Recursive GNNs for Learning Precoding Policies with Size-Generalizability

Graph neural networks (GNNs) have been shown promising in optimizing power allocation and link scheduling with good size generalizability and low training complexity. These merits are important for learning wireless policies under dynamic environments, which partially come from the matched permutation equivariance (PE) properties of the GNNs to the policies to be learned. Nonetheless, it has been noticed in literature that only satisfying the PE property of a precoding policy in multi-antenna systems cannot ensure a GNN for learning precoding to be generalizable to the unseen number of users. Incorporating models with GNNs helps improve size generalizability, which however is only applicable to specific problems, settings, and algorithms. In this paper, we propose a framework of size generalizable GNNs for learning precoding policies that are purely data-driven and can learn wireless policies including but not limited to baseband and hybrid precoding in multi-user multi-antenna systems. To this end, we first find a special structure of each iteration of two numerical algorithms for optimizing precoding, from which we identify the key characteristics of a GNN that affect its size generalizability. Then, we design size-generalizable GNNs that are with these key characteristics and satisfy the PE properties of precoding policies in a recursive manner. Simulation results show that the proposed GNNs can be well-generalized to the number of users for learning baseband and hybrid precoding policies and require much fewer samples than existing counterparts to achieve the same performance.Comment: 37 pages, 8 figure

Learning Resource Allocation Policy: Vertex-GNN or Edge-GNN?

Graph neural networks (GNNs) update the hidden representations of vertices (called Vertex-GNNs) or hidden representations of edges (called Edge-GNNs) by processing and pooling the information of neighboring vertices and edges and combining to incorporate graph topology. When learning resource allocation policies, GNNs cannot perform well if their expressive power are weak, i.e., if they cannot differentiate all input features such as channel matrices. In this paper, we analyze the expressive power of the Vertex-GNNs and Edge-GNNs for learning three representative wireless policies: link scheduling, power control, and precoding policies. We find that the expressive power of the GNNs depend on the linearity and output dimensions of the processing and combination functions. When linear processors are used, the Vertex-GNNs cannot differentiate all channel matrices due to the loss of channel information, while the Edge-GNNs can. When learning the precoding policy, even the Vertex-GNNs with non-linear processors may not be with strong expressive ability due to the dimension compression. We proceed to provide necessary conditions for the GNNs to well learn the precoding policy. Simulation results validate the analyses and show that the Edge-GNNs can achieve the same performance as the Vertex-GNNs with much lower training and inference time

Multidimensional Graph Neural Networks for Wireless Communications

Graph neural networks (GNNs) have been shown promising in improving the efficiency of learning communication policies by leveraging their permutation properties. Nonetheless, existing works design GNNs only for specific wireless policies, lacking a systematical approach for modeling graph and selecting structure. Based on the observation that the mismatched permutation property from the policies and the information loss during the update of hidden representations have large impact on the learning performance and efficiency, in this paper we propose a unified framework to learn permutable wireless policies with multidimensional GNNs. To avoid the information loss, the GNNs update the hidden representations of hyper-edges. To exploit all possible permutations of a policy, we provide a method to identify vertices in a graph. We also investigate the permutability of wireless channels that affects the sample efficiency, and show how to trade off the training, inference, and designing complexities of GNNs. We take precoding in different systems as examples to demonstrate how to apply the framework. Simulation results show that the proposed GNNs can achieve close performance to numerical algorithms, and require much fewer training samples and trainable parameters to achieve the same learning performance as the commonly used convolutional neural networks

Understanding the Performance of Learning Precoding Policy with GNN and CNNs

Learning-based precoding has been shown able to be implemented in real-time, jointly optimized with channel acquisition, and robust to imperfect channels. Yet previous works rarely explain the design choices and learning performance, and existing methods either suffer from high training complexity or depend on problem-specific models. In this paper, we address these issues by analyzing the properties of precoding policy and inductive biases of neural networks, noticing that the learning performance can be decomposed into approximation and estimation errors where the former is related to the smoothness of the policy and both depend on the inductive biases of neural networks. To this end, we introduce a graph neural network (GNN) to learn precoding policy and analyze its connection with the commonly used convolutional neural networks (CNNs). By taking a sum rate maximization precoding policy as an example, we explain why the learned precoding policy performs well in the low signal-to-noise ratio regime, in spatially uncorrelated channels, and when the number of users is much fewer than the number of antennas, as well as why GNN is with higher learning efficiency than CNNs. Extensive simulations validate our analyses and evaluate the generalization ability of the GNN