In this paper, we study the design and analysis of a class of efficient algorithms for computing the Gromov-Wasserstein (GW) distance tailored to large-scale graph learning tasks. Armed with the Luo-Tseng error bound condition (Luo & Tseng, 1992), two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) are proven to be (linearly) convergent. Upon taskspecific properties, our analysis further provides novel theoretical insights to guide how to select the best fit method. As a result, we are able to provide comprehensive experiments to validate the effectiveness of our methods on a host of tasks, including graph alignment, graph partition, and shape matching. In terms of both wall-clock time and modeling performance, the proposed methods achieve state-of-the-art results.

Gromov Wasserstein (GW) distance is a powerful tool for comparing and aligning probability distributions supported on different metric spaces. It has become the main modeling technique for aligning heterogeneous data for a wide range of graph learning tasks. However, the GW distance is known to be highly sensitive to outliers, which can result in large inaccuracies if the outliers are given the same weight as other samples in the objective function. To mitigate this issue, we introduce a new and robust version of the GW distance called RGW. RGW features optimistically perturbed marginal constraints within a ϕ-divergence based ambiguity set. To make the benefits of RGW more accessible in practice, we develop a computationally efficient algorithm, Bregman proximal alternating linearization minimization, with a theoretical convergence guarantee. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of RGW on real-world graph learning tasks, such as subgraph matching and partial shape correspondence.