Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval and Matrix Completion
[edit]
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:33453354, 2018.
Abstract
Recent years have seen a flurry of activities in designing provably efficient nonconvex optimization procedures for solving statistical estimation problems. For various problems like phase retrieval or lowrank matrix completion, stateoftheart nonconvex procedures require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. When it comes to 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 several nonconvex problems: even in the absence of explicit regularization, 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 two statistical estimation problems, i.e. solving random quadratic systems of equations and lowrank matrix completion, we establish that gradient descent achieves nearoptimal statistical and computational guarantees without explicit regularization. As a byproduct, for noisy matrix completion, we demonstrate that gradient descent enables optimal control of both entrywise and spectralnorm errors.
Related Material


