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 nk 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.
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Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form
Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. For a large class of SDPs, upon random perturbation of the cost matrix, with high probability, we show that all approximate second-order stationary points are approximate global optima for the penalty formulation of appropriately rank-constrained SDPs, as long as the number of constraints scales sub-quadratically with the desired rank. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.
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- Award ID(s):
- 1719558
- NSF-PAR ID:
- 10098817
- Date Published:
- Journal Name:
- Proceedings of the 31st Conference On Learning Theory, PMLR
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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