标题： 多层马尔可夫随机近似中的中心极限定理的渐近方差 显示英文标题

标题： Asymptotic Variance in the Central Limit Theorem for Multilevel Markovian Stochastic Approximation

摘要： 在本文中，我们考虑与反问题相关的有限维参数估计问题。在这样的情况下，人们试图最大化与贝叶斯模型相关的边缘似然。后者模型与偏微分方程或常微分方程的解相关。因此，在最大化边缘似然时存在两个主要困难：(i) 微分方程的解并不总是可以解析求解，(ii) 边缘似然也不可解析求解。通常使用微分方程的数值解来处理(i)，这会导致数值偏差；而(ii)已经在文献中被广泛研究，例如使用马尔可夫随机近似方法。众所周知，为了减少获得参数最大值的计算量，可以使用微分方程解的层次结构，并结合随机梯度方法。几种方法正是这样做的。在本文中，我们考虑与已知估计相关的中心极限定理的渐近方差，并以微分方程解的精度为基准，找到渐近方差的界限。这些界限的重要性在于它们提供了缺失的理论指导，以设定模拟参数；也就是说，这些似乎是首次有助于在实践中高效运行这些方法的数学结果。 显示英文摘要

摘要： In this note we consider the finite-dimensional parameter estimation problem associated to inverse problems. In such scenarios, one seeks to maximize the marginal likelihood associated to a Bayesian model. This latter model is connected to the solution of partial or ordinary differential equation. As such, there are two primary difficulties in maximizing the marginal likelihood (i) that the solution of differential equation is not always analytically tractable and (ii) neither is the marginal likelihood. Typically (i) is dealt with using a numerical solution of the differential equation, leading to a numerical bias and (ii) has been well studied in the literature using, for instance, Markovian stochastic approximation. It is well-known that to reduce the computational effort to obtain the maximal value of the parameter, one can use a hierarchy of solutions of the differential equation and combine with stochastic gradient methods. Several approaches do exactly this. In this paper we consider the asymptotic variance in the central limit theorem, associated to known estimates and find bounds on the asymptotic variance in terms of the precision of the solution of the differential equation. The significance of these bounds are the that they provide missing theoretical guidelines on how to set simulation parameters; that is, these appear to be the first mathematical results which help to run the methods efficiently in practice.