Stochastic Approximation with Biased MCMC for Expectation Maximization

The expectation maximization (EM) algorithm is a widespread method for empirical Bayesian inference, but its expectation step (E-step) is often intractable. Employing a stochastic approximation scheme with Markov chain Monte Carlo (MCMC) can circumvent this issue, resulting in an algorithm known as MCMC-SAEM. While theoretical guarantees for MCMC-SAEM have previously been established, these results are restricted to the case where asymptotically unbiased MCMC algorithms are used. In practice, MCMC-SAEM is often run with asymptotically biased MCMC, for which the consequences are theoretically less understood. In this work, we fill this gap by analyzing the asymptotics and non-asymptotics of SAEM with biased MCMC steps, particularly the effect of bias. We also provide numerical experiments comparing the Metropolis-adjusted Langevin algorithm (MALA), which is asymptotically unbiased, and the unadjusted Langevin algorithm (ULA), which is asymptotically biased, on synthetic and real datasets. Experimental results show that ULA is more stable with respect to the choice of Langevin stepsize and can sometimes result in faster convergence.

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. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary 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 showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.