Noise Regularizes Over-parameterized Rank One Matrix Recovery, Provably

We investigate the role of noise in optimization algorithms for learning over-parameterized models. Specifically, we consider the recovery of a rank one matrix Y^*∈ Rd× d from a noisy observation Y using an over-parameterization model. We parameterize the rank one matrix Y^* by XX^top, where X∈ Rd× d. We then show that under mild conditions, the estimator, obtained by the randomly perturbed gradient descent algorithm using the square loss function, attains a mean square error of O(σ2/d), where σ2 is the variance of the observational noise. In contrast, the estimator obtained by gradient descent without random perturbation only attains a mean square error of O(σ2). Our result partially justifies the implicit regularization effect of noise when learning over-parameterized models, and provides new understanding of training over-parameterized neural networks.