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Page 1: Near Optimal Bayesian Active Learning for Decision Making · Presentation. Decision Making Setting Hypotheses Prior probability distribution over the set of hypotheses Tests Decision

Near Optimal Bayesian Active Learning for Decision

Making

Page 2: Near Optimal Bayesian Active Learning for Decision Making · Presentation. Decision Making Setting Hypotheses Prior probability distribution over the set of hypotheses Tests Decision
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simplified setting: think of one of the following six animals

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N

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N

N

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Y

N

N

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Y

N

N

Y

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Y

N

N

N

Y

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Y

Y

Y

Y Y

N

N

N

N

N

(optimal decision tree problem)

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Robotics Query: localization trajectory

Response: distance until contact

Output: button-push trajectory

Javdani, Shervin. "Gathering Information For Decision Making In Touch Based Localization". Presentation.

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Setting

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hypothesis space

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hypothesis space

hypothesis

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hypothesis space

hypothesis

prior probability

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true hypothesis

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test

outcome

hypotheses map tests to outcomes

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run test

observe outcome

rule out inconsistent hypotheses

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run test

observe outcome

rule out inconsistent hypotheses

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continue running tests...

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run test

observe outcome

rule out inconsistent hypotheses

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true hypothesis

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evidence

consistent hypotheses

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choose tests in a systematic way in order to maximally reduce uncertainty in the hypothesis

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decision region

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decision region space

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→

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equivalence class determination (ECD) problem

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multiple hypotheses per decision region, non-disjoint

decision region determination (DRD) problem

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●●●●

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run test

observe outcome

rule out inconsistent hypotheses

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run test

observe outcome

rule out inconsistent hypotheses

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continue running tests...

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run test

observe outcome

rule out inconsistent hypotheses

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make this decision

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choose tests in a systematic way in order to drive uncertainty in the hypothesis toward a single decision

use policy to choose which tests to run

a policy maps the current evidence to the next test to run (or to stop running tests)

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a feasible policy eventually finds the decision region containing the correct hypothesis; it drives all remaining uncertainty in the hypotheses

toward a single decision region

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expected cost of feasible policy

optimal policy

is the set of tests chosen by policy when is the correct hypothesis

(the policy satisfies feasibility)

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all subregions are contained in the blue decision region

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3

For proof, see Section 7.1 from the paper

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41

23

4

For proof, see Section 7.1 from the paper

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The Hyperedge Cutting (HEC) Algorithm

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set of consistent hyperedges, given evidence

Theorem 1

All consistent hypotheses lie in some decision region if and only if all hyperedges are cut, i.e.,

to make a decision, we need to cut all hyperedges in the splitting hypergraph

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run tests rule out all hypotheses in

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run tests rule out all hypotheses in

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run tests rule out all hypotheses in

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find the policy that cuts all hyperedges with minimum cost(by running the minimum number of tests)

choose the test that cuts the most hyperedges in expectation

Goal:

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for a subregion to be consistent, at least one of its hypotheses must be consistent

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for a hyperedge to be consistent, all of its subregions must be consistent

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total weight of a collection of hyperedges

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number of tests

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Hyperedge Cutting (HEC) Algorithm

greedy

How does the greedy policy compare with the optimal policy?

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For proof, see Lemma 2 in Section 7.3 of the paper.

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asking a question (i.e. going deeper in the tree) cannot hurt you

worst case: gain no new information

can always recover previous classification, no matter what boundary we draw

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For proof, see Lemma 3 in Section 7.3 of the paper.

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we cannot gain more information from a question by moving it deeper in the tree

a classification boundary cannot get longer by adding other classification boundaries first

shortersame length

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where

Golovin, Daniel, and Andreas Krause. "Adaptive submodularity: Theory and applications in active learning and stochastic optimization." Journal of Artificial Intelligence Research (2011): 427-486.

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Demo

http://mybinder.org/repo/gvanhorn38/nobal
http://mybinder.org/repo/gvanhorn38/nobal
Page 74: Near Optimal Bayesian Active Learning for Decision Making · Presentation. Decision Making Setting Hypotheses Prior probability distribution over the set of hypotheses Tests Decision

RequirementsTo run HEC you will need to define:

● A discrete set of hypotheses.● Prior probabilities for the hypotheses.● A set of decision regions.● A discrete set of deterministic tests.

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“Naive” Implementation

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“Naive” Implementation This can be computed once per iteration.

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“Naive” Implementation

To compute this, you have to construct the splitting hypergraph for each test, hypothesis pair

This can be computed once per iteration.

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“Naive” Implementation

To compute this, you have to construct the splitting hypergraph for each test, hypothesis pair

for t in tests: for h in hypotheses: nodes, hyperedges = build_splitting_hypergraph(...) weight = compute_weight_of_hyperedges(...)

Page 79: Near Optimal Bayesian Active Learning for Decision Making · Presentation. Decision Making Setting Hypotheses Prior probability distribution over the set of hypotheses Tests Decision

“Naive” Implementation

for t in tests: for h in hypotheses: nodes, hyperedges = build_splitting_hypergraph(...) weight = compute_weight_of_hyperedges(...)

This requires enumerating every multiset of order k and checking if any decision region contains all the subregions �

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“Efficient” implementation

Intuition:

Compute the weight of all multisets (i.e. hyperedges) and iteratively subtract off the weight of invalid multisets (i.e. hyperedges whose subregions are contained within one decision region).

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“Efficient” implementation

Computing a sum of multisets, where a multiset corresponds to a product, is equivalent to computing a complete homogeneous symmetric polynomial (CHP).

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“Efficient” implementation

Subregions (nodes of our splitting hypergraph)

All multisets over group G of cardinality k̂

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“Efficient” implementation

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“Efficient” implementation

Computing a sum of multisets, where a multiset corresponds to a product, is equivalent to computing a complete homogeneous symmetric polynomial (CHP).

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“Efficient” implementation

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Efficient Implementation

��

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“Efficient” implementation

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“Efficient” implementation

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W =

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W -=

W -=

W -=

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W =

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W -=

W -=

W -=

W -=

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W =

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“Efficient” implementation

�

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Computational ComplexityThis algorithm is good at minimizing the number of tests, but at a high computational cost. The computational bottleneck for HEC lies in the construction of the hypergraph:

The computation is exponential in hyperedge cardinality k.

If your tests are expensive, then perhaps you are fine with waiting.

If your tests are cheap, then using a simpler algorithm that conducts more tests could run much faster than HEC.

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Computational ComplexityThere is a follow up paper1 where the authors attempt to mitigate this computational cost:

“... our algorithm is exponentially faster than HEC in theory, significantly faster (often by orders of magnitude) in practice, while offering similar empirical performance.”

1Chen, Yuxin, et al. "Submodular Surrogates for Value of Information." AAAI. 2015.

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Robotic Decision Making Application

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Decision Making Task

Javdani, Shervin. "Gathering Information For Decision Making In Touch Based Localization". Presentation.

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Decision Making SettingHypotheses

Prior probability distribution over the set of hypotheses

Tests

Decision regions

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Decision Making SettingHypotheses: object locations

Prior probability distribution over the set of hypotheses

Tests

Decision regions

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Decision Making SettingHypotheses: object locations

Prior probability distribution over the set of hypotheses

Tests: guarded moves

Test outcomes: distance until contact

Decisions regions

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Decision Making SettingHypotheses: object locations

Prior probability distribution over the set of hypotheses

Tests: guarded moves

Test outcomes: distance until contact

Decisions: button-push moves

Decision regions: sets of hypotheses on which decisions will succeed

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Given: Hypotheses

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true hypothesisGiven: Hypotheses

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Given: Prior Probabilities

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Given: Decisions

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Given: Decision Regions

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Hypergraph: Nodes

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Hypergraph: Cardinality

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Hypergraph: Hyperedges

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DRD: Tests

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make this decision

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HypothesesSample randomobject locations (2000)

Uncertainty due to noise fromsensors, inaccurate models,calibration error

Javdani, Shervin. "Gathering Information For Decision Making In Touch Based Localization". Presentation.

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TestsSample random start locations and orientations (150)

Move along path until contact is sensed

h(t) = o = distance travelled

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DecisionsSet of start locations (50)

Javdani, Shervin. "Gathering Information For Decision Making In Touch Based Localization". Presentation.

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Decision Regions

Javdani, Shervin. "Gathering Information For Decision Making In Touch Based Localization". Presentation.

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Javdani, Saeed, Matthew Klingensmith, J. Andrew Bagnell, Nancy S. Pollard, and Siddhartha S. Srinivasa. "Efficient touch based localization through submodularity." In Robotics and Automation (ICRA), 2013 IEEE International Conference on, pp. 1828-1835. IEEE, 2013.

http://www.youtube.com/watch?v=_HiyKKDStBE
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End

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Simon, Tong. Active Learning: Theory and Applications. Stanford, 2001.http://www.robotics.stanford.edu/~stong/papers/tong_thesis.pdf

http://www.robotics.stanford.edu/~stong/papers/tong_thesis.pdf
http://www.robotics.stanford.edu/~stong/papers/tong_thesis.pdf
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Results

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