Selecting Observations against Adversarial Objectives

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Selecting Observations against Adversarial
Objectives

• Andreas Krause
• Brendan McMahan
• Carlos Guestrin
• Anupam Gupta

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Observation selection problems
Detectcontaminationsin water networks
Place sensors forbuilding automation
Monitor rivers, lakes using robots

• Set V of possible observations (sensor
  locations,..)
• Want to pick subset A µ V such that
• For most interesting utilities F, NP-hard!

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Key observation Diminishing returns
Placement B S1,, S5
Placement A S1, S2
Adding S will help a lot!
Adding S doesnt help much
New sensor S
Formalization Submodularity For A µ B, F(A
S) F(A) F(B S) F(B)
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Submodularitywith Guestrin, Singh, Leskovec,
VanBriesen, Faloutsos, Glance

• We prove submodularity for
• Mutual information F(A) H(unobs) H(unobsA)
• UAI 05, JMLR 07 (Spatial prediction)
• Outbreak detection F(A) Impact reduction
  sensing A
• KDD 07 (Water monitoring, )
• Also submodular
• Geometric coverage F(A) area covered
• Variance reduction F(A) Var(Y) Var(YA)

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Why is submodularity useful?
Greedy Algorithm(forward selection)

• Theorem Nemhauser et al 78
• Greedy algorithm gives constant factor
  approximation
• F(Agreedy) (1-1/e) F(Aopt)
• Can get online (data dependent) bounds for any
  algorithm
• Can significantly speed up greedy algorithm
• Can use MIP / branch bound for optimal solution

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Robust observation selection

• What if
• parameters ? of model P(XV j ?) unknown /
  change?
• sensors fail?
• an adversary selects the outbreak scenario?

Morevariabilityhere now?new
Best placement forparameters ?old
Attackhere!
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Robust prediction
Confidencebands
pH value
Horizontal positions V
Low average variance (MSE) but high maximum (in
most interesting part!)
Typical objective Minimize average variance
(MSE)

• Instead minimize width of the confidence bands
• For every location s 2 V, define Fs(A) Var(s)
  Var(sA)
• Minimize width ? simultaneously maximize all
  Fs(A)
• Each Fs(A) is (often) submodular! Das Kempe
  07

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Adversarial observation selection

• Given
• Possible observations V,
• Submodular functions F1,,Fm
• Want to solve
• Can model many problems this way
• Width of confidence bands Fi is variance at
  location i
• unknown parameters Fi is info-gain with
  parameters ?i
• adversarial outbreak scenarios Fi is utility for
  scenario i
• Unfortunately, mini Fi(A) is not submodular ?

One Fi foreach location i
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How does greedy do?
Set A F1 F2 mini Fi
x 1 0 0
y 0 2 0
z ? ? ?
x,y 1 2 1
x,z 1 ? ?
y,z ? 2 ?
Greedy picks z first
Optimalsolution(k2)
Then, canchoose onlyx or y

• ? Greedy does arbitrarily badly. Is there
  something better?

Theorem The problem maxA k mini F(A) does
not admit any approximation unless PNP ?
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Alternative formulation

• If somebody told us the optimal value,
• can we recover the optimal solution A?
• Need to solve dual problem
• Is this any easier?
• Yes, if we relax the constraint A k

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Solving the alternative problem

• Trick For each Fi and c, define truncation

Fi(A)
Fi(A)
A
Set F1 F2 F1 F2 Favg,1 mini Fi
x 1 0 1 0 ½ 0
y 0 2 0 1 ½ 0
z ? ? ? ? ? ?
x,y 1 2 1 1 1 1
x,z 1 ? 1 ? (1?)/2 ?
y,z ? 2 ? 1 (1?)/2 ?
Lemma
mini Fi(A) c ? Favg,c(A) c
Favg,c(A) is submodular!
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Why is this useful?

• Can use the greedy algorithm to find
  (approximate) solution!
• Proposition Greedy algorithm finds
• AG with AG ? k and Favg,c(AG) c
• where ? 1log maxs ?i Fi(s)

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Back to our example
Set F1 F2 mini Fi Favg,1
x 1 0 0 ½
y 0 2 0 ½
z ? ? ? ?
x,y 1 2 1 1
x,z 1 ? ? (1?)/2
y,z ? 2 ? (1?)/2

• Guess c1
• First pick x
• Then pick y
• ? Optimal solution! ?
• How do we find c?

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Submodular Saturation Algorithm

• Given set V, integer k and functions F1,,Fm
• Initialize cmin0, cmax mini Fi(V)
• Do binary search c (cmincmax)/2
• Use greedy algorithm to find AG such that
  Favg,c(AG) c
• If AG gt ? k decrease cmax
• If AG ? k increase cmin
• until convergence

AG ? k? c too low
AG gt ? k? c too high
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Theoretical guarantees
Theorem The problem maxA k mini F(A) does
not admit any approximation unless PNP ?
Theorem Saturate finds a solution AS such
that mini Fi(AS) OPTk and AS ?
k where OPTk maxA k mini Fi(A) ?
1 log maxs ?i Fi(s)

• Theorem If there were a polytime algorithm
  with better constant ? lt ?, then NPµ DTIME(nlog
  log n)

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Experiments

• Minimizing maximum variance in GP regression
• Robust biological experimental design
• Outbreak detection against adversarial
  contaminations
• Goals
• Compare against state of the art
• Analyze appropriateness ofworst-case assumption

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Spatial prediction
better
Environmental monitoring
Precipitation data

• Compare to state of the art Sacks et.al. 88,
  Wiens 05,
• Highly tuned simulated annealing heuristics (7
  parameters)
• Saturate is competitive faster, better on
  larger problems

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Maximum vs. average variance
better
Environmental monitoring
Precipitation data

• Minimizing the worst-case leads to good
  average-case score, not vice versa

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Outbreak detection
better
Water networks
Water networks

• Results even more prominent on water network
  monitoring (12,527 nodes)

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Robust experimental design

• Learn parameters ? of nonlinear function
• yi f(xi,?) w
• Choose stimuli xi to facilitate MLE of ?
• Difficult optimization problem!
• Common approach linearization!
• yi ¼ f(xi,?0) rf?0(xi)T (?-?0) w
• Allows nice closed form (fractional) solution! ?
• How should we choose ?0??

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Robust experimental design

• State-of-the-art Flaherty et al., NIPS 06
• Assume perturbation on Jacobian rf?0(xi)
• Solve robust SDP against worst-case perturbation
• Minimize maximum eigenvalue of estimation error
  (E-optimality)
• This paper
• Assume perturbation of initial parameter estimate
  ?0
• Use Saturate to perform well against all initial
  parameter estimates
• Minimize MSE of parameter estimate(Bayesian
  A-optimality, typically submodular!)

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Experimental setup

• Estimate parameters of Michaelis-Menten model (to
  compare results)
• Evaluate efficiency of designs

Loss of optimal design, knowing true parameter
?true
Loss of robust design, assuming (wrong) initial
parameter ?0
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Robust design results
A
B
C
A
B
C
better
Low uncertainty in ?0
High uncertainty in ?0

• Saturate more efficient than SDP if optimizing
  for high parameter uncertainty

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Future (current) work

• Incorporating complex constraints (communication,
  etc.)
• Dealing with large numbers of objectives
• Constraint generation
• Improved guarantees for certain objectives
  (sensor failures)
• Trading off worst-case and average-case scores

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Conclusions

• Many observation selection problems require
  optimizing adversarially chosen submodular
  function
• Problem not approximable to any factor!
• Presented efficient algorithm Saturate
• Achieves optimal score, with bounded increase in
  cost
• Guarantees are best possible under reasonable
  complexity assumptions
• Saturate performs well on real-world problems
• Outperforms state-of-the-art simulated annealing
  algorithms for sensor placement, no parameters to
  tune
• Compares favorably with SDP based solutions for
  robust experimental design