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

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 tildeO(dr2) random linear measurements of a rank r positive semidefinite matrix Xstar, we can recover Xstar by parameterizing it by UU^top with U∈ mathbb Rd× d and minimizing the squared loss, even if r ll d. We prove that starting from a small initialization, gradient descent recovers Xstar in tildeO(√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.