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Efficient Strategies for Minimizing Submodular Functions and Approximating Maximization

This paper discusses methods for efficiently minimizing submodular functions, which can be solved in polynomial time, and highlights the challenges in maximizing convex functions, which are NP-hard. It emphasizes approximation guarantees for maximizing submodular functions, with practical applications such as set cover problems and sensor placement in buildings. Various examples illustrate the submodularity property, particularly in feature selection for probabilistic models. The paper combines theoretical insights with practical algorithms, providing bounds and guarantees for various optimization scenarios.

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Efficient Strategies for Minimizing Submodular Functions and Approximating Maximization

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  1. Maximizing submodular functions Minimizing convex functions: Polynomial time solvable! Minimizing submodular functions: Polynomial time solvable! Maximizing convex functions: NP hard! Maximizing submodular functions: NP hard! But can get approximation guarantees 

  2. Example: Set cover Want to cover floorplan with discs Place sensorsin building Possiblelocations V For A µ V: z(A) = “area covered by sensors placed at A” Node predicts values of positions with some radius Formally: W finite set, collection of n subsets Siµ W For A µ V={1,…,n} define z(A) = |i2 A Si|

  3. S’ S’ Set cover is submodular A={S1,S2} S1 S2 z(A[{S’})-z(A) ¸ z(B[{S’})-z(B) S1 S2 S3 S4 B = {S1,S2,S3,S4}

  4. Y“Sick” X1“Fever” X2“Rash” X3“Male” Uncertaintyafter knowing XA Uncertaintybefore knowing XA Example: Feature selection • Given random variables Y, X1, … Xn • Want to predict Y from subset XA = (Xi1,…,Xik) Want k most informative features: A* = argmax IG(XA; Y) s.t. |A| · k where IG(XA; Y) = H(Y) - H(Y | XA) Naïve BayesModel

  5. Y1 Y2 Y3 X1 X2 X3 X4 Example: Submodularity of info-gain Y1,…,Ym, X1, …, Xn discrete RVs z(A) = IG(Y; XA) = H(Y)-H(Y | XA) • z(A) is always monotonic • However, NOT always submodular Theorem [Krause & Guestrin UAI’ 05]If Xi are all conditionally independent given Y,then z(A) is submodular! Hence, greedy algorithm works! In fact, NO algorithm can do better than (1-1/e) approximation!

  6. Leanforward Leanleft Slouch Building a Sensing Chair [Mutlu, Krause, Forlizzi, Guestrin, Hodgins UIST ‘07] • People sit a lot • Activity recognition inassistive technologies • Seating pressure as user interface Equipped with 1 sensor per cm2! Costs $16,000!  Can we get similar accuracy with fewer, cheaper sensors? 82% accuracy on 10 postures! [Tan et al]

  7. How to place sensors on a chair? • Sensor readings at locations V as random variables • Predict posture Y using probabilistic model P(Y,V) • Pick sensor locations A* µ V to minimize entropy: Possible locations V Placed sensors, did a user study: Similar accuracy at <1% of the cost!

  8. Offline (Nemhauser)bound 1.4 Data-dependentbound 1.2 1 0.8 0.6 Greedysolution 0.4 0.2 0 0 5 10 15 20 Bounds on optimal solution[Krause et al., J Wat Res Mgt ’08] Submodularity gives data-dependent bounds on the performance of any algorithm Sensing quality z(A) Higher is better Water networksdata Number of sensors placed

  9. Summary (1) • Minimization of submodular functions • Submodularity and convexity • Submodular Polyhedron • Symmetric submodular functions

  10. Summary (2) • Pseudo-boolean functions • Representation (polynomial, posiform, tableau, graph cut) • Reduction to quadratic polynomial • Necessary and sufficient conditions for submodularity • Minimization of quadratic and cubic submodular functions via graph cuts • Lower bound via roof duality • LP via posiform representation • LP via linear relaxation • Max flow via symmetric graph construction

  11. Further reading • Combinatorial algorithms for submodular (and bisubmodular) function minimization • More algorithms/bounds for maximizing submodular functions • Linear and semidefinite relaxations • Matroids, greedoids, intersection of matroids, polymatroids and more • Generalized roof duality

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