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Submodularity for Distributed Sensing Problems Zeyn Saigol IR Lab, School of Computer Science University of Birmingham 6th July 2010 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 http://www.submodularity.org/) IRLab, 6 Jul 2010 Submodularity 2/16 Set functions Submodularity in AI has been popularised by Andreas Krause and Carlos Guestrin Finite set V = {1,2,…,n} Function F: 2V R Example: F(A) = IG(XA; Y) = H(Y) – H(Y | XA) = y,xA P(xA) [log P(y | xA) – log P(y)] Y Y “Sick” “Sick” X1 X2 X2 X3 “Fever” “Rash” “Rash” “Male” F({X1,X2}) = 0.9 F({X2,X3}) = 0.5 IRLab, 6 Jul 2010 Submodularity 3/16 Submodular set functions Set function F on V is called submodular if For all A,B V: F(A)+F(B) F(AB)+F(AB) + + A A BB AB Equivalent diminishing returns characterization: Submodularity: B A + S Large improvement + S Small improvement For AB, sB, F(A {s}) – F(A) F(B {s}) – F(B) IRLab, 6 Jul 2010 Submodularity 4/16 Example problem: sensor coverage Place sensors Possible in building locations V Node predicts For A V: F(A) = “area values of positions covered by sensors placed at A” with some radius Formally: W finite set, collection of n subsets Si W For A V={1,…,n} define F(A) = |iA Si| IRLab, 6 Jul 2010 Submodularity 5/16 Set coverage is submodular A={S1,S2} S1 S2 S’ S’ F(A{S’})-F(A) ≥ F(B{S’})-F(B) S1 S2 S3 S4 S’ B = {S1,S2,S3,S4} IRLab, 6 Jul 2010 Submodularity 6/16 Approximate maximization Given: finite set V, monotonic submodular function F(A) Want: A* V such that NP-hard! Y “Sick” Greedy algorithm: M Start with A0 = {}; X1 X2 X3 For i = 1 to k “Fever” “Rash” “Male” si := argmaxs F(Ai-1{s})-F(Ai-1) Ai := Ai-1{si} IRLab, 6 Jul 2010 Submodularity 7/167 Guarantee on greedy algorithm Theorem [Nemhauser et al ‘78] Given a monotonic submodular function F, F()=0, the greedy maximization algorithm returns Agreedy F(Agreedy) (1-1/e) max|A| k F(A) ~63% Sidenote: Greedy algorithm gives 1/2 approximation for maximization over any matroid C! [Fisher et al ’78] IRLab, 6 Jul 2010 Submodularity 8/16 Example: Submodularity of info-gain Y1,…,Ym, X1, …, Xn discrete RVs F(A) = IG(Y; XA) = H(Y)-H(Y | XA) F(A) is always monotonic However, NOT always submodular Theorem [Krause & Guestrin UAI’ 05] If Xi are all conditionally independent given Y, then F(A) is submodular! Y1 Y2 Y3 Hence, greedy algorithm works! X1 X2 X3 X4 In fact, NO practical algorithm can do better than (1-1/e) approximation! 9 IRLab, 6 Jul 2010 Submodularity 9/16 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)=XAC(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 F ( A X ) F ( A) C( X ) Finally choose best set A Maintains (1 − 1/e) approximation guarantee IRLab, 6 Jul 2010 Submodularity 10/16 Path planning maxA F(A) or maxA mini Fi(A) subject to So far: |A| k In practice, more complex constraints: Sensors need to communicate Locations need to be (form a routing tree) connected by paths [Krause et al., IPSN 06] [Chekuri & Pal, FOCS ’05] [Singh et al, IJCAI ’07] Lake monitoring Building monitoring IRLab, 6 Jul 2010 Submodularity 11/16 Informative path planning So far: max F(A) s.t. |A| k Most informative locations Robot needs to travel might be far apart! between selected locations Locations V nodes in a graph s4 2 C(A) = cost of cheapest path 2 1 s2 s1 1 1 s5 connecting nodes A 1 s3 s10 1 s11 1 max F(A) s.t. C(A) B IRLab, 6 Jul 2010 Submodularity 12/16 The pSPIEL Algorithm [K, Guestrin, Gupta, Kleinberg IPSN ‘06] pSPIEL: Efficient nonmyopic algorithm (padded Sensor Placements at Informative and cost- Effective Locations) Select starting and ending location s1 and sB g1,2 g2,2 g1,1 g2,1 Decompose sensing region into g1,3 g2,3 C1 C2 small, well-separated clusters S1 Solve cardinality constrained SB problem per cluster (greedy) g4,3 g3,1 g3,3 Combine solutions using C4 orienteering algorithm g4,1 g4,4 C3 g3,2 g4,2 g3,4 Smooth resulting path IRLab, 6 Jul 2010 Submodularity 13/16 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 2010 Submodularity 14/16 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(XA; XVnA)) Clustering MAP inference in Markov Random Fields IRLab, 6 Jul 2010 Submodularity 15/16 Summary Submodularity is useful! IRLab, 6 Jul 2010 Submodularity 16/16

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