1 adaptive submodularity: a new approach to active learning and stochastic optimization joint work...
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1California Institute of TechnologyCenter for the Mathematics of Information
Adaptive Submodularity:A New Approach to Active Learning and Stochastic
Optimization
Joint work with Andreas Krause
1California Institute of Technology Center for the Mathematics of Information
Daniel Golovin
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2California Institute of TechnologyCenter for the Mathematics of Information
Max K-Cover (Oil Spill Edition)
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3California Institute of TechnologyCenter for the Mathematics of Information
SubmodularityT
ime
Tim
e
Discrete diminishing returns property for set functions.
``Playing an action at an earlier stage only increases its marginal benefit''
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4California Institute of TechnologyCenter for the Mathematics of Information
The Greedy Algorithm
Theorem [Nemhauser et al ‘78]
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5California Institute of TechnologyCenter for the Mathematics of Information
Stochastic Max K-Cover
Asadpour et al. (`08): (1-1/e)-approx if sensors (independently) either work perfectly or fail completely.
Bayesian: Known failure distribution. Adaptive: Deploy a sensor and see what you get. Repeat K times.
0.5
0.2
0.3
At 1st location
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6California Institute of TechnologyCenter for the Mathematics of Information
Adaptive SubmodularityT
ime
Playing an action at an earlier stage only increases its marginal benefit
expected(taken over its outcome)
Gain moreGain less
(i.e., at an ancestor)
Select Item
StochasticOutcome
Adaptive Monotonicity:Δ(a | obs) ≥ 0, always
Δ(action | observations)
[G & Krause, 2010]
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7California Institute of TechnologyCenter for the Mathematics of Information
What’s it good for?
Allows us to generalize results to the adaptive realm, including:
(1-1/e)-approximation for Max K-Cover, submodular maximization
(ln(n)+1)-approximation for Set Cover
“Accelerated” implementation
Data-Dependent Upper Bounds on OPT
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8California Institute of TechnologyCenter for the Mathematics of Information
Recall the Greedy Algorithm
Theorem [Nemhauser et al ‘78]
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9California Institute of TechnologyCenter for the Mathematics of Information
The Adaptive-Greedy Algorithm
Theorem [G & Krause, COLT ‘10]
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10California Institute of TechnologyCenter for the Mathematics of Information
[Adapt-monotonicity] - -
( ) - [Adapt-submodularity]
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11California Institute of TechnologyCenter for the Mathematics of Information
…
The world-state dictates which path in the tree we’ll take.
1. For each node at layer i+1, 2. Sample path to layer j, 3. Play the resulting layer j action at layer i+1.
How to play layer j at layer i+1
By adapt. submod.,playing a layer earlieronly increases it’s marginal benefit
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12California Institute of TechnologyCenter for the Mathematics of Information
[Adapt-monotonicity] - -
( ) - ( ) -
[Def. of adapt-greedy]
( ) - [Adapt-submodularity]
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13California Institute of TechnologyCenter for the Mathematics of Information
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14California Institute of TechnologyCenter for the Mathematics of Information
2
1 3
Stochastic Max Cover is Adapt-Submod
1 3
Gain moreGain less
adapt-greedy is a (1-1/e) ≈ 63% approximation to the adaptive optimal solution.
Random sets distributedindependently.
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15California Institute of TechnologyCenter for the Mathematics of Information
Influence in Social Networks
Who should get free cell phones?V = {Alice, Bob, Charlie, Daria, Eric, Fiona}F(A) = Expected # of people influenced when targeting A
0.5
0.30.5 0.4
0.2
0.2 0.5
Prob. ofinfluencing
Alice
Bob
Charlie
Daria Eric
Fiona
[Kempe, Kleinberg, & Tardos, KDD `03]
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16California Institute of TechnologyCenter for the Mathematics of Information
Alice
Bob
Charlie
Daria Eric
Fiona0.5
0.30.5 0.4
0.2
0.2 0.5
Key idea: Flip coins c in advance “live” edgesFc(A) = People influenced under outcome c (set cover!)F(A) = c P(c) Fc(A) is submodular as well!
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17California Institute of TechnologyCenter for the Mathematics of Information
0.40.50.2
0.2Daria
Prob. ofinfluencing
Eric
Fiona0.5
0.30.5
Alice
Bob
Charlie
Adaptively select promotion targets, see which of their friends are influenced.
Adaptive Viral Marketing
?
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18California Institute of TechnologyCenter for the Mathematics of Information
Adaptive Viral Marketing
Alice
Bob
Charlie
DariaEric
Fiona0.5
0.30.5 0.4
0.2
0.2 0.5
Objective adapt monotone & submodular. Hence, adapt-greedy is a (1-1/e) ≈ 63%
approximation to the adaptive optimal solution.
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19California Institute of TechnologyCenter for the Mathematics of Information
Stochastic Min Cost Cover Adaptively get a threshold amount of value. Minimize expected number of actions. If objective is adapt-submod and
monotone, we get a logarithmic approximation.
[Goemans & Vondrak, LATIN ‘06][Liu et al., SIGMOD ‘08]
[Feige, JACM ‘98]
[Guillory & Bilmes, ICML ‘10]c.f., Interactive Submodular Set Cover
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20California Institute of TechnologyCenter for the Mathematics of Information
Optimal Decision Trees
x1
x2
x3
1
1
0
0 0
=
=
=
Garey & Graham, 1974; Loveland, 1985; Arkin et al., 1993; Kosaraju et al., 1999; Dasgupta, 2004; Guillory & Bilmes, 2009; Nowak, 2009; Gupta et al., 2010
“Diagnose the patient as cheaply as possible (w.r.t. expected cost)”
1
10
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21California Institute of TechnologyCenter for the Mathematics of Information
Objective = probability mass of hypotheses you have ruled out.
It’s Adaptive Submodular.
Outcome = 1Outcome = 0
Test x
Test wTest v
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22California Institute of TechnologyCenter for the Mathematics of Information
Generate upper bounds on Use them to avoid some evaluations.
Accelerated Greedy
time
Saved evaluations
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23California Institute of TechnologyCenter for the Mathematics of Information
Generate upper bounds on Use then to avoid some evaluations.
Accelerated Greedy
Empirical Speedups we obtained:
- Temperature Monitoring: 2 - 7x
- Traffic Monitoring: 20 - 40x
- Speedup often increases with
instance size.
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24California Institute of TechnologyCenter for the Mathematics of Information
Ongoing work Active learning with noise
With Andreas Krause & Debajyoti Ray, to appear NIPS ‘10
Edges between any twodiseases in distinct groups
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25California Institute of TechnologyCenter for the Mathematics of Information
Active Learning of Groups via Edge Cutting
Edge Cutting Objective is Adaptive Submodular
First approx-result for noisy observations
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26California Institute of TechnologyCenter for the Mathematics of Information
Conclusions
New structural property useful for design & analysis of adaptive algorithms
Recovers and generalizes many known results in a unified manner. (We can also handle costs)
Tight analyses & optimal-approx factors in many cases. “Accelerated” implementation yields significant speedups.
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x1
x2
x3
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0 010.5
0.30.5 0.4
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0.2 0.5
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27California Institute of TechnologyCenter for the Mathematics of Information
Q&A2
1 30.5
0.30.5 0.4
0.2
0.2 0.5
x1
x2
x3
1
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0
0 01