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

Thomas Pumir, Samy Jelassi, Nicolas Boumal

Abstract: We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome the scalability issues arising for large instances, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n×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, 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, almost all second-order stationary points (SOSPs) are optimal. Nevertheless, in practice, one can only compute points which approximately satisfy necessary optimality conditions, so that it is crucial to know whether such points are also approximately optimal. To this end, and under similar assumptions, we use smoothed analysis to show that ASOSPs for a randomly perturbed objective function are approximate global optima, as long as the number of constraints scales sub-quadratically with the desired rank of the optimal solution. In this setting, an approximate optimum Y maps to the approximate optimum X=YY∗ of the SDP. We particularize our results to SDP relaxations of phase retrieval.