## Probability Seminar: Guo-Jhen Wu, Brown University

**Guo-Jhen Wu
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**Brown University**

Temperature selection for the infinite swapping algorithm

Parallel tempering, also known as replica exchange, is an algorithm used to speed up the convergence of slowly converging Markov processes (corresponding to lower temperatures for models from the physical sciences). By constructing other processes with higher temperature and allowing Metropolis type swaps between the different processes, the original process is able to explore the state space more efficiently via the swapping mechanism. It has been proven that by sending the swap rate to infinity, the sampling properties reach optimality in a certain sense. Moreover, this “infinite swapping limit” is attained by process with symmetrized dynamics, which when combined with a weighted empirical measure provide approximations to the original problem. After discussing the construction, we focus on optimizing variance with respect to selection of the temperatures. As will be discussed, there are two main contributions of variance reduction. The first one comes from a lowering of energy barriers and consequent improved communication properties. The second and less obvious source is because of the weights appearing in the weighted empirical measure. These two variance reduction mechanisms behave in opposite ways as the temperatures vary. Based on a large deviations analysis of the variance, we are able to identify the best temperature sequence for certain models when the lowest temperature is sent to zero, i.e., when sampling is most difficult.

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**Refreshments will be served at 3:45 in the 3rd floor lounge of Hanes Hall**