Using entropy to drive search in occupancy grids - Zeyn Saigol

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					Submodularity for Distributed Sensing

                 Zeyn Saigol
       IR Lab, School of Computer Science
            University of Birmingham

                  6th July 2010

     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 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
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                    Submodular set functions
   Set function F on V is called submodular if
        For all A,B  V: F(A)+F(B)  F(AB)+F(AB)

                    +                                     +
                     A A BB
   Equivalent diminishing returns characterization:

Submodularity:          B A                    + S   Large improvement
                                               + S   Small improvement

For AB, sB, F(A {s}) – F(A)  F(B {s}) – F(B)
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               Example problem: sensor coverage
Place sensors
in building

  Node predicts                 For A  V: F(A) = “area
values of positions          covered by sensors placed at A”
 with some radius
                       W finite set, collection of n subsets Si  W
                       For A  V={1,…,n} define F(A) = |iA Si|
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                    Set coverage is submodular

             S1          S2

                                    S’                  F(A{S’})-F(A)


              S1         S2

                                     S’         B = {S1,S2,S3,S4}
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                    Approximate maximization

  Given: finite set V, monotonic submodular function F(A)
  Want:     A*  V such that

                    NP-hard!                      Y
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}

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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 Agreedy
           F(Agreedy)  (1-1/e) max|A| k F(A)


                                   Sidenote: Greedy algorithm gives
                                   1/2 approximation for
                                   maximization over any matroid C!
                                   [Fisher et al ’78]

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             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
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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AC(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

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                             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
                               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
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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
                                                                  Decompose sensing region into
           g1,3                                       g2,3
                  C1                             C2               small, well-separated clusters
   S1                                                             Solve cardinality constrained
                                                                  problem per cluster (greedy)
                   g4,3          g3,1            g3,3
                                                                  Combine solutions using
                   C4                                             orienteering algorithm
g4,1       g4,4                                  C3
                   g4,2                   g3,4                    Smooth resulting path

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                         Limitations of pSPEIL

      Requires locality property – far apart observations
       are independent
      Adaptive algorithm [Singh, Krause, Kaiser
              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

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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(XA; XVnA))
              Clustering
              MAP inference in Markov Random Fields

IRLab, 6 Jul 2010                    Submodularity            15/16

        Submodularity is useful!

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