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Fast Semidifferential-Based Submodular Function Optimization Fast Semidifferential-based Submodular Function Optimization: Extended Version Rishabh Iyer [email protected] University of Washington, Seattle, WA 98195, USA Stefanie Jegelka [email protected] University of California, Berkeley, CA 94720, USA Jeff Bilmes [email protected] University of Washington, Seattle, WA 98195, USA Abstract family of feasible solution sets. The set C could express, We present a practical and powerful new for example, that solutions must be an independent set framework for both unconstrained and con- in a matroid, a limited budget knapsack, or a cut (or strained submodular function optimization spanning tree, path, or matching) in a graph. Without based on discrete semidifferentials (sub- and making any further assumptions about f, the above super-differentials). The resulting algorithms, problems are trivially worst-case exponential time and which repeatedly compute and then efficiently moreover inapproximable. optimize submodular semigradients, offer new If we assume that f is submodular, however, then in and generalize many old methods for sub- many cases the above problems can be approximated modular optimization. Our approach, more- and in some cases solved exactly in polynomial time. A over, takes steps towards providing a unifying function f : 2V ! R is said to be submodular (Fujishige, paradigm applicable to both submodular min- 2005) if for all subsets S; T ⊆ V , it holds that f(S) + imization and maximization, problems that f(T ) ≥ f(S [ T ) + f(S \ T ). Defining f(jjS) , f(S [ historically have been treated quite distinctly. j) − f(S) as the gain of j 2 V with respect to S ⊆ V , The practicality of our algorithms is impor- then f is submodular if and only if f(jjS) ≥ f(jjT ) tant since interest in submodularity, owing to for all S ⊆ T and j2 = T . Traditionally, submodularity its natural and wide applicability, has recently has been a key structural property for problems in been in ascendance within machine learning. combinatorial optimization, and for applications in We analyze theoretical properties of our al- econometrics, circuit and game theory, and operations gorithms for minimization and maximization, research. More recently, submodularity's popularity in and show that many state-of-the-art maxi- machine learning has been on the rise. mization algorithms are special cases. Lastly, we complement our theoretical analyses with On the other hand, a potential stumbling block is supporting empirical experiments. that machine learning problems are often large (e.g., \big data") and are getting larger. For general uncon- strained submodular minimization, the computational 1. Introduction complexity often scales as a high-order polynomial. These algorithms are designed to solve the most general In this paper, we address minimization and maximiza- case and the worst-case instances are often contrived tion problems of the following form: and unrealistic. Typical-case instances are much more benign, so simpler algorithms (e.g., graph-cut) might Problem 1: min f(X); Problem 2: max f(X) X2C X2C suffice. In the constrained case, however, the problems often become NP-complete. Algorithms for submod- V where f : 2 ! R is a discrete set function on subsets ular maximization are very different in nature from V of a ground set V = f1; 2; ··· ; ng, and C ⊆ 2 is a their submodular minimization cohorts, and their com- th plexity too varies depending on the problem. In any Proceedings of the 30 International Conference on Ma- case, there is an urgent need for efficient, practical, and chine Learning, Atlanta, Georgia, USA, 2013. JMLR: W&CP volume 28. Copyright 2013 by the author(s). scalable algorithms for the aforementioned problems if Fast Semidifferential-based Submodular Function Optimization submodularity is to have a lasting impact on the field restating the problem as a submodular vertex cover, of machine learning. i.e., Problem 1 where C is the set of all vertex covers in a graph. In this paper, we address the issue of scalability and simultaneously draw connections across the apparent gap between minimization and maximization problems. Clustering: Variants of submodular minimization We demonstrate that many algorithms for submodu- have been successfully applied to clustering problems lar maximization may be viewed as special cases of a (Narasimhan et al., 2006; Nagano et al., 2010). generic minorize-maximize framework that relies on discrete semidifferentials. This framework encompasses Limited Vocabulary Speech Corpora: The prob- state-of-the-art greedy and local search techniques, and lem of finding a maximum size speech corpus with provides a rich class of very practical algorithms. In bounded vocabulary (Lin & Bilmes, 2011a) can be addition, we show that any approximate submodular posed as submodular function minimization subject maximization algorithm can be seen as an instance of to a size constraint. Alternatively, cardinality can our framework. be treated as a penalty, reducing the problem to un- constrained submodular minimization (Jegelka et al., We also present a complementary majorize-minimize 2011). framework for submodular minimization that makes two contributions. For unconstrained minimization, we obtain new nontrivial bounds on the lattice of minimiz- Size constraints: The densest k-subgraph and size- ers, thereby reducing the possible space of candidate constrained graph cut problems correspond to submod- minimizers. This method easily integrates into any ular minimization with cardinality constraints, prob- other exact minimization algorithm as a preprocessing lems that are very hard (Svitkina & Fleischer, 2008). step to reduce running time. In the constrained case, Specialized algorithms for cardinality and related con- we obtain practical algorithms with bounded approx- straints were proposed e.g. in (Svitkina & Fleischer, imation factors. We observe these algorithms to be 2008; Nagano et al., 2011). empirically competitive to more complicated ones. Minimum Power Assignment: In wireless net- As a whole, the semidifferential framework offers a new works, one seeks a connectivity structure that maintains unifying perspective and basis for treating submodu- connectivity at a minimum energy consumption. This lar minimization and maximization problems in both problem is equivalent to finding a suitable structure the constrained and unconstrained case. While it has (e.g., a spanning tree) minimizing a submodular cost long been known (Fujishige, 2005) that submodular function (Wan et al., 2002). functions have tight subdifferentials, our results rely on a recently discovered property (Iyer & Bilmes, 2012a; Jegelka & Bilmes, 2011b) showing that submodular Transportation: Costs in real-world transportation functions also have superdifferentials. Furthermore, our problems are often non-additive. For example, it may approach is entirely combinatorial, thus complementing be cheaper to take a longer route owned by one carrier (and sometimes obviating) related relaxation methods. rather than a shorter route that switches carriers. Such economies of scale, or \right of usage" properties are captured in the \Categorized Bottleneck Path Prob- 2. Motivation and Background lem" { a shortest path problem with submodular costs Submodularity's escalating popularity in machine learn- (Averbakh & Berman, 1994). Similar costs have been ing is due to its natural applicability. Indeed, instances considered for spanning tree and matching problems. of Problems 1 and 2 are seen in many forms, to wit: Summarization/Sensor placement: Submodular MAP inference/Image segmentation: Markov maximization also arises in many subset extraction Random Fields with pairwise attractive potentials are problems. Sensor placement (Krause et al., 2008), docu- important in computer vision, where MAP inference ment summarization (Lin & Bilmes, 2011b) and speech is identical to unconstrained submodular minimization data subset selection (Lin & Bilmes, 2009), for example, solved via minimum cut (Boykov & Jolly, 2001). A are instances of submodular maximization. richer higher-order model can be induced for which MAP inference corresponds to Problem 1 where V is Determinantal Point Processes: The Determi- a set of edges in a graph, and C is a set of cuts in this nantal Point Processes (DPPs) which have found nu- graph | this was shown to significantly improve many merous applications in machine learning (Kulesza & image segmentation results (Jegelka & Bilmes, 2011b). Taskar, 2012) are known to be log-submodular distri- Moreover, Delong et al.(2012) efficiently solve MAP butions. In particular, the MAP inference problem is inference in a sparse higher-order graphical model by a form of non-monotone submodular maximization. Fast Semidifferential-based Submodular Function Optimization Indeed, there is strong motivation for solving Problems 1 and 2 but, as mentioned above, these problems come not without computational difficulties. Much work has therefore been devoted to developing optimal or near optimal algorithms. Among the several algorithms (McCormick, 2005) for the unconstrained variant of Problem 1, where C = 2V , the best complexity to date is O(n5γ + n6)(Orlin, 2009)(γ is the cost of evaluating f). This has motivated studies on faster, possibly special case or approximate, methods (Stobbe Figure 1. Illustration of the chain of sets and permutation & Krause, 2010; Jegelka et al., 2011). Constrained sigma minimization problems, even for simple constraints such as a cardinality lower bound, are mostly NP-hard, and not approximable to
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