Convergence Rate of Block-Coordinate Maximization Burer-Monteiro Method for Solving Large SDPs

TL;DR

This paper introduces a coordinate ascent method for solving low-rank non-convex formulations of large SDPs, with provable convergence guarantees and an optimal approximation ratio under the unique games conjecture.

Contribution

It provides the first convergence analysis for block-coordinate maximization on the Burer-Monteiro formulation, including linear convergence near local maxima and an optimal approximation guarantee.

Findings

Global convergence to first-order stationary points with sublinear rate.

Linear convergence around local maxima under quadratic decay.

Achieves $1-O(1/r)$ approximation ratio with explicit iteration bounds.

Abstract

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro proposed to reduce the dimension of the problem by appealing to a low-rank factorization and solve the subsequent non-convex problem instead. In this paper, we present coordinate ascent based methods to solve this non-convex problem with provable convergence guarantees. More specifically, we prove that the block-coordinate maximization algorithm applied to the non-convex Burer-Monteiro method globally converges to a first-order stationary point with a sublinear rate without any assumptions on the problem. We further show that this algorithm converges linearly around a local maximum provided that the objective function exhibits quadratic decay. We establish that this condition generically holds when the rank of the factorization is sufficiently large. Furthermore, incorporating Lanczos method to the block-coordinate maximization, we propose an algorithm that is guaranteed to return a solution that provides $1-O(1/r)$ approximation to the original SDP without any assumptions, where $r$ is the rank of the factorization. This approximation ratio is known to be optimal (up to constants) under the unique games conjecture, and we can explicitly quantify the number of iterations to obtain such a solution.