Bayesian Posterior Sampling via Stochastic Gradient Fisher Scoring, Proceedings of the 29th International Conference on Machine Learning, ICML 2012, 2012. ,
S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations, SIAM Journal on Scientific Computing, vol.30, pp.997-1014, 2008. ,
Zero-variance principle for Monte Carlo algorithms, Physical review letters, vol.83, p.4682, 1999. ,
S-ROCK methods for stiff Ito SDEs, Commun. Math. Sci, vol.6, pp.845-868, 2008. ,
An introduction to MCMC for machine learning, Machine learning, vol.50, pp.5-43, 2003. ,
Calculus: Multi Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, 1969. ,
A comparative study of Monte Carlo methods for efficient evaluation of marginal likelihood, Computational Statistics & Data Analysis, vol.56, pp.3398-3414, 2012. ,
Sampling normalizing constants in high dimensions using inhomogeneous diffusions, 2016. ,
Distributed Stochastic Gradient MCMC, Proceedings of the 31st International Conference on Machine Learning, vol.32, pp.1044-1052, 2014. ,
An Adaptive Version for the Metropolis Adjusted Langevin Algorithm with a Truncated Drift, Methodology and Computing in Applied Probability, vol.8, pp.1573-7713, 2006. ,
A simple proof of the Poincaré inequality for a large class of probability measures, Electronic Communications in Probability, vol.13, pp.60-66, 2008. ,
Control Variates for Stochastic Gradient MCMC, 2017. ,
, From microphysics to macrophysics: methods and applications of statistical physics, vol.1, 2007.
, On Markov chain
, Monte Carlo methods for tall data, Journal of Machine Learning Research, vol.18, pp.1-43, 2017.
Finite-time Analysis of Projected Langevin Monte Carlo, Proceedings of the 28th International Conference on Neural Information Processing Systems. NIPS'15, pp.1243-1251, 2015. ,
Error bounds and normalising constants for sequential Monte Carlo samplers in high dimensions, Adv. in Appl. Probab, vol.46, issue.1, pp.279-306, 2014. ,
The Fundamental Incompatibility of Scalable Hamiltonian Monte Carlo and Naive Data Subsampling, Proceedings of the 32nd International Conference on Machine Learning, vol.37, pp.533-540, 2015. ,
Coupling and Convergence for Hamiltonian Monte Carlo, 2018. ,
Tuning tempered transitions, Statistics and Computing, vol.22, issue.1, pp.65-78, 2012. ,
Analysis and geometry of Markov diffusion operators, Grundlehren der Mathematischen Wissenschaften, vol.348 ,
URL : https://hal.archives-ouvertes.fr/hal-00929960
, , 2014.
Nonasymptotic mixing of the MALA algorithm, IMA Journal of Numerical Analysis, vol.33, pp.80-110, 2013. ,
On Classical Limit Theorems for Diffusions, Sankhy?: The Indian Journal of Statistics, Series A, vol.44, p.581572, 1961. ,
Concentration inequalities: a nonasymptotic theory of independence, 2013. ,
URL : https://hal.archives-ouvertes.fr/hal-00794821
Latent dirichlet allocation, Journal of machine Learning research, vol.3, pp.993-1022, 2003. ,
Notes on the Implicit Function Theorem, p.2013 ,
, Geometry of isotropic convex bodies, vol.196, 2014.
The tamed unadjusted Langevin algorithm, Stochastic Processes and their Applications, pp.304-4149, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01648667
Randomized Hamiltonian Monte Carlo, Ann. Appl. Probab, vol.27, pp.2159-2194, 2017. ,
Geometric integrators and the Hamiltonian Monte Carlo method, Acta Numerica, vol.27, pp.113-206, 2018. ,
Mathematics in Science and Engineering. The discrete time case, Stochastic optimal control, vol.139, 1978. ,
Convex optimization, 2004. ,
Pathwise accuracy and ergodicity of metropolized integrators for SDEs, Communications on Pure and Applied Mathematics, vol.63, pp.1097-0312, 2010. ,
Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA 9, vol.2, pp.337-382, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00585271
Chernoff-Type Bounds for the Gaussian Error Function, IEEE Transactions on Communications, vol.59, issue.11, pp.90-6778, 2011. ,
On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators, Advances in Neural Information Processing Systems 28, pp.2278-2286, 2015. ,
Regularization in Regression: Comparing Bayesian and Frequentist Methods in a Poorly Informative Situation, Bayesian Anal, vol.7, issue.2, pp.477-502, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00943727
Stochastic gradient hamiltonian Monte Carlo, Proceedings of the 31st International Conference on Machine Learning, pp.1683-1691, 2014. ,
On the Theory of Variance Reduction for Stochastic Gradient Monte Carlo, 2018. ,
A Convergence Analysis for A Class of Practical Variance-Reduction Stochastic Gradient MCMC, 2017. ,
Underdamped Langevin MCMC: A non-asymptotic analysis, Proceedings of the 31st Conference On Learning Theory. Ed. by Sébastien Bubeck, Vianney Perchet, and Philippe Rigollet, vol.75, pp.300-323, 2018. ,
Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors, Statistics and Computing, vol.22, issue.4, pp.1573-1375, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00460914
MCMC methods for functions: modifying old algorithms to make them faster, In: Statist. Sci, vol.28, issue.3, pp.883-4237, 2013. ,
Monte Carlo methods in Bayesian computation, 2000. ,
Monte Carlo methods in Bayesian computation, 2012. ,
Computation of the volume of convex bodies, 2015. ,
Bypassing KLS: Gaussian cooling and an O * (n 3 ) volume algorithm, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pp.539-548, 2015. ,
Bayes model selection with path sampling: factor models and other examples, Statistical Science, vol.28, pp.95-115, 2013. ,
Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent, Proceedings of the 2017 Conference on Learning Theory. Ed. by Satyen Kale and Ohad Shamir, vol.65, pp.678-689, 2017. ,
Theoretical guarantees for approximate sampling from smooth and log-concave densities, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.79, pp.1467-9868, 2017. ,
Splitting for rare event simulation: A large deviation approach to design and analysis, Stochastic Processes and their Applications, vol.119, pp.304-4149, 2009. ,
Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01565514
Multilevel sequential Monte Carlo samplers for normalizing constants, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01593880
Genealogical and interacting particle systems with applications, Moral. Feynman-Kac formulae. Probability and its Applications, pp.0-387, 2004. ,
Computing the volume of convex bodies: a case where randomness provably helps, Probabilistic combinatorics and its applications, vol.44, pp.123-170, 1991. ,
Subgeometric rates of convergence of f-ergodic strong Markov processes, Stochastic Processes and their Applications, vol.119, pp.304-4149, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00077681
Geometric ergodicity of the bouncy particle sampler, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01839335
Bayesian Sampling Using Stochastic Gradient Thermostats, Proceedings of the 27th International Conference on Neural Information Processing Systems, vol.2, pp.3203-3211, 2014. ,
Control variates for estimation based on reversible Markov chain Monte Carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.74, pp.1467-9868, 2012. ,
User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient, 2017. ,
User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient, Stochastic Processes and their Applications, 2019. ,
Variance Reduction Using Nonreversible Langevin Samplers, Journal of Statistical Physics, vol.163, issue.3, pp.1572-9613, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01164466
High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-01304430
Nonasymptotic convergence analysis for the unadjusted Langevin algorithm, Ann. Appl. Probab, vol.27, issue.3, pp.1551-1587, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01176132
Analysis of Langevin Monte Carlo via convex optimization, 2018. ,
Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau, SIAM Journal on Imaging Sciences, vol.11, issue.1, pp.473-506, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01267115
On the convergence of Hamiltonian Monte Carlo, 2017. ,
Markov Chains, Springer Series in Operations Research and Financial Engineering, pp.978-3319977034, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-02022651
Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation, 2017. ,
On sampling from a log-concave density using kinetic Langevin diffusions, 2018. ,
Sparse regression learning by aggregation and Langevin Monte-Carlo, J. Comput. System Sci, vol.78, pp.1423-1443, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00362471
Hybrid Monte Carlo, Physics Letters B, vol.195, pp.370-2693, 1987. ,
Variance Reduction in Stochastic Gradient Langevin Dynamics, Advances in Neural Information Processing Systems, vol.29, pp.1154-1162, 2016. ,
Log-concave sampling: Metropolis-Hastings algorithms are fast, Proceedings of the 31st Conference On Learning Theory. Ed. by Sébastien Bubeck, Vianney Perchet, and Philippe Rigollet, vol.75, pp.793-797, 2018. ,
Measure theory and fine properties of functions, 2015. ,
Couplings and quantitative contraction rates for Langevin dynamics, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01484275
Quantitative contraction rates for Markov chains on general state spaces, 2018. ,
Global Non-convex Optimization with Discretized Diffusions, Advances in Neural Information Processing Systems 31, pp.9671-9680, 2018. ,
A computer simulation of charged particles in solution. I. Technique and equilibrium properties, The Journal of Chemical Physics, vol.62, pp.4189-4196, 1975. ,
Error analysis of the transport properties of Metropolized schemes, ESAIM: Proc, vol.48, pp.341-363, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00952768
Improving power posterior estimation of statistical evidence, Statistics and Computing, vol.24, pp.1573-1375, 2014. ,
Batch means and spectral variance estimators in Markov chain Monte Carlo, Ann. Statist, vol.38, issue.2, pp.1034-1070, 2010. ,
Stochastic differential equations and applications. Courier Corporation, 2012. ,
Estimating the evidence-a review, Statistica Neerlandica, vol.66, pp.288-308, 2012. ,
On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces, Mathematische Nachrichten 147.1 (, pp.185-203 ,
Bayesian data analysis, vol.2 ,
Representations of knowledge in complex systems, J. Roy. Statist. Soc. Ser. B, vol.56, pp.35-9246, 1994. ,
A Liapounov bound for solutions of the Poisson equation, In: Ann. Probab, vol.24, issue.2, pp.916-931, 1996. ,
Simulating normalizing constants: From importance sampling to bridge sampling to path sampling, Statistical science, pp.163-185, 1998. ,
Measuring Sample Quality with Diffusions, 2016. ,
, Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products, 2014.
Tutorial in pattern theory, Division of Applied Mathematics, 1983. ,
Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling, Journal of the American Statistical Association, vol.87, pp.523-532, 1992. ,
Elliptic partial differential equations of second order, 2015. ,
Distributed Bayesian Learning with Stochastic Natural Gradient Expectation Propagation and the Posterior Server, Journal of Machine Learning Research, vol.18, pp.1-37, 2017. ,
Variance reduction via an approximating markov process, 1997. ,
Martingale limit theory and its application, 2014. ,
Accelerating diffusions, Ann. Appl. Probab, vol.15, issue.2, pp.1433-1444, 2005. ,
Accelerating Gaussian Diffusions, Ann. Appl. Probab, vol.3, issue.3, pp.897-913, 1993. ,
Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, vol.236, 1112. ,
Unbiased Hamiltonian Monte Carlo with couplings, 2017. ,
Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, pp.1364-5021, 2011. ,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab, vol.22, pp.1611-1641, 2012. ,
Yet another look at Harris' ergodic theorem for Markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI, vol.63, pp.109-117, 2011. ,
Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, SIAM Journal on Numerical Analysis, vol.40, pp.1041-1063, 2002. ,
Mirrored Langevin Dynamics, Advances in Neural Information Processing Systems 31, pp.2878-2887, 2018. ,
Approximation algorithms for the normalizing constant of Gibbs distributions, Ann. Appl. Probab, vol.25, issue.2, pp.974-985, 2015. ,
, Stochastic Differential Equations and Diffusion Processes. North-Holland Mathematical Library, 1989.
SciPy: Open source scientific tools for Python, 2001. ,
Ordinal data modeling, 2006. ,
Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Physical Review E, vol.56, p.5018, 1997. ,
Multilevel particle filters: normalizing constant estimation, Statistics and Computing, pp.1573-1375, 2016. ,
Markov chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling, In: Statist. Sci, vol.20, issue.1, pp.50-67, 2005. ,
Curvature, concentration and error estimates for Markov chain Monte Carlo, Ann. Probab, vol.38, issue.6, pp.2418-2442, 2010. ,
Random generation of combinatorial structures from a uniform distribution, Theoretical Computer Science, vol.43, pp.304-3975, 1986. ,
On weighted parallel volumes, Beiträge Algebra Geom, vol.50, pp.495-519, 2009. ,
Austerity in MCMC Land: cutting the Metropolis-hastings Budget, Proceedings of the 31st International Conference on International Conference on Machine Learning, vol.32, pp.181-189, 2014. ,
Survival analysis: techniques for censored and truncated data, 2005. ,
Special Issue in Honour of William J. (Bill) Fitzgerald, Digital Signal Processing, vol.47, pp.1051-2004, 2015. ,
Weak backward error analysis for overdamped Langevin processes, IMA Journal of Numerical Analysis, vol.35, pp.583-614, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00905684
Introduction to geometric probability, 1997. ,
Brownian Motion and Stochastic Calculus ,
, Graduate Texts in Mathematics, p.9780387976556, 1991.
Information theory and statistics. Reprint of the second, pp.0-486, 1968. ,
Scalable MCMC for mixed membership stochastic blockmodels, Artificial Intelligence and Statistics, pp.723-731, 2016. ,
Kinetic energy choice in Hamiltonian/hybrid Monte Carlo, 2017. ,
Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks, Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence. AAAI'16, pp.1788-1794, 2016. ,
Monte Carlo strategies in scientific computing, 2008. ,
An adaptive Euler-Maruyama scheme for SDEs: convergence and stability, IMA Journal of Numerical Analysis, vol.27, pp.479-506, 2007. ,
The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics, IMA Journal of Numerical Analysis, vol.36, issue.1, p.13, 2016. ,
URL : https://hal.archives-ouvertes.fr/hal-00854791
Markov chains and mixing times, vol.107, 2017. ,
, Statistics of random Processes: I. general Theory, vol.5, 2013.
Sampling constrained probability distributions using Spherical Augmentation, 2015. ,
Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, vol.25, pp.681-880, 2016. ,
The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume, p.31, 1990. ,
, Annual Symposium on Foundations of Computer Science, vol.1, pp.346-354, 1990.
Random walks in a convex body and an improved volume algorithm, Random structures & algorithms, vol.4, pp.359-412, 1993. ,
Free energy computations: A mathematical perspective, 2010. ,
Hit-and-Run from a Corner, SIAM Journal on Computing, vol.35, issue.4, pp.985-1005, 2006. ,
The Geometry of Logconcave Functions and Sampling Algorithms, Random Struct, vol.30, pp.1042-9832, 2007. ,
A Complete Recipe for Stochastic Gradient MCMC, Advances in Neural Information Processing Systems 28, pp.2917-2925, 2015. ,
Sequential Monte Carlo Samplers, Journal of the Royal Statistical Society. Series B (Statistical Methodology), vol.68, issue.3, p.13697412, 2006. ,
URL : https://hal.archives-ouvertes.fr/hal-01593880
Control techniques for complex networks, 2008. ,
Bayesian core: a practical approach to computational Bayesian statistics, 2007. ,
URL : https://hal.archives-ouvertes.fr/hal-00450489
Importance sampling methods for Bayesian discrimination between embedded models, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00424475
Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions, 2017. ,
Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, In: Stochastic Process. Appl, vol.101, issue.2, pp.304-4149, 2002. ,
Zero variance Markov chain Monte Carlo for Bayesian estimators, Statistics and Computing, vol.23, pp.1573-1375, 2013. ,
Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations, SIAM Journal on Numerical Analysis, vol.48, pp.552-577, 2010. ,
Markov Chains and Stochastic Stability. 2nd, p.9780521731829, 2009. ,
Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab, vol.25, pp.1-8678, 1993. ,
On the Poisson equation for MetropolisHastings chains, 2015. ,
Asymptotic variance for Random Walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation, 2017. ,
Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo, Advances in Neural Information Processing Systems 31, pp.6027-6037, 2018. ,
The True Cost of Stochastic Gradient Langevin Dynamics, 2017. ,
Annealed importance sampling, Statistics and Computing, vol.11, pp.1573-1375, 2001. ,
Robust Stochastic Approximation Approach to Stochastic Programming, SIAM Journal on Optimization, vol.19, pp.1574-1609, 2009. ,
URL : https://hal.archives-ouvertes.fr/hal-00976649
Introductory lectures on convex optimization: A basic course, vol.87, 2013. ,
cubature: Adaptive Multivariate Integration over Hypercubes. R package version, vol.1, pp.3-6, 2016. ,
Fixed precision MCMC estimation by median of products of averages, J. Appl. Probab, vol.46, issue.2, pp.309-329, 2009. ,
Convergence Rates for a Class of Estimators Based on Stein's Method, Accepted in Bernoulli, 2018. ,
Control Functionals for Quasi-Monte Carlo Integration, Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, vol.51, pp.56-65, 2016. ,
Control functionals for Monte Carlo integration, Journal of the Royal Statistical Society: Series B (Statistical Methodology, pp.1467-9868, 2016. ,
The Controlled Thermodynamic Integral for Bayesian Model Evidence Evaluation, Journal of the American Statistical Association, vol.111, pp.634-645, 2016. ,
Correlation functions and computer simulations, Nuclear Physics B, vol.180, pp.378-384, 1981. ,
Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, vol.60, 2014. ,
Bayesian Nonnegative Matrix Factorization with Stochastic Variational Inference, Handbook of Mixed Membership Models and Their Applications, pp.205-224, 2014. ,
The Bayesian lasso, J. Amer. Statist. Assoc, vol.103, pp.162-1459, 2008. ,
Scikit-learn: Machine Learning in Python, Journal of Machine Learning Research, vol.12, pp.2825-2830, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00650905
Maximum-a-posteriori estimation with Bayesian confidence regions, 2016. ,
Zero Variance Differential Geometric Markov Chain Monte Carlo Algorithms, Bayesian Anal, vol.9, issue.1, pp.97-128, 2014. ,
Exact hamiltonian monte carlo for truncated multivariate gaussians, Journal of Computational and Graphical Statistics, vol.23, pp.518-542, 2014. ,
Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability, Journal of Multivariate Analysis, vol.6, pp.47-259, 1976. ,
Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex, Advances in Neural Information Processing Systems 26, pp.3102-3110, 2013. ,
On the Poisson Equation and Diffusion Approximation. I, Ann. Probab, vol.29, issue.3, pp.1061-1085, 2001. ,
R: a language and environment for statistical computing. R Foundation for Statistical Computing, 2018. ,
Monte Carlo statistical methods. Second. Springer Texts in Statistics, pp.0-387, 2004. ,
Brownian dynamics as smart Monte Carlo simulation, The Journal of Chemical Physics, vol.69, pp.4628-4633, 1978. ,
Efficient Gibbs sampling of truncated multivariate normal with application to constrained linear regression, Unpublished manuscript, 2004. ,
On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion), In: Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol.59, pp.1467-9868, 1997. ,
Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Applied Prob, vol.7, pp.110-120, 1997. ,
Simulation and the Monte Carlo method, Wiley Series in Probability and Statistics ,
The Bayesian choice: from decision-theoretic foundations to computational implementation, 2007. ,
, Convex analysis, 2015.
Optimal Scaling of Discrete Approximations to Langevin Diffusions, Journal of the Royal Statistical Society. Series B (Statistical Methodology), vol.60, issue.1, p.13697412, 1998. ,
Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis, Proceedings of the 2017 Conference on Learning Theory, vol.65, pp.1674-1703, 2017. ,
Irreversible Langevin samplers and variance reduction: a large deviations approach, Nonlinearity, vol.28, issue.7, pp.2081-2103, 2015. ,
Variance reduction for irreversible Langevin samplers and diffusion on graphs, Electron. Commun. Probab, vol.20, p.pp, 2015. ,
A perturbative approach to control variates in molecular dynamics, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01709265
Exponential convergence of Langevin distributions and their discrete approximations, vol.2, pp.1350-7265, 1996. ,
Variational analysis, Grundlehren der Mathematischen Wissenschaften, vol.317 ,
, , pp.3-540, 1998.
A note on tamed Euler approximations, In: Electron. Commun. Probab, vol.18, p.pp, 2013. ,
Convex bodies: the Brunn-Minkowski theory, vol.151, 2013. ,
, Conformational Dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules, 1999.
Approximation Analysis of Stochastic Gradient Langevin Dynamics by using Fokker-Planck Equation and Ito Process, Proceedings of the 31st International Conference on Machine Learning, vol.32, pp.982-990, 2014. ,
Multidimensional diffusion processes, 2007. ,
Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Analysis and Applications, vol.8, pp.483-509, 1990. ,
URL : https://hal.archives-ouvertes.fr/inria-00075490
Consistency and fluctuations for stochastic gradient Langevin dynamics, The Journal of Machine Learning Research, vol.17, pp.193-225, 2016. ,
Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation, 2017. ,
Monte Carlo Estimation of the Free Energy by Multistage Sampling, The Journal of Chemical Physics, vol.57, pp.5457-5462, 1972. ,
Optimal transport : old and new. Grundlehren der mathematischen Wissenschaften, pp.978-981, 2009. ,
Exploration of the (Non-)Asymptotic Bias and Variance of Stochastic Gradient Langevin Dynamics, Journal of Machine Learning Research, vol.17, pp.1-48, 2016. ,
Attaining the Optimal Gaussian Diffusion Acceleration, Journal of Statistical Physics, vol.155, issue.3, pp.1572-9613, 2014. ,
Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem, Proceedings of the 31st Conference On Learning Theory. Ed. by Sébastien Bubeck, Vianney Perchet, and Philippe Rigollet, vol.75, pp.2093-3027, 2018. ,
Bayesian Learning via Stochastic Gradient Langevin Dynamics, Proceedings of the 28th International Conference on International Conference on Machine Learning. ICML'11, pp.978-979, 2011. ,
Regression analysis, vol.14, 1959. ,
Estimating the statistical evidence -a review, 2011. ,
Towards automatic model comparison: an adaptive sequential Monte Carlo approach, Journal of Computational and Graphical Statistics just-accepted, 2015. ,
A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics, Proceedings of the 2017 Conference on Learning Theory. Ed. by Satyen Kale and Ohad Shamir, vol.65, pp.1980-2022, 2017. ,
Co-Coercivity and Its Role In the Convergence of Iterative Schemes For Solving Variational Inequalities, SIAM J. on Optimization, vol.6, issue.3, pp.1052-6234, 1996. ,
,
M ? 1} in the example of a logistic regression. The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i (g i ), p.20 ,
, 46 3.2 Computation of the volume of the cube with MYULA and hit-and-run algorithm
, Boxplots of ? 1 , ? 2 , ? 3 for the truncated Gaussian variable in dimension 10, p.53
, Boxplots of ? 1 , ? 2 , ? 3 for the truncated Gaussian variable in dimension 100. 54 3.5 Lasso path for the Gibbs sampler, Wall HMC and MYULA algorithms, p.55
, Boxplots of the error on the first moment for the multivariate Gaussian (first coordinate) in dimension 1000 starting at 0 for different step sizes, p.74
, Boxplots of the error on the first moment for the double well in dimension 100 starting at (100, 0 ?99 ) for different step sizes
, Boxplots of the error on the second moment for the double well in dimension 100 starting at 0 for different step sizes
, Boxplots of the error on the first moment for the Ginzburg-Landau model in dimension 1000 starting at (100, 0 ?999 ) for different step sizes, p.77
, Boxplots of the error for TULAc on the first and second moments for the badly conditioned Gaussian variable in dimension 100 starting at 0 for different step sizes
Boxplots of the logarithm of the normalizing constants of a multivariate Gaussian distribution in dimension d ? {10, 25, 50}, p.113 ,
, 2 Boxplots of the log evidence for the two models on the Gaussian regression, vol.115
The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i, M ? 1} in the example of the Gaussian regression (model M 1 ), p.116 ,
, Boxplots of the log evidence for the two models on the logistic regression. The methods are the Laplace method (L)
The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i, M ? 1} in the example of the logistic regression (model M 1 ), p.118 ,
, Boxplot of the log evidence for the mixture of Gaussian distributions, p.119
, MALA and RWM algorithms for the logistic regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimators of [PMG14] using a first (ZV-1) or second (ZV-2) order polynomial bases. The plots in the second column are
, 177 the ULA, MALA and RWM algorithms for the logistic regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zerovariance estimator of [PMG14] using a first (ZV-1) or second (ZV-2) order polynomial basis
, x 3 using the ULA, MALA and RWM algorithms for the probit regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimator of, p.179
, 3 using the ULA, MALA and RWM algorithms for the probit regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimator of, p.180
, 7 in the asymptotic N ? +?.?,? SGD ,? LMC ,? FP and? SGLD are the means under the stationary distributions ?, ? SGD , ? LMC , ? FP and ? SGLD , respectively. The associated circles indicate the order of magnitude of the covariance matrix. While LMC and SGLDFP concentrate to the posterior mean ? with a covariance matrix of the order 1/N , SGLD and SGD are at a distance of order ? 1 of? and do not concentrate as N ? +?, Figure illustrating the definitions of cone(0, ? ? ), b(z ?1 ), c(z ?1 ) and ?(z ?1 ). 186 xx 7.1 Illustration of Proposition 7.5, Theorem 7.6 and Theorem 7
,
, Variance of the stochastic gradients of SGLD, SGLDFP and SGD function of N , in logarithmic scale
, Negative loglikelihood on the test dataset for SGLD, SGLDFP and SGD function of the number of iterations for different values of N ? 10 3, vol.10