Reviews: Estimating Convergence of Markov chains with L-Lag Couplings

The authors generalize 1-lag coupling of the chains to L-lag coupling and provide upper bounds on some distribution distances including the total variation and 1-Wasserstein distance. This bound serves as a convergence check for MCMC, e.g., to stop the burn-in phase. The main contributions of the paper are 1) deriving a computable bound of the distribution distance between two (L-lagged) chains, and 2) presenting algorithms (e.g., Coupled Random-Walk Metropolis-Hastings, Coupled HMC, etc.) using the bound as a stopping criterion for burn-in. Unfortunately, the second part together with the proof of the bound is in the supplementary material. The presented bound and method to compute it is, to the best of knowledge, novel and significantly extends the state-of-the-art.