Approximate Bregman Proximal Gradient Algorithm for Relatively Smooth Nonconvex Optimization

In this paper, we propose the approximate Bregman proximal gradient algorithm (ABPG) for solving composite nonconvex optimization problems. ABPG employs a new distance that approximates the Bregman distance, making the subproblem of ABPG simpler to solve compared to existing Bregman-type algorithms. The subproblem of ABPG is often expressed in a closed form. Similarly to existing Bregman-type algorithms, ABPG does not require the global Lipschitz continuity for the gradient of the smooth part. Instead, assuming the smooth adaptable property, we establish the global subsequential convergence under standard assumptions. Additionally, assuming that the Kurdyka--{\L}ojasiewicz property holds, we prove the global convergence for a special case. Our numerical experiments on the $\ell_p$ regularized least squares problem, the $\ell_p$ loss problem, and the nonnegative linear system show that ABPG outperforms existing algorithms especially when the gradient of the smooth part is not globally Lipschitz or even local Lipschitz continuous.Comment: 26 pages, 10 figure

Fast and Provably Convergent Algorithms for Gromov-Wasserstein in Graph Data

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~\citep{luo1992error}, two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) enjoy the convergence guarantees. Upon task-specific 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

A Variational Perspective on Accelerated Methods in Optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.Comment: 38 pages. Subsumes an earlier working draft arXiv:1509.0361