On L^q convergence of the Hamiltonian Monte Carlo

• Autorzy: Ghosh Soumyadip , Lu Yingdong , Nowicki Tomasz

Identyfikatory

DOI

10.1515/jaa-2022-1006

• Języki publikacji: EN

Abstrakty

EN

We establish L^q convergence for Hamiltonian Monte Carlo (HMC) algorithms. More specifically, under mild conditions for the associated Hamiltonian motion, we show that the outputs of the algorithms converge (strongly for 2 ≤ q < ∞ and weakly for 1 < q < 2) to the desired target distribution. In addition, we establish a general convergence rate for an L^q convergence given a convergence rate at a specific q^∗, and apply this result to conclude geometric convergence in the Euclidean space for HMC with uniformly strongly logarithmic concave target and auxiliary distributions.We also present the results of experiments to illustrate convergence in L^q.

Słowa kluczowe

EN

convergence; Lq spaces; Hamiltonian Monte Carlo

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2023
• Tom: Vol. 29, nr 1
• Strony: 161--169
• Opis fizyczny: Bibliogr. 7 poz., wykr.

Twórcy

autor

Ghosh Soumyadip

• Mathematics of AI, IBM T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, New York, USA

autor

Lu Yingdong

• Mathematics of AI, IBM T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, New York, USA

• https://orcid.org/0000-0002-3672-0865

autor

Nowicki Tomasz

• Mathematics of AI, IBM T.J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, New York, USA

• https://orcid.org/0000-0002-8260-0231

Bibliografia

• [1] M. Ankerst, M. M. Breunig, H.-P. Kriegel and J. Sander, Optics: Ordering points to identify the clustering structure, SIGMOD Rec. 28 (1999), no. 2, 49-60.
• [2] S. Duane, A. D. Kennedy, B. J. Pendleton and D. Roweth, Hybrid Monte Carlo, Phys. Lett. B 195 (1987), no. 2, 216-222.
• [3] S. Ghosh, Y. Lu and T. Nowicki, Hamiltonian Monte Carlo with asymmetrical momentum distributions, preprint (2021), https://arxiv.org/abs/2110.12907.
• [4] S. Ghosh, Y. Lu and T. Nowicki, On L₂ convergence of the Hamiltonian Monte Carlo, Appl. Math. Lett. 127 (2022), Article ID 107811.
• [5] T. Hastie, R. Tibshirani and M. Wainwright, Statistical Learning with Sparsity: The Lasso and Generalizations, Chapman & Hall/CRC Monogr. Statist. Appl. Probab., Taylor & Francis, New York, 2015.
• [6] S. Livingstone, M. Betancourt, S. Byrne and M. Girolami, On the geometric ergodicity of Hamiltonian Monte Carlo, Bernoulli 25 (2019), no. 4A, 3109-3138.
• [7] L. Verlet, Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev. 159 (1967), 98-103.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).