Newton methods for structured nonsmooth optimization


Speciality : Mathématiques Appliquées

2/12/2022 - 13:45 Gilles Bareilles (Univ. Grenoble Alpes) Maison Jean Kuntzmann

Keywords :
  • statistical learning
  • manifold identification
In this talk, we discuss the optimization of structured nonsmooth functions. Such functions appear in machine learning and signal processing and encompass e.g., low-complexity regression problems, such as lasso, and matrix optimization problems, with eigenvalues and the nuclear norm. We exploit the structure of these nonsmooth functions to build algorithms that converge fast – at the speed of Newton's method, thus yielding high precision estimates of nonsmooth minimizers.

We consider successively functions that write as sums and compositions of smooth functions and simple nonsmooth functions. Their nondifferentiability points organize in smooth manifolds, such that the nonsmooth function is smooth along the manifold and nonsmooth across it. In both cases, we propose optimization algorithms that detect and exploit these structure manifolds. The two key ingredients in our approach are (i) subtle geometrical properties of the proximal operator, and (ii) algorithmic tools from smooth constrained programming. We operate without assuming knowledge of the optimal manifold, and use variational analysis tools to cover both convex and nonconvex settings. We provide numerical illustrations on classical problems.


Nadia Brauner ()


  • Jérôme Malick
  • Franck Iutzeler


  • Jalal Fadili
  • Claudia Sagastizábal


  • Jean-Charles Gilbert
  • Mathurin Massias