Submodular Functions: from Discrete to Continous Domains

TL;DR

This paper extends the theory of submodular functions from discrete set-functions to continuous domains, establishing convexity properties, optimization equivalences, and practical algorithms for minimization.

Contribution

It generalizes submodularity-convexity links to continuous domains, introduces new convex extensions, and develops algorithms with convergence guarantees.

Findings

Submodular functions can be extended to convex functions on probability measures.

Minimizing submodular functions is equivalent to solving convex optimization problems.

New algorithms for submodular minimization with proven convergence rates.

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.