How well do local algorithms solve semidefinite programs?

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-Renyi 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.