Discrete & Computational Geometry

, Volume 59, Issue 4, pp 757–783 | Cite as

Sampling from a Log-Concave Distribution with Projected Langevin Monte Carlo

  • Sébastien Bubeck
  • Ronen Eldan
  • Joseph Lehec


We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected stochastic gradient descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in \(\widetilde{O}(n^7)\) steps (where n is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of \(\widetilde{O}(n^4)\) was proved by Lovász and Vempala.


Langevin Monte Carlo Sampling and optimization Log-concave measures Rapidly-mixing random walks 

Mathematics Subject Classification

47N10 68W20 68W25 


  1. 1.
    Ahn, S., Korattikara, A., Welling, M.: Bayesian posterior sampling via stochastic gradient Fisher scoring. In: Proceedings of the 29th International Conference on Machine Learning (ICML’12), pp. 782–846. IMLS (2012)Google Scholar
  2. 2.
    Bach, F., Moulines, E.: Non-strongly-convex smooth stochastic approximation with convergence rate \({{\rm O}}(1/n)\). In: Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS’13), vol. 1, pp. 773–781. Curran Associates (2013)Google Scholar
  3. 3.
    Cousins, B., Vempala, S.: Bypassing KLS: Gaussian cooling and an \({{\rm O}}^*(n^3)\) volume algorithm. In: Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC’15), pp. 539–548. ACM, New York (2015)Google Scholar
  4. 4.
    Dalalyan, A.S.: Theoretical guarantees for approximate sampling from a smooth and log-concave densities. J. R. Stat. Soc. Stat. Methodol. Ser. B.
  5. 5.
    Dyer, M., Frieze, A., Kannan, R.: A random polynomial-time algorithm for approximating the volume of convex bodies. J. Assoc. Comput. Mach. 38(1), 1–17 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kannan, R., Narayanan, H.: Random walks on polytopes and an affine interior point method for linear programming. Math. Oper. Res. 37, 1–20 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 23. Springer, Berlin (1991)Google Scholar
  8. 8.
    Lehec, J.: Representation formula for the entropy and functional inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 49(3), 885–899 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  10. 10.
    Lindvall, T., Rogers, L.C.G.: Coupling of multidimensional diffusions by reflection. Ann. Probab. 14(3), 860–872 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lovász, L., Vempala, S.: Hit-and-run from a corner. SIAM J. Comput. 35(4), 985–1005 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lovász, L., Vempala, S.: The geometry of logconcave functions and sampling algorithms. Random Struct. Algorithm. 30(3), 307–358 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nemirovsky, A.S., Yudin, D.B.: Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience Series in Discrete Mathematics. Wiley, New York (1983)Google Scholar
  14. 14.
    Pflug, G.Ch.: Stochastic minimization with constant step-size: asymptotic laws. SIAM J. Control Optim. 24(4), 655–666 (1986)Google Scholar
  15. 15.
    Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22, 400–407 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6(3), 264–274 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9(1), 163–177 (1979)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient Langevin dynamics. In: Proceedings of the 28th International Conference on International Conference on Machine Learning (ICML’11), pp. 681–688. Omnipress (2011)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael
  3. 3.CEREMADE, Université Paris-DauphineParisFrance

Personalised recommendations