Exponential error rates of SDP for block models: Beyond Grothendieck's inequality

TL;DR

This paper demonstrates that semidefinite programming (SDP) for the stochastic block model achieves exponentially decaying error rates, surpassing previous polynomial bounds, and works effectively in both sparse and dense graph regimes.

Contribution

The authors prove that SDP attains exponential error decay in community detection, extending results beyond Grothendieck's inequality and demonstrating robustness under various model perturbations.

Findings

SDP achieves exponential error decay in community detection.

Error bounds hold in sparse, dense, and heterogeneous regimes.

SDP solution alone suffices for optimal error rates without additional processing.

Abstract

In this paper we consider the cluster estimation problem under the Stochastic Block Model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees, as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the Censored Block Model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing, and improves upon existing polynomially-decaying error bounds proved using the Grothendieck\textquoteright s inequality. Our analysis has two key ingredients: (i) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges, and (ii) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation.