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Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions

Authors Zongchen Chen, Santosh S. Vempala

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Design and analysis of algorithms
Keywords
  • logconcave distribution
  • sampling
  • Hamiltonian Monte Carlo
  • spectral gap
  • strong convexity

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Abstract

We study Hamiltonian Monte Carlo (HMC) for sampling from a strongly logconcave density proportional to e^{-f} where f:R^d -> R is mu-strongly convex and L-smooth (the condition number is kappa = L/mu). We show that the relaxation time (inverse of the spectral gap) of ideal HMC is O(kappa), improving on the previous best bound of O(kappa^{1.5}); we complement this with an example where the relaxation time is Omega(kappa). When implemented using a nearly optimal ODE solver, HMC returns an epsilon-approximate point in 2-Wasserstein distance using O~((kappa d)^{0.5} epsilon^{-1}) gradient evaluations per step and O~((kappa d)^{1.5}epsilon^{-1}) total time.

Cite As Get BibTex

Zongchen Chen and Santosh S. Vempala. Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 64:1-64:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.64

Author Details

Zongchen Chen
  • School of Computer Science, Georgia Institute of Technology, USA
Santosh S. Vempala
  • School of Computer Science, Georgia Institute of Technology, USA

Funding

  • Chen, Zongchen: This work was supported in part by CCF-1563838 and CCF-1617306.
  • Vempala, Santosh S.: This work was supported in part by CCF-1563838, CCF-1717349 and DMS-1839323.

Acknowledgements

We are grateful to Yin Tat Lee for helpful discussions.

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