## OptAzur seminars : Emmanuel Soubies et Luc Pronzato

lundi 3 juin 2024

The next and last (before the summer) session of the OptAzur seminars will take place on Monday June 3rd 2024 at 2pm in the conference room 007 of the I3S laboratory in Sophia-Antipolis.

We will have the pleasure to welcome Emmanuel Soubies (CNRS, IRIT) and Luc Pronzato (CNRS, i3S), please see below for titles and abstracts.

**14h** - Emmanuel Soubies (CNRS, IRIT, ENSEEIHT)

**Exact Continuous Relaxations of L0-Regularized Generalized Linear Models**

Sparse generalized linear models are widely used in fields such as statistics, computer vision, signal/image processing and machine learning. The natural sparsity promoting regularizer is the l0 pseudo-norm which is discontinuous and non-convex. In this talk, we will present the l0-Bregman relaxation (B-Rex), a general framework to compute exact continuous relaxations of such l0-regularized criteria. Although in general still non-convex, these continuous relaxations are qualified as exact in the sense that they let unchanged the set of global minimizer while enjoying a better optimization landscape. In particular, we will show that some local minimizers of the initial functional are eliminated by these relaxations. Finally, these properties will be illustrated on both sparse Kullback-Leibler regression and sparse logistic regression problems.

**15h15** - Luc Pronzato (CNRS, I3S)

**Safe screening rules in convex optimisation: some examples in optimal design of experiments**

Many convex optimisation problems can be formulated as the minimisation of a convex function of a probability measure over a given set. Typical examples include determining the ellipsoid of minimum volume, or the smallest ball, containing a set of points. When it is known in advance that the optimal measure will be supported by a small number of points, it is advantageous to eliminate unnecessary points (candidates) in order to simplify the problem. Safe screening rules aim to eliminate such points: a rule defines a test to be applied to the candidates in order to eliminate those that are useless; a rule is safe when no point supporting an optimal measure is eliminated. The aim is to apply the screening rule during optimisation, regardless of the optimisation algorithm used. Usually, the efficiency of elimination increases when approaching the optimum, so the rule should be applied several (many) times and be as simple as possible. In addition to the construction of ellipsoids and balls of minimal volume, I will present the construction of screening rules for different criteria in optimal design of experiments, some of them with a link to (quadratic) Lasso. Parts of this work are based on collaborations with Radoslav Harman (Comenius University, Bratislava) and Guillaume Sagnol (TU Berlin).