Toward a Graph Foundation Model: Pre-Training Transformers With Random Walks

A foundation model like GPT elicits many emergent abilities, owing to the pre-training with broad inclusion of data and the use of the powerful Transformer architecture. While foundation models in natural languages are prevalent, can we build similar models for graphs? This paper describes an approach toward a graph foundation model that is pre-trained with diverse graph datasets by adapting the Transformer backbone. A central challenge toward this end is how a sequence model encodes graphs of varying sizes and from different domains. We propose representing a node as multiple random walks, such that the Transformer can extract node representations from sequences, which in turn form edge and graph representations. We develop a novel context prediction loss for these random walks and theoretically analyze their expressive power in distinguishing neighborhoods and graphs. We also demonstrate the pre-training of our model and its adaptation to downstream tasks, showcasing its potential as a foundation for processing and reasoning with graph-structured data.

An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian

Nonlinear acceleration methods are powerful techniques to speed up fixed-point iterations. However, many acceleration methods require storing a large number of previous iterates and this can become impractical if computational resources are limited. In this paper, we propose a nonlinear Truncated Generalized Conjugate Residual method (nlTGCR) whose goal is to exploit the symmetry of the Hessian to reduce memory usage. The proposed method can be interpreted as either an inexact Newton or a quasi-Newton method. We show that, with the help of global strategies like residual check techniques, nlTGCR can converge globally for general nonlinear problems and that under mild conditions, nlTGCR is able to achieve superlinear convergence. We further analyze the convergence of nlTGCR in a stochastic setting. Numerical results demonstrate the superiority of nlTGCR when compared with several other competitive baseline approaches on a few problems. Our code will be available in the future.