论文标题
关于拉格朗日蒙特卡洛的数值集成商的几篇评论
Several Remarks on the Numerical Integrator in Lagrangian Monte Carlo
论文作者
论文摘要
Riemannian歧管汉密尔顿蒙特卡洛(RMHMC)是一种强大的贝叶斯推理方法,它利用后分布的基础几何信息,以便有效地遍历参数空间。但是,哈密顿式的形式需要复杂的数值集成符,例如广泛的跨越方法,它可以保留详细的平衡条件。这些数值积分器的区别特征是它们涉及解决方案的解决方案。 Lagrangian Monte Carlo(LMC)提议通过从汉密尔顿形式主义过渡到Lagrangian Dynamics来消除固定点的迭代,其中提供了完全显式的集成器。这项工作对LMC中使用的数值集成商做出了一些贡献。首先,在文献中据称,集成商仅对于拉格朗日运动方程而言仅是一阶准确的。相反,我们表明LMC集成商享有二阶精度。其次,当前的LMC概念需要在每个步骤中进行四个决定性计算,以维持详细的平衡;我们对LMC中的集成过程进行了简单的修改,以便将确定性计算的数量从四个减少到两个,同时仍保留完全显式的数值集成方案。第三,我们证明LMC集成仪具有与人类错误的某种鲁棒性,而该错误与广义的LeapFrog Integrator没有共享,这在后一种情况下可能会使详细的平衡无效。我们在几个基准贝叶斯推理任务的背景下讨论这些贡献。
Riemannian manifold Hamiltonian Monte Carlo (RMHMC) is a powerful method of Bayesian inference that exploits underlying geometric information of the posterior distribution in order to efficiently traverse the parameter space. However, the form of the Hamiltonian necessitates complicated numerical integrators, such as the generalized leapfrog method, that preserve the detailed balance condition. The distinguishing feature of these numerical integrators is that they involve solutions to implicitly defined equations. Lagrangian Monte Carlo (LMC) proposes to eliminate the fixed point iterations by transitioning from the Hamiltonian formalism to Lagrangian dynamics, wherein a fully explicit integrator is available. This work makes several contributions regarding the numerical integrator used in LMC. First, it has been claimed in the literature that the integrator is only first-order accurate for the Lagrangian equations of motion; to the contrary, we show that the LMC integrator enjoys second order accuracy. Second, the current conception of LMC requires four determinant computations in every step in order to maintain detailed balance; we propose a simple modification to the integration procedure in LMC in order to reduce the number of determinant computations from four to two while still retaining a fully explicit numerical integration scheme. Third, we demonstrate that the LMC integrator enjoys a certain robustness to human error that is not shared with the generalized leapfrog integrator, which can invalidate detailed balance in the latter case. We discuss these contributions within the context of several benchmark Bayesian inference tasks.