Purchase submodular functions and optimization, volume 47 1st edition. Submodular functions and optimization, volume 47 1st edition. In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. Journal of machine learning research open source software paper. Submodular programs unconstrained optimization 216 a minimizing submodular functions 217 b minimizing modular functions 223 7. Machine learning and convex optimization with submodular. Extended tutorial slides, updated july 6 2008 pdf, ppt software, materials and references. Submodular function minimization sfm is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in. In continuous domains, applying convex optimization techniques enables ef. We also prove negative results, in particular a query complexity result matching our algorithmic result in the symmetric case. Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including max cut in directedundirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Submodular functions and optimization second edition annals of discrete mathematicsseries editor.
Smkp is a natural extension of both multiple knapsack and the problem of monotone submodular maximization subject to a knapsack constraint. This often simpli es the problem, replacing complicated constraints. This is the first book devoted to the exposition of the theory of submodular functions from an elementary technical level to an advanced one. Curvature and optimal algorithms for learning and minimizing submodular functions rishabh iyer y, stefanie jegelka, jeff bilmes yuniversity of washington, dept. Maximizing submodular or monotone approximately submodular. Nov 28, 2011 submodular functions are relevant to machine learning for at least two reasons.
Curvature and optimal algorithms for learning and minimizing. While these results are wellknown in the optimization community, to our knowledge. You may be thinking you dont have to be a genius to make that inference so ill expand a little. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics, algorithmic computer science, and algorithmic economics, and. In this section we will see several examples of submodular functions and optimization. Nonetheless, it is known that if f gis monotone then minf. Online submodular minimization for combinatorial structures. The lovasz extensions of submodular functions 211 7. These techniques allow to minimize efficiently certain energy functions corresponding to pairwise markov random fields mrfs. Unlike the case of minimization, maximizing a submodular functions is nphard even in the unconstrained setting. Submodular optimization and approximation algorithms satoru iwata rims, kyoto university submodular functions cut capacity functions. Our new algorithms are based on simple iterative methods for the minimumnorm problem, such as the conditional gradient and the fujishigewolfe algorithms. Moreover, analogous to the fast methods for convex function minimization, it turns out that submodular functions can also be minimized fast, viz.
Submodular functions are relevant to machine learning for at least two reasons. Submodular functions and optimization volume 58 annals of. According to lov asz lov asz, 1983a submodular functions occur in many mathematical models in economics, engineering, science, management science, and other sciences. In section 3 we present three classes of these problems. Safe element screening for submodular function minimization. Learning and optimization with submodular functions. We prove that this set function is submodular for some popular neural network text classifiers under simplifying assumption.
Discrete adversarial attacks and submodular optimization. Think about functions applied to a set of sets of elements, you want to find t. Let f1 and f2 be two set functions, with f2, but not f1, parametrized by a string of random bits r. We present sfo, a toolbox for use in matlab or octave that implements algorithms for minimization and maximization of submodular functions. Submodular functions and optimization sciencedirect.
Annals of discrete mathematics submodular functions and. I in order to produce product e2eit is necessary to set up the machines needed to manufacture e, and this costs money. Submodular functions and optimization download here. An introduction to submodular functions and optimization. Highperformance implementation of the minimum norm point algorithm for submodular function minimization with several applications matlab toolbox for submodular function optimization maintained by andreas krause. On the other hand, other authors have started realizing explicitly the value of submodular functions for summarization lin and bilmes, 2010. For instance max cut is a special case even when the function is required only to be nonnegative.
Such functions are also referred to as additive or modular. Creating submodular functions from other submodular functions is also possible, using sfo fn lincomb for nonnegative linear combinations, and sfo fn trunc for truncation. An online algorithm for maximizing submodular functions. Pdf learning and optimization with submodular functions. Custom submodular functions can be used either by inheriting from sfo fn, or by using the sfo fn wrapper function, which wraps a. Geometric rescaling algorithms for submodular function. Theneedsoftheapplications,andinparticularthesheer bulk of large data sets, have brought into focus the development of fast algorithms for. Motivating \business school example i suppose that you manage a factory that is capable of making any one of a large nite set eof products. Since the publication of the 1st edition of this book fifteen years ago, submodular functions have been showing further increasing importance in optimization, combinatorics, discrete mathematics, algorithmic computer science, and algorithmic economics, and there have been made remarkable developments of theory and algorithms in submodular. Several algorithms for submodular optimization described in this survey are. Submodular cuts and approximating submodular functions 217 used for obtaining all of our lower bounds. This is the first book devoted to the exposition of the theory of submodular functions from an elementary technical level to an advanc. For this reason, an optimization problem which concerns optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints.
I the setup cost isnonlinear, and it depends on which other products you choose to produce. In many naturally occurring optimization problems one needs to ensure that the definition of the optimization problem lends itself to. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. In order to make the book useful to a wider audience, the material on electrical networks and that on submodular functions are presented independently of each other. Bruce shepherd %b proceedings of the 36th international conference on machine learning %c proceedings of machine learning research %d 2019 %e kamalika chaudhuri %e ruslan salakhutdinov %f pmlrv97santiago19a %i pmlr %j proceedings of machine learning research. Considering that the objective function of many combinatorial optimization problems can be nonmonotone not necessarily symmetric or non submodular, we study the performance of eas on the general problem classes, maximizing submodular functions withwithout a size constraint and maximizing monotone approximately submodular functions with a. One class, which contains the location problem, arises from matroids, another from the assignment problem and a third from boolean polynomials. Streaming algorithms for submodular function maximization. Submodular functions and optimization pdf free download. Structured sparsityinducing norms relaxation of the penalization of supports by submodular functions.
Submodular programs constrained optimization 228 a lagrangian functions and optimality conditions 229 b related problems 234 b. Submodular functions and optimization 1st edition isbn. Algorithms and lower bounds for submodular cuts and title. Learnability, structure, and optimization mariaflorina balcanyand nicholas j.
While these algorithms only have pseudopolynomial running time. A toolbox for submodular function optimization the. Optimization techniques based on graph cuts have become a standard tool for many vision applications. Submodular functions and optimization ebook by satoru. Submodularity is a very natural property of various functions occurring in such models. Its a field dedicated to the optimization of submodular functions. Optimization of submodular functions tutorial lecture i jan vondrak1 1ibm almaden research center san jose, ca jan vondrak ibm almaden submodular optimization tutorial 1 1. Submodular functions and optimization, volume 58 2nd edition. We design several constantfactor approximation algorithms for maximization of nonnegative submodular functions. Pdf submodular functions are relevant to machine learning for at least two reasons. Submodular functions and optimization volume 58 annals. The concept of a submodular function in discrete optimization appears to be in several respects analogous to that of a convex function in continuous optimization.
The unconstrained problem can be shown to be inapproximable if it is allowed to be negative. Oct 31, 2016 maximizing submodular functions under cardinality constraints lies at the core of numerous data mining and machine learning applications, including data diversification, data summarization, and. Submodular functions arise in a variety of contexts, often in optimization settings. Fujishige, on the polyhedral approach to submodular functions. Those works are mainly on submodular function optimization. To motivate the representation of combinatorial optimization. It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optim. Read submodular functions and optimization second edition by satoru fujishige available from rakuten kobo. A class of submodular functions for document summarization. Submodular functions are discrete functions that model laws of diminishing returns and enjoy numerous algorithmic applications. In this paper we formulate the attacks with discrete input on a set function as an optimization task. More recently, there is a large interest in constrained submodular function optimization driven both by theoretical progress and a variety of applicationsincomputerscience. We present a new class of polynomialtime algorithms for submodular function minimization sfm, as well as a unified framework to obtain strongly polynomial sfm algorithms.
Submodular optimization and approximation algorithms. Submodular functions share many properties in common with convex functions, one of which. Optimization of submodular functions tutorial lecture i. Minimizing nonsubmodular functions with graph cuts a.
The present 2nd edition is still a unique book on submodular functions, which is essential to students and researchers interested in combinatorial optimization, discrete mathematics, and discrete algorithms in the fields of mathematics, operations research, computer science, and economics. Currently, there is an accepted view within the computer vision community that graph cuts can only be used for optimizing a limited class of. Submodular functions have properties which are very similar to convex and concave functions. A unifying view of the theory is shown by means of base polyhedra and duality for submodular and supermodular systems. In this monograph, we present the theory of submodular functions from a convex.
Indeed, limited prior work has shown improved results for constrained minimization. We demonstrate the relevance of these results to vision on the problem of binary texture restoration. The submodular sparsest and balanced cut problems considered in this paper use sub modular functions to generalize two wellstudied problems in computer science. Submodular functions and optimization by dylanbrunson issuu. Submodular functions and optimization by satoru fujishige. An analysis of approximations for maximizing submodular set. Submodular function maximization is a central problem in combinatorial optimization, generalizing many im. It has widely been recognized that submodular functions play essential roles in efficiently solvable combinatorial optimization problems. Jun 16, 20 submodular functions and optimization download here. Maximizing nonmonotone submodular functions under matroid. Submodular functions and optimization by fujishige, s. Submodular minimization nonsmooth convex optimization parallel algorithm for special case 3.
While these results are wellknown in the optimization community, to our knowledge they were not used in the context of computer vision and mrf optimization. For example, the problem of finding a maximum weight independent. Submodular functions are relevant to machine learning for mainly two reasons. A tutorial script illustrates the application of submodularity to machine learning and ai problems such as feature selection, clustering, inference and optimized information gathering. The importance of submodular functions has been widely recognized in recent years in combinatorial optimization. Submodular programs constrained optimization 228 a lagrangian functions and optimality conditions 229 b. Finally, anyone looking for an additional level of abstraction from submodular functions should look at murotas monograph on. Several hard combinatorial optimization problems can be posed in this framework. They have been used in many areas, including combinatorial optimization, machine learning, and.863 180 974 373 1007 1210 1265 52 942 174 1002 683 599 822 700 210 1465 448 4 801 1255 1279 915 1291 659 1418 1213 508 910 1549 1254 1545 1506 5 984 323 741 1022 676 259 196 968 830 239