Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity
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Martin J. Wainwright | Peter L. Bartlett | Nicolas Flammarion | Wenlong Mou | P. Bartlett | M. Wainwright | Nicolas Flammarion | Wenlong Mou
[1] A. Bovier,et al. Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times , 2004 .
[2] Lester W. Mackey,et al. Measuring Sample Quality with Diffusions , 2016, The Annals of Applied Probability.
[3] A. Eberle,et al. Coupling and convergence for Hamiltonian Monte Carlo , 2018, The Annals of Applied Probability.
[4] C. Villani,et al. Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .
[5] Hyunjoong Kim,et al. Functional Analysis I , 2017 .
[6] Santosh S. Vempala,et al. Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices , 2019, NeurIPS.
[7] Jian Peng,et al. Accelerating Nonconvex Learning via Replica Exchange Langevin diffusion , 2019, ICLR.
[8] Michael I. Jordan,et al. Sampling can be faster than optimization , 2018, Proceedings of the National Academy of Sciences.
[9] Maxim Raginsky,et al. Local Optimality and Generalization Guarantees for the Langevin Algorithm via Empirical Metastability , 2018, COLT.
[10] D. Bakry,et al. A simple proof of the Poincaré inequality for a large class of probability measures , 2008 .
[11] Martin J. Wainwright,et al. Log-concave sampling: Metropolis-Hastings algorithms are fast! , 2018, COLT.
[12] Andre Wibisono,et al. Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem , 2018, COLT.
[13] Alain Durmus,et al. High-dimensional Bayesian inference via the unadjusted Langevin algorithm , 2016, Bernoulli.
[14] Alain Durmus,et al. Convergence of diffusions and their discretizations: from continuous to discrete processes and back , 2019, 1904.09808.
[15] A. J. Stam. Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon , 1959, Inf. Control..
[16] B. Davis,et al. Integral Inequalities for Convex Functions of Operators on Martingales , 2011 .
[17] É. Moulines,et al. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm , 2015, 1507.05021.
[18] R. Tweedie,et al. Exponential convergence of Langevin distributions and their discrete approximations , 1996 .
[19] M. V. Tretyakov,et al. Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.
[20] C. Villani,et al. Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities , 2005 .
[21] Arnak S. Dalalyan,et al. On sampling from a log-concave density using kinetic Langevin diffusions , 2018, Bernoulli.
[22] Peter L. Bartlett,et al. Convergence of Langevin MCMC in KL-divergence , 2017, ALT.
[23] E. Vanden-Eijnden,et al. Pathwise accuracy and ergodicity of metropolized integrators for SDEs , 2009, 0905.4218.
[24] D. Stroock,et al. Logarithmic Sobolev inequalities and stochastic Ising models , 1987 .
[25] B. Jourdain,et al. Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme , 2014, 1405.7007.
[26] Santosh S. Vempala,et al. Algorithmic Theory of ODEs and Sampling from Well-conditioned Logconcave Densities , 2018, ArXiv.
[27] M. Yor,et al. Continuous martingales and Brownian motion , 1990 .
[28] D. Talay,et al. Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .
[29] Giuseppe Toscani,et al. Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation , 1999 .
[30] Desmond J. Higham,et al. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..
[31] Andrej Risteski,et al. Beyond Log-concavity: Provable Guarantees for Sampling Multi-modal Distributions using Simulated Tempering Langevin Monte Carlo , 2017, NeurIPS.
[32] A. Veretennikov,et al. On polynomial mixing bounds for stochastic differential equations , 1997 .
[33] Michael B. Giles,et al. Multilevel Monte Carlo method for ergodic SDEs without contractivity , 2018, Journal of Mathematical Analysis and Applications.
[34] Denis Talay. Simulation and numerical analysis of stochastic differential systems : a review , 1990 .
[35] Nisheeth K. Vishnoi,et al. Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo , 2018, NeurIPS.
[36] Mateusz B. Majka,et al. Quantitative contraction rates for Markov chains on general state spaces , 2018, Electronic Journal of Probability.
[37] Matus Telgarsky,et al. Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis , 2017, COLT.
[38] Thomas M. Cover,et al. Elements of Information Theory , 2005 .
[39] Jonathan C. Mattingly,et al. Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , 2002 .
[40] A. Dalalyan. Theoretical guarantees for approximate sampling from smooth and log‐concave densities , 2014, 1412.7392.
[41] Michael I. Jordan,et al. Sharp Convergence Rates for Langevin Dynamics in the Nonconvex Setting , 2018, ArXiv.
[42] C. Villani,et al. ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS , 2004 .
[43] A. Eberle. Couplings, distances and contractivity for diffusion processes revisited , 2013 .
[44] Michael I. Jordan,et al. Underdamped Langevin MCMC: A non-asymptotic analysis , 2017, COLT.
[45] Andrew M. Stuart,et al. Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations , 2009, SIAM J. Numer. Anal..
[46] Mateusz B. Majka,et al. Nonasymptotic bounds for sampling algorithms without log-concavity , 2018, The Annals of Applied Probability.
[47] A. Iserles. A First Course in the Numerical Analysis of Differential Equations: Gaussian elimination for sparse linear equations , 2008 .
[48] Tengyuan Liang,et al. Statistical inference for the population landscape via moment‐adjusted stochastic gradients , 2017, Journal of the Royal Statistical Society: Series B (Statistical Methodology).
[49] M. Talagrand. A new isoperimetric inequality and the concentration of measure phenomenon , 1991 .
[50] Yuchen Zhang,et al. A Hitting Time Analysis of Stochastic Gradient Langevin Dynamics , 2017, COLT.
[51] Ohad Shamir,et al. Global Non-convex Optimization with Discretized Diffusions , 2018, NeurIPS.
[52] M. Ledoux,et al. Logarithmic Sobolev Inequalities , 2014 .
[53] T. Faniran. Numerical Solution of Stochastic Differential Equations , 2015 .
[54] G. Pavliotis. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations , 2014 .
[55] Arnak S. Dalalyan,et al. User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient , 2017, Stochastic Processes and their Applications.
[56] R. Khasminskii. Stochastic Stability of Differential Equations , 1980 .
[57] Arnak S. Dalalyan,et al. Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets , 2019, J. Mach. Learn. Res..
[58] Espen Bernton,et al. Langevin Monte Carlo and JKO splitting , 2018, COLT.
[59] E. Vanden-Eijnden,et al. Non-asymptotic mixing of the MALA algorithm , 2010, 1008.3514.