Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization

• URL: http://arxiv.org/abs/2504.09708v1
• Date: Sun, 13 Apr 2025 20:06:49 GMT
• Title: Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization
• Authors: Gavin Zhang, Salar Fattahi, Richard Y. Zhang,
• Abstract summary: In practical instances of nonspecified matrix factorization, the rank of the true solutionrstar$ is often unknown, so the rankr$ of the model can be singular as $r>rstar$.<n>We propose an inexpensive suber for matrix sensing variant non matrix factorization that restores the convergence factor back to linear, even in agnosticized case.<n>Our numerical experiments find that PrecGD works equally well in restoring the convergence of other variants non matrix factorization.
• Score: 19.32160757444549
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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