In this paper, we propose a variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance. We focus on the particular case when control variates are represented as deep neural networks. We derive the optimal convergence rate of the asymptotic variance under various ergodicity assumptions on the underlying Markov chain. The proposed approach relies upon recent results on the stochastic errors of variance reduction algorithms and function approximation theory.

翻译：本文提出了一种基于加性控制变量及其渐近方差的适当估计最小化的马尔可夫链方差缩减方法。我们重点关注控制变量以深度神经网络表示的特殊情况。在底层马尔可夫链的各种遍历性假设下，我们推导了渐近方差的最优收敛速度。所提出的方法依赖于方差缩减算法随机误差的最新理论成果以及函数逼近理论。