🤖 AI Summary
In overparameterized nonconvex matrix factorization—where the specified rank $r$ exceeds the true rank $r^*$—gradient descent suffers sublinear convergence, severely limiting efficiency. This paper proposes PrecGD, a lightweight preconditioned gradient descent method that restores linear convergence without requiring prior knowledge of $r^*$. Key contributions include: (i) the first theoretical demonstration that $ell_2$ regularization, within a specific damping range, effectively mitigates ill-conditioning of the factor matrices; and (ii) a novel adaptive damping strategy, computed cheaply from current iterates, which robustly handles the conditioning of the ground-truth solution. PrecGD maintains linear convergence even under noise and achieves the information-theoretically optimal estimation error bound. Experiments across diverse overparameterized matrix sensing and factorization tasks confirm substantial improvements in both convergence speed and reconstruction accuracy.
📝 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.