We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of \(\mathcal {O}(\varepsilon )\) is achieved with \(\mathcal {O}(\varepsilon ^{-2})\) complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity \(\mathcal {O}(\varepsilon ^{-2}|\log {\varepsilon }|^{3})\), in contrast to the complexity \(\mathcal {O}(\varepsilon ^{-3})\) of currently available methods. Numerical experiments confirm our theoretical findings.

中文翻译：

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随机微分方程不变测度的多级蒙特卡罗方法

我们开发了一个框架，该框架允许使用多级蒙特卡洛（MLMC）方法（Giles in Acta Numer。24：259-328，2015. https://doi.org/10.1017/S096249291500001X）来计算期望值遍历SDE的不变度量。在这种情况下，我们研究具有强凹势的（过度阻尼的）Langevin方程。我们表明，当有适当的用于数字积分器的收缩耦合时，与MLMC文献中的大多数结果相比，人们可以获得MLMC方差的及时估计。结果，通过\（\ mathcal {O}（\ varepsilon ^ {-2}）\）达到\（\ mathcal {O}（\ varepsilon）\）的均方根误差。复杂度与Markov Chain Monte Carlo（MCMC）方法相当，但是将其应用于大型数据集时可能需要大量计算。最后，我们为大型数据集应用程序介绍了最近引入的随机梯度Langevin动力学方法的多层次版本（Welling和Teh，于：第28 ICML会议论文集，2011年）。我们证明这是第一个具有复杂度\（\ mathcal {O}（\ varepsilon ^ {-2} | \ log {\ varepsilon} | ^ {3}）\）的随机梯度MCMC方法，与复杂度\ （\ mathcal {O}（\ varepsilon ^ {-3}）\）当前可用的方法。数值实验证实了我们的理论发现。