The evaluation tests ePCNet under Rician fading and varied transmitter–receiver geometries. It continues to outperform competing schemes across these channel settings.

  • Rician fading: ePCNet and WMMSE outperform RR and GBPC by combining multiple local optima.Under relatively low noise power, ePCNet achieves better performance than WMMSE.
  • Rician fading: ePCNet remains effective under Rician fading, indicating applicability beyond Rayleigh channels.The Rician evaluation uses a 0 dB K-factor while retaining the Rayleigh-experiment configurations.
  • Different geometries: The geometry experiments model channel gains using small-scale fading and distance-dependent pathloss.The channel gain combines the fading coefficient with a pathloss function based on transmitter–receiver distance.
  • Different geometries: ePCNet can still outperform other schemes when different transmitter–receiver geometries are modeled.Transmitters and receivers are uniformly randomly distributed within a 10 meter × 10 meter area.

VII. NUMERICAL EVALUATIONS FOR SRM-QC

The SRM-QC evaluation compares PCNet/ePCNet with geometric programming under minimum-rate constraints. ePCNet performs strongly when constraints are moderate, while GP is more effective as feasibility becomes tightly restricted.

  • Evaluation setup: The SRM-QC comparison uses successive convex approximation to solve the non-convex problem iteratively via geometric programming.The implementation compares PCNet/ePCNet against a well-known GP-based SRM-QC method.
  • Sum-rate comparison: ePCNet achieves higher sum rate than GP when the minimum-rate constraint is not strong.With rmin = (0.5, 0.5, 0, 0, 0), ePCNet with M = 10 outperforms GP by 23%.
  • Sum-rate comparison: When the rate constraint becomes tight, the GP-based method performs more efficiently.A tighter constraint shrinks the feasible solution space, which may contain fewer local optima.
  • Feasibility and hit rate: ePCNet’s hit rate decreases as the rate constraint becomes stronger.For rmin = (0.5, 0.5, 0.5, 0, 0), one PCNet reaches 82%, increasing to 94% with ten networks.
  • Feasibility and hit rate: GP achieves a 100% hit rate because it searches the feasible solution space directly.The comparison assumes simulated samples have feasible solutions and uses 10^4 channel samples for the high-complexity GP evaluation.

C. The impact of λ

The scaling factor λ controls the trade-off between satisfying minimum-rate constraints and maximizing sum rate. Its effect depends on both the ensemble size and constraint regime.

  • Single-PCNet behavior: λ = 0 causes severe performance degradation for a single PCNet, showing that penalty terms are necessary.The penalty terms address minimum-rate constraints during unsupervised training.
  • Constraint handling: Penalty scaling can degrade performance because the network cannot constrain its search space to the feasible region for every training sample.The model may keep minimizing penalty terms instead of optimizing sum rate when constraint violations vary across samples.
  • Ensemble-size trade-off: λ = 10 provides the best reported performance across the tested ensemble sizes for rmin = (0.5, 0.5, 0.5, 0, 0).The comparison evaluates ePCNet with different ensemble sizes and λ values.
  • Ensemble-size trade-off: A larger λ may improve sum rate by increasing hit rate but can reduce ePCNet’s ability to combine networks for higher sum rate.The trade-off is visible when comparing λ = 5 and λ = 30 across ensemble sizes.
  • Broader conclusions: PCNet and ePCNet use channel coefficients as inputs, directly maximize sum rate during unsupervised training, and ensemble independently trained PCNets.Adding noise power as an input yields PCNet+/ePCNet+ with desirable generalization capacity.
  • Broader conclusions: The framework may extend to other non-convex wireless optimization problems, but discrete objectives and general constraints remain difficult for neural networks.The paper identifies generalization beyond the studied problems as a potential direction rather than an established result.
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