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Hybrid Downlink Beamforming with Outage Constraints under Imperfect CSI using Model-Driven Deep Learning

Lukas Schynol, Marius Pesavento

TL;DR

This work tackles energy-efficient hybrid downlink beamforming with probabilistic outage constraints under imperfect CSI. It introduces a model-aided deep unrolling framework that integrates a greedy analog beam selection with instance-adaptive error modeling via a graph neural network, and employs an adaptive annealing-based differentiable outage loss to enforce constraints. The approach yields a gradient-based training pipeline with provable gradient existence and demonstrates strong generalization across numbers of users and varying CSI quality, while reducing transmit power relative to benchmarks. The results indicate practical potential for robust, efficient hybrid beamforming in realistic, data-constrained wireless systems, with fast training and adaptability across channel conditions. Overall, the method provides a scalable, interpretable alternative to conic optimization for outage-constrained, energy-efficient hybrid downlink design.

Abstract

We consider energy-efficient multi-user hybrid downlink beamforming (BF) and power allocation under imperfect channel state information (CSI) and probabilistic outage constraints. In this domain, classical optimization methods resort to computationally costly conic optimization problems. Meanwhile, generic deep network (DN) architectures lack interpretability and require large training data sets to generalize well. In this paper, we therefore propose a lightweight model-aided deep learning architecture based on a greedy selection algorithm for analog beam codewords. The architecture relies on an instance-adaptive augmentation of the signal model to estimate the impact of the CSI error. To learn the DN parameters, we derive a novel and efficient implicit representation of the nested constrained BF problem and prove sufficient conditions for the existence of the corresponding gradient. In the loss function, we utilize an annealing-based approximation of the outage compared to conventional quantile-based loss terms. This approximation adaptively anneals towards the exact probabilistic constraint depending on the current level of quality of service (QoS) violation. Simulations validate that the proposed DN can achieve the nominal outage level under CSI error due to channel estimation and channel compression, while allocating less power than benchmarks. Thereby, a single trained model generalizes to different numbers of users, QoS requirements and levels of CSI quality. We further show that the adaptive annealing-based loss function can accelerate the training and yield a better power-outage trade-off.

Hybrid Downlink Beamforming with Outage Constraints under Imperfect CSI using Model-Driven Deep Learning

TL;DR

This work tackles energy-efficient hybrid downlink beamforming with probabilistic outage constraints under imperfect CSI. It introduces a model-aided deep unrolling framework that integrates a greedy analog beam selection with instance-adaptive error modeling via a graph neural network, and employs an adaptive annealing-based differentiable outage loss to enforce constraints. The approach yields a gradient-based training pipeline with provable gradient existence and demonstrates strong generalization across numbers of users and varying CSI quality, while reducing transmit power relative to benchmarks. The results indicate practical potential for robust, efficient hybrid beamforming in realistic, data-constrained wireless systems, with fast training and adaptability across channel conditions. Overall, the method provides a scalable, interpretable alternative to conic optimization for outage-constrained, energy-efficient hybrid downlink design.

Abstract

We consider energy-efficient multi-user hybrid downlink beamforming (BF) and power allocation under imperfect channel state information (CSI) and probabilistic outage constraints. In this domain, classical optimization methods resort to computationally costly conic optimization problems. Meanwhile, generic deep network (DN) architectures lack interpretability and require large training data sets to generalize well. In this paper, we therefore propose a lightweight model-aided deep learning architecture based on a greedy selection algorithm for analog beam codewords. The architecture relies on an instance-adaptive augmentation of the signal model to estimate the impact of the CSI error. To learn the DN parameters, we derive a novel and efficient implicit representation of the nested constrained BF problem and prove sufficient conditions for the existence of the corresponding gradient. In the loss function, we utilize an annealing-based approximation of the outage compared to conventional quantile-based loss terms. This approximation adaptively anneals towards the exact probabilistic constraint depending on the current level of quality of service (QoS) violation. Simulations validate that the proposed DN can achieve the nominal outage level under CSI error due to channel estimation and channel compression, while allocating less power than benchmarks. Thereby, a single trained model generalizes to different numbers of users, QoS requirements and levels of CSI quality. We further show that the adaptive annealing-based loss function can accelerate the training and yield a better power-outage trade-off.
Paper Structure (38 sections, 2 theorems, 68 equations, 3 figures, 4 tables)

This paper contains 38 sections, 2 theorems, 68 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Paper Structure
  5. Notation
  6. System Model and Problem Formulation
  7. System Model
  8. Energy-efficient Hybrid Beamforming with Exact CSI
  9. Energy-efficient Hybrid Beamforming with Inexact CSI
  10. Hybrid Downlink Beamforming and Power Allocation for Perfect CSI
  11. Robust Unrolled Hybrid Downlink Beamforming and Power Allocation
  12. Proposed Model Architecture
  13. Initialization
  14. Model Gradient
  15. Beam Selection
  16. ...and 23 more sections

Key Result

Lemma 4.1

The point $(\breve{\bm{B}}_i^\star, q_i^\star)_i$ is a solution to Problem eq:problem_dl_convex for some $\underline{\widehat{\bm{\psi}}}$ if and only if $(\breve{\bm{b}}_i^\star,q_i^\star)_i$ with $\breve{\bm{B}}_i^\star= \breve{\bm{b}}_i^\star(\breve{\bm{b}}_i^\star)^\mathsf{H}$ is a dual feasible

Figures (3)

  • Figure 1: Block diagram of the proposed architecture $\mathcal{M}(\mathcal{S}; \bm{\theta})$ for outage-constrained hybrid . $\mathcal{F}_{\mathsf{A}}$ represent one analog beamformer selection step, while $\mathcal{F}_\mathsf{Ul}$ represents solving the virtual uplink problem \ref{['eq:problem_ul_inexactcsi_digital']}.
  • Figure 2: Allocated power over empirical outage probability for different targets $P_{\mathsf{out}}$: $0.09$ (circles), $0.10$ (triangles), $0.11$ (squares). 5 folds each.
  • Figure 3: Empirical outage probability (validation) and dual variables corresponding to data groups of different quality $\xi_i=P_{\mathsf{pil},i}$ for the training run of fold 1. Nominal outage $P_{\mathsf{out}}=0.1$.

Theorems & Definitions (4)

  • Lemma 4.1
  • Theorem 4.2
  • proof : Proof of Theorem \ref{['th:dlbf_derivative']}
  • Remark