The Annals of Applied Probability

Fast Langevin based algorithm for MCMC in high dimensions

Alain Durmus, Gareth O. Roberts, Gilles Vilmart, and Konstantinos C. Zygalakis

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Abstract

We introduce new Gaussian proposals to improve the efficiency of the standard Hastings–Metropolis algorithm in Markov chain Monte Carlo (MCMC) methods, used for the sampling from a target distribution in large dimension $d$. The improved complexity is $\mathcal{O}(d^{1/5})$ compared to the complexity $\mathcal{O}(d^{1/3})$ of the standard approach. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate (with asymptotical value 0.704), independently of the target distribution. Numerical experiments confirm our theoretical findings.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 4 (2017), 2195-2237.

Dates
Received: July 2015
Revised: October 2016
First available in Project Euclid: 30 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1504080030

Digital Object Identifier
doi:10.1214/16-AAP1257

Mathematical Reviews number (MathSciNet)
MR3693524

Zentralblatt MATH identifier
1373.60053

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 65C05: Monte Carlo methods

Keywords
Weak convergence Markov chain Monte Carlo diffusion limit exponential ergodicity

Citation

Durmus, Alain; Roberts, Gareth O.; Vilmart, Gilles; Zygalakis, Konstantinos C. Fast Langevin based algorithm for MCMC in high dimensions. Ann. Appl. Probab. 27 (2017), no. 4, 2195--2237. doi:10.1214/16-AAP1257. https://projecteuclid.org/euclid.aoap/1504080030


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References

  • [1] Abdulle, A., Vilmart, G. and Zygalakis, K. C. (2014). High order numerical approximation of the invariant measure of ergodic SDEs. SIAM J. Numer. Anal. 52 1600–1622.
  • [2] Beskos, A. and Stuart, A. (2009). MCMC methods for sampling function space. In ICIAM 076th International Congress on Industrial and Applied Mathematics 337–364. Eur. Math. Soc., Zürich.
  • [3] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [4] Christensen, O. F., Roberts, G. O. and Rosenthal, J. S. (2005). Scaling limits for the transient phase of local Metropolis–Hastings algorithms. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 253–268.
  • [5] Cotter, S. L., Roberts, G. O., Stuart, A. M. and White, D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster. Statist. Sci. 28 424–446. DOI:10.1214/13-STS421.
  • [6] Durmus, A., Roberts, G. O., Vilmart, G. and Zygalakis, K. C. (2017). Supplement to “Fast Langevin based algorithm for MCMC in high dimensions.” DOI:10.1214/16-AAP1257SUPP.
  • [7] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [8] Fathi, M. and Stoltz, G. (2017). Improving dynamical properties of metropolized discretizations of overdamped Langevin dynamics. Numer. Math. 136 545–602.
  • [9] Girolami, M. and Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 123–214.
  • [10] Hansen, N. R. (2003). Geometric ergodicity of discrete-time approximations to multivariate diffusions. Bernoulli 9 725–743.
  • [11] Has’minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis 7. Sijthoff & Noordhoff, Alphen aan den Rijn. Translated from the Russian by D. Louvish.
  • [12] Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • [13] Jourdain, B., Lelièvre, T. and Miasojedow, B. (2014). Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior. Bernoulli 20 1930–1978.
  • [14] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
  • [15] Liu, J. S. (2008). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • [16] Mattingly, J. C., Stuart, A. M. and Tretyakov, M. V. (2010). Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 48 552–577.
  • [17] Mengersen, K. L. and Tweedie, R. L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
  • [18] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [19] Ozaki, T. (1992). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach. Statist. Sinica 2 113–135.
  • [20] Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • [21] Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7 110–120.
  • [22] Roberts, G. O. and Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 255–268. DOI:10.1111/1467-9868.00123.
  • [23] Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling for various Metropolis–Hastings algorithms. Statist. Sci. 16 351–367.
  • [24] Roberts, G. O. and Stramer, O. (2002). Langevin diffusions and Metropolis–Hastings algorithms. Methodol. Comput. Appl. Probab. 4 337–357.
  • [25] Roberts, G. O. and Tweedie, R. L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2 341–363. DOI:10.2307/3318418.
  • [26] Roberts, G. O. and Tweedie, R. L. (1996). Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110. DOI:10.1093/biomet/83.1.95.
  • [27] Talay, D. and Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509.
  • [28] Wolfram Research, Inc. (2014). Mathematica.

Supplemental materials

  • Supplement to “Fast Langevin based algorithm for MCMC in high dimensions”. Mathematica notebooks.