Randomized Sensing in Adversarial Environments
Andreas Krause
Joint work with Daniel Golovin and Alex Roper
International Joint Conference on Artificial Intelligence 2011
Motivation
Want to manage sensing resources to enable robust monitoring under uncertainty
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Roboticenvironmental
monitoring
Detectsurvivors after
disaster
Coordinatecameras to
detect intrusions
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Select two cameras to query, in order to detect the most people.
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People Detected:
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Duplicates only counted
once
A Sensor Selection Problem
Set V of sensors, |V| = nSelect a set of k sensors Sensing quality model
NP-hard…
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A Sensor Selection Problem
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SubmodularityDiminishing returns property for adding more sensors.
Many objectives are submodular [K, Guestrin ‘07]Detection, coverage, mutual information, and others
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+1
For all , and a sensor ,
Greedy algorithm
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Lets choose sensors S = {v1 , … , vk} greedily
[Nemhauser et al ‘78] If F is submodular, greedy algorithm gives constant factor approx.:
i Fi({3}) Fi({5})1 0 12 1 0
Sensing in Adversarial Environments
Set I of m intrusion scenariosFor scenario i: Fi(A) is sensing utility when selecting A
Intruder chooses worst-case scenario, knowing the sensors7
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Deterministic minimax solution
One approach: Want to solve
[K, McMahan, Guestrin, Gupta ’08]:NP-hardGreedy algorithm fails arbitrarily badlySATURATE algorithm provides near-optimal solution
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Disadvantage of minimax approach
Suppose we pick {3} and {5} with probability 1/2
Randomization can perform arbitrarily better!
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1
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i Fi({3}) Fi({5})1 1 02 0 1
The randomized sensing problem
Given submodular functions F1,…,Fm, want to find
NP-hard!
Even representing the optimal solution may require exponential space!
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Existing approachesMany techniques for solving matrix games
Typically don’t scale to combinatorially large strategy sets
Security games [Tambe et al]Solve large scale Stackelberg games for security applicationsCannot capture general submodular objective functions
LP based approach [Halvorson et al ‘09]Double oracle with approximate best responseNo polynomial time convergence convergence guarantee
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Randomized sensingDefine
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Thus, can minimize over q instead of over p!
Distributionover sensing
actions
Distribution over intrusions
Equivalent problem: Finding q*Want to solve
Use multiplicative update algorithm [Freund & Schapire ‘99]
InitializeFor t = 1:T
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NP-hard But submodular!
The RSENSE algorithm
Initialize
For t=1:TUse greedy algorithm to compute
based on objective function
Update
Return
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Performance guarantee
Theorem: Let Suppose RSENSE runs for iterations. For the resulting distribution it holds that
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Handling more general constraintsSo far: wanted
Many application may require more complex constraints:
Examples:Informative path planning:Controlling PTZ cameras:Nonuniform cost:
Can replace greedy algorithm by - best response RSENSE guarantees
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Example: Lake monitoringMonitor pH values using robotic sensor
Position s along transect
pH v
alue
Observations A
True (hidden) pH values
Prediction at unobservedlocations
transect
Where should we sense to minimize our maximum error?
Use probabilistic model(Gaussian processes)
to estimate prediction error
(often) submodular[Das & Kempe ’08]
Var(s | A)
Experimental results
Randomized sensing outperforms deterministic solutions 18
Running time
RSENSE outperforms existing LP based method 19
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pSPIEL Results: Search & RescueMap from Robocup Research Challenge
Coordination of multiple mobile sensors to detect survivors of major urban disasterBuildings obstruct viewfield of cameraFi(A) = Expected # of people detected at location i
Detection Range
Detected Survivors
Experimental results
Randomization outperforms deterministic solutionRSENSE finds solution faster than existing methods
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Worst- vs. average caseGiven: Possible locations V, submodular functions F1,…,Fm
Average-case score Worst-case score
Strong assumptions! Very pessimistic!
Want to optimize both average- and worst-case score!
Can modify RSENSE to solve this problem!Compute best response to
Tradeoff results
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Wors
t case
sco
re
Average case score
Knee intradeoff
curve
Search &rescue Wors
t case
sco
reAverage case score
Envtl. monitoring
Can find good compromise between average- and worst-case score!
ConclusionsWish to find randomized strategy for maximizing an adversarially-chosen submodular functionDeveloped RSENSE, which provides near-optimal performancePerforms well on two real applications
Search and rescueEnvironmental monitoring
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