# Jing Dong: Replica-Exchange Langevin Diffusion and its Application to Optimization and Sampling

**Time: **
Mon 2020-11-16 15.15 - 16.15

**Location: **
Zoom, meeting ID: 621 4469 8204

**Lecturer: **
Jing Dong, Columbia University

### Abstract

In this talk, we investigate the application of replica-exchange Langevin diffusion (LD) for two tasks: solving nonconvex optimization problems and sampling from multimodal target distributions. For the nonconvex optimization problem, we use replica-exchange to facilitate the collaboration between gradient descent (LD with zero temperature) and LD. We show that this algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in the online setting. For the Markov chain Monte Carlo problem, LD is one of the main workhorses. However, its convergence rate can be significantly reduced if the target distribution has multiple isolated modes. Replica exchange allows us to add another LD sampling a high-temperature version of the target density to facilitate faster convergence to stationary. When the target density is a mixture of log-concave densities, we quantify the spectral gap of replica-exchange LD and show that the algorithm with properly chosen temperature and the exchange intensity can achieve constant or even better convergence rates. We further quantify the benefit of replica-exchange for multiple LDs sampling at different temperatures.

**Zoom notes:** The passcode for this meeting is 321777. This meeting ID —
621 4469 8204
— will be the recurring meeting for the Statistics and Probability Seminar.