A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

We prove that with high probability over the choice of a random graph G from the Erdös-Rényi distribution G(n,1/2), the n
^O(d)
-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n
^{1/2-c(d/log n)1/2}
for some constant c > 0. This yields a nearly tight n
^1/2-o(1)
bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum-of-Squares lower bounds. This framework is inspired by taking a computational analogue 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.