Ergodic variational flows

This work presents a new class of variational family -- ergodic variational flows -- that not only enables tractable i.i.d. sampling and density evaluation, but also comes with MCMC-like convergence guarantees. Ergodic variational flows consist of a mixture of repeated applications of a measure-preserving and ergodic map to an initial reference distribution. We provide mild conditions under which the variational distribution converges weakly and in total variation to the target as the number of steps in the flow increases; this convergence holds regardless of the value of variational parameters, though different parameter values may result in faster or slower convergence. We develop a practical implementation of the flow family using Hamiltonian dynamics combined with deterministic momentum refreshment, including a tunable step size to optimize the trade-off between simulation fidelity and computational cost. Simulated and real data experiments provide an empirical verification of the convergence theory, and demonstrate that the method provides more reliable posterior approximations than several black-box normalizing flows, as well as samples of comparable quality to those obtained from state-of-the-art MCMC methods.