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Numerous empirical evidences have corroborated the importance of noise innonconvex optimization problems. The theory behind such empirical observations,however, is still largely unknown. This paper studies this fundamental problemthrough 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 theinitialization. In contrast, we show that a perturbed form of GD with anarbitrary initialization converges to a global optimum that is uniquelydetermined by the injected noise. Our result implies that the noise imposesimplicit bias towards certain optima. Numerical experiments are provided tosupport our theory.

Noisy Gradient Descent Converges to Flat Minima for Nonconvex Matrix Factorization