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.
翻译:我们为Nesterov加速梯度流动提供了一个框架,用于在概率空间中为巴伊西亚反向问题设计高效的中位马尔科夫链Monte Carlo(MCMC)算法。这里还考虑了四个信息度量的例子,包括Fisher-Rao 度量、Wasserstein-2 度量、Kalman-Wasserstein 度量和Stein 度量。Fisher-Rao 度量、Wasserstein-2 度量和Stein 度量度。对于Fish-Rao和Wasserstein-2 度量度,我们证明了加速梯度流的趋同性。在执行过程中,我们建议为Wasserstein-2、Kalman-Wasserstein和Stein 加速梯度流采用重开技术,采用抽样高效的离散时间算法。我们还制定了一个内核带带宽带宽选择法,从布朗动样品中了解密度的梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度的梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度梯度。