Accelerating the diffusion-based ensemble sampling by non-reversible dynamics

Part of Proceedings of the International Conference on Machine Learning 1 pre-proceedings (ICML 2020)

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Authors

Futoshi Futami, Issei Sato, Masashi Sugiyama

Abstract

Posterior distribution approximation is a central task in Bayesian inference. Stochastic gradient Langevin dynamics (SGLD) and its extensions have been widely used practically and studied theoretically. While SGLD updates a single particle at a time, ensemble methods that update multiple particles simultaneously have been recently gathering attention. Compared with the naive parallel-chain SGLD that updates multiple particles independently, ensemble methods update particles with their interactions. Thus, these methods are expected to be more particle-efficient than the naive parallel-chain SGLD because particles can be aware of other particles’ behavior through their interactions. Although ensemble methods demonstrated their superior performance numerically, no theoretical guarantee exists to assure such particle-efficiency and it is unclear whether those ensemble methods are really superior to the naive parallel-chain SGLD in the non-asymptotic settings. To cope with this problem, we propose a novel ensemble method that uses a non-reversible Markov chain for the interaction, and we present a non-asymptotic theoretical analysis for our method. Our analysis shows that, for the first time, the interaction causes a faster convergence rate than the naive parallel-chain SGLD in the non-asymptotic setting if the discretization error is appropriately controlled. Numerical experiments show that we can control the discretization error by tuning the interaction appropriately.