High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
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发表时间：
2019-08
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通讯作者：
Wenlong Mou;Yian Ma;Yi-An Ma;M. Wainwright;P. Bartlett;Michael I. Jordan
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作者：
Wenlong Mou;Yian Ma;Yi-An Ma;M. Wainwright;P. Bartlett;Michael I. Jordan
我们提出了基于三阶Langevin动力学的马尔可夫链蒙特卡洛（MCMC）算法，用于从具有对数孔和光滑密度的分布中采样。高阶动力学允许更灵活的离散方案，我们开发了一种特定的方法，将分裂与更准确的集成结合在一起。对于由广义线性模型引起的广泛类别的$ d $维分布，我们证明，由此产生的三阶算法会产生来自$ \ varepsilon> 0 $在Wasserstein距离的分布中的样品o \ left（\ frac {d^{1/3}}} {\ varepsilon^{2/3}}} \右）$ step。此结果只需要梯度上的Lipschitz条件。对于具有$ \ alpha $ -th订单平滑度的一般强烈凸电势，我们证明混合时间缩放为$ o \ left（\ frac {d^{1/3}}} \ frac {d^{1/2}}} {\ varepsilon^{1/（\ alpha-- 1）}} \ right）$。
We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/3}}{ \varepsilon^{2/3}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $\alpha$-th order smoothness, we prove that the mixing time scales as $O \left(\frac{d^{1/3}}{\varepsilon^{2/3}} + \frac{d^{1/2}}{\varepsilon^{1/(\alpha - 1)}} \right)$.