The Burer-Monteiro SDP method can fail even above the Barvinok-Pataki boundDownload PDF

Published: 31 Oct 2022, Last Modified: 03 Jul 2024NeurIPS 2022 AcceptReaders: Everyone
Keywords: non-convex optimization, semidefinite programming, Burer-Monteiro, manifold optimization, low-rank SDP
TL;DR: We prove that the Burer-Monteiro method for solving semidefinite programs can fail even above the Barvinok-Pataki bound by constructing spurious local minima even when the rank is linear.
Abstract: The most widely used technique for solving large-scale semidefinite programs (SDPs) in practice is the non-convex Burer-Monteiro method, which explicitly maintains a low-rank SDP solution for memory efficiency. There has been much recent interest in obtaining a better theoretical understanding of the Burer-Monteiro method. When the maximum allowed rank $p$ of the SDP solution is above the Barvinok-Pataki bound (where a globally optimal solution of rank at most \(p\) is guaranteed to exist), a recent line of work established convergence to a global optimum for generic or smoothed instances of the problem. However, it was open whether there even exists an instance in this regime where the Burer-Monteiro method fails. We prove that the Burer-Monteiro method can fail for the Max-Cut SDP on $n$ vertices when the rank is above the Barvinok-Pataki bound ($p \ge \sqrt{2n}$). We provide a family of instances that have spurious local minima even when the rank $p = n/2$. Combined with existing guarantees, this settles the question of the existence of spurious local minima for the Max-Cut formulation in all ranges of the rank and justifies the use of beyond worst-case paradigms like smoothed analysis to obtain guarantees for the Burer-Monteiro method.
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