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 Erd\H os-Rényi distribution G(n, 1/2), the n^O(d⁾-time degree d sum-of-squares (SOS) semidefinite programming relaxation for the clique problem will give a value of at least n¹/2 - c(d/ log n⁾¹/² for some constant c > 0. This yields a nearly tight n¹/2 - 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 pseudocalibration to construct SOS lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudodistributions (i.e., dual certificates for the SOS semidefinite program) and sheds further light on the ways in which this hierarchy differs from others.