Stochastic Approximation with Biased MCMC for Expectation Maximization
This work addresses a theoretical gap for practitioners using biased MCMC in EM algorithms, though it is incremental as it builds on existing MCMC-SAEM frameworks.
The paper tackles the problem of analyzing the expectation maximization algorithm when using biased Markov chain Monte Carlo methods, which are common in practice but lack theoretical understanding, by providing asymptotic and non-asymptotic analyses and showing that biased methods like ULA can be more stable and sometimes converge faster than unbiased ones like MALA in experiments.
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.