# Van Dantzig Seminar

#### nationwide series of lectures in statistics

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## Van Dantzig Seminar: 27 January 2017

#### Programme: (click names or scroll down for titles and abstracts)

 14:00 - 14:05 Opening 14:05 - 15:05 Cun-Hui Zhang (Rutgers University) 15:05 - 15:25 Break 15:25 - 16:25 Eric Moulines (École Polytechnique) 16:30 - 17:30 Reception
 Location: University of Amsterdam, Science Park 904, Room G2.10 (Check directions for G building, unusual location)

## Titles and abstracts

• Cun-Hui Zhang

Beyond Gaussian Approximation: Bootstrap in Large Scale Simultaneous Inference

The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on ad hoc bases. Recently, Gaussian approximation was used to justify bootstrap adjustments in large scale simultaneous inference in some general settings where the multiplicity of the inference problem, p, is allowed to be of greater order than the sample size n. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high-dimension. We reduce the sample size requirement to the five seventh of the existing theory for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov-Smirnov distance, and to the usual $$n \gg \log(p)$$ for certain approximately optimal bootstrap adjustments. New comparison and anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable.

• Eric Moulines

The Langevin MCMC: theory and methods.

In machine learning literature, a large number of problems amount to simulate a density which is log-concave (at least in the tails) and perhaps non smooth. Most of the research efforts so far has been devoted to the Maximum A posteriori problem, which amounts to solve a high-dimensional convex (perhaps non smooth) program. The purpose of this lecture is to understand how we can use ideas which have proven very useful in machine learning community to solve large scale optimization problems to design efficient sampling algorithms, with convergence guarantees (and possibly usable convergence bounds).

In high dimension, only first order method (exploiting exclusively gradient information) is a must. Most of the efficient algorithms know so far may be seen as variants of the gradient descent algorithms, most often coupled with partial updates (coordinates descent algorithms). This of course suggests to study methods derived from Euler discretization of the Langevin diffusion, which may be seen as a noisy version of the gradient descent. Partial updates may in this context as Gibbs steps where some components are frozen. This algorithm may be generalized in the non-smooth case by regularizing the objective function. The Moreau-Yosida inf-convolution algorithm is an appropriate candidate in such case, because it does not modify the minimum value of the criterion while transforming a non smooth optimization problem in a smooth one. We will prove convergence results for these algorithms with explicit convergence bounds both in Wasserstein distance and in total variation.

Joint work with Alain Durmus.