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 selection. Specifically, GIBE unifies the current explanation methods via formulating the optimization objective of these two phases. It further explores the roles of regularization in two phases respectively. Guided by these explorations, we reveal the essence of regularization and propose four intriguing propositions in terms of it. We believe a better theory of regularization is fundamental: • The essence of regularization: Regularization in GNNs explainability is essentially the tradeoff scheme to pursue the balance between the phases of feature attribution and selection (Section 3.2). • On Sparsity: The optimal coefficients of regularization are proportional to the sparsity of the explanation, that is, high sparsity should require large regularization and vice versa (Section 4.1). • On stochastic mechanism: Existing methods imply an implicit regularization effect of stochastic mechanism, which endows GNNs explainability with better compressibility (Section 4.2). • On OOD issue: The contradictory effects of regularization on two phases are responsible for the OOD issue in the post-hoc explainability (Section 4.3). Furthermore, based on these findings, we propose two common optimization methods, which can bolster the performance of current explainers via sparsity-adaptive and OOD-resistant regularization schemes. Extensive empirical studies validate our findings and proposed methods in Section 5. (Luo et al., 2020; Miao et al., 2022) . For example, GSAT (in its post-hoc working mode) (Miao et al., 2022) trains a parameterized predictor to generate the stochastic attention for each edge as their importance; • Mask scores of structural features (Ying et al., 2019; Wang et al., 2021) . For example, GNNExplainer (Ying et al., 2019) adds soft masks to the input features and trains them by maximizing the mutual information between the masked outcome and target prediction; • Prediction changes on structure perturbations (Yuan et al., 2021; Lin et al., 2021) . For example, PGMExplainer (Vu & Thai, 2020) collects the prediction change on the random node perturbations and learns a Bayesian network from these observations.

2. PRELIMINARY AND RELATED WORK

Feature selection. With attribution scores of features, input features are sequentially filled into the set of salient features to generate the explanatory subgraph according to their importance rank. Many regularization terms are introduced to guide this process. For example, sparsity constraints (Ying et al., 2019; Schlichtkrull et al., 2021) typically leverage the l 1 norm to guarantee that the selected subgraph remains within a prescribed size; connective constraints (Luo et al., 2020; Wang et al., 2021) give more selective probabilities to the edges connecting with the part selected already; more recently, information bottleneck constraints (Miao et al., 2022) are proposed to squeeze the mutual information between the input graph and the selected subgraph.

3. RETHINKING THE ROLE OF REGULARIZATION

In this section, we rethink the role of regularization in GNNs explainers. We start with a new form of graph information bottleneck tailored for explainability and the formulation of the feature attribution and selection (Section 3.1). Guided by the above theory, we analyze the effect of regularization in two phases, respectively (Section 3.2).

