Smoothed analysis of the low-rank approach for smooth semidefinite programs
This provides theoretical guarantees for practical optimization in SDPs, addressing scalability for researchers and practitioners in optimization and machine learning, though it is incremental as it builds on prior smoothed analysis work.
The paper tackles the non-convexity issue in the Burer-Monteiro factorized approach for semidefinite programs (SDPs) by showing that approximate second-order stationary points (SOSPs) for a randomly perturbed objective are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors), and bounds the optimality gap for the original problem, as applied to an SDP relaxation of phase retrieval.
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 $n$ by $k$ such that $X = YY^*$ 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, the problem 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 this end, and 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.