Estimating Convergence of Markov chains with L-Lag Couplings

Markov chain Monte Carlo (MCMC) methods generate samples that are asymptotically distributed from a target distribution of interest as the number of iterations goes to infinity. Various theoretical results provide upper bounds on the distance between the target and marginal distribution after a fixed number of iterations. These upper bounds are on a case by case basis and typically involve intractable quantities, which limits their use for practitioners. We introduce L-lag couplings to generate computable, non-asymptotic upper bound estimates for the total variation or the Wasserstein distance of general Markov chains. We apply L-lag couplings to three tasks: (i) determining MCMC burn-in with guarantees; (ii) comparison of different MCMC algorithms targeting the same distribution; (iii) comparison of exact and approximate MCMC algorithms such as Metropolis adjusted Langevin and unadjusted Langevin.

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