[Paper Review] Matrix Completion has No Spurious Local Minimum

Rong Ge, Jason D. Lee+1 more|arXiv (Cornell University)|May 24, 2016

Sparse and Compressive Sensing Techniques32 references186 citations

TL;DR

The paper proves that the non-convex objective for symmetric, positive semidefinite matrix completion has no spurious local minima; all local minima are global minima, enabling SGD/gradient methods from arbitrary initialization to find the underlying low-rank matrix, even with noisy observations.

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 extit{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 extit{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.

Motivation & Objective

• Investigate why practical non-convex matrix completion methods succeed with random or arbitrary initialization.
• Characterize the geometry of the non-convex objective in PSD matrix completion under partial/noisy observations.
• Establish conditions under which all local minima are global minima and provide convergence guarantees for gradient-based methods.

Proposed method

• Analyze the regularized non-convex objective f(X) with observation set Omega and a regularizer R(X) that enforces incoherence.
• Show that every local minimum of the regularized objective corresponds to the true low-rank solution ZZ^T, i.e., f(X)=0 and XX^T=ZZ^T=M.
• Use first- and second-order optimality conditions and concentration inequalities to relate partial observations to full-observation behavior.
• Introduce a simple, generalizable proof strategy by focusing on inequalities that are linear in the observation indicator 1_Omega (a technique robust to sampling).
• Provide formal rank-1 analysis first, then extend to general rank r with analogous arguments and a tau-relaxed second-order condition to guarantee convergence.

Experimental results

Research questions

• RQ1 Under what conditions do local minima of the non-convex PSD matrix completion objective coincide with the global optimum?
• RQ2 How does partial/noisy observation affect the landscape, and can regularization enforce desirable geometry to avoid spurious minima?
• RQ3 Can gradient-based methods (including SGD) converge to the global minimum from arbitrary initialization in polynomial time?
• RQ4 How does the proposed analysis extend from rank-1 to general rank r in PSD matrix completion?

Key findings

• Local minima of the regularized objective occur only at the true factorization XX^T=ZZ^T=M under appropriate sampling (p) and incoherence parameters.
• With high probability, SGD/gradient descent from any initialization converges to the global minimum in polynomial time.
• The results are robust to noise; all local minima remain close to the true solution when observations are corrupted by Gaussian noise of controlled magnitude.
• A tau-relaxed second-order condition suffices to guarantee convergence to a global minimum via known optimization theory results.
• The proof strategy generalizes beyond matrix completion to other problems with partial or noisy observations.