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Many problems in the physical sciences, machine learning, and statistical inference necessitate sampling from a high-dimensional, multimodal probability distribution. Markov Chain Monte Carlo (MCMC) algorithms, the ubiquitous tool for this task, typically rely on random local updates to propagate configurations of a given system in a way that ensures that generated configurations will be distributed according to a target probability distribution asymptotically. In high-dimensional settings with multiple relevant metastable basins, local approaches require either immense computational effort or intricately designed importance sampling strategies to capture information about, for example, the relative populations of such basins. Here, we analyze an adaptive MCMC, which augments MCMC sampling with nonlocal transition kernels parameterized with generative models known as normalizing flows. We focus on a setting where there are no preexisting data, as is commonly the case for problems in which MCMC is used. Our method uses 1) an MCMC strategy that blends local moves obtained from any standard transition kernel with those from a generative model to accelerate the sampling and 2) the data generated this way to adapt the generative model and improve its efficacy in the MCMC algorithm. We provide a theoretical analysis of the convergence properties of this algorithm and investigate numerically its efficiency, in particular in terms of its propensity to equilibrate fast between metastable modes whose rough location is known a priori but respective probability weight is not. We show that our algorithm can sample effectively across large free energy barriers, providing dramatic accelerations relative to traditional MCMC algorithms. 

Neural networks with a large number of units ad- mit a mean-field description, which has recently served as a theoretical explanation for the favor- able training properties of “overparameterized” models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appro- priate assumptions. In this work, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We im- plement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean- field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of units. We illustrate our algorithms with two models to provide intuition for the mech- anism through which convergence is accelerated