## Learning with Submodular Functions: A Convex Optimization Perspective

**Prof. Francis Bach**

*France, Ecole Normale Supérieure de Paris*

Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions, and (2) the Lovász extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In this talk, I will present the theory of sub- modular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, I will show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples

of submodular functions, I will review various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions.

of submodular functions, I will review various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions.

This talk will be based on the recent monograph: F. Bach. Learning with Submodular Functions: A Convex Optimization Perspective. Foundations and Trends in Machine Learning, 6(2-3):145-373, 2013.