## turn-key and scalable synchronous distributed MCMC algorithm

Last week, I attended a Lagrange seminar where Vincent Plassier presented a ICML²¹ paper he had co-authored with Maxime Vono, Alain Durmus, and Eric Moulines. Aiming at distributed MCMC algorithms that operate on several machines, with a target distribution that breaks as a target

$\int\prod_{i=1}^b \pi_i(\theta,z_i)\,\text d\mathbf{z}=\prod_{i=1}^b e^{U_i(A_i\theta)}$

where θ is common to all terms. And each term in the product can (only) be computed locally. This setup is obviously the same as for the embarrassingly parallel approaches of Neiswanger et al. (2014) and Scott et al. (2016). And it follows an earlier proposal of Vono et al. (2020), which appears as a full Gibbs algorithm on the augmented parameters (θ,z), assuming each term is a conditional density in the latent z’s. Which requires constant communications between the b workers and the central “master” node when θ is concerned. The ICML²¹ paper overcomes this difficulty by defining an approximate target with a Normal component in z. Meaning that the (approximate) conditional distribution of θ given the latent z is Normal, i.e. considering the augmented joint

$\prod_{i=1}^b\exp\left\{u_i(z_i)-\rho_i||z_i-A_i\theta||^2\right\}$

but despite the Gaussian aspect, this is not always practical:

“When d [is large], this Gibbs sampling scheme unfortunately leads to prohibitive computational costs and hence prevents its practical use for general Bayesian inference problems.”

The authors then move to simulating from several Langevin step, more specifically running one move of the Euler-Maruyama discretisation scheme of the overdamped Langevin stochastic differential equation. Communication with the central node is then reduced. The paper proposes a proof of convergence in this unusual (since overdamped) setup. As well as bounds on the bias due to the inclusion of the latent variables. They also manage to find the required scaling of the various parameters involved (Normal variance, discretisation scale, Langevin runs) to achieve convergence, which I find rather remarkable. The table at the top illustrates the comparison with earlier methods, whenever available.

### One Response to “turn-key and scalable synchronous distributed MCMC algorithm”

1. Thanks Xian for this review of our paper. Indeed, although we used « only » unadjusted overdamped Langevin steps and a seemingly naive Gaussian prior with diagonal covariance matrix, deriving mixing time bounds in this scenario was quite involved.
An interesting follow-up would be to derive a similar distributed sampling scheme for distributions with discrete support.

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