High-dimensional Bayesian inference via the unadjusted Langevin algorithm
High-dimensional Bayesian inference via the unadjusted Langevin algorithm
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DOI：
10.3150/18-bej1073
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发表时间：
2019-11-01
期刊：
影响因子：
1.5
通讯作者：
Moulines, Eric
中科院分区：
文献类型：
--
作者：
Durmus, Alain;Moulines, Eric
在本文中，我们考虑的问题，采样的高维概率分布PI具有密度w.r.t. R-d上勒贝格测度，已知直到归一化常数x右箭头pi（x）= e（-U（x）（））/ integral（Rd）e（-U（y））dy。这样的问题自然发生在例如贝叶斯推理和机器学习中。在U是连续可微的，del U是全局Lipschitz的，U是强凸的假设下，我们得到了基于Langevin随机微分方程Euler离散化的抽样方法在2阶Wasserstein距离和全变差距离下收敛于平稳性的非渐近界，包括常数步长和递减步长.这些边界的状态空间的维数的依赖是明确的。适当加权的经验措施的收敛性也进行了研究，均方误差和指数偏差不等式的界报告的功能是可测的和有界的。一个例子来支持我们的主张，贝叶斯推理的二元回归。
We consider in this paper the problem of sampling a high-dimensional probability distribution pi having a density w.r.t. the Lebesgue measure on R-d, known up to a normalization constant x bar right arrow pi(x) = e(-U(x)()) / integral(Rd)e(-U(y)) dy. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that U is continuously differentiable, del U is globally Lipschitz and U is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order 2 and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.