FlexiDrop: Theoretical Insights and Practical Advances in Random Dropout Method on GNNs

Graph Neural Networks (GNNs) are powerful tools for handling graph-type data. Recently, GNNs have been widely applied in various domains, but they also face some issues, such as overfitting, over-smoothing and non-robustness. The existing research indicates that random dropout methods are an effective way to address these issues. However, random dropout methods in GNNs still face unresolved problems. Currently, the choice of dropout rate, often determined by heuristic or grid search methods, can increase the generalization error, contradicting the principal aims of dropout. In this paper, we propose a novel random dropout method for GNNs called FlexiDrop. First, we conduct a theoretical analysis of dropout in GNNs using rademacher complexity and demonstrate that the generalization error of traditional random dropout methods is constrained by a function related to the dropout rate. Subsequently, we use this function as a regularizer to unify the dropout rate and empirical loss within a single loss function, optimizing them simultaneously. Therefore, our method enables adaptive adjustment of the dropout rate and theoretically balances the trade-off between model complexity and generalization ability. Furthermore, extensive experimental results on benchmark datasets show that FlexiDrop outperforms traditional random dropout methods in GNNs.

Graph Neural Aggregation-diffusion with Metastability

Continuous graph neural models based on differential equations have expanded the architecture of graph neural networks (GNNs). Due to the connection between graph diffusion and message passing, diffusion-based models have been widely studied. However, diffusion naturally drives the system towards an equilibrium state, leading to issues like over-smoothing. To this end, we propose GRADE inspired by graph aggregation-diffusion equations, which includes the delicate balance between nonlinear diffusion and aggregation induced by interaction potentials. The node representations obtained through aggregation-diffusion equations exhibit metastability, indicating that features can aggregate into multiple clusters. In addition, the dynamics within these clusters can persist for long time periods, offering the potential to alleviate over-smoothing effects. This nonlinear diffusion in our model generalizes existing diffusion-based models and establishes a connection with classical GNNs. We prove that GRADE achieves competitive performance across various benchmarks and alleviates the over-smoothing issue in GNNs evidenced by the enhanced Dirichlet energy.

Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theory

This paper presents a novel study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Diverging from extant approaches grounded in random walk analysis or particle systems, we approach this problem through operator semigroup theory. This theoretical framework allows us to rigorously prove that oversmoothing is intrinsically linked to the ergodicity of the diffusion operator. This finding further poses a general and mild ergodicity-breaking condition, encompassing the various specific solutions previously offered, thereby presenting a more universal and theoretically grounded approach to mitigating oversmoothing in diffusion-based GNNs. Additionally, we offer a probabilistic interpretation of our theory, forging a link with prior works and broadening the theoretical horizon. Our experimental results reveal that this ergodicity-breaking term effectively mitigates oversmoothing measured by Dirichlet energy, and simultaneously enhances performance in node classification tasks.

Improved Naive Bayes with Mislabeled Data

Labeling mistakes are frequently encountered in real-world applications. If not treated well, the labeling mistakes can deteriorate the classification performances of a model seriously. To address this issue, we propose an improved Naive Bayes method for text classification. It is analytically simple and free of subjective judgements on the correct and incorrect labels. By specifying the generating mechanism of incorrect labels, we optimize the corresponding log-likelihood function iteratively by using an EM algorithm. Our simulation and experiment results show that the improved Naive Bayes method greatly improves the performances of the Naive Bayes method with mislabeled data.

Analysis of Graph Neural Networks with Theory of Markov Chains

In this paper, we provide a theoretical tool for the interpretation and analysis of \emph{graph neural networks} (GNNs). We use Markov chains on graphs to mathematically model the forward propagation processes of GNNs. The graph neural networks are divided into two classes of operator-consistent and operator-inconsistent based on whether the Markov chains are time-homogeneous. Based on this, we study \emph{over-smoothing} which is an important problem in GNN research. We attribute the over-smoothing problem to the convergence of an arbitrary initial distribution to a stationary distribution. We prove the effectiveness of the previous methods for alleviating the over-smoothing problem. Further, we give the conclusion that operator-consistent GNN cannot avoid over-smoothing at an exponential rate in the Markovian sense. For operator-inconsistent GNN, we theoretically give a sufficient condition for avoiding over-smoothing. Based on this condition, we propose a regularization term which can be flexibly added to the training of the neural network. Finally, we design experiments to verify the effectiveness of this condition. Results show that our proposed sufficient condition not only improves the performance but also alleviates the over-smoothing phenomenon.