AND/OR Search for Marginal MAP

Marginal MAP problems are known to be very difficult tasks for graphical models and are so far solved exactly by systematic search guided by a join-tree upper bound. In this paper, we develop new AND/OR branch and bound algorithms for marginal MAP that use heuristics extracted from weighted mini-buckets enhanced with message-passing updates. We demonstrate the effectiveness of the resulting search algorithms against previous join-tree based approaches, which we also extend to accommodate high induced width models, through extensive empirical evaluations. Our results show not only orders-of-magnitude improvements over the state-of-the-art, but also the ability to solve problem instances well beyond the reach of previous approaches.

[1]  Vibhav Gogate,et al.  Join-Graph Propagation Algorithms , 2010, J. Artif. Intell. Res..

[2]  Rina Dechter,et al.  AND/OR Branch-and-Bound search for combinatorial optimization in graphical models , 2009, Artif. Intell..

[3]  Qiang Cheng,et al.  Approximating the Sum Operation for Marginal-MAP Inference , 2012, AAAI.

[4]  Denis Deratani Mauá,et al.  Anytime Marginal MAP Inference , 2012, ICML.

[5]  Rina Dechter,et al.  Generalized best-first search strategies and the optimality of A* , 1985, JACM.

[6]  Rina Dechter,et al.  A Comparison of Time-Space Schemes for Graphical Models , 2007, IJCAI.

[7]  Richard E. Korf,et al.  Linear-Space Best-First Search , 1993, Artif. Intell..

[8]  Michael J. Quinn,et al.  Taking Advantage of Stable Sets of Variables in Constraint Satisfaction Problems , 1985, IJCAI.

[9]  Javier Larrosa,et al.  Unifying tree decompositions for reasoning in graphical models , 2005, Artif. Intell..

[10]  Denis Deratani Mauá.  Equivalences between maximum a posteriori inference in Bayesian networks and maximum expected utility computation in influence diagrams , 2016, Int. J. Approx. Reason..

[11]  Rina Dechter,et al.  From Exact to Anytime Solutions for Marginal MAP , 2016, AAAI.

[12]  Rina Dechter,et al.  AND/OR search spaces for graphical models , 2007, Artif. Intell..

[13]  Yair Weiss,et al.  MAP Estimation, Linear Programming and Belief Propagation with Convex Free Energies , 2007, UAI.

[14]  Dan Roth,et al.  On the Hardness of Approximate Reasoning , 1993, IJCAI.

[15]  Hal Daumé,et al.  Message-Passing for Approximate MAP Inference with Latent Variables , 2011, NIPS.

[16]  Roni Stern,et al.  Using Lookaheads with Optimal Best-First Search , 2010, AAAI.

[17]  Rina Dechter.  Reasoning with Probabilistic and Deterministic Graphical Models: Exact Algorithms , 2013, Reasoning with Probabilistic and Deterministic Graphical Models: Exact Algorithms.

[18]  Lars Otten,et al.  Join-graph based cost-shifting schemes , 2012, UAI.

[19]  Rina Dechter,et al.  Pushing Forward Marginal MAP with Best-First Search , 2015, IJCAI.

[20]  Nils J. Nilsson,et al.  Principles of Artificial Intelligence , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[21]  Rina Dechter,et al.  Applying Marginal MAP Search to Probabilistic Conformant Planning: Initial Results , 2014, AAAI Workshop: Statistical Relational Artificial Intelligence.

[22]  Wei Ping,et al.  Decomposition Bounds for Marginal MAP , 2015, NIPS.

[23]  Qiang Liu,et al.  Bounding the Partition Function using Holder's Inequality , 2011, ICML.

[24]  Adnan Darwiche,et al.  Solving MAP Exactly using Systematic Search , 2002, UAI.

[25]  Qiang Liu,et al.  Variational algorithms for marginal MAP , 2011, J. Mach. Learn. Res..

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

[27]  Solomon Eyal Shimony,et al.  Finding MAPs for Belief Networks is NP-Hard , 1994, Artif. Intell..

[28]  Martin J. Wainwright,et al.  A new class of upper bounds on the log partition function , 2002, IEEE Transactions on Information Theory.

[29]  A. Darwiche,et al.  Complexity Results and Approximation Strategies for MAP Explanations , 2011, J. Artif. Intell. Res..

[30]  Blai Bonet,et al.  A Concise Introduction to Models and Methods for Automated Planning , 2013, A Concise Introduction to Models and Methods for Automated Planning.

[31]  R. Dechter,et al.  Unifying Cluster-Tree Decompositions for Reasoning in Graphical models ∗ , 2005 .

[32]  Rina Dechter,et al.  Weighted heuristic anytime search: new schemes for optimization over graphical models , 2015, Annals of Mathematics and Artificial Intelligence.

[33]  Rina Dechter,et al.  Systematic vs. Non-systematic Algorithms for Solving the MPE Task , 2003, UAI.

[34]  James D. Park,et al.  MAP Complexity Results and Approximation Methods , 2002, UAI.

[35]  Ira Pohl,et al.  Heuristic Search Viewed as Path Finding in a Graph , 1970, Artif. Intell..

[36]  Tommi S. Jaakkola,et al.  Approximate inference using conditional entropy decompositions , 2007, AISTATS.

[37]  Simon de Givry,et al.  Anytime Hybrid Best-First Search with Tree Decomposition for Weighted CSP , 2015, CP.

[38]  Changhe Yuan,et al.  Efficient Computation of Jointree Bounds for Systematic MAP Search , 2009, IJCAI.

[39]  Akihiro Kishimoto,et al.  Recursive Best-First AND/OR Search for Optimization in Graphical Models , 2014, UAI.

[40]  Rina Dechter,et al.  Mini-buckets: a general scheme for approximating inference , 2002 .

[41]  Lars Otten,et al.  Anytime AND/OR depth-first search for combinatorial optimization , 2011, AI Commun..

[42]  Raymond Reiter,et al.  Characterizing Diagnoses , 1990, AAAI.

[43]  Solomon Eyal Shimony,et al.  A new algorithm for finding MAP assignments to belief networks , 1990, UAI.

[44]  Rina Dechter,et al.  Bucket Elimination: A Unifying Framework for Reasoning , 1999, Artif. Intell..

[45]  Javier Larrosa,et al.  A General Scheme for Multiple Lower Bound Computation in Constraint Optimization , 2001, CP.

[46]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[47]  Rina Dechter,et al.  Mini-buckets: A general scheme for bounded inference , 2003, JACM.

[48]  Rina Dechter,et al.  Memory intensive AND/OR search for combinatorial optimization in graphical models , 2009, Artif. Intell..