BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Langevin MCMC: theory and methods - Eric François Moulines (Tél écom ParisTech) DTSTART:20170707T080000Z DTEND:20170707T084500Z UID:TALK73182@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:Nicolas Brosse\, Ecole Polytechnique\, Paris
Al ain Durmus\, Telecom ParisTech and Ecole Normale Supé\;rieure Paris- Saclay
Marcelo Pereira\, Herriot-Watt University\, Edinburgh


The complexity and sheer size of modern datasets\, to whichever increasin gly demanding questions are posed\, give rise to major challenges. Traditi onal simulation methods often scale poorly with data size and model comple xity and thus fail for the most complex of modern problems.
We a re considering the problem of sampling from a log-concave distribution. Ma ny problems in machine learning fall into this framework\,
such as lin ear ill-posed inverse problems with sparsity-inducing priors\, or large sc ale Bayesian binary regression.



The purpose of this lectur e is to explain how we can use ideas which have proven very useful in mach ine learning community to
solve large-scale optimization problems to de sign efficient sampling algorithms.
Most of the efficient algorithms k now so far may be seen as variants of the gradient descent algorithms\, most often coupled with «\; partial updates »\; (coordinates d escent algorithms). This\, of course\, suggests studying methods derived f rom Euler discretization of the Langevin diffusion. Partial updates may in this context as «\; Gibbs steps »\;This algorithm may be genera lized in the non-smooth case by «\; regularizing »\; the objecti ve function. The Moreau-Yosida inf-convolution algorithm is an appropriate candidate in such case.

We will prove convergence results fo r these algorithms with explicit convergence bounds both in Wasserstein di stance and in total variation. Numerical illustrations will be presented ( on the computation of Bayes factor for model choice\, Bayesian \;analy sis of high-dimensional regression\, aggregation of estimators) to illustr ate our results. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR