Randomized Sensing in Adversarial Environments
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Randomized Sensing in Adversarial Environments
Andreas Krause
Joint work with Daniel Golovin and Alex Roper
International Joint Conference on Artificial Intelligence 2011
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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
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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 ,
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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.:
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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 AIntruder chooses worst-case scenario, knowing the sensors
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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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i Fi({3}) Fi({5})1 1 02 0 1
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The randomized sensing problemGiven 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
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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!
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The RSENSE algorithmInitialize
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)
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Experimental results
Randomized sensing outperforms deterministic solutions 18
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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
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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
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Tradeoff results
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Wor
st c
ase
scor
e
Average case score
Knee intradeoff
curve
Search &rescue Wor
st c
ase
scor
eAverage case score
Envtl. monitoring
Can find good compromise between average- and worst-case score!
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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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