Noise Regularizes Over-parameterized Rank One Matrix Recovery, Provably

• URL: http://arxiv.org/abs/2202.03535v1
• Date: Mon, 7 Feb 2022 21:53:51 GMT
• Title: Noise Regularizes Over-parameterized Rank One Matrix Recovery, Provably
• Authors: Tianyi Liu, Yan Li, Enlu Zhou and Tuo Zhao
• Abstract summary: We parameterize the rank one matrix $Y*$ by $XXtop$, where $Xin Rdtimes d$. 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 $O(sigma2/d)$. In contrast, the estimator obtained by gradient descent without random perturbation only attains a mean square error of $O(sigma2)$.
• Score: 42.427869499882206
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate the role of noise in optimization algorithms for learning over-parameterized models. Specifically, we consider the recovery of a rank one matrix $Y^*\in R^{d\times d}$ from a noisy observation $Y$ using an over-parameterization model. We parameterize the rank one matrix $Y^*$ by $XX^\top$, where $X\in R^{d\times d}$. 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 $O(\sigma^2/d)$, where $\sigma^2$ 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 $O(\sigma^2)$. 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.

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