Optimization assisted MCMC

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Markov Chain Monte Carlo (MCMC) sampling methods are widely used but often encounter either slow convergence or biased sampling when applied to multimodal high dimensional distributions. In this paper, we present a general framework of improving classical MCMC samplers by employing a global optimization method. The global optimization method first reduces a high dimensional search to an one dimensional geodesic to find a starting point close to a local mode. The search is accelerated and completed by using a local search method such as BFGS. We modify the target distribution by extracting a local Gaussian distribution aound the found mode. The process is repeated to find all the modes during sampling on the fly. We integrate the optimization algorithm into the Wormhole Hamiltonian Monte Carlo (WHMC) method. Experimental results show that, when applied to high dimensional, multimodal Gaussian mixture models and the network sensor localization problem, the proposed method achieves much faster convergence, with relative error from the mean improved by about an order of magnitude than WHMC in some cases.