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Designing Heterogeneous GNNs with Desired Permutation Properties for Wireless Resource Allocation

Jianyu Zhao, Chenyang Yang, Tingting Liu

TL;DR

The paper tackles learning wireless policies with graph neural networks by enforcing complex permutation properties through systematic HetGNN design. It introduces a principled graph-construction approach that captures related sets and their permutations, plus sufficient conditions for HetGNN architectures to preserve permutation equivariance under layerwise updates. Through two representative problems—power allocation and hybrid precoding—it demonstrates that permutation priors substantially reduce training sample and parameter space requirements, while topology priors complement generalization. The work shows that combining permutation-aware graph design with principled HetGNN architectures yields improved learning efficiency and robust size generalization, offering practical impact for scalable wireless policy learning with heterogeneous graph structures.

Abstract

Graph neural networks (GNNs) have been designed for learning a variety of wireless policies, i.e., the mappings from environment parameters to decision variables, thanks to their superior performance, and the potential in enabling scalability and size generalizability. These merits are rooted in leveraging permutation prior, i.e., satisfying the permutation property of the policy to be learned (referred to as desired permutation property). Many wireless policies are with complicated permutation properties. To satisfy these properties, heterogeneous GNNs (HetGNNs) should be used to learn such policies. There are two critical factors that enable a HetGNN to satisfy a desired permutation property: constructing an appropriate heterogeneous graph and judiciously designing the architecture of the HetGNN. However, both the graph and the HetGNN are designed heuristically so far. In this paper, we strive to provide a systematic approach for the design to satisfy the desired permutation property. We first propose a method for constructing a graph for a policy, where the edges and their types are defined for the sake of satisfying complicated permutation properties. Then, we provide and prove three sufficient conditions to design a HetGNN such that it can satisfy the desired permutation property when learning over an appropriate graph. These conditions suggest a method of designing the HetGNN with desired permutation property by sharing the processing, combining, and pooling functions according to the types of vertices and edges of the graph. We take power allocation and hybrid precoding policies as examples for demonstrating how to apply the proposed methods and validating the impact of the permutation prior by simulations.

Designing Heterogeneous GNNs with Desired Permutation Properties for Wireless Resource Allocation

TL;DR

The paper tackles learning wireless policies with graph neural networks by enforcing complex permutation properties through systematic HetGNN design. It introduces a principled graph-construction approach that captures related sets and their permutations, plus sufficient conditions for HetGNN architectures to preserve permutation equivariance under layerwise updates. Through two representative problems—power allocation and hybrid precoding—it demonstrates that permutation priors substantially reduce training sample and parameter space requirements, while topology priors complement generalization. The work shows that combining permutation-aware graph design with principled HetGNN architectures yields improved learning efficiency and robust size generalization, offering practical impact for scalable wireless policy learning with heterogeneous graph structures.

Abstract

Graph neural networks (GNNs) have been designed for learning a variety of wireless policies, i.e., the mappings from environment parameters to decision variables, thanks to their superior performance, and the potential in enabling scalability and size generalizability. These merits are rooted in leveraging permutation prior, i.e., satisfying the permutation property of the policy to be learned (referred to as desired permutation property). Many wireless policies are with complicated permutation properties. To satisfy these properties, heterogeneous GNNs (HetGNNs) should be used to learn such policies. There are two critical factors that enable a HetGNN to satisfy a desired permutation property: constructing an appropriate heterogeneous graph and judiciously designing the architecture of the HetGNN. However, both the graph and the HetGNN are designed heuristically so far. In this paper, we strive to provide a systematic approach for the design to satisfy the desired permutation property. We first propose a method for constructing a graph for a policy, where the edges and their types are defined for the sake of satisfying complicated permutation properties. Then, we provide and prove three sufficient conditions to design a HetGNN such that it can satisfy the desired permutation property when learning over an appropriate graph. These conditions suggest a method of designing the HetGNN with desired permutation property by sharing the processing, combining, and pooling functions according to the types of vertices and edges of the graph. We take power allocation and hybrid precoding policies as examples for demonstrating how to apply the proposed methods and validating the impact of the permutation prior by simulations.
Paper Structure (34 sections, 2 theorems, 11 equations, 6 figures, 4 tables)

This paper contains 34 sections, 2 theorems, 11 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Motivation and Contributions
  4. Permutation Prior
  5. Wireless Policy, Permutation Prior, and Set
  6. Example 1 (Power Allocation)
  7. Power Allocation Policy
  8. Sets and Permutation Property
  9. Example 2 (Hybrid Precoding)
  10. Hybrid Precoding Policy
  11. Sets and Permutation Property
  12. Procedure of Design for Exploiting Permutation Prior
  13. Constructing Heterogeneous Graphs with Desired Permutation Properties
  14. Constructing Graphs From Optimization Problems
  15. Principle to Construct Graph
  16. ...and 19 more sections

Key Result

Proposition 1

A VertexGNN will satisfy Property 1 if the three functions of the update equation in eqn:update-vertex satisfy the following conditions: (C1) $q_{v_1, n_1}(\cdot)$ and $q_{v_2, n_2}(\cdot)$ are identical if (i) the $v_1$-th and the $v_2$-th vertices are with the same type, (ii) the $n_1$-th and the

Figures (6)

  • Figure 1: Schematic diagram of designing GNN to exploit permutation prior
  • Figure 2: Graphs for two simple examples, (a) and (b) are graphs for D2D communications with two Txs and Rxs, (c) and (d) are graphs for CoMP-JT system with two base stations (BSs), the vertices or the edges with the same color belong to the same type
  • Figure 3: Graphs constructed for the power allocation problem and hybrid precoding problem with our new principle, the vertices or the edges with the same color belong to the same type
  • Figure 4: Relational priors exploited in each DNN
  • Figure 5: Impact of relational priors on the learning performance of different policies with different numbers of training samples
  • ...and 1 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proof 1
  • Proposition 2