Stochastic Approximation with Biased MCMC for Expectation Maximization
Samuel Gruffaz Kyurae Kim Alain Oliviero Durmus Jacob R. Gardner
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.