Accelerated Information Gradient flow
This work addresses the need for faster and more efficient sampling methods in Bayesian inference, particularly for inverse problems, though it appears incremental as it builds on existing gradient flow and MCMC techniques.
The paper tackles the problem of designing efficient mean-field MCMC algorithms for Bayesian inverse problems by introducing a framework for Nesterov's accelerated gradient flows in probability space, using various information metrics, and demonstrates improved performance in numerical experiments like Bayesian logistic regression and neural networks.
We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.