We present a novel technique for learning the mass matrices in samplers
obtained from discretized dynamics that preserve some energy function. Existing
adaptive samplers use Riemannian preconditioning techniques, where the mass
matrices are functions of the parameters being sampled. This leads to
significant complexities in the energy reformulations and resultant dynamics,
often leading to implicit systems of equations and requiring inversion of
high-dimensional matrices in the leapfrog steps. Our approach provides a
simpler alternative, by using existing dynamics in the sampling step of a Monte
Carlo EM framework, and learning the mass matrices in the M step with a novel
online technique. We also propose a way to adaptively set the number of samples
gathered in the E step, using sampling error estimates from the leapfrog
dynamics. Along with a novel stochastic sampler based on Nos\'{e}-Poincar\'{e}
dynamics, we use this framework with standard Hamiltonian Monte Carlo (HMC) as
well as newer stochastic algorithms such as SGHMC and SGNHT, and show strong
performance on synthetic and real high-dimensional sampling scenarios; we
achieve sampling accuracies comparable to Riemannian samplers while being
significantly faster.

该研究提出了一种新颖的学习从离散化动力学中保留某些能量函数的采样器中的质量矩阵的技术。采用以前的动态学习 Monte Carlo EM 框架中的采样步骤中使用的现有动力学，并使用一种新颖的在线技术在 M 步中学习质量矩阵。此外，还提出一种通过从跳字动力学中的采样误差估计自适应设置 E 步中收集的样本数量的方法，使用标准的哈密顿蒙特卡罗 (HMC) 以及新的随机算法，如 SGHMC 和 SGNHT，在合成和真实高维采样场景中表现出强大的性能。