Learning with Submodular Functions: A Convex Optimization Perspective

Bach, Francis

doi:10.1561/2200000039

Cited by 210 publications

(237 citation statements)

References 165 publications

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“…In this case, if one chooses λ = 1 our algorithm performsÕ ε n 3 /2 value queries, but onlyÕ ε (n) independence oracle queries. 3 However, if one chooses λ = k the algorithm performs onlyÕ ε (n) value queries while the number of independence oracle queries grows toÕ ε n 3 /2 . This allows flexibility when the two types of queries have different time complexities in the application at hand.…”

Section: Our Resultsmentioning

confidence: 99%

“…Many well-known problems in combinatorial optimization are in fact submodular maximization problems, including: Max Cut [23,26,28,30,40], Max DiCut [16,23,24], Generalized Assignment [9,11,18,20], Max k-Coverage [15,31], Max Bisection [2,21], and Max Facility Location [1,12,13]. Furthermore, practical applications of submodular maximization problems are common in social networks [25,29], vision [5,27], machine learning [32,33,34,35,36] (the reader is referred to a comprehensive survey by Bach [3]), and many other areas. Elegant algorithmic techniques were developed in the course of this line of research which achieved provable, and in some cases even tight, approximation guarantees.…”

Section: Introductionmentioning

confidence: 99%

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Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

2017

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Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently Badanidiyuru and Vondrák [4] were the first to explicitly look for such algorithms in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The algorithm of Badanidiyuru and Vondrák matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm performs.Our main result is a new algorithm for this general case which establishes a surprising tradeoff between two seemingly unrelated quantities: the number of value oracle queries and the number of matroid independence queries performed by the algorithm. Specifically, one can decrease the former by increasing the latter and vice versa, while maintaining the best possible approximation guarantee. Such a tradeoff is very useful since various applications might incur significantly different costs in querying the value and matroid independence oracles. Furthermore, in case the rank of the matroid is O(n c ), where n is the size of the ground set and c is an absolute constant smaller than 1, the total number of oracle queries our algorithm uses can be made to have a smaller magnitude compared to that needed by [4]. We also provide even faster algorithms for the well studied special cases of a cardinality constraint and a partition matroid independence constraint, both of which capture many real-world applications and have been widely studied both theorically and in practice.

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Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

2017

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“…At its turn, function w ψ(x n , y) was proven to be submodular in [8]. Given that the sum of submodular functions is also submodular [9], the proposition follows.…”

Section: Ii-b Learning With the V-jaune Lossmentioning

confidence: 89%

Minimum-Risk Structured Learning of Video Summarization

Hussein

Piccardi

2017

2017 IEEE International Symposium on Multimedia (ISM)

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Video summarization is an important multimedia task for applications such as video indexing and retrieval, video surveillance, human-computer interaction and video "storyboarding". In this paper, we present a new approach for automatic summarization of video collections that leverages a structured minimum-risk classifier and efficient submodular inference. To test the accuracy of the predicted summaries we utilize a recently-proposed measure (V-JAUNE) that considers both the content and frame order of the original video. Qualitative and quantitative tests over two action video datasets -the ACE and the MSR DailyActivity3D datasets -show that the proposed approach delivers more accurate summaries than the compared minimum-risk and syntactic approaches.

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“…For functions f : A × B → R with A, B ⊆ R n and f (a, ·) being either convex or concave and f (·, b) being either convex or concave for any a ∈ A and b ∈ B, tightest convex extensions have been characterized by Ballerstein (2013). Tightest convex extensions of pseudo-Boolean functions f : {0, 1} n → R are known as convex closures (Bach, 2013). Convex closures of submodular functions are Lovász extensions.…”

Section: Convex Extensionsmentioning

confidence: 99%

Convexification of Learning from Constraints

Shcherbatyi

Andres

2016

Lecture Notes in Computer Science

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Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, the problem is resistant to standard optimization techniques. We construct MIPs with the same solutions whose objective functions are convex. Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. Computing values of this tightest convex extension is NP-hard. However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. For some decompositions, common loss and regularization functions, we derive a closed form.

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Learning with Submodular Functions: A Convex Optimization Perspective

Cited by 210 publications

References 165 publications

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

Minimum-Risk Structured Learning of Video Summarization

Convexification of Learning from Constraints

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