Skip to main content

Smoothed analysis of the low-rank approach for smooth semidefinite programs

Author(s): Pumir, Thomas; Jelassi, Samy; Boumal, Nicolas

To refer to this page use:
Abstract: We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size nxk such that X = Y Y* is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval.
Publication Date: 2018
Citation: Pumir, Thomas, Jelassi, Samy, Boumal, Nicolas. (2018). Smoothed analysis of the low-rank approach for smooth semidefinite programs. ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 31 (NIPS 2018), 31
ISSN: 1049-5258
Type of Material: Journal Article
Version: Author's manuscript

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.