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Non-myopic Informative Path Planning in Spatio-Temporal
Models
Alexandra Meliou
Andreas Krause
Carlos Guestrin
Joe Hellerstein
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Collection Tours
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Approximate Queries Approximate representation of the world:
Discrete locations Lossy communication Noisy measurements
Applications do not expect accurate values (tolerance to noise)
Monitored phenomena usually demonstrate strong correlationsCorrelation makes approximation cheap
Example: Return the temperature at all locations ±1C, with 95% confidence
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Optimizing Information
: sensing nodes on path
Approximate answers
Search for most informative paths
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Continuous Queries
Repeated at periodic intervals Finite horizon
Example: Return the temperature at all locations ±1C, with 95% confidence,
every 10 minutes for the next 5 hours.
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Myopic vs Nonmyopic tradeoff
Myopic approach: repeat optimization for every timestep
Timestep 1Timestep 2
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Myopic vs Nonmyopic tradeoff
Nonmyopic approach: optimize for all timesteps
Timestep 1Timestep 2 No work! Extra node
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Quantify Informativeness
Entropy [Shewry & Wynn ‘87]
Mutual Information [Caselton & Zidek ‘84]
Reduction of predictive variance [Chaloner & Verdinelli ‘95]
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Measuring Information
1
4
3
5
2
Observing 1 gives information on 3 and 4
Observing 2 gives information on 3 and 5
After observing 2, observing 3 becomes less useful
Diminishing Returns
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Submodular Functions
)()()()( BFXBFAFXAF −∪≥−∪BA⊆
BA
X
X
+
+
More reward
Less reward
Entropy, mutual information and reduction of predictive variance are all submodular.
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Non-myopic Spatio-Temporal Path Planning (NSTP)
Given: A collection of submodular functions ft
• ft only depends on data collected at times 1..t
A set of accuracy constraints kt
Find: A collection of paths Pt with
( )( ) ttt
T
ttP
kPfts
PCP
≥
= ∑=
:1
1
*
..
minarg
Minimize cost
Subject to reward constraints
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Planning for multiple timesteps
Harder than planning for one
First idea : Solve an equivalent single step problem
instead!
obviously
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Nonmyopic Planning Graph
t=1 t=2 t=3
A solution path on the NPG = collection of paths for multiple timesteps
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Solve the single step problem
NP hard No good known approximation guarantees
Dual: Submodular Orienteering Problem
€
P* = argmaxP f P( )
s.t. C P( ) ≤ B
€
P* = argminP C P( )
s.t. f P( ) ≥ K
dual: primal:Maximize reward
Subject to budget constraints
Minimize cost
Subject to reward constraints
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Good News
The dual algorithm [Chekuri & Pal ’05] provides an O(logn) factor approximation
€
f ( ˆ P ) ≥f (OPT)
logn
(where n is the size of the network)
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Covering Algorithm
Transform a dual blackbox solution to a primal solution
€
P* = argmaxP f P( )
s.t. C P( ) ≤ B
€
P* = argminP C P( )
s.t. f P( ) ≥ K
dual:
primal:
Reward required to “cover”
(with α approximation factor)
Call with BOPT
Return solution with reward ≥K/α
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CoveringAlgorithm
Transform a dual blackbox solution to a primal solution
Reward required to “cover”
• Call SOP for increasing budgets
• Guaranteed to cover K/α reward when called for BOPT
• Update chosen set and repeat for uncovered reward
• Terminate when ε portion left
Guaranteed to use at most budget
€
2logε
log 1−1
α
⎛
⎝ ⎜
⎞
⎠ ⎟BOPT
• Call for budget 1 : insufficient reward• Call for budget 2• Call for budget BOPT: reward sufficient!
uncovered reward
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Bad News
On the unrolled graph the Chekuri-Pal guarantee becomes O(log(nT))
The running time on the unrolled graph is O((BnT)log(nT))
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Addressing Computation Complexity
DP Algorithm Algorithm details in proceedings Bug in proof of guarantees. Not fixed (yet)
New algorithm: Nonmyopic Greedy Details on my webpage… Guaranteed to provide O(logn) approximation
Better than the previous O(log(nT))
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Approach
Replace expensive blackbox, with cheaper blackbox
Covering transformation
Chekuri-Pal
SOP on NPG
Blackbox for dual
Nonmyopic greedy
algorithm
Blackbox for dual
More efficient:Nonmyopic greedy calls the dual on the smaller network graph instead of the unrolled
graph
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Nonmyopic Greedy
Time
Bud
get
dual(b,Gt)
R = 2C = 1
R = 1C = 1
R = 1C = 1
R = 3C = 2
R = 5C = 4
R = 4C = 2
R = 3C = 2
R = 6C = 4
R = 3C = 2
R = 5C = 3
R = 4C = 3
R = 5C = 4
budget
P1
Cost = 2
Time = 2
Best greedy choice condition on A1
A2
R = 2C = 1
R = 1C = 1
R = 1C = 1
R = 2C = 2
X
R = 1C = 2
R = 1C = 2
XX X XX
P2
Cost = 1
Time = 1
R = 0C = 1
R = 0C = 1
R = 1C = 1
X X XP3
Cost = 1
Time = 3
Best ratio R/C1. Condition on picked data2. Recompute matrix
A1
Return best of A1, A2
dual(budget=4,time=1)
dual(budget=1,time=3)
For border cases were A1 is bad, A2 is guaranteed to be good
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Nonmyopic Greedy Guarantees
Nonmyopic greedy Chekuri-Pal on NPG
O(B2T(nB)logn) O((nBT)log(nT))
runn
ing
tim
eap
prox
imat
ion
€
f (P) ≥1− e−1
2log nf (OPT)
€
f (P) ≥1
log(nT)f (OPT)
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Myopic and Nonmyopic evaluation
Varying Constraints
Setup: 46 nodes on the Intel Berkeley Lab deployment 7 days of data (5 for learning, 2 for testing)
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Cost and Runtime
Varying Horizon
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Effect of greedy parameters
Varying budget levels
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Conclusions Transform any blackbox solution to
nonmyopic
Obtain primal from dual
Nonmyopic greedy provides significant runtime improvements and better theoretical guarantees