Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification
This addresses a computational bottleneck in optimization for machine learning and related fields, offering a practical improvement for gradient descent methods in overparameterized settings.
The paper tackles the problem of slow convergence in overparameterized nonconvex Burer-Monteiro factorization by proposing a preconditioner that restores linear convergence rates, achieving this while remaining agnostic to ill-conditioning in the global minimizer.
We consider using gradient descent to minimize the nonconvex function $f(X)=φ(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $φ$ 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 $φ$ 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 $X^{\star}$.