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Minimizing the inclusive Kullback-Leibler (KL) divergence with stochasticgradient descent (SGD) is challenging since its gradient is defined as anintegral over the posterior. Recently, multiple methods have been proposed torun SGD with biased gradient estimates obtained from a Markov chain. This paperprovides the first non-asymptotic convergence analysis of these methods byestablishing their mixing rate and gradient variance. To do this, wedemonstrate that these methods-which we collectively refer to as Markov chainscore ascent (MCSA) methods-can be cast as special cases of the Markov chaingradient descent framework. Furthermore, by leveraging this new understanding,we develop a novel MCSA scheme, parallel MCSA (pMCSA), that achieves a tighterbound on the gradient variance. We demonstrate that this improved theoreticalresult translates to superior empirical performance.

Markov Chain Score Ascent: A Unifying Framework of Variational Inference with Markovian Gradients