Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution

• Published in: Foundations of computational mathematics 2020-06, Vol.20 (3), p.451-632
• Main Authors: Ma, Cong, Wang, Kaizheng, Chi, Yuejie, Chen, Yuxin
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; Computer Science; Convergence; Deconvolution; Economics; Empirical analysis; Error analysis; Estimation theory; Functions, Implicit; Iterative algorithms; Iterative methods; Linear and Multilinear Algebras; Linear programming; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Methods; Numerical Analysis; Optimization; Perturbation; Phase retrieval; Regularization; Statistical models; Trajectory analysis
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-019-09429-9

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g., trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex optimization: even in the absence of explicit regularization, gradient descent enforces proper regularization implicitly under various statistical models. In fact, gradient descent follows a trajectory staying within a basin that enjoys nice geometry, consisting of points incoherent with the sampling mechanism. This “implicit regularization” feature allows gradient descent to proceed in a far more aggressive fashion without overshooting, which in turn results in substantial computational savings. Focusing on three fundamental statistical estimation problems, i.e., phase retrieval, low-rank matrix completion, and blind deconvolution, we establish that gradient descent achieves near-optimal statistical and computational guarantees without explicit regularization. In particular, by marrying statistical modeling with generic optimization theory, we develop a general recipe for analyzing the trajectories of iterative algorithms via a leave-one-out perturbation argument. As a by-product, for noisy matrix completion, we demonstrate that gradient descent achieves near-optimal error control—measured entrywise and by the spectral norm—which might be of independent interest.