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

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.