Spectral gap of replica exchange Langevin diffusion on mixture distributions

Langevin diffusion (LD) is one of the main workhorses for sampling problems. However, its convergence rate can be significantly reduced if the target distribution is a mixture of multiple densities, especially when each component density concentrates around a different mode. Replica exchange Langevin diffusion (ReLD) is a sampling method that can circumvent this issue. This approach can be further extended to multiple replica exchange Langevin diffusion (mReLD). While ReLD and mReLD have been used extensively in statistics, molecular dynamics, and other applications, there is limited existing analysis on its convergence rate and choices of the temperatures. This paper addresses these problems assuming the target distribution is a mixture of log-concave densities. We show ReLD can obtain constant or better convergence rates. We also show mReLD with K additional LDs can achieve the same results while the exchange frequency only needs to be (1/K)-th power of the one in ReLD.