Nonlinear Hamiltonian Monte Carlo its Particle Approximation
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We present a nonlinear (in the sense of McKean) generalization of Hamiltonian Monte Carlo (HMC) termed nonlinear HMC (nHMC) capable of sampling from nonlinear probability measures of mean-field type. When the underlying confinement potential is K-strongly convex and L-gradient Lipschitz, and the underlying interaction potential is gradient Lipschitz, nHMC can produce an ε-accurate approximation of a d-dimensional nonlinear probability measure in L^1-Wasserstein distance using O((L/K) log(1/ε)) steps. Owing to a uniform-in-steps propagation of chaos phenomenon, and without further regularity assumptions, unadjusted HMC with randomized time integration for the corresponding particle approximation can achieve ε-accuracy in L^1-Wasserstein distance using O( (L/K)^5/3 (d/K)^4/3 (1/ε)^8/3log(1/ε) ) gradient evaluations. These mixing/complexity upper bounds are a specific case of more general results developed in the paper for a larger class of non-logconcave, nonlinear probability measures of mean-field type.
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