We present a coupling framework to upper bound the total variation mixing time of various Metropolis-adjusted, gradient-based Markov kernels in the ‘high acceptance regime’. The approach uses a localization argument to boost local mixing of the underlying unadjusted kernel to mixing of the adjusted kernel when the acceptance rate is suitably high. As an application, mixing time guarantees are developed for a non-reversible, adjusted Markov chain based on the kinetic Langevin diffusion, where little is currently understood.
References
J. M. Altschuler and S. Chewi, Faster high-accuracy log-concave sampling via algorithmic warm starts, arXiv preprint arXiv: 2302.10249 (2023). MR4720376J. M. Altschuler and S. Chewi, Faster high-accuracy log-concave sampling via algorithmic warm starts, arXiv preprint arXiv: 2302.10249 (2023). MR4720376
C. Andrieu, A. Lee, and S. Livingstone, A general perspective on the metropolis-hastings kernel, arXiv preprint arXiv: 2012.14881 (2020).C. Andrieu, A. Lee, and S. Livingstone, A general perspective on the metropolis-hastings kernel, arXiv preprint arXiv: 2012.14881 (2020).
A. Beskos, F. J. Pinski, J. M. Sanz-Serna, and A. M. Stuart, Hybrid Monte-Carlo on Hilbert spaces, Stochastic Processes and their Applications 121 (2011), no. 10, 2201–2230. MR2822774A. Beskos, F. J. Pinski, J. M. Sanz-Serna, and A. M. Stuart, Hybrid Monte-Carlo on Hilbert spaces, Stochastic Processes and their Applications 121 (2011), no. 10, 2201–2230. MR2822774
L. J. Billera and P. Diaconis, A Geometric Interpretation of the Metropolis-Hastings Algorithm, Statistical Science 16 (2001), no. 4, 335 – 339. MR1888448L. J. Billera and P. Diaconis, A Geometric Interpretation of the Metropolis-Hastings Algorithm, Statistical Science 16 (2001), no. 4, 335 – 339. MR1888448
N. Bou-Rabee and A. Eberle, Two-scale coupling for preconditioned hamiltonian monte carlo in infinite dimensions, Stochastics and Partial Differential Equations: Analysis and Computations 9 (2021), no. 1, 207–242. MR4218791N. Bou-Rabee and A. Eberle, Two-scale coupling for preconditioned hamiltonian monte carlo in infinite dimensions, Stochastics and Partial Differential Equations: Analysis and Computations 9 (2021), no. 1, 207–242. MR4218791
N. Bou-Rabee and A. Eberle, Couplings for Andersen Dynamics in High Dimension, Ann. Inst. H. Poincaré Probab. Statist 58 (2022), no. 2, 916–944. MR4421613N. Bou-Rabee and A. Eberle, Couplings for Andersen Dynamics in High Dimension, Ann. Inst. H. Poincaré Probab. Statist 58 (2022), no. 2, 916–944. MR4421613
N. Bou-Rabee, A. Eberle, and R. Zimmer, Coupling and convergence for hamiltonian monte carlo, Ann. Appl. Probab. 30 (2020), no. 3, 1209–1250. MR4133372N. Bou-Rabee, A. Eberle, and R. Zimmer, Coupling and convergence for hamiltonian monte carlo, Ann. Appl. Probab. 30 (2020), no. 3, 1209–1250. MR4133372
N. Bou-Rabee and K. Schuh, Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models, Electronic Journal of Probability 28 (2023), no. none, 1 – 40. MR4610714N. Bou-Rabee and K. Schuh, Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models, Electronic Journal of Probability 28 (2023), no. none, 1 – 40. MR4610714
N. Bou-Rabee and E. Vanden-Eijnden, Pathwise accuracy and ergodicity of Metropolized integrators for SDEs, Comm Pure and Appl Math 63 (2010), 655–696. MR2583309N. Bou-Rabee and E. Vanden-Eijnden, Pathwise accuracy and ergodicity of Metropolized integrators for SDEs, Comm Pure and Appl Math 63 (2010), 655–696. MR2583309
O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, The Annals of Applied Probability 24 (2014), no. 2, 526 – 552. MR3178490O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, The Annals of Applied Probability 24 (2014), no. 2, 526 – 552. MR3178490
E. Camrud, A. Durmus, P. Monmarché, and G. Stoltz, Second order quantitative bounds for unadjusted generalized hamiltonian monte carlo, arXiv preprint arXiv: 2306.09513 (2023).E. Camrud, A. Durmus, P. Monmarché, and G. Stoltz, Second order quantitative bounds for unadjusted generalized hamiltonian monte carlo, arXiv preprint arXiv: 2306.09513 (2023).
Yu Cao, Jianfeng Lu, and Lihan Wang, On explicit -convergence rate estimate for underdamped langevin dynamics, Archive for Rational Mechanics and Analysis 247 (2023), 90. MR4632836Yu Cao, Jianfeng Lu, and Lihan Wang, On explicit -convergence rate estimate for underdamped langevin dynamics, Archive for Rational Mechanics and Analysis 247 (2023), 90. MR4632836
Y. Chen, R. Dwivedi, M. J. Wainwright, and B. Yu, Fast mixing of metropolized hamiltonian monte carlo: Benefits of multi-step gradients, Journal of Machine Learning Research 21 (2020), no. 92, 1–72. MR4119160Y. Chen, R. Dwivedi, M. J. Wainwright, and B. Yu, Fast mixing of metropolized hamiltonian monte carlo: Benefits of multi-step gradients, Journal of Machine Learning Research 21 (2020), no. 92, 1–72. MR4119160
Z. Chen and K. Gatmiry, When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?, arXiv preprint arXiv: 2304.04724 (2023).Z. Chen and K. Gatmiry, When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?, arXiv preprint arXiv: 2304.04724 (2023).
X. Cheng, N. S. Chatterji, Y. Abbasi-Yadkori, P. L. Bartlett, and M. I. Jordan, Sharp convergence rates for langevin dynamics in the nonconvex setting, arXiv preprint arXiv: 1805.01648 (2018).X. Cheng, N. S. Chatterji, Y. Abbasi-Yadkori, P. L. Bartlett, and M. I. Jordan, Sharp convergence rates for langevin dynamics in the nonconvex setting, arXiv preprint arXiv: 1805.01648 (2018).
X. Cheng, N. S. Chatterji, P. L. Bartlett, and M. I. Jordan, Underdamped Langevin MCMC: A non-asymptotic analysis, Conference On Learning Theory, 2018, pp. 300–323.X. Cheng, N. S. Chatterji, P. L. Bartlett, and M. I. Jordan, Underdamped Langevin MCMC: A non-asymptotic analysis, Conference On Learning Theory, 2018, pp. 300–323.
S. Chewi, C. Lu, K. Ahn, X. Cheng, T. L. Gouic, and P. Rigollet, Optimal dimension dependence of the metropolis-adjusted langevin algorithm, arXiv preprint arXiv: 2012.12810 (2020).S. Chewi, C. Lu, K. Ahn, X. Cheng, T. L. Gouic, and P. Rigollet, Optimal dimension dependence of the metropolis-adjusted langevin algorithm, arXiv preprint arXiv: 2012.12810 (2020).
A. S. Dalalyan, Theoretical guarantees for approximate sampling from smooth and log-concave densities, Journal of the Royal Statistical Society Series B: Statistical Methodology 79 (2017), no. 3, 651–676. MR3641401A. S. Dalalyan, Theoretical guarantees for approximate sampling from smooth and log-concave densities, Journal of the Royal Statistical Society Series B: Statistical Methodology 79 (2017), no. 3, 651–676. MR3641401
A. S. Dalalyan and L. Riou-Durand, On sampling from a log-concave density using kinetic langevin diffusions, Bernoulli 26 (2020), no. 3, 1956–1988. MR4091098A. S. Dalalyan and L. Riou-Durand, On sampling from a log-concave density using kinetic langevin diffusions, Bernoulli 26 (2020), no. 3, 1956–1988. MR4091098
V. De Bortoli and A. Durmus, Convergence of diffusions and their discretizations: from continuous to discrete processes and back, arXiv preprint arXiv: 1904.09808 (2019).V. De Bortoli and A. Durmus, Convergence of diffusions and their discretizations: from continuous to discrete processes and back, arXiv preprint arXiv: 1904.09808 (2019).
G. Deligiannidis, D. Paulin, A. Bouchard-Côté, and A. Doucet, Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates, The Annals of Applied Probability 31 (2021), no. 6, 2612 – 2662. MR4350970G. Deligiannidis, D. Paulin, A. Bouchard-Côté, and A. Doucet, Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates, The Annals of Applied Probability 31 (2021), no. 6, 2612 – 2662. MR4350970
P. Diaconis and L. Saloff-Coste, What do we know about the metropolis algorithm?, Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, 1995, pp. 112–129.P. Diaconis and L. Saloff-Coste, What do we know about the metropolis algorithm?, Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, 1995, pp. 112–129.
A. Durmus and A. Eberle, Asymptotic bias of inexact markov chain monte carlo methods in high dimension, arXiv: 2108.00682 [math.PR], 2021.A. Durmus and A. Eberle, Asymptotic bias of inexact markov chain monte carlo methods in high dimension, arXiv: 2108.00682 [math.PR], 2021.
A. Durmus, G. Fort, and E. Moulines, Subgeometric rates of convergence in Wasserstein distance for Markov chains, Ann. Inst. H. Poincaré Probab. Statist 52 (2016), no. 4, 1799 – 1822. MR3573296A. Durmus, G. Fort, and E. Moulines, Subgeometric rates of convergence in Wasserstein distance for Markov chains, Ann. Inst. H. Poincaré Probab. Statist 52 (2016), no. 4, 1799 – 1822. MR3573296
A. Durmus and E. Moulines, Nonasymptotic convergence analysis for the unadjusted langevin algorithm, Annals of Applied Probability 27 (2017), no. 3, 1551–1587. MR3678479A. Durmus and E. Moulines, Nonasymptotic convergence analysis for the unadjusted langevin algorithm, Annals of Applied Probability 27 (2017), no. 3, 1551–1587. MR3678479
A. Durmus and E. Moulines, High-dimensional bayesian inference via the unadjusted langevin algorithm, Bernoulli 25 (2019), no. 4A, 2854–2882. MR4003567A. Durmus and E. Moulines, High-dimensional bayesian inference via the unadjusted langevin algorithm, Bernoulli 25 (2019), no. 4A, 2854–2882. MR4003567
A. Durmus and E. Moulines, On the geometric convergence for MALA under verifiable conditions, arXiv preprint arXiv: 2201.01951 (2022).A. Durmus and E. Moulines, On the geometric convergence for MALA under verifiable conditions, arXiv preprint arXiv: 2201.01951 (2022).
A. Durmus, E. Moulines, and E. Saksman, Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo, The Annals of Statistics 48 (2020), no. 6, 3545 – 3564. MR4185819A. Durmus, E. Moulines, and E. Saksman, Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo, The Annals of Statistics 48 (2020), no. 6, 3545 – 3564. MR4185819
A. Eberle, Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions, The Annals of Applied Probability 24 (2014), no. 1, 337 – 377. MR3161650A. Eberle, Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions, The Annals of Applied Probability 24 (2014), no. 1, 337 – 377. MR3161650
A. Eberle, A. Guillin, and R. Zimmer, Couplings and quantitative contraction rates for Langevin dynamics, Ann. Probab. 47 (2019), no. 4, 1982–2010. MR3980913A. Eberle, A. Guillin, and R. Zimmer, Couplings and quantitative contraction rates for Langevin dynamics, Ann. Probab. 47 (2019), no. 4, 1982–2010. MR3980913
A. Eberle, A. Guillin, and R. Zimmer, Quantitative harris-type theorems for diffusions and mckean–vlasov processes, Transactions of the American Mathematical Society 371 (2019), no. 10, 7135–7173. MR3939573A. Eberle, A. Guillin, and R. Zimmer, Quantitative harris-type theorems for diffusions and mckean–vlasov processes, Transactions of the American Mathematical Society 371 (2019), no. 10, 7135–7173. MR3939573
A. Eberle and M. B. Majka, Quantitative contraction rates for Markov chains on general state spaces, Electronic Journal of Probability 24 (2019), no. none, 1 – 36. MR3933205A. Eberle and M. B. Majka, Quantitative contraction rates for Markov chains on general state spaces, Electronic Journal of Probability 24 (2019), no. none, 1 – 36. MR3933205
M. A. Erdogdu and R. Hosseinzadeh, On the convergence of langevin monte carlo: The interplay between tail growth and smoothness, Conference on Learning Theory, PMLR, 2021, pp. 1776–1822.M. A. Erdogdu and R. Hosseinzadeh, On the convergence of langevin monte carlo: The interplay between tail growth and smoothness, Conference on Learning Theory, PMLR, 2021, pp. 1776–1822.
M. Hairer and J. C. Mattingly, Yet another look at harris’ ergodic theorem for markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI: Centro Stefano Franscini, Ascona, May 2008, Springer, 2011, pp. 109–117. MR2857021M. Hairer and J. C. Mattingly, Yet another look at harris’ ergodic theorem for markov chains, Seminar on Stochastic Analysis, Random Fields and Applications VI: Centro Stefano Franscini, Ascona, May 2008, Springer, 2011, pp. 109–117. MR2857021
M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of harris’ theorem with applications to stochastic delay equations, Probability theory and related fields 149 (2011), 223–259. MR2773030M. Hairer, J. C. Mattingly, and M. Scheutzow, Asymptotic coupling and a general form of harris’ theorem with applications to stochastic delay equations, Probability theory and related fields 149 (2011), 223–259. MR2773030
M. Hairer, A. M. Stuart, and S. J. Vollmer, Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions, Ann. Appl. Probab. 24 (2014), no. 6, 2455–2490. MR3262508M. Hairer, A. M. Stuart, and S. J. Vollmer, Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions, Ann. Appl. Probab. 24 (2014), no. 6, 2455–2490. MR3262508
P. E. Jacob, J. O’Leary, and Y. F. Atchadé, Unbiased markov chain monte carlo with couplings (with discussion), J. R. Stat. Soc. Ser. B 82 (2020), 543–600. MR4112777P. E. Jacob, J. O’Leary, and Y. F. Atchadé, Unbiased markov chain monte carlo with couplings (with discussion), J. R. Stat. Soc. Ser. B 82 (2020), 543–600. MR4112777
T. S. Kleppe, Connecting the dots: Numerical randomized hamiltonian monte carlo with state-dependent event rates, Journal of Computational and Graphical Statistics 31 (2022), no. 4, 1238–1253. MR4513384T. S. Kleppe, Connecting the dots: Numerical randomized hamiltonian monte carlo with state-dependent event rates, Journal of Computational and Graphical Statistics 31 (2022), no. 4, 1238–1253. MR4513384
Y. T. Lee, R. Shen, and K. Tian, Logsmooth gradient concentration and tighter runtimes for Metropolized Hamiltonian Monte Carlo, Proceedings of Thirty Third Conference on Learning Theory, vol. 125, 2020, pp. 2565–2597.Y. T. Lee, R. Shen, and K. Tian, Logsmooth gradient concentration and tighter runtimes for Metropolized Hamiltonian Monte Carlo, Proceedings of Thirty Third Conference on Learning Theory, vol. 125, 2020, pp. 2565–2597.
Y. T. Lee, R. Shen, and K. Tian, Lower bounds on metropolized sampling methods for well-conditioned distributions, Advances in Neural Information Processing Systems, vol. 34, Curran Associates, Inc., 2021, pp. 18812–18824.Y. T. Lee, R. Shen, and K. Tian, Lower bounds on metropolized sampling methods for well-conditioned distributions, Advances in Neural Information Processing Systems, vol. 34, Curran Associates, Inc., 2021, pp. 18812–18824.
B. Leimkuhler, D. Paulin, and P. A. Whalley, Contraction and convergence rates for discretized kinetic langevin dynamics, arXiv preprint arXiv: 2302.10684 (2023). MR4748799B. Leimkuhler, D. Paulin, and P. A. Whalley, Contraction and convergence rates for discretized kinetic langevin dynamics, arXiv preprint arXiv: 2302.10684 (2023). MR4748799
S. Livingstone, M. Betancourt, S. Byrne, and M. Girolami, On the geometric ergodicity of Hamiltonian Monte Carlo, Bernoulli 25 (2019), no. 4A, 3109–3138. MR4003576S. Livingstone, M. Betancourt, S. Byrne, and M. Girolami, On the geometric ergodicity of Hamiltonian Monte Carlo, Bernoulli 25 (2019), no. 4A, 3109–3138. MR4003576
N. Madras and D. Sezer, Quantitative bounds for markov chain convergence: Wasserstein and total variation distances, Bernoulli 16 (2010), no. 3, 882–908. MR2730652N. Madras and D. Sezer, Quantitative bounds for markov chain convergence: Wasserstein and total variation distances, Bernoulli 16 (2010), no. 3, 882–908. MR2730652
J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stoch. Proc. Appl. 101 (2002), no. 2, 185–232. MR1931266J. C. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stoch. Proc. Appl. 101 (2002), no. 2, 185–232. MR1931266
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equations of state calculations by fast computing machines, J Chem Phys 21 (1953), 1087–1092.N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equations of state calculations by fast computing machines, J Chem Phys 21 (1953), 1087–1092.
S. P. Meyn and R. L. Tweedie, Computable Bounds for Geometric Convergence Rates of Markov Chains, The Annals of Applied Probability 4 (1994), no. 4, 981 – 1011. MR1304770S. P. Meyn and R. L. Tweedie, Computable Bounds for Geometric Convergence Rates of Markov Chains, The Annals of Applied Probability 4 (1994), no. 4, 981 – 1011. MR1304770
P. Monmarché, High-dimensional mcmc with a standard splitting scheme for the underdamped langevin diffusion., Electronic Journal of Statistics 15 (2021), no. 2, 4117–4166. MR4309974P. Monmarché, High-dimensional mcmc with a standard splitting scheme for the underdamped langevin diffusion., Electronic Journal of Statistics 15 (2021), no. 2, 4117–4166. MR4309974
P. Monmarché, An entropic approach for hamiltonian monte carlo: the idealized case, arXiv preprint arXiv: 2209.13405 (2022). MR4728169P. Monmarché, An entropic approach for hamiltonian monte carlo: the idealized case, arXiv preprint arXiv: 2209.13405 (2022). MR4728169
P. Monmarché, Hmc and langevin united in the unadjusted and convex case, arXiv preprint arXiv: 2202.00977 (2022).P. Monmarché, Hmc and langevin united in the unadjusted and convex case, arXiv preprint arXiv: 2202.00977 (2022).
R. Montenegro and P. Tetali, Mathematical aspects of mixing times in Markov chains, Found. Trends Theor. Comput. Sci. 1 (2006), no. 3, x+121. MR2341319R. Montenegro and P. Tetali, Mathematical aspects of mixing times in Markov chains, Found. Trends Theor. Comput. Sci. 1 (2006), no. 3, x+121. MR2341319
J. O’Leary and G. Wang, Metropolis-hastings transition kernel couplings, arXiv preprint arXiv: 2102.00366 (2021).J. O’Leary and G. Wang, Metropolis-hastings transition kernel couplings, arXiv preprint arXiv: 2102.00366 (2021).
L. Riou-Durand and J. Vogrinc, Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo, arXiv preprint arXiv: 2202.13230 (2022).L. Riou-Durand and J. Vogrinc, Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo, arXiv preprint arXiv: 2202.13230 (2022).
G. Roberts and J. Rosenthal, One-shot coupling for certain stochastic recursive sequences, Stochastic processes and their applications 99 (2002), no. 2, 195–208. MR1901153G. Roberts and J. Rosenthal, One-shot coupling for certain stochastic recursive sequences, Stochastic processes and their applications 99 (2002), no. 2, 195–208. MR1901153
G. O. Roberts and R. L. Tweedie, Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli 2 (1996), 341–363. MR1440273G. O. Roberts and R. L. Tweedie, Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli 2 (1996), 341–363. MR1440273
G. O. Roberts and R. L. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms, Biometrika 1 (1996), 95–110. MR1399158G. O. Roberts and R. L. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms, Biometrika 1 (1996), 95–110. MR1399158
J. S. Rosenthal, Minorization conditions and convergence rates for markov chain monte carlo, Journal of the American Statistical Association 90 (1995), no. 430, 558–566. MR1340509J. S. Rosenthal, Minorization conditions and convergence rates for markov chain monte carlo, Journal of the American Statistical Association 90 (1995), no. 430, 558–566. MR1340509
J. M. Sanz-Serna and K. C. Zygalakis, Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations, The Journal of Machine Learning Research 22 (2021), no. 1, 11006–11042. MR4329821J. M. Sanz-Serna and K. C. Zygalakis, Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations, The Journal of Machine Learning Research 22 (2021), no. 1, 11006–11042. MR4329821
A. Scemama, T. Lelièvre, G. Stoltz, E. Cancés, and M. Caffarel, An efficient sampling algorithm for variational Monte Carlo, J Chem Phys 125 (2006), 114105.A. Scemama, T. Lelièvre, G. Stoltz, E. Cancés, and M. Caffarel, An efficient sampling algorithm for variational Monte Carlo, J Chem Phys 125 (2006), 114105.
D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Processes and Related Fields 8 (2002), 1–36. MR1924934D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Processes and Related Fields 8 (2002), 1–36. MR1924934
G. Wang, J. O’Leary, and P. Jacob, Maximal couplings of the metropolis-hastings algorithm, International Conference on Artificial Intelligence and Statistics, PMLR, 2021, pp. 1225–1233.G. Wang, J. O’Leary, and P. Jacob, Maximal couplings of the metropolis-hastings algorithm, International Conference on Artificial Intelligence and Statistics, PMLR, 2021, pp. 1225–1233.
K. Wu, S. Schmidler, and Y. Chen, Minimax mixing time of the Metropolis-adjusted Langevin algorithm for log-concave sampling, J. Mach. Learn. Res. 23 (2022). MR4577709K. Wu, S. Schmidler, and Y. Chen, Minimax mixing time of the Metropolis-adjusted Langevin algorithm for log-concave sampling, J. Mach. Learn. Res. 23 (2022). MR4577709
J. Yang and J. S. Rosenthal, Complexity results for MCMC derived from quantitative bounds, The Annals of Applied Probability 33 (2023), no. 2, 1459 – 1500. MR4564431J. Yang and J. S. Rosenthal, Complexity results for MCMC derived from quantitative bounds, The Annals of Applied Probability 33 (2023), no. 2, 1459 – 1500. MR4564431
