Self-Avoiding Random Dynamics on Integer Complex Systems

This article introduces a new specialized algorithm for equilibrium Monte Carlo sampling of binary-valued systems, which allows for large moves in the state space. This is achieved by constructing self-avoiding walks (SAWs) in the state space. As a consequence, many bits are flipped in a single MCMC step. We name the algorithm SARDONICS, an acronym for Self-Avoiding Random Dynamics on Integer Complex Systems. The algorithm has several free parameters, but we show that Bayesian optimization can be used to automatically tune them. SARDONICS performs remarkably well in a broad number of sampling tasks: toroidal ferromagnetic and frustrated Ising models, 3D Ising models, restricted Boltzmann machines and chimera graphs arising in the design of quantum computers.

[1]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[2]  B. Berg,et al.  Multicanonical algorithms for first order phase transitions , 1991 .

[3]  Nando de Freitas,et al.  Bayesian optimization for adaptive MCMC , 2011, 1110.6497.

[4]  Geoffrey E. Hinton,et al.  Reducing the Dimensionality of Data with Neural Networks , 2006, Science.

[5]  Jun S. Liu,et al.  Bayesian Clustering with Variable and Transformation Selections , 2003 .

[6]  Donald R. Jones,et al.  Global versus local search in constrained optimization of computer models , 1998 .

[7]  Nando de Freitas,et al.  Large-Flip Importance Sampling , 2007, UAI.

[8]  Nando de Freitas,et al.  A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning , 2010, ArXiv.

[9]  Gareth O. Roberts,et al.  Examples of Adaptive MCMC , 2009 .

[10]  Nando de Freitas,et al.  Intracluster Moves for Constrained Discrete-Space MCMC , 2010, UAI.

[11]  Nando de Freitas,et al.  Hot Coupling: A Particle Approach to Inference and Normalization on Pairwise Undirected Graphs , 2005, NIPS.

[12]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[13]  Michael I. Jordan,et al.  Latent Dirichlet Allocation , 2001, J. Mach. Learn. Res..

[14]  Naomichi Hatano,et al.  The multicanonical Monte Carlo method , 2000, Computing in Science & Engineering.

[15]  James L. McClelland,et al.  Parallel distributed processing: explorations in the microstructure of cognition, vol. 1: foundations , 1986 .

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

[17]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Andreas Krause,et al.  Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting , 2009, IEEE Transactions on Information Theory.

[19]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[20]  Nando de Freitas,et al.  A tutorial on stochastic approximation algorithms for training Restricted Boltzmann Machines and Deep Belief Nets , 2010, 2010 Information Theory and Applications Workshop (ITA).

[21]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[22]  Nando de Freitas,et al.  Inductive Principles for Restricted Boltzmann Machine Learning , 2010, AISTATS.

[23]  Nando de Freitas,et al.  Adaptive MCMC with Bayesian Optimization , 2012, AISTATS.

[24]  S J Mitchell,et al.  Rejection-free Monte Carlo algorithms for models with continuous degrees of freedom. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .

[26]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[27]  Honglak Lee,et al.  Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations , 2009, ICML '09.

[28]  D. Lizotte.  Practical bayesian optimization , 2008 .

[29]  Ruslan Salakhutdinov,et al.  On the quantitative analysis of deep belief networks , 2008, ICML '08.

[30]  Daan Frenkel,et al.  Configurational bias Monte Carlo: a new sampling scheme for flexible chains , 1992 .

[31]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[32]  Christian P. Robert,et al.  Monte Carlo Statistical Methods (Springer Texts in Statistics) , 2005 .

[33]  David S. Leslie,et al.  Optimistic Bayesian Sampling in Contextual-Bandit Problems , 2012, J. Mach. Learn. Res..

[34]  J. Mockus,et al.  The Bayesian approach to global optimization , 1989 .

[35]  Adam D. Bull,et al.  Convergence Rates of Efficient Global Optimization Algorithms , 2011, J. Mach. Learn. Res..

[36]  Geoffrey E. Hinton,et al.  Generating more realistic images using gated MRF's , 2010, NIPS.

[37]  HamzeFiras,et al.  Self-Avoiding Random Dynamics on Integer Complex Systems , 2013 .

[38]  Kenny Q. Ye.  Orthogonal Column Latin Hypercubes and Their Application in Computer Experiments , 1998 .

[39]  D. Landau,et al.  Efficient, multiple-range random walk algorithm to calculate the density of states. , 2000, Physical review letters.

[40]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[41]  Fabián A. Chudak,et al.  The Ising model : teaching an old problem new tricks , 2010 .

[42]  A. W. Rosenbluth,et al.  MONTE CARLO CALCULATION OF THE AVERAGE EXTENSION OF MOLECULAR CHAINS , 1955 .

[43]  Nando de Freitas,et al.  From Fields to Trees , 2004, UAI.

[44]  Gerard T. Barkema,et al.  Monte Carlo Methods in Statistical Physics , 1999 .

[45]  Pierre Del Moral,et al.  Sequential Monte Carlo for rare event estimation , 2012, Stat. Comput..

[46]  C. Robert,et al.  Controlled MCMC for Optimal Sampling , 2001 .

[47]  Geoffrey E. Hinton,et al.  Learning to Represent Spatial Transformations with Factored Higher-Order Boltzmann Machines , 2010, Neural Computation.

[48]  S. Duane,et al.  Hybrid Monte Carlo , 1987 .

[49]  Robert B. Gramacy,et al.  Parameter space exploration with Gaussian process trees , 2004, ICML.

[50]  Michael W Deem,et al.  Parallel tempering: theory, applications, and new perspectives. , 2005, Physical chemistry chemical physics : PCCP.

[51]  D. Welsh.  Complexity: Knots, Colourings and Counting: Link polynomials and the Tait conjectures , 1993 .

[52]  Fred W. Glover,et al.  Tabu Search - Part I , 1989, INFORMS J. Comput..

[53]  Matti Vihola,et al.  Grapham: Graphical models with adaptive random walk Metropolis algorithms , 2008, Comput. Stat. Data Anal..

[54]  D. Finkel,et al.  Direct optimization algorithm user guide , 2003 .

[55]  F. Barahona.  On the computational complexity of Ising spin glass models , 1982 .

[56]  Nando de Freitas,et al.  Learning about Individuals from Group Statistics , 2005, UAI.

[57]  Geoffrey E. Hinton,et al.  How to generate realistic images using gated MRF ’ s , 2010 .

[58]  Nando de Freitas,et al.  Fast Krylov Methods for N-Body Learning , 2005, NIPS.

[59]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[60]  Paul Smolensky,et al.  Information processing in dynamical systems: foundations of harmony theory , 1986 .

[61]  Thomas Stützle,et al.  Stochastic Local Search: Foundations & Applications , 2004 .

[62]  Giorgio Parisi,et al.  Numerical Simulations of Spin Glass Systems , 1997 .

[63]  Dirk P. Kroese,et al.  Efficient Monte Carlo simulation via the generalized splitting method , 2012, Stat. Comput..

[64]  H. Banks.  Center for Research in Scientific Computationにおける研究活動 , 1999 .

[65]  C. Geyer.  Markov Chain Monte Carlo Maximum Likelihood , 1991 .

[66]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[67]  P. Diggle,et al.  Model‐based geostatistics , 2007 .

[68]  Geoffrey E. Hinton,et al.  A Learning Algorithm for Boltzmann Machines , 1985, Cogn. Sci..

[69]  Nando de Freitas,et al.  Portfolio Allocation for Bayesian Optimization , 2010, UAI.

[70]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[71]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[72]  Roman Garnett,et al.  Active Data Selection for Sensor Networks with Faults and Changepoints , 2010, 2010 24th IEEE International Conference on Advanced Information Networking and Applications.

[73]  Radford M. Neal.  MCMC Using Hamiltonian Dynamics , 2011, 1206.1901.

[74]  Kotagiri Ramamohanarao,et al.  Sparse Bayesian Learning for Regression and Classification using Markov Chain Monte Carlo , 2002, ICML.

[75]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.