Matrix Completion has No Spurious Local Minimum

TL;DR

This paper proves that for positive semidefinite matrix completion, the non-convex objective has no spurious local minima, enabling simple algorithms to find global solutions from arbitrary initializations.

Contribution

It establishes the absence of spurious local minima in the non-convex formulation of positive semidefinite matrix completion, explaining the success of simple algorithms in practice.

Findings

All local minima are global in the non-convex objective.

Gradient descent algorithms can solve the problem from arbitrary initializations.

The proof extends to noisy observed entries.

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 \textit{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 \textit{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.