Hamiltonian Monte-Carlo (HMC) and its auto-tuned variant, the No U-Turn Sampler (NUTS) can struggle to accurately sample distributions with complex geometries, e.g., varying curvature, due to their constant step size for leapfrog integration and fixed mass matrix. In this work, we develop a strategy to locally adapt the step size parameter of HMC at every iteration by evaluating a low-rank approximation of the local Hessian and estimating its largest eigenvalue. We combine it with a strategy to similarly adapt the trajectory length by monitoring the no U-turn condition, resulting in an adaptive sampler, ATLAS: adapting trajectory length and step-size. We further use a delayed rejection framework for making multiple proposals that improves the computational efficiency of ATLAS, and develop an approach for automatically tuning its hyperparameters during warmup. We compare ATLAS with state-of-the-art samplers like NUTS on a suite of synthetic and real world examples, and show that i) unlike NUTS, ATLAS is able to accurately sample difficult distributions with complex geometries, ii) it is computationally competitive to NUTS for simpler distributions, and iii) it is more robust to the tuning of hyperparamters.

翻译：哈密顿蒙特卡洛（HMC）及其自动调参变体——无 U 型转向采样器（NUTS），由于在蛙跳积分中使用固定步长和固定质量矩阵，在采样具有复杂几何结构（如曲率变化）的分布时可能难以保证精度。本文提出一种策略，通过计算局部 Hessian 矩阵的低秩近似并估计其最大特征值，在每次迭代中局部自适应地调整 HMC 的步长参数。我们进一步结合一种通过监测无 U 型转向条件来自适应调整轨迹长度的策略，从而得到一个自适应采样器 ATLAS（自适应轨迹长度与步长）。此外，我们采用延迟拒绝框架生成多重提案，以提高 ATLAS 的计算效率，并开发了一种在预热阶段自动调整其超参数的方法。我们将 ATLAS 与 NUTS 等先进采样器在一系列合成及实际案例上进行比较，结果表明：i) 与 NUTS 不同，ATLAS 能够准确采样具有复杂几何结构的困难分布；ii) 对于较简单的分布，其计算效率与 NUTS 相当；iii) 其对超参数调优具有更强的鲁棒性。