Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification

TL;DR

This paper introduces a preconditioner for gradient descent that accelerates convergence in overparameterized nonconvex Burer--Monteiro problems, enabling efficient global optimality certification despite ill-conditioning.

Contribution

The authors propose an inexpensive preconditioner that restores linear convergence rates in overparameterized settings, independent of ill-conditioning and overparameterization.

Findings

Preconditioned gradient descent achieves linear convergence in overparameterized cases.

The method is robust to ill-conditioning of the global minimizer.

Overparameterization no longer slows convergence with the proposed preconditioner.

Abstract

We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally rank deficient, then its rank deficiency certifies it as being globally optimal. This way of certifying global optimality necessarily requires the search rank $r$ of the current iterate $X$ to be overparameterized with respect to the rank $r^{\star}$ of the global minimizer $X^{\star}$. Unfortunately, overparameterization significantly slows down the convergence of gradient descent, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$, even when $\phi$ is strongly convex. In this paper, we propose an inexpensive preconditioner that restores the convergence rate of gradient descent back to linear in the overparameterized case, while also making it agnostic to possible ill-conditioning in the global minimizer $X^{\star}$.