  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.
