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.
@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2019.64, author = {Chen, Zongchen and Vempala, Santosh S.}, title = {{Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {64:1--64:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.64}, URN = {urn:nbn:de:0030-drops-112790}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.64}, annote = {Keywords: logconcave distribution, sampling, Hamiltonian Monte Carlo, spectral gap, strong convexity} }
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