Paper: Enhancing Mutual Information Estimation in Self-Interpretable Graph Neural Networks (ICLR-2024)

Reviewer comment (Theory):
For the predictive term in Eqn. 3, the appropriate formula derivation is: $I(G_{sub};Y)=E_{p(G_{sub},Y)}\log{\frac{p(Y|G_{sub})}{p(Y)}}\geq E_{p(G_{sub},Y)}q_{\omega}(Y|G_{sub})+H(Y)$.

Author response:
Thanks for your comments. We apologize for the error in Eqn. 3, where the denominator of the first term of line 1 should be $p(Y)$ instead of $p(G_{sub})$. We have revised it in the paper. Please note, however, that this mistake does not impact the following derivations and conclusions presented in the paper.

Reviewer comment (Theory):
Does $p_{\phi}(G_{sub}|G)$ and $q_{\theta}(G_{sub}|G)$ share the same subgraph generator? If so, what is the intuition behind using $q_{\theta}(G_{sub}|G)$ to approach $p_{\phi}(G_{sub}|G)$?

Author response:
Thanks for your question! $p_\phi(G_{sub}|G)$ is the probability that $G_{sub}$ is sampled by $G$ based on a underlying subgraph generation process parameterized by $\phi$. However, learning $\phi$ is not enough to calculate the exact value of $p_\phi(G_{sub}|G)$, it should also include a subgraph matching step to compute $p_\phi(G_{sub}|G)$, leading to a NP-hard complexity. Previous works (e.g., GSAT) made some assumptions to ignore the subgraph matching step. Nonetheless, this has resulted in an imprecise learning of $\phi$ and an inaccurate calculation of $I(G_{sub};G)$.

Instead, we simultaneously learn $\phi$ and approximate $p_\phi$ by a variational distribution $q_\theta$. Based on Eq.7 in the paper, the maximization of $E_{p(G, G_{\text{sub}})} [ \log q_\theta( G_{\text{sub}}|G) ]$ is equivalent to minimizing $E_{p(G)}[KL(p_\phi(G_{\text{sub}}|G) || q_\theta(G_{\text{sub}}|G))]$. Thus, we learn $q_\theta$ in Eq.8 with a fixed $\phi$. The optimization of $\phi$ is achieved by minimizing the total GIB objective.
Please refer to the ***Common concern 1*** for a rigorous analysis.


In the detailed implementation, the subgraph generator $g_\phi$ is used to instantiate the subgraph generation process, and hence determines the distribution $p_\phi(G_{sub}|G)$. $q_\theta$ is implemented as a neural network taking $G_{sub}$ and $G$ as inputs and outputs a probability. The network $q_\theta$ includes two GNNs to extract representations of $G_{sub}$ and $G$ respectively and then calcutate a probability. The parameters in $g_\phi$ and $q_\theta$ are learned iteratively, as depicted in the pseudo-code in Appendix.

Reviewer comment (Novelty):
This work closely follows GSAT[1]. Its main theoretical contribution is the addition of the information bottleneck (IB) upper bound loss $L_{GCLUB}$ to the objective of GSAT[1], which is based on the idea of variational CLUB[2].

Author response:
Thanks for your comments. The novelty of the paper is that we point out the limitation of GSAT and propose a framework to approximate $p_\phi(G_{sub}|G)$ by the variational distribution $q_\theta(G_{sub}|G)$. 


In detail, GSAT assumed the generated subgraph $G_{sub}$ is an edge-weighted variant of $G$, where the nodes in $G_{sub}$ and $G$ are one-to-one correspondence, and the edges in $G$ are sampled to obtain $G_{sub}$. However, these works overlook the scenario where the number of nodes in $G_{sub}$ is less than that in $G$. Even if the number of nodes in $G_{sub}$ equates to that of $G$, the correspondence between nodes in $G_{sub}$ and those in $G$ may be nonunique. In such cases, it is necessary to perform subgraph matching before calculating the probability $p_\phi(G_{sub}|G)$. Since subgraph matching is NP-hard, it becomes extremely difficult to accurately compute the value of $p_\phi(G_{sub}|G)$. Given the learned $\phi$, previous studies have not been able to compute an effective value of $p_\phi(G_{sub}|G)$.
Moreover, computing the prior distribution $r(G_{sub})$ over graphs is equally challenging as $p_\phi(G_{sub}|G)$. GSAT involves the term $r(G_{sub})$ to compute the upper bound value of $I(G_{sub};G)$. However, the calculation of $r(G_{sub})$ requires to designate a random graph sampling process and hence encounters similar difficulties as when calculating $p_\phi(G_{sub}|G)$.

Instead, our proposed method approximates $p_\phi$ by a variational distribution $q_\theta$. Based on Eq.7 in the paper, the maximization of $E_{p(G, G_{\text{sub}})} \left[ \log q_\theta( G_{\text{sub}}|G) \right]$ is equivalent to minimizing $E_{p(G)}[KL(p_\phi(G_{\text{sub}}|G)$ || $q_\theta(G_{\text{sub}}|G))]$. Thus, we learn $q_\theta$ in Eq.8. In addition, we use the CLUB to calculate the upper bound of $I(G_\text{sub};G)$, which avoids considering the prior $r(G_\text{sub})$.

Reviewer comment (Theory):
Does the proposed model's ability to remove the spurious correlation come from the framework of GSAT? Can GEMINI provide a theoretical guarantee for the removal of spurious correlations?

Author response:
Theoretically justifying the capability of the proposed method for removing spurious signals can be challenging, since we need to mathematically formalize the *removal*  of spurious signals. Therefore, we conducted an experiment to empirically compare the resistance to spurious signals between the proposed method and GSAT. We have also added a section in Appendix A.6.

Specifically, we select three checkpoints from SPMotif (0.9) and SPMotif (0.7) for both methods and randomly generate 3000 spurious graphs to evaluate the **spurious score** and the **predictive score**. Each spurious graph is generated by randomly selecting a motif graph and a base graph and then assembling the two. For each spurious graph $g_i$, we denote its label as $y_i$ (determined by its motif graph) and its spurious-aligned class as $s_i$ (determined by its base graph). The spurious-aligned class is the label of motif graph with which a spurious base graph is correlated in the training set SPMotif (0.9) and SPMotif (0.7). For example, in SPMotif (0.9), a spurious base graph *tree* has a probability of 0.9 to appear with a motif graph *house* together. Since the label for *house* is 0, the spurious-aligned class for *tree* is 0.

In the evaluation dataset, we ensure $y_i \neq s_i$ for each $g_i$. Referring to the established work (Post hoc explainers may be ineffective for detecting unknown spurious correlation), we define the **spurious score** as 

$
SpurScore = \frac{1}{N}\sum_{i=1}^{N} f(g_i)[s_i].
$

where $f(g_i)$ is the output probabilities of a model, i.e., a three-dimensional vector in our setting. $N$ is the number of evaluation samples, i.e., 3000 in this experiment. The spurious score indicates the average probability of a model mistakenly predicting a graph to its spurious class.
Similarly, we calculate the **predictive score** as

$
PredScore = \frac{1}{N}\sum_{i=1}^{N} f(g_i)[y_i].
$

The predictive score indicates the average probability of a model correctly predicting a graph to its ground truth class.

The results are shown in the table:
|        |  SPMotif (0.9) |                  |  SPMotif (0.7) |                  |
|:------:|:--------------:|:----------------:|:--------------:|:----------------:|
|        | Spurious Score | Predictive Score | Spurious Score | Predictive Score |
|  GSAT  |     0.0985     |      0.7517      |     0.0108     |      0.7711      |
| GEMINI |     0.0708     |      0.7776      |     0.0029     |      0.9123      |

We find that for models trained from both datasets, the spurious score of GEMINI is smaller than that of GSAT. Moreover, the predictive score of GEMINI is significantly better than that of GSAT, especially when the spurious signal is moderate, e.g., SPMotif (0.7). These results verify the superior resistance and robustness against spurious signals compared to GSAT.

Reviewer comment (Reproducibility):
Some of the numerical results reported in Table 1 and Table 2 are quite different from those reported in GSAT. The differences are particularly noticeable with the numbers that involve MNIST-75sp in Table 1 and those associated with SPMotif in Table 2, relating to GIN and GSAT. It would be helpful if the authors could provide further details about their implementations and explanations for these differences.

Author response:
Thanks for your comments! For the SPMotif results in Table 2, there is a parameter to control the size of the base (spurious) graph in the generation of SPMotif. In GSAT, the size is relatively larger, e.g., 20 for tree, 60 for wheel, in the test set. We find that a large base graphs prone to result in distribution shift. We slightly adjust the parameter to be smaller, e.g., 15 for tree, 30 for wheel, to relieve the distribution shift and facilitate comparison. Because we mainly aim to focus on the spurious correlation and interpretation performance, instead of distribution shifts. Moreover, we generate more samples for SPMotif compared with the implementation of GSAT. We generate 30000 samples for a SPMotif dataset, while the number is 3000 in GSAT according to their source code. Hence, the predictive performance of GSAT on SMPotifs in Table 2 is better than their originally reported result.

For the MNIST-75sp results in Table 1, We adopt the strictly identical dataset settings as with the GSAT baseline. The difference may come from some model parameters, adopted loss coefficients, and randomness, since the dataset size of MNIST-75sp is only 20000 and we only run three times to obtain the averaged results, which may also affect the evaluation numbers.

Reviewer comment (Theory):
On page 3, in the last sentence before Eq.3, should it be a “lower” bound of $I(G_{sub};Y)$?

Author response:
Yes, you are correct. We are sorry for this typo and have made a revision.

Reviewer comment (Experiments):
The experimental results are not convincing enough. SOTA explainers for GNNs should be set as baselines. Moreover, the proposed model did not exhibit a significant improvement compared to these baselines.

Author response:
Thank you for your feedback. Indeed, GSAT is the state-of-the-art baseline with both predictive and interpretable capabilities. 
Additionally, GNNExplainer and PGExplainer are frequently employed as post-hoc explanation methods for GNNs in prior research. As demonstrated in Table 1 & 2, our proposed method maintains comparable predictive performance with mainstream GNNs (e.g., GIN), and improves the interpretation AUC over the second-best baseline across five datasets. The average improvement is about 3%$\sim$ 5% over the state-of-the-art self-interpretable baseline. Thus, we believe our model exhibits a significant advancement in interpretability, contributing to the existing body of research on GNNs.

Reviewer comment (Evaluation):
The authors assert that the proposed method can generate sparse subgraphs even without the need for sparse regularization. However, it is evident that L_sp is introduced as a subgraph term to regulate graph sparsity. In the ablation study, the authors argue that this term is essential, which appears to be inconsistent with their initial claim in the introduction.

Author response:
Thanks for your comments. In the first section of the ablation study (Effect of the $L_{GCLUB}$ term), we remove the $L_{sp}$ term and rely solely on the $L_{GCLUB}$ regularization. Figure 2(b) demonstrates that $L_{GCLUB}$ could generate sparse graphs, e.g., the average edge/node probability is about $0.6\sim 0.7$. We claim $L_{sp}$ is essential for generating even ***sparser*** graphs, e.g., average edge/node probability of $0.3\sim 0.5$. The term $L_{sp}$ is used to provide users with the opportunity to achieve a desired level of subgraph sparsity based on their preferences.

Reviewer comment (Theory):
What is the benefit of modeling arbitrary subgraphs and graphs? Since G_{sub}^1 should be sampled from G_1 and should not be related to G_2.

Author response:
Thanks for your questions! Different from the previous works, we provide a more accurate calculation of $p_\phi(G_{sub}|G)$ and $I(G;G_{sub})$. In detail, we approximate $p_\phi(G_{sub}|G)$ by a variational distribution $q_\theta(G_{sub}|G)$, and use the CLUB upper bound for $I(G;G_{sub})$.  

When adopting our $I_{GCLUB}$ bound to calculate the upper bound of $I(G;G_{sub})$, a subgraph instance of $G_{sub}$ may not be directly sampled from a graph instance of $G$. According to the CLUB bound in Eq. 5 of the paper, for the second term $E_{p(G)} E_{p(G_{sub}) } [ \log p_\phi( G_{\text{sub}}|G )]$, $G_{sub}$ and $G$ come from their own marginal distributions. Hence when we calculate this term, we need to calculate $p_\phi( g_{\text{sub}}^2 |g^1 )$ where $g^1$ is sampled from $p(G)$, and $g_{\text{sub}}^2$ is sampled from $p(G_{\text{sub}})$. $g_{\text{sub}}^2$ may not come from $g^1$.

Reviewer comment (Theory):
Why is the MI upper bound approximation proposed in the method better than previous methods?

Author response:
Thanks for your questions! The reason for the utilization of CLUB bound is to avoid the estimation of the prior distribution $r(G_{sub})$. Computing $r(G_\text{sub})$ over graphs is equally challenging as $p_\phi(G_{sub}|G)$. For example, GSAT regarded the prior probability of a subgraph instance $r(g_{sub})$ as the product of edge probabilities on $g$, where the existence of each edge follows an independent Bernoulli distribution. However, $r(g_{sub})$ is essentially $r(g_{sub}|g)$, which relies on the parent graph $g$ (as we have stated, in GSAT and other works, $g_{sub}$ must be a weighted version of $g$ and the matching between node sets of $g_{sub}$ and $g$ is implicitly assumed to be known), and is not a *real* prior distribution over graphs. A real graph prior $r(g_{sub})$ should be able to be calculated for an arbitrary graph $g_{sub}$ without knowing its *parent graph* $g$. Hence, we should utilize an MI bound that does not depend on  $r(G_{sub})$.