A Nearly Tight Sum of Squares Lower Bound for Planted Clique

Abstract: We prove that with high probability over the draw of a graph from the Erdos-Renyi distribution G(n,1/2), the ~n^d time, degree d sum of squares relaxation for the clique problem gives a value of ~n^{1/2-o(1)} for any d = o(\log(n)). This yields a nearly tight n^{1/2-o(1)} lower bound on the value of this program for any degree d=o(logn).

Key to the proof is a new framework that we call pseudo-calibration to construct Sum of Squares lower bounds. This framework is inspired by taking a computational analog of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.