# Convergence Rates of Smooth Message Passing with Rounding in Entropy-Regularized MAP Inference

## TL;DR

This paper analyzes the convergence rates of smooth message passing algorithms with rounding for entropy-regularized MAP inference in graphical models, providing theoretical guarantees on iteration complexity for recovering the true MAP solution.

## Contribution

It offers the first theoretical analysis of convergence rates for entropy-regularized message passing algorithms in MAP inference, including conditions for exact recovery.

## Key findings

- Convergence rates depend on regularization parameters and problem structure.
- Under certain conditions, the algorithm guarantees recovery of the true MAP solution.
- Provides bounds on the number of iterations needed for $ ext{epsilon}$-optimality.

## Abstract

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.