On Nonconvex Optimization for Machine Learning

Journal Article (Journal Article)

Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient-their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.

Full Text

Duke Authors

Cited Authors

  • Jin, C; Netrapalli, P; Ge, R; Kakade, SM; Jordan, MI

Published Date

  • March 1, 2021

Published In

Volume / Issue

  • 68 / 2

Electronic International Standard Serial Number (EISSN)

  • 1557-735X

International Standard Serial Number (ISSN)

  • 0004-5411

Digital Object Identifier (DOI)

  • 10.1145/3418526

Citation Source

  • Scopus