Matrix Completion has No Spurious Local Minimum
This addresses a foundational issue in machine learning for collaborative filtering and recommender systems, providing theoretical guarantees for practical optimization methods.
The paper tackles the problem of why random initialization works for non-convex matrix completion algorithms by proving that the objective function has no spurious local minima, ensuring global convergence with arbitrary initialization in polynomial time, even with noisy observations.
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.