# Using entropy to drive search in occupancy grids - Zeyn Saigol

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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(AB)+F(AB)

+                                     +
A A BB
AB
   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)
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) = |iA 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)=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

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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