A Grothendieck-type inequality for local maxima by Andrea Montanari

April 25 (Monday) at 1:30 pm GHC-8102

Speaker: David Choi

Abstract: A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few hundreds variables (in the underlying combinatorial problem). On the other hand, it has been observed empirically that an effective strategy amounts to introducing a (non-convex) rank constraint, and solving the resulting smooth optimization problem by ascent methods. This non-convex problem has --generically-- a large number of local maxima, and the reason for this success is therefore unclear. This paper provides rigorous support for this approach. For the problem of maximizing a linear functional over the elliptope, we prove that all local maxima are within a small gap from the SDP optimum. In several problems of interest, arbitrarily small relative error can be achieved by taking the rank constraint k to be of order one, independently of the problem size.

PS: I might also mention little bits from this paper, just to motivate the other one:
Reference: "Community detection in sparse networks via Grothendieck's inequality", Olivier Gu├ędon and Roman Vershynin,http://arxiv.org/abs/1411.4686.

This is based on joint work with Michael Mahoney. Reference: http://arxiv.org/abs/1505.06659.