## turn-key and scalable synchronous distributed MCMC algorithm

Posted in Statistics, University life with tags , , , , , on April 29, 2022 by xi'an

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.

## communication-efficient distributed statistical learning

Posted in Books, Statistics, University life with tags , , , , , , , , on June 10, 2016 by xi'an

Michael Jordan, Jason Lee, and Yun Yang just arXived a paper with their proposal on handling large datasets through distributed computing, thus contributing to the currently very active research topic of approximate solutions in large Bayesian models. The core of the proposal is summarised by the screenshot above, where the approximate likelihood replaces the exact likelihood with a first order Taylor expansion. The first term is the likelihood computed for a given subsample (or a given thread) at a ratio of one to N and the difference of the gradients is only computed once at a good enough guess. While the paper also considers M-estimators and non-Bayesian settings, the Bayesian part thus consists in running a regular MCMC when the log-target is approximated by the above. I first thought this proposal amounted to a Gaussian approximation à la Simon Wood or to an INLA approach but this is not the case: the first term of the approximate likelihood is exact and hence can be of any form, while the scalar product is linear in θ, providing a sort of first order approximation, albeit frozen at the chosen starting value.

Assuming that each block of the dataset is stored on a separate machine, I think the approach could further be implemented in parallel, running N MCMC chains and comparing the output. With a post-simulation summary stemming from the N empirical distributions thus produced. I also wonder how the method would perform outside the fairly smooth logistic regression case, where the single sample captures well-enough the target. The picture above shows a minor gain in a misclassification rate that is already essentially zero.