Submodular functions: from discrete to continuous domains

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.

Near-optimal discrete optimization for experimental design: a regret minimization approach

“…One notable exception is the D-optimality f D : Σ → (det Σ) −1/p , which does not satisfy (B1) because f D is not convex. For this particular objective, it is a well-established practice to consider the negative log-determinant log f D : Σ → − 1 p log det Σ, which is convex (see for example [9]). This motivates us to consider an alternative set of assumptions that concern the log f function: Input: function min ω F λ (ω) defined in Eq (3.2); its Lipschitz constant L λ ; and T number of iterations.…”

Near-optimal Approximate Discrete and Continuous Submodular Function Minimization

“…In order to extend our results to the domain [k] n , we use the continuous extension of a submodular function f developed by Bach [Bac19], which is the analogue of the Lovasz extension. We show that our algorithms extend to this setting.…”

“…Previous work [Bac19] has considered more general domains for submodular functions, instead of the standard {0, 1} n . Such a domain that the definition of submodularity can be extended to is functions f : [k] n → R. We call a function f : [k] n → R submodular if for all x, y ∈ [k] n we have that…”

Algorithmic Social Intervention

“…Submodularity captures diminishing returns and appears in application domains ranging from viral marketing [23], to machine learning [28], to auction theory [46]. We analyze submodular functions in two settings: Continuous: Continuous submodularity, which has lately received increasing attention [4,5,42] generalizes the notion of a submodular set function to continuous domains. Many well-known discrete problems (e.g., sensor placement, influence maximization, or facility location) admit natural extensions where resources are divided in a continuous manner.…”