Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations
Yuanzhi Li, Tengyu Ma, Hongyang Zhang · arXiv (Cornell University) · 2017
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 $\\tilde{O}(dr^{2})$ random linear measurements of a rank $r$ positive semidefinite matrix $X^{\\star}$, we can recover $X^{\\star}$ by parameterizing it by $UU^\\top$ 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 $X^{\\star}$ 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.