ON REGULARIZATION FOR EXPLAINING GRAPH NEURAL NETWORKS: AN INFORMATION THEORY PERSPECTIVE

Abstract

This work studies the explainability of graph neural networks (GNNs), which is important for the credibility of GNNs in practical usage. Existing work mostly follows the two-phase paradigm to interpret a prediction: feature attribution and selection. However, another important component -regularization, which is crucial to facilitate the above paradigm -has been seldom studied. In this work, we explore the role of regularization in GNNs explainability from the perspective of information theory. Our main findings are: 1) regularization is essentially pursuing the balance between two phases, 2) its optimal coefficient is proportional to the sparsity of explanations, 3) existing methods imply an implicit regularization effect of stochastic mechanism, and 4) its contradictory effects on two phases are responsible for the out-of-distribution (OOD) issue in post-hoc explainability. Based on these findings, we propose two common optimization methods, which can bolster the performance of the current explanation methods via sparsity-adaptive and OOD-resistant regularization schemes. Extensive empirical studies validate our findings and proposed methods. Code is available at https://anonymous.4open.science/r/Rethink_Reg-07F0.

1. INTRODUCTION

Graph Neural Networks (GNNs) (Dwivedi et al., 2020; Wu et al., 2019) have achieved remarkable progress on various graph-related tasks (Mahmud et al., 2021; Zhao et al., 2021; Guo & Wang, 2021) . However, GNNs usually work as a black box, making the decision-making process obscure and hard to interpret (Ribeiro et al., 2016) . Hence, answering the question: "What knowledge does the GNN use to make a certain prediction?" is becoming crucial. To solve this question, most prior studies (Yu et al., 2021; Miao et al., 2022) realize post-hoc explainability by extracting the informative yet sparse subgraphs as explanations, following the principle of graph information bottleneck (GIB) (Wu et al., 2020) . The common paradigm of these explainers can be summarized as the relay race of feature attribution and selection. Specifically, feature attribution distributes the prediction to the input features and traces their importance, and feature selection sequentially fills features into the explanatory subgraph according to the importance rank, where regularization terms are introduced to constrain subgraph properties like size and connectivity. However, existing explainers allocate little attention to the role of regularization in them, but is the focus of our work. On the one hand, without digging deeper into regularization theoretically, we hardly acquire a plain picture of how regularization specifically affects the process of feature attribution and selection. Furthermore, some regularization in existing explainers lacks concrete theoretical support and is seemingly not more than an empirical trick. For example, GNNExplainer (Ying et al., 2019) leverages the l 1 norm to constrain the magnitude of masks and selects the edge with larger importance (i.e., larger mask). The key here is not the absolute magnitude of the mask (i.e., l 1 norm), but rather the relative magnitude between the masks. Thus, we argue that the necessity of l 1 norm needs more theoretical support. In sight of this, we endeavor to rethink the role of regularization in GNNs explainability from the perspective of information theory. Before starting, we first reshape the principle of GIB as GIBE (i.e., new GIB form tailored for GNNs Explainability) in the language of feature attribution and

