  Maximum a posteriori (MAP) inference is a fundamental computational paradigm
for statistical inference. In the setting of graphical models, MAP inference
entails solving a combinatorial optimization problem to find the most likely
configuration of the discrete-valued model. Linear programming (LP) relaxations
in the Sherali-Adams hierarchy are widely used to attempt to solve this
problem, and smooth message passing algorithms have been proposed to solve
regularized versions of these LPs with great success. This paper leverages
recent work in entropy-regularized LPs to analyze convergence rates of a class
of edge-based smooth message passing algorithms to $\epsilon$-optimality in the
relaxation. With an appropriately chosen regularization constant, we present a
theoretical guarantee on the number of iterations sufficient to recover the
true integral MAP solution when the LP is tight and the solution is unique.
