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Does there exist an optimal swapping rate function for replica-exchange diffusions that minimizes asymptotic variance uniformly across all target distributions with a given mixing time?

Related: Peskun ordering theorem for Markov chains, Holley-Stroock perturbation lemma for spectral gaps, Diaconis-Holmes-Neal theorem on optimal tempering schedules

Replica-exchange Monte Carlo methods work by running multiple diffusion processes at different temperatures and randomly swapping their states to help each process escape local traps in the probability landscape. The swapping mechanism paper establishes a rigorous framework for continuous-time interacting diffusions and derives exact solutions only in special tractable cases. The open problem is whether one can characterize a universally optimal swapping rate, meaning a rate that does not depend on the specific target distribution but only on coarse properties like mixing time or spectral gap, such that the resulting interacting system minimizes the asymptotic variance of Monte Carlo estimators across an entire class of distributions simultaneously.

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