Understanding the Performance of Learning Precoding Policies with Graph and Convolutional Neural Networks

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.