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Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints (SCSC/ SCSK) Rishabh Iyer Jeff Bilmes University of Washington, Seattle

  1. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints (SCSC/ SCSK) Rishabh Iyer Jeff Bilmes University of Washington, Seattle NIPS-2013 MELODI M achin E L earning, O ptimization, & D ata I nterpretation @ UW Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 0 / 20

  2. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Outline Introduction to Submodular Functions 1 Problem Formulation of SCSC/ SCSK 2 Algorithmic Framework 3 Empirical Results 4 Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 1 / 20

  3. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Set functions f : 2 V → R { , } , , , V = , , , , V is a finite “ground” set of objects. A set function f : 2 V → R produces a value for any subset A ⊆ V . Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 2 / 20

  4. } Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Set functions f : 2 V → R { , } , , A = , For example, f ( A ) = 22, Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 2 / 20

  5. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  6. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  7. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Gain = 1 Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  8. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Gain = 1 Gain = 0 Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  9. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Gain = 1 Gain = 0 Monotonicity: f ( A ) ≤ f ( B ) , if A ⊆ B . Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  10. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Set Functions Special class of set functions. f ( A ∪ v ) − f ( A ) ≥ f ( B ∪ v ) − f ( B ) , if A ⊆ B (1) Gain = 1 Gain = 0 Monotonicity: f ( A ) ≤ f ( B ) , if A ⊆ B . Modular function f ( X ) = � i ∈ X f ( i ) analogous to linear functions. Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 3 / 20

  11. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Two Sides of Submodularity Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 4 / 20

  12. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Two Sides of Submodularity Submodular Minimization Solve min { f ( X ) | X ⊆ V } . Polynomial-time. Relation to convexity. Models cooperation. Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 4 / 20

  13. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Two Sides of Submodularity Submodular Maximization Submodular Minimization Solve max { g ( X ) | X ⊆ V } . Solve min { f ( X ) | X ⊆ V } . Constant-factor approximable. Polynomial-time. Relation to concavity. Relation to convexity. Models diversity and coverage. Models cooperation. Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 4 / 20

  14. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Two Sides of Submodularity Submodular Maximization Submodular Minimization Solve max { g ( X ) | X ⊆ V } . Solve min { f ( X ) | X ⊆ V } . Constant-factor approximable. Polynomial-time. Relation to concavity. Relation to convexity. Models diversity and coverage. Models cooperation. Sometimes we want to simultaneously maximize coverage/ diversity ( g ) while minimizing cooperative costs ( f ). Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 4 / 20

  15. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Two Sides of Submodularity Submodular Maximization Submodular Minimization Solve max { g ( X ) | X ⊆ V } . Solve min { f ( X ) | X ⊆ V } . Constant-factor approximable. Polynomial-time. Relation to concavity. Relation to convexity. Models diversity and coverage. Models cooperation. Sometimes we want to simultaneously maximize coverage/ diversity ( g ) while minimizing cooperative costs ( f ). Often these naturally occur as budget or cover constraints (for example, maximize diversity subject to a budget constraint on the submodular cost). Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 4 / 20

  16. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  17. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  18. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  19. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). We introduce the following, which is often more natual anyway: Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  20. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). We introduce the following, which is often more natual anyway: Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  21. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). We introduce the following, which is often more natual anyway: Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  22. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). We introduce the following, which is often more natual anyway: While DS optimization is NP hard to approximate, SCSC and SCSK however, retain approximation guarantees! Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  23. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Submodular Optimization with Submodular Constraints Historically: DS optimization Unfortunately, NP hard to approximate (Iyer-Bilmes’12). We introduce the following, which is often more natual anyway: While DS optimization is NP hard to approximate, SCSC and SCSK however, retain approximation guarantees! Throughout this talk, assume f and g are monotone. Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 5 / 20

  24. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results Our Main Contributions Show how SCSC/SCSK subsume a number of important optimization problems. Provide a unifying algorithmic framework for these. Provide a complete characterization of the hardness of these problems. Emphasize the scalability and practicality of some of our algorithms! Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 6 / 20

  25. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results I - Submodular Set Cover and Submodular Knapsack Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 7 / 20

  26. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results I - Submodular Set Cover and Submodular Knapsack Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 7 / 20

  27. Submodular Functions Problem Formulation Algorithmic Framework Empirical Results I - Submodular Set Cover and Submodular Knapsack Data Subset Selection Sensor Placement (Wei et al’13) Document Summarization (Krause et al’08) (Lin-Bilmes’11) Iyer & Bilmes, 2013 (UW, Seattle) SCSC/SCSK NIPS-2013 7 / 20

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