Snake: a Stochastic Proximal Gradient Algorithm for Regularized Problems over Large Graphs

TL;DR

The paper introduces 'Snake', a stochastic proximal gradient algorithm designed for regularized optimization problems on large, complex graphs, leveraging simple path-based methods for efficient computation and proven convergence.

Contribution

It proposes a novel stochastic proximal gradient algorithm that efficiently handles regularized problems on large graphs by exploiting simple path structures.

Findings

Algorithm converges under general conditions

Effective in trend filtering and graph inpainting

Performs well on large graphs in experiments

Abstract

A regularized optimization problem over a large unstructured graph is studied, where the regularization term is tied to the graph geometry. Typical regularization examples include the total variation and the Laplacian regularizations over the graph. When applying the proximal gradient algorithm to solve this problem, there exist quite affordable methods to implement the proximity operator (backward step) in the special case where the graph is a simple path without loops. In this paper, an algorithm, referred to as "Snake", is proposed to solve such regularized problems over general graphs, by taking benefit of these fast methods. The algorithm consists in properly selecting random simple paths in the graph and performing the proximal gradient algorithm over these simple paths. This algorithm is an instance of a new general stochastic proximal gradient algorithm, whose convergence is proven. Applications to trend filtering and graph inpainting are provided among others. Numerical experiments are conducted over large graphs.