Monte Carlo Sampling Methods Based On Dynamics

In machine learning and Bayesian inference,the complex probabilistic model need to calculate the intractable high-dimension integral.Markov chain Monte Carlo(MCMC)is one of the commonly used approximation method of Bayesian probabilistic models.MCMC methods utilize the samples generated from the Markov chain to approximate the complex probability distribution.When the number of the samples is large enough,MCMC methods are able to provide the asymptotically unbiased estimation for the probabilistic models.Dynamics based sampling method is one of the most popular MCMC methods.Hamiltonian Monte Carlo(HMC)is the most typical representative of dynamics based MCMC methods.It exploits Hamiltonian dynamics to construct the Markov chain to sample from the target distribution with high efficiency.However,HMC may have trouble in autocorrelation of samples and multi-modal sampling problems.The purpose of this work is to study the dynamics based MCMC samplers,and design the samplers with high performance,which can sample from the multi-modal distributions.First,in order to improve the performance of the dynamics based MCMC sampler,which is to improve the convergence and decline the autocorrelation of samples,this work proposes the neural networks Langevin Monte Carlo(NNLMC).It makes full use of the flexibility of neural networks and the high efficiency of Langevin dynamics to construct a novel MCMC sampler.We propose the new update functions to generate samples and exploit the appropriate loss functions to further improve the performance of NNLMC during the process of sampling.To verify the performance of the proposed sampler,we sample from various challenging distributions and posterior distributions of the datasets.The results demonstrate that NNLMC is able to sample from the target distribution efficiently and independently.Its performance is superior to the state-of-the-art MCMC methods.In addition,in order to sample from the multi-modal distributions,this work further proposes Langevin normalization flows Monte Carlo(NFLMC),which introduces normalization to Langevin diffusions to design the sampler.We propose the appropriate loss functions to ensure that the samples generated from NFLMC are able to converge to the target distribution.We then study the unnormalized target distributions.Finally,we conduct the experiments on multi-modal distributions,various challenging distributions and the posterior distributions of the datasets.The results show that the proposed method can not only deal with complex distributions and posterior distributions of the real datasets,but also sample from the multi-modal distributions accurately with low autocorrelation of samples.Its performance is superior to the state-of-the-art MCMC methods.Finally,in order to sample from the distant multi-modal distributions,this paper further propose the variational hybrid Monte Carlo(VHMC).We explore the new mode through constructing an appropriate variational distribution and design the new acceptance rate of samples to ensure that the samples are able to converge to the target distribution.In order to verify the performance of the sampler,we conduct the experiments on various Gaussian mixtures distributions.The results demonstrate that our method can sample from the distant multi-modal distributions accurately and efficiently.To sum up,NNLMC,NFLMC,and VHMC can generate samples efficiently and independently for different target distributions.As a high-efficiency sampling method,NNLMC exploits neural networks to construct flexible Markov chain.Compared with the traditional dynamics based samplers,NNLMC can generate samples with lower autocorrelation.As a sampling method for multi-modal distributions,NFLMC proposes Langevin normalization flows to construct the mapping between different distributions.Compared with NNLMC,NFLMC is able to sample from the multi-modal distributions accurately and efficiently,meanwhile,eliminating the autocorrelation of samples.As a sampling method for distant multi-modal distributions,VHMC exploits the variational distribution of the target distribution to explore the new mode.Compared with NFLMC,VHMC can further sample from the distant multi-modal distributions accurately and efficiently.The experiments conducted on the various complex distributions,multi-modal distributions,and the posterior distributions prove the efficiency and the accuracy of NNLMC,NFLMC,and VHMC.