Stochastic approximation with biased mcmc for expectation maximization

Samuel Gruffaz, Kyurae Kim, Alain Oliviero Durmus, Jacob R. Gardner

Abstract:

The expectation maximization (EM) algo-
rithm is a widespread method for empiri-
cal Bayesian inference, but its expectation
step (E-step) is often intractable. Employ-
ing a stochastic approximation scheme with
Markov chain Monte Carlo (MCMC) can cir-
cumvent this issue, resulting in an algorithm
known as MCMC-SAEM. While theoretical
guarantees for MCMC-SAEM have previ-
ously been established, these results are re-
stricted to the case where asymptotically un-
biased MCMC algorithms are used. In prac-
tice, MCMC-SAEM is often run with asymp-
totically biased MCMC, for which the conse-
quences 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 numeri-
cal experiments comparing the Metropolis-
adjustedLangevinalgorithm(MALA),which
is asymptotically unbiased, and the unad-
justed 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 re-
sult in faster convergence.