Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization
Abstract
Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that is uniquely determined by the injected noise. Our result implies that the noise imposes implicit bias towards certain optima. Numerical experiments are provided to support our theory.
Cite
@article{arxiv.2102.12430,
title = {Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization},
author = {Tianyi Liu and Yan Li and Song Wei and Enlu Zhou and Tuo Zhao},
journal= {arXiv preprint arXiv:2102.12430},
year = {2021}
}