Hybrid analog and digital beamforming (HBF) is an essential technique for terahertz (THz) communications to support high spectral efficiency with affordable cost. Optimizing hybrid beamforming with deep learning can improve system performance and enhance robustness to imperfect channels. However, pure data-driven neural networks suffer from high training complexity and weak interpretability, while the performance of existing model-based approaches (e.g., deep unfolding) is limited by the algorithm itself. In this paper, we propose a model-based neural network, namely HBF-NN, to optimize hybrid beamforming for multi-antenna multi-carrier THz systems, which consists of two jointly trained modules for optimizing analog and digital beamforming matrices, respectively. To simplify the function to be learned, we propose to optimize the analog beamforming in angle domain. To learn the digital beamforming efficiently, we conceive a graph neural network structure by harnessing the permutation property, recursive property, and the structure of a commonly-used algorithm of singular value decomposition. Simulation results show that the proposed HBF-NN achieves higher spectral efficiency than numerical algorithms, while requiring significantly fewer training samples, free parameters, and less training time than existing data-driven counterpart to achieve the same performance.

Deep learning has been widely used for wireless optimization. In most existing studies, a deep neural network (DNN) is trained for a particular optimization problem and then tested on the same problem with samples drawn from the same distribution as the training data. Practical systems, however, typically involve multiple optimization problems with conflicting objective functions. In this paper, we study the learning for multi-objective optimization (MOO) by using beamforming in multi-user multi-antenna system as an illustrative example. We seek to learn a beamforming policy for two distinct optimization problems: power-constrained sum rate maximization and signal-to-interference-plus-noise ratio-constrained power minimization. Different from conventional MOO methods, which primarily find Pareto-optimal solutions to balance the conflicting objective functions, we resort to transfer learning (TL) and model-agnostic meta-learning (MAML) to learn an adaptive beamforming policy, allowing efficient fine-tuning of trained DNNs with limited samples for either of the two optimization problems. Simulation results demonstrate that both TL and MAML enable the trained DNNs to efficiently adapt to the optimization problems, and graph neural network is a promising network architecture for learning adaptive beamforming policies.

Learning-based precoding has been shown able to be implemented in real-time, jointly optimized with channel acquisition, and robust to imperfect channels. Nonetheless, existing works rarely explain when and why a deep neural network (DNN) for learning precoding policy can perform well. In this paper, we strive to understand the learning performance by taking baseband precoding as an example, for which the optimal precoding matrices of several problems such as sum rate maximization have mathematical structure. Toward this goal, we design a graph neural network (GNN) with edge-update mechanism, whose inductive bias matches to the precoding policy, and analyze its connection to the commonly used convolutional neural networks (CNNs). Noticing that the learning performance can be decomposed into approximation and estimation errors, which depend on the smoothness of a policy and the inductive bias of a DNN, we analyze in which system settings the precoding policy is harder to be approximated by a DNN and how the inductive biases introduced by parameter sharing affect estimation errors. We proceed to derive the estimation error bounds of the DNNs. Simulations validate our analyses and verify the gain of GNN over CNNs in terms of reducing sample complexity.

Deep learning has been introduced to optimize wireless policies such as precoding for enabling real-time implementation. Yet prevalent studies assume that training and test samples are drawn from the same distribution, which is not true in dynamic wireless environments. As a result, a well-trained deep neural network (DNN) may require retraining to adapt to new environments, incurring the overhead of data collection. The required training samples for adaptation can be reduced by introducing inductive biases into DNNs, which can be learned automatically by meta-learning or embedded in DNNs by designing graph neural networks (GNNs). Almost all previous works on meta-learning overlooked the prior-known permutation equivariance (PE) properties, which widely exist in wireless policies and can be harnessed to reduce the hypothesis space of a DNN. In this paper, we strive to answer the following question: which way of introducing inductive biases is more effective in reducing samples for retraining, GNNs or meta-learning? We take the sum-rate maximization precoding problem as an example to answer the question. Simulation results show that the GNNs are more efficient than meta-learning, and meta-learning for precoding cannot adapt to new scenarios where the number of users differs from the training scenario.

Reconfigurable intelligent surface (RIS) is capable of controlling environment smartly for improving the performance of wireless communications. To reduce the pilot overhead of estimating the high-dimensional channels in RIS-aided systems, deep neural networks have been introduced to learn the beam-forming policy with received pilot sequences in an end-to-end (E2E) manner. However, existing works either ignore or only consider part of the permutation equivariant (PE) properties of the E2E policy. As a result, the designed neural networks suffer from high sample complexity. In this paper, we analyze the PE property of an E2E active and passive beamforming policy in a RIS-aided multi-user multi-antenna system, and design a graph neural network (GNN) architecture with matched inductive bias to learn the policy. By taking sum rate maximization problem as an example, simulation results demonstrate the benefits of the proposed GNN in terms of reducing the sample complexity to achieve the expected sum rate.

Optimizing precoding with deep learning enables its real-time implementation. In addition to the learning perfor-mance such as sum rate, training complexity is also important since neural networks (NNs) have to be re-trained in time-varying channels. By leveraging the prior-known property for a policy to be learned, inductive biases can be introduced to the structure of NNs to balance the learning performance and training com-plexity. Most existing works use convolutional neural networks for learning precoding policy, without considering whether their inductive biases match the precoding task. In this paper, we first show that full-digital precoding policy exhibits permutation equivariance property and introduce graph NN (GNN) to learn the policy. We then analyze and show the connections between the structures and inductive biases of several NNs. Simulation results show that the inductive bias of the GNN is well-matched to the precoding policy, and hence achieves higher sum-rate with given number of training samples and needs lower training complexity to achieve the same sum-rate than other NNs.