# Stochastic Approximation with Biased MCMC for Expectation Maximization

arXiv: https://arxiv.org/abs/2402.17870 · HTML: https://mlsanity.com/paper/2402.17870/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 63 references that are in the map.
- First posted on arXiv 27 Feb 2024.

## Builds on (most cited first, 9 of 9)

- [A High Probability Analysis of Adaptive SGD with Momentum](https://mlsanity.com/paper/2007.14294/) (2020)
- [Non-asymptotic Analysis of Biased Stochastic Approximation Scheme](https://mlsanity.com/paper/1902.00629/) (2019)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Asymptotic Bias of Stochastic Gradient Search](https://mlsanity.com/paper/1709.00291/) (2017)
- [Interacting Particle Langevin Algorithm for Maximum Marginal Likelihood Estimation](https://mlsanity.com/paper/2303.13429/) (2023)
- [Stochastic Approximation Beyond Gradient for Signal Processing and Machine Learning](https://mlsanity.com/paper/2302.11147/) (2023)
- [A General Method for Robust Bayesian Modeling](https://mlsanity.com/paper/1510.05078/) (2015)
- [Convergence of First-Order Methods for Constrained Nonconvex Optimization with Dependent Data](https://mlsanity.com/paper/2203.15797/) (2022)
- [Constrained Stochastic Nonconvex Optimization with State-dependent Markov Data](https://mlsanity.com/paper/2206.11346/) (2022)

## Built on by (most cited first, 5 of 5)

- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [Learning Latent Variable Models via Jarzynski-adjusted Langevin Algorithm](https://mlsanity.com/paper/2505.18427/) (2025)
- [Scalable Vertical Federated Learning via Data Augmentation and Amortized Inference](https://mlsanity.com/paper/2405.04043/) (2024)
- [Efficient Stochastic Optimisation via Sequential Monte Carlo](https://mlsanity.com/paper/2601.22003/) (2026)
- [Particle-based Generalised Stochastic Optimisation](https://mlsanity.com/paper/2608.02844/) (2026)

## Abstract (arXiv)

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.