Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization

TL;DR

This paper introduces PrecGD, a preconditioned gradient descent method that restores linear convergence in over-parameterized nonconvex matrix factorization, even with ill-conditioned ground truth, by using a specific regularization-based preconditioning.

Contribution

The paper proposes a novel preconditioning technique for nonconvex matrix factorization that achieves linear convergence in over-parameterized settings, addressing issues of slow convergence and ill-conditioning.

Findings

PrecGD restores linear convergence in over-parameterized matrix factorization.

The method is stable under noise and converges to an optimal error bound.

Numerical experiments confirm PrecGD's effectiveness across variants.

Abstract

In practical instances of nonconvex matrix factorization, the rank of the true solution $r^{\star}$ is often unknown, so the rank $r$ of the model can be overspecified as $r>r^{\star}$. This over-parameterized regime of matrix factorization significantly slows down the convergence of local search algorithms, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$. We propose an inexpensive preconditioner for the matrix sensing variant of nonconvex matrix factorization that restores the convergence rate of gradient descent back to linear, even in the over-parameterized case, while also making it agnostic to possible ill-conditioning in the ground truth. Classical gradient descent in a neighborhood of the solution slows down due to the need for the model matrix factor to become singular. Our key result is that this singularity can be corrected by $\ell_{2}$ regularization with a specific range of values for the damping parameter. In fact, a good damping parameter can be inexpensively estimated from the current iterate. The resulting algorithm, which we call preconditioned gradient descent or PrecGD, is stable under noise, and converges linearly to an information theoretically optimal error bound. Our numerical experiments find that PrecGD works equally well in restoring the linear convergence of other variants of nonconvex matrix factorization in the over-parameterized regime.