Control the GNN: Utilizing Neural Controller with Lyapunov Stability for Test-Time Feature Reconstruction

The performance of graph neural networks (GNNs) is susceptible to discrepancies between training and testing sample distributions. Prior studies have attempted to enhance GNN performance by reconstructing node features during the testing phase without modifying the model parameters. However, these approaches lack theoretical analysis of the proximity between predictions and ground truth at test time. In this paper, we propose a novel node feature reconstruction method grounded in Lyapunov stability theory. Specifically, we model the GNN as a control system during the testing phase, considering node features as control variables. A neural controller that adheres to the Lyapunov stability criterion is then employed to reconstruct these node features, ensuring that the predictions progressively approach the ground truth at test time. We validate the effectiveness of our approach through extensive experiments across multiple datasets, demonstrating significant performance improvements.

Distribution shift mitigation at test time with performance guarantees

Due to inappropriate sample selection and limited training data, a distribution shift often exists between the training and test sets. This shift can adversely affect the test performance of Graph Neural Networks (GNNs). Existing approaches mitigate this issue by either enhancing the robustness of GNNs to distribution shift or reducing the shift itself. However, both approaches necessitate retraining the model, which becomes unfeasible when the model structure and parameters are inaccessible. To address this challenge, we propose FR-GNN, a general framework for GNNs to conduct feature reconstruction. FRGNN constructs a mapping relationship between the output and input of a well-trained GNN to obtain class representative embeddings and then uses these embeddings to reconstruct the features of labeled nodes. These reconstructed features are then incorporated into the message passing mechanism of GNNs to influence the predictions of unlabeled nodes at test time. Notably, the reconstructed node features can be directly utilized for testing the well-trained model, effectively reducing the distribution shift and leading to improved test performance. This remarkable achievement is attained without any modifications to the model structure or parameters. We provide theoretical guarantees for the effectiveness of our framework. Furthermore, we conduct comprehensive experiments on various public datasets. The experimental results demonstrate the superior performance of FRGNN in comparison to mainstream methods.

Probability-Dependent Gradient Decay in Large Margin Softmax

In the past few years, Softmax has become a common component in neural network frameworks. In this paper, a gradient decay hyperparameter is introduced in Softmax to control the probability-dependent gradient decay rate during training. By following the theoretical analysis and empirical results of a variety of model architectures trained on MNIST, CIFAR-10/100 and SVHN, we find that the generalization performance depends significantly on the gradient decay rate as the confidence probability rises, i.e., the gradient decreases convexly or concavely as the sample probability increases. Moreover, optimization with the small gradient decay shows a similar curriculum learning sequence where hard samples are in the spotlight only after easy samples are convinced sufficiently, and well-separated samples gain a higher gradient to reduce intra-class distance. Based on the analysis results, we can provide evidence that the large margin Softmax will affect the local Lipschitz constraint of the loss function by regulating the probability-dependent gradient decay rate. This paper provides a new perspective and understanding of the relationship among concepts of large margin Softmax, local Lipschitz constraint and curriculum learning by analyzing the gradient decay rate. Besides, we propose a warm-up strategy to dynamically adjust Softmax loss in training, where the gradient decay rate increases from over-small to speed up the convergence rate.