In this paper, we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete-time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithms. The numerical performance of the new method is illustrated for the Langevin-type Markov chain Monte Carlo (MCMC) methods.

中文翻译：

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通过鞅表示的马尔可夫链加性泛函的方差减少

在本文中，我们针对马尔可夫链的加性泛函提出了一种有效的方差减少方法，该方法依赖于一种新的离散时间鞅表示。我们的方法是完全非渐近的，不需要了解平稳分布（甚至任何类型的遍历性）或底层密度的特定结构。通过严格分析所提出算法的收敛特性，我们表明其成本-方差乘积确实小于其中一种朴素算法。对于朗之万型马尔可夫链蒙特卡罗（MCMC）方法，说明了新方法的数值性能。