Reviews: Fast Convergence of Belief Propagation to Global Optima: Beyond Correlation Decay

The major contributions of this paper are that it proves the global convergence of BP(Theorem 1.3) and VI(Theorem 1.2) on ferromagnetic Ising model with a specific initialization, i.e., to initialize variables to be 1. The proof of Theorem 1.2 is based on the fact that the mean-field free energy function, i.e., \Phi(x) is concave on the set S obtained by the update rule, and then we can use Holder's inequality to expand the \Phi(x*) - Phi(x_t) and get the upper bounds. The proof of Theorem 1.3 is based on the fact that the norm of \Phi(v)'s gradient is less than 1(Lemma 3.2), and the properties of variable \mu sandwiched between v 0 and final v T(Lemma 3.5 and Lemma F.1). Other minor contributions include that it provides examples to empirically show the convergence(appendix G) and it shows how to use ellipsoid method to optimize the beliefs(appendix H). I have to admit that I am not familiar with this area, so can only go through a part of the proof, and I am not able to evaluate the originality and quality of this work.