Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations

Abstract: We show that the gradient descent algorithm provides an implicit regularization effect in the learning of over-parameterized matrix factorization models and one-hidden-layer neural networks with quadratic activations. Concretely, we show that given $\tilde{O}(dr<sup>{2})$ random linear measurements of a rank $r$ positive semidefinite matrix $X<sup>{\star}$, we can recover $X<sup>{\star}$ by parameterizing it by $UU<sup>\top$ with $U\in \mathbb R^{d\times d}$ and minimizing the squared loss, even if $r \ll d$. We prove that starting from a small initialization, gradient descent recovers $X<sup>{\star}$ in $\tilde{O}(\sqrt{r})$ iterations approximately. The results solve the conjecture of Gunasekar et al.'17 under the restricted isometry property. The technique can be applied to analyzing neural networks with one-hidden-layer quadratic activations with some technical modifications.