Matrix completion has no spurious local minimum

Rong Ge, Jason D. Lee, Tengyu Ma

Research output: Contribution to journalConference article

130 Scopus citations

Abstract

Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent progress in proving various non-convex algorithms converge from a good initial point, it remains unclear why random or arbitrary initialization suffices in practice. We prove that the commonly used non-convex objective function for positive semidefinite matrix completion has no spurious local minima - all local minima must also be global. Therefore, many popular optimization algorithms such as (stochastic) gradient descent can provably solve positive semidefinite matrix completion with arbitrary initialization in polynomial time. The result can be generalized to the setting when the observed entries contain noise. We believe that our main proof strategy can be useful for understanding geometric properties of other statistical problems involving partial or noisy observations.

Original languageEnglish (US)
Pages (from-to)2981-2989
Number of pages9
JournalAdvances in Neural Information Processing Systems
StatePublished - Jan 1 2016
Externally publishedYes
Event30th Annual Conference on Neural Information Processing Systems, NIPS 2016 - Barcelona, Spain
Duration: Dec 5 2016Dec 10 2016

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing

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