SFO: A Toolbox for Submodular Function Optimization

In recent years, a fundamental problem structure has emerged as very useful in a variety of machine learning applications: Submodularity is an intuitive diminishing returns property, stating that adding an element to a smaller set helps more than adding it to a larger set. Similarly to convexity, submodularity allows one to efficiently find provably (near-) optimal solutions for large problems. We present SFO, a toolbox for use in MATLAB or Octave that implements algorithms for minimization and maximization of submodular functions. A tutorial script illustrates the application of submodularity to machine learning and AI problems such as feature selection, clustering, inference and optimized information gathering

Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201

Structured Sparsity: Discrete and Convex approaches

Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated {\em structured} sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure