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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 \emph{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 \emph{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}$$.
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Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization
In practical instances of nonconvex matrix factorization, the rank of the true solution r^{\star} is often unknown, so the rank rof the model can be over-specified 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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- Award ID(s):
- 2047462
- PAR ID:
- 10314750
- Date Published:
- Journal Name:
- Advances in Neural Information Processing Systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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