发表机构
Department of Computer Science, Yale University; Department of Electrical Engineering and Computer Science, MIT（耶鲁大学计算机科学系; 麻省理工学院电气工程与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机哈密顿蒙特卡罗算法在从对数凹概率分布采样时的加速混合时间，通过使用特定随机积分时间，证明其在KL散度中指数收敛，给出不同情况下达到误差ε的总积分时间的尺度，分析依赖于平均KL散度的界。
AI 中文摘要
我们证明随机哈密顿蒙特卡罗（RHMC）算法在从对数凹概率分布采样时具有加速混合时间保证。RHMC通过对一些随机积分时间重复模拟连续时间哈密顿动力学，并在每次模拟之间将速度重置为独立高斯随机变量来进行。我们表明，当目标分布是对数凹且满足α - 塔拉格兰德不等式时（例如，如果目标分布是α - 强对数凹），若使用均值为Θ(α^(-1/2))的三角分布或指数分布的随机积分时间，RHMC在KL散度中指数快速收敛，达到KL散度误差ε的总积分时间为O(α^(-1/2) log(ε^(-1)))。我们还表明，当目标分布是对数凹时，若使用均值指数增长的三角分布的随机积分时间序列，达到KL散度误差ε的总积分时间为O(ε^(-1/2))。我们的分析依赖于沿哈密顿动力学的平均KL散度的界，这受到基于哈密顿动力学的加速优化方法的类似结果的启发。
英文摘要
We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an $α$-Talagrand inequality (for example, if the target distribution is $α$-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean $Θ(α^{-1/2})$, then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(α^{-1/2} \log(\varepsilon^{-1}))$. We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\varepsilon^{-1/2})$. Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.
Comments74 pages
