Accelerated Sampling on Discrete Spaces with Non-Reversible Markov Processes

We consider the task of MCMC sampling from a distribution defined on a discrete space. Building on recent insights provided in [Zan19], we devise a class of efficient continuous-time, non-reversible algorithms which make active use of the structure of the underlying space. Particular emphasis is placed on how symmetries and other group-theoretic notions can be used to improve exploration of the space. We test our algorithms on a range of examples from statistics, computational physics, machine learning, and cryptography, which show improvement on alternative algorithms. We provide practical recommendations on how to design and implement these algorithms, and close with remarks on the outlook for both discrete sampling and continuous-time Monte Carlo more broadly.

[1]  Vijay V. Vazirani,et al.  Approximation algorithms for metric facility location and k-Median problems using the primal-dual schema and Lagrangian relaxation , 2001, JACM.

[2]  P. Peskun,et al.  Optimum Monte-Carlo sampling using Markov chains , 1973 .

[3]  Francis Bach,et al.  Submodular functions: from discrete to continuous domains , 2015, Mathematical Programming.

[4]  Craig Gentry,et al.  Trapdoors for hard lattices and new cryptographic constructions , 2008, IACR Cryptol. ePrint Arch..

[5]  Thomas B. Berrett,et al.  The conditional permutation test for independence while controlling for confounders , 2018, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[6]  Nicolas Chopin,et al.  Sequential Monte Carlo on large binary sampling spaces , 2011, Statistics and Computing.

[7]  Christian P. Robert,et al.  Coordinate sampler: a non-reversible Gibbs-like MCMC sampler , 2018, Stat. Comput..

[8]  Thomas B. Berrett,et al.  The conditional permutation test , 2018, 1807.05405.

[9]  Faster Monte Carlo simulations at low temperatures. The waiting time method , 2001, cond-mat/0107475.

[10]  A. B. Bortz,et al.  A new algorithm for Monte Carlo simulation of Ising spin systems , 1975 .

[11]  Nando de Freitas,et al.  Self-Avoiding Random Dynamics on Integer Complex Systems , 2011, TOMC.

[12]  Marina Vannucci,et al.  Gene selection: a Bayesian variable selection approach , 2003, Bioinform..

[13]  Giacomo Zanella,et al.  Informed Proposals for Local MCMC in Discrete Spaces , 2017, Journal of the American Statistical Association.

[14]  Radford M. Neal,et al.  ANALYSIS OF A NONREVERSIBLE MARKOV CHAIN SAMPLER , 2000 .

[15]  T. J. Mitchell,et al.  Bayesian Variable Selection in Linear Regression , 1988 .

[16]  Mark C. Fuhs,et al.  A Spin Glass Model of Path Integration in Rat Medial Entorhinal Cortex , 2006, The Journal of Neuroscience.

[17]  E. George,et al.  APPROACHES FOR BAYESIAN VARIABLE SELECTION , 1997 .

[18]  I-Cheng Yeh,et al.  Modeling of strength of high-performance concrete using artificial neural networks , 1998 .

[19]  P. Wolynes,et al.  Spin glasses and the statistical mechanics of protein folding. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Sudipto Guha,et al.  Improved combinatorial algorithms for the facility location and k-median problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[21]  Carlo Baldassi.  A method to reduce the rejection rate in Monte Carlo Markov Chains on Ising spin models , 2016 .

[22]  Manuel Laguna,et al.  Tabu Search , 1997 .

[23]  Fang Chen,et al.  Lifting Markov chains to speed up mixing , 1999, STOC '99.

[24]  A. Doucet,et al.  Piecewise-Deterministic Markov Chain Monte Carlo , 2017, 1707.05296.

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

[26]  Folding thermodynamics and kinetics of imprinted renaturable heteropolymers , 1994 .

[27]  Paul Krugman,et al.  Complex landscapes in economic geography , 1994 .

[28]  János Folláth.  Gaussian Sampling in Lattice Based Cryptography , 2014 .

[29]  Claudio Rebbi,et al.  Lattice Gauge Theories And Monte Carlo Simulations , 1983 .

[30]  Liam Paninski,et al.  Auxiliary-variable Exact Hamiltonian Monte Carlo Samplers for Binary Distributions , 2013, NIPS.

[31]  Oliver Johnson.  A discrete log-Sobolev inequality under a Bakry-Emery type condition , 2015, ArXiv.

[32]  John O'Leary,et al.  Unbiased Markov chain Monte Carlo with couplings , 2017, 1708.03625.

[33]  Ayesha Khalid,et al.  On Practical Discrete Gaussian Samplers for Lattice-Based Cryptography , 2018, IEEE Transactions on Computers.

[34]  G. Casella,et al.  Rao-Blackwellisation of sampling schemes , 1996 .

[35]  Christopher Yau,et al.  The Hamming Ball Sampler , 2015, Journal of the American Statistical Association.

[36]  E. Vanden-Eijnden,et al.  Applied Stochastic Analysis , 2019, Graduate Studies in Mathematics.

[37]  Ben Taskar,et al.  Determinantal Point Processes for Machine Learning , 2012, Found. Trends Mach. Learn..

[38]  Christian P. Robert,et al.  A vanilla Rao--Blackwellization of Metropolis--Hastings algorithms , 2009, 0904.2144.

[39]  Marija Vucelja,et al.  Lifting -- A nonreversible Markov chain Monte Carlo Algorithm , 2014, 1412.8762.

[40]  Cong Ling,et al.  Achieving AWGN Channel Capacity With Lattice Gaussian Coding , 2014, IEEE Transactions on Information Theory.

[41]  M. Stephens.  Bayesian analysis of mixture models with an unknown number of components- an alternative to reversible jump methods , 2000 .

[42]  O. Cappé,et al.  Reversible jump, birth‐and‐death and more general continuous time Markov chain Monte Carlo samplers , 2003 .

[43]  Giacomo Zanella,et al.  The Barker proposal: combining robustness and efficiency in gradient-based MCMC , 2019 .

[44]  P. Fearnhead,et al.  The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data , 2016, The Annals of Statistics.

[45]  Léo Ducas,et al.  Lattice Signatures and Bimodal Gaussians , 2013, IACR Cryptol. ePrint Arch..

[46]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[47]  Michael Chertkov,et al.  Irreversible Monte Carlo Algorithms for Efficient Sampling , 2008, ArXiv.