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Submodularity for Distributed Sensing Problems Zeyn Saigol IR Lab, School of Computer Science University of Birmingham 6 th July 2010

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Outline 1. What is submodularity? 2. Non-myopic maximisation of information gain 3. Path planning 4. Robust maximisation and minimisation 5. Summary * These slides are borrowed from Andreas Krause and Carlos Guestrin (see IRLab, 6 Jul 2010Submodularity2/16

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Set functions Submodularity in AI has been popularised by Andreas Krause and Carlos Guestrin Finite set V = {1,2,…,n} Function F: 2 V R Example: F(A) = IG(X A ; Y) = H(Y) – H(Y | X A ) = y,x A P(x A ) [log P(y | x A ) – log P(y)] IRLab, 6 Jul 2010 Submodularity Y “Sick” X 1 “Fever” X 2 “Rash” F({X 1,X 2 }) = 0.9 Y “Sick” X 2 “Rash” X 3 “Male” F({X 2,X 3 }) = 0.5 3/16

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Set function F on V is called submodular if For all A,B V: F(A)+F(B) F(A B)+F(A B) Equivalent diminishing returns characterization: Submodular set functions IRLab, 6 Jul 2010Submodularity S B A S + + Large improvement Small improvement For A B, s B, F(A {s}) – F(A) F(B {s}) – F(B) Submodularity: B A A B ABAB + + 4/16

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Example problem: sensor coverage IRLab, 6 Jul 2010Submodularity Node predicts values of positions with some radius For A V: F(A) = “area covered by sensors placed at A” Formally: W finite set, collection of n subsets S i W For A V={1,…,n} define F(A) = | i A S i | Possible locations V Place sensors in building 5/16

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Set coverage is submodular IRLab, 6 Jul 2010Submodularity S1S1 S2S2 S1S1 S2S2 S3S3 S4S4 S’ A={S 1,S 2 } B = {S 1,S 2,S 3,S 4 } F(A {S’})-F(A) F(B {S’})-F(B) S’ ≥ 6/16

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Approximate maximization IRLab, 6 Jul 2010Submodularity 7 Given: finite set V, monotonic submodular function F(A) Want: A * V such that NP-hard! Greedy algorithm: Start with A 0 = {} ; For i = 1 to k s i := argmax s F(A i-1 {s})-F(A i-1 ) A i := A i-1 {s i } Y “Sick” X 1 “Fever” X 2 “Rash” X 3 “Male” M 7/16

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Guarantee on greedy algorithm Theorem [Nemhauser et al ‘78] Given a monotonic submodular function F, F( )=0, the greedy maximization algorithm returns A greedy F(A greedy ) (1-1/e) max |A| k F(A) IRLab, 6 Jul 2010Submodularity ~63% Sidenote: Greedy algorithm gives 1/2 approximation for maximization over any matroid C! [Fisher et al ’78] 8/16

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Y 1,…,Y m, X 1, …, X n discrete RVs F(A) = IG(Y; X A ) = H(Y)-H(Y | X A ) F(A) is always monotonic However, NOT always submodular Theorem [Krause & Guestrin UAI’ 05] If X i are all conditionally independent given Y, then F(A) is submodular! Example: Submodularity of info-gain IRLab, 6 Jul 2010Submodularity 9 Y1Y1 X1X1 Y2Y2 X2X2 Y3Y3 X4X4 X3X3 Hence, greedy algorithm works! In fact, NO practical algorithm can do better than (1-1/e) approximation! 9/16

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Information gain with cost Instead of each sensor having the same measurement cost, variable cost C(X) for each node Aim: max F(A) s.t. C(A) B where C(A)= X A C(X) In this case, construct every possible 3-element subset of V, and run greedy algorithm on each Greedy algorithm selects additional nodes X by maximising Finally choose best set A Maintains (1 − 1/e) approximation guarantee IRLab, 6 Jul 2010Submodularity10/16

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Path planning IRLab, 6 Jul 2010Submodularity max A F(A) or max A min i F i (A) subject to So far: |A| k In practice, more complex constraints: Locations need to be connected by paths [Chekuri & Pal, FOCS ’05] [Singh et al, IJCAI ’07] Lake monitoring Sensors need to communicate (form a routing tree) [Krause et al., IPSN 06] Building monitoring 11/16

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Informative path planning IRLab, 6 Jul 2010Submodularity So far: max F(A) s.t. |A| k s1s1 s2s2 s4s4 s5s5 s3s s 10 s Most informative locations might be far apart! Robot needs to travel between selected locations Locations V nodes in a graph C(A) = cost of cheapest path connecting nodes A max F(A) s.t. C(A) B 12/16

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The pSPIEL Algorithm [K, Guestrin, Gupta, Kleinberg IPSN ‘06] IRLab, 6 Jul 2010Submodularity pSPIEL: Efficient nonmyopic algorithm (padded Sensor Placements at Informative and cost- Effective Locations) Select starting and ending location s 1 and s B Decompose sensing region into small, well-separated clusters Solve cardinality constrained problem per cluster (greedy) Combine solutions using orienteering algorithm Smooth resulting path C1C1 C2C2 C3C3 C4C4 S1S1 SBSB g 4,2 g 4,1 g 2,2 g 2,1 g 3,1 g 3,2 g 3,3 g 1,1 g 1,2 g 1,3 g 2,3 g 3,4 g 4,4 g 4,3 13/16

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Limitations of pSPEIL Requires locality property – far apart observations are independent Adaptive algorithm [Singh, Krause, Kaiser IJCAI‘09] Just re-plans on every timestep Often this is near-optimal; if it isn’t, have to add an adaptivity gap term to the objective function U to encourage exploration IRLab, 6 Jul 2010Submodularity14/16

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Submodular optimisation spectrum Maximization: A* = argmax F(A) Sensor placement Informative path planning Active learning … Optimise for worst case: Minimization: A* = argmin F(A) Structure learning (A* = argmin I(X A ; X V n A )) Clustering MAP inference in Markov Random Fields IRLab, 6 Jul 2010Submodularity15/16

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Summary Submodularity is useful! IRLab, 6 Jul 2010Submodularity16/16

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