Submodular Functions: from Discrete to Continous Domains

Francis Bach 1, 2, *
* Auteur correspondant
2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, ENS Paris - École normale supérieure - Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : Submodular set-functions have many applications in combinatorial optimization, as they can be minimized and approximately maximized in polynomial time. A key element in many of the algorithms and analyses is the possibility of extending the submodular set-function to a convex function, which opens up tools from convex optimization. Submodularity goes beyond set-functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains, where it corresponds essentially to cross second-derivatives being nonpositive. In this paper, we show that most results relating submodularity and convexity for set-functions can be extended to all submodular functions. In particular, (a) we naturally define a continuous extension in a set of probability measures, (b) show that the extension is convex if and only if the original function is submodular, (c) prove that the problem of minimizing a submodular function is equivalent to a typically non-smooth convex optimization problem, and (d) propose another convex optimization problem with better computational properties (e.g., a smooth dual problem). Most of these extensions from the set-function situation are obtained by drawing links with the theory of multi-marginal optimal transport, which provides also a new interpretation of existing results for set-functions. We then provide practical algorithms to minimize generic submodular functions on discrete domains, with associated convergence rates.
Type de document :
Pré-publication, Document de travail
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Contributeur : Francis Bach <>
Soumis le : mardi 23 février 2016 - 17:26:05
Dernière modification le : jeudi 11 janvier 2018 - 06:28:04


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  • HAL Id : hal-01222319, version 2
  • ARXIV : 1511.00394



Francis Bach. Submodular Functions: from Discrete to Continous Domains. 2016. 〈hal-01222319v2〉



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