Message-Passing Algorithms for MAP Estimation Using DC Programming

We address the problem of finding the most likely assignment or MAP estimation in a Markov random field. We analyze the linear programming formulation of MAP through the lens of di!erence of convex functions (DC) programming, and use the concaveconvex procedure (CCCP) to develop e"cient message-passing solvers. The resulting algorithms are guaranteed to converge to a global optimum of the well-studied local polytope, an outer bound on the MAP marginal polytope. To tighten the outer bound, we show how to combine it with the mean-field based inner bound and, again, solve it using CCCP. We also identify a useful relationship between the DC formulations and some recently proposed algorithms based on Bregman divergence. Experimentally, this hybrid approach produces optimal solutions for a range of hard OR problems and nearoptimal solutions for standard benchmarks.

[1]  Tommi S. Jaakkola,et al.  Learning Efficiently with Approximate Inference via Dual Losses , 2010, ICML.

[2]  E. M. L. Beale,et al.  Nonlinear Programming: A Unified Approach. , 1970 .

[3]  Tomás Werner,et al.  High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF) , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[4]  Martin J. Wainwright,et al.  MAP estimation via agreement on (hyper)trees: Message-passing and linear programming , 2005, ArXiv.

[5]  Tomás Werner,et al.  A Linear Programming Approach to Max-Sum Problem: A Review , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[6]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[7]  Vladimir Kolmogorov,et al.  Convergent Tree-Reweighted Message Passing for Energy Minimization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  Yair Weiss,et al.  Linear Programming Relaxations and Belief Propagation - An Empirical Study , 2006, J. Mach. Learn. Res..

[9]  I JordanMichael,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008 .

[10]  W. Zangwill.  Nonlinear programming : a unified approach , 1972 .

[11]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[12]  Nikos Komodakis,et al.  MRF Energy Minimization and Beyond via Dual Decomposition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  D. Sontag.  1 Introduction to Dual Decomposition for Inference , 2010 .

[14]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[15]  Franz Rendl,et al.  Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations , 2009, Math. Program..

[16]  Pradeep Ravikumar,et al.  Quadratic programming relaxations for metric labeling and Markov random field MAP estimation , 2006, ICML.

[17]  Martin J. Wainwright,et al.  MAP estimation via agreement on trees: message-passing and linear programming , 2005, IEEE Transactions on Information Theory.

[18]  Tommi S. Jaakkola,et al.  Introduction to dual composition for inference , 2011 .

[19]  Martin J. Wainwright,et al.  Message-passing for Graph-structured Linear Programs: Proximal Methods and Rounding Schemes , 2010, J. Mach. Learn. Res..

[20]  Tommi S. Jaakkola,et al.  New Outer Bounds on the Marginal Polytope , 2007, NIPS.

[21]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[22]  Tommi S. Jaakkola,et al.  Tightening LP Relaxations for MAP using Message Passing , 2008, UAI.

[23]  WeissYair,et al.  Linear Programming Relaxations and Belief Propagation -- An Empirical Study , 2006 .

[24]  Gert R. G. Lanckriet,et al.  On the Convergence of the Concave-Convex Procedure , 2009, NIPS.

[25]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[26]  Sebastian Nowozin,et al.  Tighter Relaxations for MAP-MRF Inference: A Local Primal-Dual Gap based Separation Algorithm , 2011, AISTATS.

[27]  Zoubin Ghahramani,et al.  On the Convergence of Bound Optimization Algorithms , 2002, UAI.

[28]  Tommi S. Jaakkola,et al.  Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations , 2007, NIPS.

[29]  Alan L. Yuille,et al.  The Concave-Convex Procedure , 2003, Neural Computation.

[30]  Alan L. Yuille,et al.  CCCP Algorithms to Minimize the Bethe and Kikuchi Free Energies: Convergent Alternatives to Belief Propagation , 2002, Neural Computation.

[31]  Stephen Gould,et al.  Accelerated dual decomposition for MAP inference , 2010, ICML.