We consider in this work small random perturbations (of multiplicative noise type) of the gradient flow. We prove that under mild conditions, when the potential function is a Morse function with additional strong saddle condition, the perturbed gradient flow converges to the neighborhood of local minimizers in $O(\ln (\varepsilon^{-1}))$ time on the average, where $\varepsilon$ is the scale of the random perturbation. Under a change of time scale, this indicates that for the diffusion process that approximates the stochastic gradient method, it takes (up to logarithmic factor) only a linear time of inverse stepsize to evade from all saddle points. This can be regarded as a manifestation of fast convergence of the discrete-time stochastic gradient method, the latter being used heavily in modern statistical machine learning.
Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme
CNRS - Université Lille 1（法国国家科学研究中心-里尔第一大学）；Laboratoire de mathématiques Paul Painlevé（保罗·潘勒韦数学实验室）；LAMAV Université de Valenciennes et du Hainaut-Cambrésis（瓦朗谢纳与康布雷-汉诺威大学拉马夫研究所）
On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
An improved central limit theorem and fast convergence rates for entropic transportation costs
IMUVA, Universidad de Valladolid（巴利亚多利德大学IMUVA数学研究所）；IMT, Université de Toulouse（图卢兹大学图卢兹数学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）
