截断对数凹抽样的反射哈密顿蒙特卡罗方法
Truncated Log-concave Sampling with Reflective Hamiltonian Monte Carlo
摘要：
我们引入反射哈密顿蒙特卡罗（ReHMC），HMC为基础的算法，从一个对数凹分布限制到凸体采样。我们证明，从一个温暖的开始，步行混合到一个对数凹目标分布$π(x) \propto e^{-f(x)}$，其中$f$是$L$ -光滑和$m$ -强凸，在精确度$\varepsilon$后，$\widetilde O(κd^2 \ell^2 \log (1 / \varepsilon))$步骤的一个良好的圆形凸体，其中$κ= L / m$是负对数密度的条件数，$d$是尺寸，$\ell$是反射数的上限，并且$\varepsilon$是准确度参数。我们还开发了一个高效的开源ReHMC实现，并在各种高维数据集上进行了实验研究。实验表明，ReHMC outperfroms打了就跑和协调打了就跑的时间，它需要产生一个独立的样本，并介绍了实际的截断抽样在数千个维度。
英文摘要：
We introduce Reflective Hamiltonian Monte Carlo (ReHMC), an HMC-based algorithm, to sample from a log-concave distribution restricted to a convex body. We prove that, starting from a warm start, the walk mixes to a log-concave target distribution $π(x) \propto e^{-f(x)}$, where $f$ is $L$-smooth and $m$-strongly-convex, within accuracy $\varepsilon$ after $\widetilde O(κd^2 \ell^2 \log (1 / \varepsilon))$ steps for a well-rounded convex body where $κ= L / m$ is the condition number of the negative log-density, $d$ is the dimension, $\ell$ is an upper bound on the number of reflections, and $\varepsilon$ is the accuracy parameter. We also developed an efficient open source implementation of ReHMC and we performed an experimental study on various high-dimensional data-sets. The experiments suggest that ReHMC outperfroms Hit-and-Run and Coordinate-Hit-and-Run regarding the time it needs to produce an independent sample and introduces practical truncated sampling in thousands of dimensions.
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