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How well do local algorithms solve semidefinite programs?

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How well do local algorithms solve semidefinite programs?

Content Provider	ACM Digital Library
Author	Fan, Zhou Montanari, Andrea
Abstract	Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing and yet poorly understood dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erdos-RÃ©nyi random graphs with bounded average degree d > 1, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor 2d^2/(2d^2 + d - 1) of the upper bound. In particular, the local algorithm is at most 8/9 suboptimal, and 1 + O(d^-1) suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erdos-RÃ©nyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model. .
Starting Page	604
Ending Page	614
Page Count	11
File Format	PDF
ISBN	9781450345286
DOI	10.1145/3055399.3055451
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-06-19
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Semidefinite program relaxation Community detection Local algorithm Non-backtracking spectrum
Content Type	Text
Resource Type	Article

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