Deep Bucket Elimination

Bucket Elimination (BE) is a universal inference scheme that can solve most tasks over probabilistic and deterministic graphical models exactly. However, it often requires exponentially high levels of memory (in the induced-width) preventing its execution. In the spirit of exploiting Deep Learning for inference tasks, in this paper, we will use neural networks to approximate BE. The resulting Deep Bucket Elimination (DBE) algorithm is developed for computing the partition function. We provide a proof-of-concept empirically using instances from several different benchmarks, showing that DBE can be a more accurate approximation than current state-of-the-art approaches for approximating BE (e.g. the mini-bucket schemes), especially when problems are sufficiently hard.

[1]  Qiang Liu,et al.  Belief Propagation for Structured Decision Making , 2012, UAI.

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

[3]  Alex Graves,et al.  Playing Atari with Deep Reinforcement Learning , 2013, ArXiv.

[4]  Vibhav Gogate,et al.  SampleSearch: Importance sampling in presence of determinism , 2011, Artif. Intell..

[5]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[6]  Pierre Baldi,et al.  From Reinforcement Learning to Deep Reinforcement Learning: An Overview , 2017, Braverman Readings in Machine Learning.

[7]  Lisa Zhang,et al.  Inference in Probabilistic Graphical Models by Graph Neural Networks , 2018, 2019 53rd Asilomar Conference on Signals, Systems, and Computers.

[8]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[9]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[10]  Adnan Darwiche,et al.  Modeling and Reasoning with Bayesian Networks , 2009 .

[11]  Dan Geiger,et al.  Maximum Likelihood Haplotyping for General Pedigrees , 2005, Human Heredity.

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

[13]  Rina Dechter,et al.  Scaling Up AND/OR Abstraction Sampling , 2020, IJCAI.

[14]  Samuel S. Schoenholz,et al.  Neural Message Passing for Quantum Chemistry , 2017, ICML.

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

[16]  Volume 26 , 2002 .

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

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

[19]  Peter Henderson,et al.  An Introduction to Deep Reinforcement Learning , 2018, Found. Trends Mach. Learn..

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

[21]  David J. Spiegelhalter,et al.  Sequential updating of conditional probabilities on directed graphical structures , 1990, Networks.

[22]  John W. Fisher,et al.  Probabilistic Variational Bounds for Graphical Models , 2015, NIPS.

[23]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

[24]  William T. Freeman,et al.  Constructing free-energy approximations and generalized belief propagation algorithms , 2005, IEEE Transactions on Information Theory.

[25]  Rina Dechter,et al.  Bayesian Inference in the Presence of Determinism , 2003, AISTATS.

[26]  Rina Dechter,et al.  Hybrid Processing of Beliefs and Constraints , 2001, UAI.

[27]  Rina Dechter,et al.  Join Graph Decomposition Bounds for Influence Diagrams , 2018, UAI.

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

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

[30]  Shane Legg,et al.  Human-level control through deep reinforcement learning , 2015, Nature.

[31]  Nir Friedman,et al.  Probabilistic Graphical Models - Principles and Techniques , 2009 .

[32]  Rina Dechter,et al.  A Weighted Mini-Bucket Bound for Solving Influence Diagram , 2019, UAI.

[33]  Constance de Koning,et al.  Editors , 2003, Annals of Emergency Medicine.

[34]  Thomas Schiex,et al.  Russian Doll Search for Solving Constraint Optimization Problems , 1996, AAAI/IAAI, Vol. 1.

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

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

[37]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[38]  Adnan Darwiche,et al.  Compiling Bayesian Networks Using Variable Elimination , 2007, IJCAI.

[39]  Rina Dechter,et al.  AND/OR Multi-Valued Decision Diagrams (AOMDDs) for Graphical Models , 2008, J. Artif. Intell. Res..

[40]  Rina Dechter,et al.  AND/OR Multi-Valued Decision Diagrams (AOMDDs) for Weighted Graphical Models , 2007, UAI.

[41]  Vibhav Gogate,et al.  Look Ma, No Latent Variables: Accurate Cutset Networks via Compilation , 2019, ICML.

[42]  Ah Chung Tsoi,et al.  The Graph Neural Network Model , 2009, IEEE Transactions on Neural Networks.

[43]  Lars Hertel,et al.  Sherpa: Robust Hyperparameter Optimization for Machine Learning , 2020, SoftwareX.

[44]  Daniel Tarlow,et al.  Learning to Pass Expectation Propagation Messages , 2013, NIPS.

[45]  Michele Lombardi,et al.  Injecting domain knowledge in neural networks: a controlled experiment on a constrained problem , 2020, NeHuAI@ECAI.

[46]  Pierre Baldi,et al.  Solving the Rubik’s cube with deep reinforcement learning and search , 2019, Nature Machine Intelligence.

[47]  WorkshopJohn LangfordO tober.  NIPS 2000 , 2000 .

[48]  Qiang Yang,et al.  A Survey on Transfer Learning , 2010, IEEE Transactions on Knowledge and Data Engineering.

[49]  Pedro M. Domingos,et al.  Structured Message Passing , 2013, UAI.

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

[51]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..