communication-efficient distributed statistical learning

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?, providing a sort of first order approximation, albeit frozen at the chosen starting value.