**A**rnak Dalayan and Avetik Karagulyan (CREST) arXived a paper the other week on a focussed study of the Langevin algorithm [not MALA] when the gradient of the target is incorrect. With the following improvements *[quoting non-verbatim from the paper]*:

- a varying-step Langevin that reduces the number of iterations for a given Wasserstein precision, compared with recent results by e.g. Alan Durmus and Éric Moulines;
- an extension of convergence results for error-prone evaluations of the gradient of the target (i.e., the gradient is replaced with a noisy version, under some moment assumptions that do not include unbiasedness);
- a new second-order sampling algorithm termed LMCO’, with improved convergence properties.

What is particularly interesting to me in this setting is the use in all these papers of a discretised Langevin diffusion (a.k.a., random walk with a drift induced by the gradient of the log-target) without the original Metropolis correction. The results rely on an assumption of [strong?] log-concavity of the target, with “user-friendly” bounds on the Wasserstein distance depending on the constants appearing in this log-concavity constraint. And so does the adaptive step. (In the case of the noisy version, the bias and variance of the noise also matter. As pointed out by the authors, there is still applicability to scaling MCMC for large samples. Beyond pseudo-marginal situations.)

“…this, at first sight very disappointing behavior of the LMC algorithm is, in fact, continuously connected to the exponential convergence of the gradient descent.”

The paper concludes with an interesting mise en parallèle of Langevin algorithms and of gradient descent algorithms, since the convergence rates are the same.