Uniform-in-time weak error analysis for stochastic gradient descent algorithms via diffusion approximation


Journal Article

© 2020 International Press. Diffusion approximation provides weak approximation for stochastic gradient descent algorithms in a finite time horizon. In this paper, we introduce new tools motivated by the backward error analysis of numerical stochastic differential equations into the theoretical framework of diffusion approximation, extending the validity of the weak approximation from finite to infinite time horizon. The new techniques developed in this paper enable us to characterize the asymptotic behavior of constant-step-size SGD algorithms near a local minimum around which the objective functions are locally strongly convex, a goal previously unreachable within the diffusion approximation framework. Our analysis builds upon a truncated formal power expansion of the solution of a Kolmogorov equation arising from diffusion approximation, where the main technical ingredient is uniform-in-time bounds controlling the long-term behavior of the expansion coefficient functions near the local minimum. We expect these new techniques to bring new understanding of the behaviors of SGD near local minimum and greatly expand the range of applicability of diffusion approximation to cover wider and deeper aspects of stochastic optimization algorithms in data science.

Full Text

Duke Authors

Cited Authors

  • Feng, Y; Gao, T; Li, L; Liu, JG; Lu, Y

Published Date

  • January 1, 2020

Published In

Volume / Issue

  • 18 / 1

Start / End Page

  • 163 - 188

Electronic International Standard Serial Number (EISSN)

  • 1945-0796

International Standard Serial Number (ISSN)

  • 1539-6746

Digital Object Identifier (DOI)

  • 10.4310/CMS.2020.v18.n1.a7

Citation Source

  • Scopus