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 $\varepsilon$-accurate approximation of a $d$-dimensional nonlinear probability measure in $L^1$-Wasserstein distance using $O((L/K) \log(1/\varepsilon))$ 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 $\varepsilon$-accuracy in $L^1$-Wasserstein distance using $O( (L/K)^{5/3} (d/K)^{4/3} (1/\varepsilon)^{8/3} \log(1/\varepsilon) )$ 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.

翻译：我们提出了一种非线性（在McKean意义上）的哈密顿蒙特卡洛（HMC）泛化方法，称为非线性HMC（nHMC），能够对平均场类型的非线性概率测度进行采样。当底层约束势函数为$K$-强凸且$L$-梯度利普希茨，且底层相互作用势函数为梯度利普希茨时，nHMC可以使用$O((L/K) \log(1/\varepsilon))$步，在$L^1$-Wasserstein距离下生成$d$维非线性概率测度的$\varepsilon$-精确近似。基于步长一致混沌传播现象，且无需额外正则性假设，采用随机时间积分的未调整HMC进行相应粒子近似时，仅需$O( (L/K)^{5/3} (d/K)^{4/3} (1/\varepsilon)^{8/3} \log(1/\varepsilon) )$次梯度评估，即可在$L^1$-Wasserstein距离下达到$\varepsilon$-精度。这些混合/复杂度上界是本文针对更广泛非对数凹、平均场型非线性概率测度所建立的更一般结论的特例。