On the convergence of dynamic implementations of Hamiltonian Monte Carlo and no U-turn samplers

There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the no U-turn sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. We consider a general class of MCMC algorithms we call dynamic HMC. First, we show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second and most importantly, we present the first ergodicity result for NUTS and prove that the NUTS variant currently implemented in major software packages is ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC and show that the method is ergodic without any boundedness condition on the stepsize or the number of leapfrog steps in the case where the target is a perturbation of a Gaussian distribution.