Newton-type Methods for Inference in Higher-Order Markov Random Fields

TL;DR

This paper explores the use of Newton-type methods for solving the Lagrangian dual in MAP inference for higher-order Markov Random Fields, showing improved convergence and handling of ill-conditioned problems.

Contribution

It introduces a provably convergent Newton-based framework with strategies for efficient Hessian computation, damping, and preconditioning tailored for MAP inference.

Findings

Newton methods outperform first-order methods in convergence speed.

The proposed framework efficiently handles ill-conditioned problems.

Results demonstrate effectiveness on higher-order Markov Random Fields.

Abstract

Linear programming relaxations are central to {\sc map} inference in discrete Markov Random Fields. The ability to properly solve the Lagrangian dual is a critical component of such methods. In this paper, we study the benefit of using Newton-type methods to solve the Lagrangian dual of a smooth version of the problem. We investigate their ability to achieve superior convergence behavior and to better handle the ill-conditioned nature of the formulation, as compared to first order methods. We show that it is indeed possible to efficiently apply a trust region Newton method for a broad range of {\sc map} inference problems. In this paper we propose a provably convergent and efficient framework that includes (i) excellent compromise between computational complexity and precision concerning the Hessian matrix construction, (ii) a damping strategy that aids efficient optimization, (iii) a truncation strategy coupled with a generic pre-conditioner for Conjugate Gradients, (iv) efficient sum-product computation for sparse clique potentials. Results for higher-order Markov Random Fields demonstrate the potential of this approach.