Active LearningCOMS 6998-4:

Learning and Empirical Inference

Irina RishIBM T.J. Watson Research Center

2

Outline

Motivation Active learning approaches

Membership queries Uncertainty Sampling Information-based loss functions Uncertainty-Region Sampling Query by committee

Applications Active Collaborative Prediction Active Bayes net learning

3

Standard supervised learning modelGiven m labeled points, want to learn a classifier with misclassification rate <, chosen from a hypothesis class H with VC dimension d < 1.

VC theory: need m to be roughly d/, in the realizable case.

4

Active learning

In many situations – like speech recognition and document retrieval – unlabeled data is easy to come by, but there is a charge for each label.

What is the minimum number of labels needed to achieve the target error rate?

5

6

What is Active Learning?

Unlabeled data are readily available; labels are expensive

Want to use adaptive decisions to choose which labels to acquire for a given dataset

Goal is accurate classifier with minimal cost

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Active learning warning

Choice of data is only as good as the model itself Assume a linear model, then two data points are sufficient What happens when data are not linear?

8

Active Learning Flavors

Selective Sampling Membership Queries

Pool Sequential

Myopic Batch

9

Active Learning Approaches

Membership queries Uncertainty Sampling Information-based loss functions Uncertainty-Region Sampling Query by committee

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12

Problem

Many results in this framework, even for complicated hypothesis classes.

[Baum and Lang, 1991] tried fitting a neural net to handwritten characters.Synthetic instances created were incomprehensible to humans!

[Lewis and Gale, 1992] tried training text classifiers.“an artificial text created by a learning algorithm is unlikely to be a legitimate natural language expression, and probably would be uninterpretable by a human teacher.”

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14

15

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Uncertainty Sampling

Query the event that the current classifier is most uncertain about

Used trivially in SVMs, graphical models, etc.

x x x x x x xxxx

If uncertainty is measured in Euclidean distance:

[Lewis & Gale, 1994]

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Information-based Loss Function

Maximize KL-divergence between posterior and prior

Maximize reduction in model entropy between posterior and prior

Minimize cross-entropy between posterior and prior

All of these are notions of information gain

[MacKay, 1992]

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Query by Committee

Prior distribution over hypotheses Samples a set of classifiers from distribution Queries an example based on the degree of

disagreement between committee of classifiers

[Seung et al. 1992, Freund et al. 1997]

x x x x x x xxxx

A B C

Infogain-based Active Learning

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Notation

We Have:

1. Dataset, D

2. Model parameter space, W

3. Query algorithm, q

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t Sex Age Test A Test B Test C Disease

0 M 40-50 0 1 1 ?1 F 50-60 0 1 0 ?2 F 30-40 0 0 0 ?3 F 60+ 1 1 1 ?4 M 10-20 0 1 0 ?5 M 40-50 0 0 1 ?6 F 0-10 0 0 0 ?7 M 30-40 1 1 0 ?8 M 20-30 0 0 1 ?

Dataset (D) Example

22

Notation

We Have:

1. Dataset, D

2. Model parameter space, W

3. Query algorithm, q

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Model Example

St Ot

Probabilistic Classifier

Notation

T : Number of examples

Ot : Vector of features of example t

St : Class of example t

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Model Example

Patient state (St)St : DiseaseState

Patient Observations (Ot)Ot1 : GenderOt2 : AgeOt3 : TestAOt4 : TestBOt5 : TestC

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Possible Model Structures

S

Age

Gender

TestA

TestB

TestC

S

Age

Gender

TestA

TestB

TestC

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Model Space

St Ot

P(St)

Model:

Model Parameters: P(Ot|St)

Generative Model:Must be able to compute P(St=i, Ot=ot | w)

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Model Parameter Space (W)

• W = space of possible parameter values

• Prior on parameters:

• Posterior over models:

T

ttt WPWOSP

WPWDPDWP

)()|,(

)()|()|(

)(WP

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Notation

We Have:

1. Dataset, D

2. Model parameter space, W

3. Query algorithm, q

q(W,D) returns t*, the next sample to label

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Game

while NotDone

• Learn P(W | D)• q chooses next example to label• Expert adds label to D

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O1

S1

O2

S2

O3

S3

Simulation

O4

S4

O5

S5

O6

S6

O7

S7

?

S2=false

S7=false

S5=true

?

?

q

hmm…

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Active Learning Flavors

• Pool

(“random access” to patients)

• Sequential

(must decide as patients walk in the door)

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q?

• Recall: q(W,D) returns the “most interesting” unlabelled example.

• Well, what makes a doctor curious about a patient?

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1994

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Score Function

))|(y(uncertaint)(score tttuncert OSPS

)( tSH

i

tt iSPiSP )(log)(

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t Sex Age Test A

Test B

Test C

St

1 M 20-30

0 1 1 ?

2 F 20-30

0 1 0 ?

3 F 30-40

1 0 0 ?

4 F 60+ 1 1 0 ?

5 M 10-20

0 1 0 ?

6 M 20-30

1 1 1 ?

P(St)

0.02

0.01

0.05

0.12

0.01

0.96

H(St)

0.043

0.024

0.086

0.159

0.024

0.073

Uncertainty Sampling Example

FALSE

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t Sex Age Test A

Test B

Test C

St

1 M 20-30

0 1 1 ?

2 F 20-30

0 1 0 ?

3 F 30-40

1 0 0 ?

4 F 60+ 1 1 0 ?

5 M 10-20

0 1 0 ?

6 M 20-30

1 1 1 ?

P(St)

0.01

0.02

0.04

0.00

0.06

0.97

H(St)

0.024

0.043

0.073

0.00

0.112

0.059

Uncertainty Sampling Example

FALSE

TRUE

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Uncertainty Sampling

GOOD: couldn’t be easierGOOD: often performs pretty well

BAD: H(St) measures information gain about the samples, not the model

Sensitive to noisy samples

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Can we do better thanuncertainty sampling?

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1992

Informative with respect to what?

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Model Entropy

WW W

P(W|D) P(W|D)P(W|D)

H(W) = high H(W) = 0…better…

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Information-Gain

• Choose the example that is expected to most reduce H(W)

• I.e., Maximize H(W) – H(W | St)

Expected model space entropy if we learn St

Current model space entropy

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Score Function

)|()(

);()(score

t

ttIG

SWHWH

WSMIS

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We usually can’t just sum over all models to get H(St|W)

…but we can sample from P(W | D)

Cc

w

cPcP

CHWH

dwwPwPWH

)(log)(

)()(

)(log)()(

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Conditional Model Entropy

dwwPwPWHw )(log)()(

dwiSwPiSwPiSWHw ttt )|(log)|()|(

i

w tttt dwiSwPiSwPiSPSWH )|(log)|()()|(

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Score Function

)|()()(score ttIG SCHCHS

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t Sex Age Test A

Test B

Test C

St

1 M 20-30

0 1 1 ?

2 F 20-30

0 1 0 ?

3 F 30-40

1 0 0 ?

4 F 60+ 1 1 1 ?

5 M 10-20

0 1 0 ?

6 M 20-30

0 0 1 ?

P(St)

0.02

0.01

0.05

0.12

0.01

0.02

Score =

H(C) - H(C|St)

0.53

0.58

0.40

0.49

0.57

0.52

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Score Function

)|()(

)|()()(score

CSHSH

SCHCHS

tt

ttIG

Familiar?

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Uncertainty Sampling & Information Gain

)|()()(score

)()(score

CSHSHS

SHS

tttInfoGain

ttUncertain

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But there is a problem…

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“the expected information gain of an unlabeled sample is NOT a sufficient criterion for constructing good queries”

If our objective is to reduce the prediction error, then

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Strategy #2:Query by Committee

Temporary Assumptions:

Pool Sequential

P(W | D) Version Space

Probabilistic Noiseless

QBC attacks the size of the “Version space”

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O1

S1

O2

S2

O3

S3

O4

S4

O5

S5

O6

S6

O7

S7

Model #1 Model #2

FALSE!

FALSE!

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O1

S1

O2

S2

O3

S3

O4

S4

O5

S5

O6

S6

O7

S7

Model #1 Model #2

TRUE!TRUE!

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O1

S1

O2

S2

O3

S3

O4

S4

O5

S5

O6

S6

O7

S7

Model #1 Model #2

TRUE!FALSE!

Ooh, now we’re going to learnsomething for sure!

One of them is definitely wrong.

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The Original QBCAlgorithm

As each example arrives…

1. Choose a committee, C, (usually of size 2) randomly from Version Space

2. Have each member of C classify it

3. If the committee disagrees, select it.

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1992

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Infogain vs Query by Committee

[Seung, Opper, Sompolinsky, 1992; Freund, Seung, Shamir, Tishby 1997]

First idea: Try to rapidly reduce volume of version space?

Problem: doesn’t take data distribution into account.H:

Which pair of hypotheses is closest? Depends on data distribution P.Distance measure on H: d(h,h’) = P(h(x) h’(x))

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Query-by-committee

First idea: Try to rapidly reduce volume of version space?

Problem: doesn’t take data distribution into account.

H:

To keep things simple, say d(h,h’) = Euclidean distance

Error is likely to remain large!

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Query-by-committee

Elegant scheme which decreases volume in a manner which is sensitive to the data distribution.

Bayesian setting: given a prior on H

H1 = HFor t = 1, 2, …

receive an unlabeled point xt drawn from P[informally: is there a lot of disagreement about

xt in Ht?]choose two hypotheses h,h’ randomly from (,

Ht)if h(xt) h’(xt): ask for xt’s labelset Ht+1

Problem: how to implement it efficiently?

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Query-by-committee

For t = 1, 2, …receive an unlabeled point xt drawn from Pchoose two hypotheses h,h’ randomly from (, Ht)if h(xt) h’(xt): ask for xt’s labelset Ht+1

Observation: the probability of getting pair (h,h’) in the inner loop (when a query is made) is proportional to (h) (h’) d(h,h’).

Ht

vs.

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Query-by-committee

Label bound: For H = {linear separators in Rd}, P = uniform distribution, just d log 1/ labels to reach a hypothesis with error < .

Implementation: need to randomly pick h according to (, Ht).

e.g. H = {linear separators in Rd}, = uniform distribution:

Ht

How do you pick a random point from a convex body?

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Sampling from convex bodies

By random walk!1. Ball walk2. Hit-and-run

[Gilad-Bachrach, Navot, Tishby 2005] Studies random walks and also ways to kernelize QBC.

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Some challenges

[1] For linear separators, analyze the label complexity for some distribution other than uniform!

[2] How to handle nonseparable data?Need a robust base learner

true boundary+-

Active Collaborative Prediction

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Approach: Collaborative Prediction (CP)

QoS measure (e.g. bandwidth)

?7033009Client4

1688Client3

567187Client2

18 3674Client1

Server3Server2Server1

Given previously observed ratings R(x,y), where

X is a “user” and Y is a “product”, predict

unobserved ratings

4 Raja

109Irina

24Gerry

31 ?Alina

Shrek GeishaMatrix

Movie Ratings

- will Alina like “The Matrix”? (unlikely )

- will Client 86 have fast download from Server 39?

- will member X of funding committee approve our project Y?

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10

0

clie

nts

100 servers

Collaborative Prediction = Matrix Approximation

• Important assumption: matrix entries are NOT independent, e.g. similar users have similar tastes

• Approaches: mainly factorized models assuming hidden ‘factors’ that affect ratings (pLSA, MCVQ, SVD, NMF, MMMF, …)

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User’s ‘weights’ associated with

‘factors’’

Factors

Assumptions: - there is a number of (hidden) factors behind the user preferences that relate to (hidden) movie properties

2 4 5 1 4 2

- movies have intrinsic values associated with such factors - users have intrinsic weights with such factors; user ratings a weighted (linear) combinations of movie’s values

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2 4 5 1 4 2

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2 4 5 1 4 2

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2 4 5 1 4 2

3 1 2 2 5

4 2 4 1 3 1

3 3 4 2

2 3 1 4 3 2

2 2 1 4

1 3 1 1 4Y XObjective: find a factorizable X=UV’ that approximates Y

),'(minarg'

YXLossXX

7 2 5 4 5 3 1 4 2

3 1 2 4 2 2 7 5 6

4 3 2 2 4 1 4 3 1

3 1 2 3 4 3 2 4 5

2 3 2 1 3 4 3 5 2

8 2 2 9 1 8 3 4 5

1 2 3 5 1 1 5 6 4

=

and satisfies some “regularization” constraints (e.g. rank(X) < k)Loss functions: depends on the nature of your problem

rank k

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Matrix Factorization Approaches

Singular value decomposition (SVD) – low-rank approximation Assumes fully observed Y and sum-squared loss

In collaborative prediction, Y is only partially observed Low-rank approximation becomes non-convex problem w/ many local minima

Furthermore, we may not want sum-squared loss, but instead accurate predictions (0/1 loss, approximated by hinge loss) cost-sensitive predictions (missing a good server vs suggesting a bad one) ranking cost (e.g., suggest k ‘best’ movies for a user)

NON-CONVEX PROBLEMS!

Use instead the state-of-art Max-Margin Matrix Factorization [Srebro 05] replaces bounded rank constraint by bounded norm of U, V’ vectors convex optimization problem! – can be solved exactly by semi-definite programming strongly relates to learning max-margin classifiers (SVMs)

Exploit MMMF’s properties to augment it with active sampling!

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Key Idea of MMMF Rows – feature vectors, Columns – linear classifiers

Linear classifiersweight vectors

Featu

re v

ect

ors

f1

v2

“marg

in”

Xij = signij x marginij

Predictorij = signij

-1

If signij > 0, classify as +1,Otherwise classify as -1

“margin” here = Dist(sample, line)

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MMMF: Simultaneous Search for Low-norm Feature Vectors and Max-margin Classifiers

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Margin-based heuristics:

min-margin (most-uncertain) min-margin positive (“good” uncertain) max-margin (‘safe choice’ but no info)

-0.3

-0.4

0.6

-0.9

0.1

-0.5

0.8

0.5

0.7

0.9

-0.2

-0.5

-0.5

-0.9 -0.6

-0.7-0.5

-0.1

-0.5

-0.9

-0.9 -0.8 -0.5

-0.6

-0.5

-0.5

-0.5 -0.5

-0.4

-0.8

-0.1

-0.5-0.5

-0.5

0.1

0.3 0.4

0.6

0.3

0.8 0.2

0.7

0.9

0.2

0.5 0.2 0.3

0.6 0.6

0.9

0.6

0.7

0.30.8

0.6

0.4 0.5

0.30.9

0.2

0.1

0.6

-0.2

Active Learning with MMMF

- We extend MMMF to Active-MMMF using margin-based active sampling

- We investigate exploitation vs exploration trade-offs imposed by different heuristics

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A-MMMF(M,s)1. Given s sparse matrix Y, learn approximation X = MMMF(Y)

2. Using current predictions, actively select “best s” samples and request their labels (e.g., test client/server pair via ‘enforced’ download)

3. Add new samples to Y

4. Repeat 1-3

Active Max-Margin Matrix Factorization

Issues: Beyond simple greedy margin-based heuristics?

Theoretical guarantees? not so easy with non-trivial learning methods and non-trivial data distributions

(any suggestions??? )

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Empirical Results

Network latency prediction

Bandwidth prediction (peer-to-peer)

Movie Ranking Prediction

Sensor net connectivity prediction

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Empirical Results: Latency Prediction

P2Psim data NLANR-AMP data

Active sampling with most-uncertain (and most-uncertain positive) heuristics provide consistent improvement over random and least-uncertain-next sampling

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Movie Rating Prediction (MovieLens)

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Sensor Network Connectivity

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Introducing Cost: Exploration vs Exploitation

Active sampling lower prediction errors at lower costs (saves 100s of samples) (better prediction better server assignment decisions faster downloads

Active sampling achieves a good exploration vs exploitation trade-off: reduced decision cost AND information gain

DownloadGrid: bandwidth prediction

PlanetLab: latency prediction

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Conclusions

Common challenge in many applications: need for cost-efficient sampling

This talk: linear hidden factor models with active sampling

Active sampling improves predictive accuracy while keeping sampling complexity low in a wide variety of applications

Future work:

Better active sampling heuristics?

Theoretical analysis of active sampling performance?

Dynamic Matrix Factorizations: tracking time-varying matrices

Incremental MMMF? (solving from scratch every time is too costly)

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ReferencesSome of the most influential papers

• Simon Tong, Daphne Koller. Support Vector Machine Active Learning with Applications to Text Classification. Journal of Machine Learning Research. Volume 2, pages 45-66. 2001.

• Y. Freund, H. S. Seung, E. Shamir, N. Tishby. 1997. Selective sampling using the query by committee algorithm. Machine Learning, 28:133—168

• David Cohn, Zoubin Ghahramani, and Michael Jordan. Active learning with statistical models, Journal of Artificial Intelligence Research, (4): 129-145, 1996.

• David Cohn, Les Atlas and Richard Ladner. Improving generalization with active learning, Machine Learning 15(2):201-221, 1994.

• D. J. C. Mackay. Information-Based Objective Functions for Active Data Selection. Neural Comput., vol. 4, no. 4, pp. 590--604, 1992.

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NIPS papers• Francis Bach. Active learning for misspecified generalized linear models. NIPS-06

• Ran Gilad-Bachrach, Amir Navot, Naftali Tishby. Query by Committee Made Real. NIPS-05

• Brent Bryan, Jeff Schneider, Robert Nichol, Christopher Miller, Christopher Genovese, Larry Wasserman . Active Learning For Identifying Function Threshold Boundaries . NIPS-05

• Rui Castro, Rebecca Willett, Robert Nowak. Faster Rates in Regression via Active Learning. NIPS-05

• Sanjoy Dasgupta. Coarse sample complexity bounds for active learning. NIPS-05

• Masashi Sugiyama. Active Learning for Misspecified Models. NIPS-05

• Brigham Anderson, Andrew Moore. Fast Information Value for Graphical Models. NIPS-05

• Dan Pelleg, Andrew W. Moore. Active Learning for Anomaly and Rare-Category Detection. NIPS-04

• Sanjoy Dasgupta. Analysis of a greedy active learning strategy. NIPS-04

• T. Jaakkola and H. Siegelmann. Active Information Retrieval. NIPS-01

• M. K. Warmuth et al. Active Learning in the Drug Discovery Process. NIPS-01

• Jonathan D. Nelson, Javier R. Movellan. Active Inference in Concept Learning. NIPS-00

• Simon Tong, Daphne Koller. Active Learning for Parameter Estimation in Bayesian Networks. NIPS-00

• Thomas Hofmann and Joachim M. Buhnmnn. Active Data Clustering. NIPS-97

• K. Fukumizu. Active Learning in Multilayer Perceptrons. NIPS-95

• Anders Krogh, Jesper Vedelsby. NEURAL NETWORK ENSEMBLES, CROSS VALIDATION, AND ACTIVE LEARNING. NIPS-94

• Kah Kay Sung, Partha Niyogi. ACTIVE LEARNING FOR FUNCTION APPROXIMATION. NIPS-94

• David Cohn, Zoubin Ghahramani, Michael I. Jordan. ACTIVE LEARNING WITH STATISTICAL MODELS. NIPS-94

• Sebastian B. Thrun and Knut Moller. Active Exploration in Dynamic Environments. NIPS-91

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ICML papers• Maria-Florina Balcan, Alina Beygelzimer, John Langford. Agnostic Active Learning. ICML-06

• Steven C. H. Hoi, Rong Jin, Jianke Zhu, Michael R. Lyu. Batch Mode Active Learning and Its Application to Medical Image Classification. ICML-06

• Sriharsha Veeramachaneni, Emanuele Olivetti, Paolo Avesani. Active Sampling for Detecting Irrelevant Features. ICML-06

• Kai Yu, Jinbo Bi, Volker Tresp. Active Learning via Transductive Experimental Design. ICML-06

• Rohit Singh, Nathan Palmer, David Gifford, Bonnie Berger, Ziv Bar-Joseph. Active Learning for Sampling in Time-Series Experiments With Application to Gene Expression Analysis. ICML-05

• Prem Melville, Raymond Mooney. Diverse Ensembles for Active Learning. ICML-04

• Klaus Brinker. Active Learning of Label Ranking Functions. ICML-04

• Hieu Nguyen, Arnold Smeulders. Active Learning Using Pre-clustering. ICML-04

• Greg Schohn and David Cohn. Less is More: Active Learning with Support Vector Machines, ICML-00

• Simon Tong, Daphne Koller. Support Vector Machine Active Learning with Applications to Text Classification. ICML-00.

• COLT papers

• S. Dasgupta, A. Kalai, and C. Monteleoni. Analysis of perceptron-based active learning. COLT-05.

• H. S. Seung, M. Opper, and H. Sompolinski. 1992. Query by committee. COLT-92, pages 287--294.

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Journal Papers• Antoine Bordes, Seyda Ertekin, Jason Weston, Leon Bottou. Fast Kernel Classifiers with Online

and Active Learning. Journal of Machine Learning Research (JMLR), vol. 6, pp. 1579-1619, 2005.

• Simon Tong, Daphne Koller. Support Vector Machine Active Learning with Applications to Text Classification. Journal of Machine Learning Research. Volume 2, pages 45-66. 2001.

• Y. Freund, H. S. Seung, E. Shamir, N. Tishby. 1997. Selective sampling using the query by committee algorithm. Machine Learning, 28:133--168

• David Cohn, Zoubin Ghahramani, and Michael Jordan. Active learning with statistical models, Journal of Artificial Intelligence Research, (4): 129-145, 1996.

• David Cohn, Les Atlas and Richard Ladner. Improving generalization with active learning, Machine Learning 15(2):201-221, 1994.

• D. J. C. Mackay. Information-Based Objective Functions for Active Data Selection. Neural Comput., vol. 4, no. 4, pp. 590--604, 1992.

• Haussler, D., Kearns, M., and Schapire, R. E. (1994). Bounds on the sample complexity of Bayesian learning using information theory and the VC dimension. Machine Learning, 14, 83--113

• Fedorov, V. V. 1972. Theory of optimal experiment. Academic Press.

• Saar-Tsechansky, M. and F. Provost. Active Sampling for Class Probability Estimation and Ranking. Machine Learning 54:2 2004, 153-178.

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Workshops

• http://domino.research.ibm.com/comm/research_projects.nsf/pages/nips05workshop.index.html

89

Appendix

Active Learning of Bayesian Networks

91

Entropy Function• A measure of information in random

event X with possible outcomes {x1,…,xn}

• Comments on entropy function:– Entropy of an event is zero when the

outcome is known

– Entropy is maximal when all outcomes are equally likely

• The average minimum yes/no questions to answer some question (connection to binary search)

H(x) = - i p(xi) log2 p(xi)

[Shannon, 1948]

92

Kullback-Leibler divergence

• P is the true distribution; Q distribution is used to encode data instead of P

• KL divergence is the expected extra message length per datum that must be transmitted using Q

• Measure of how “wrong” Q is with respect to true distribution P

DKL(P || Q) = i P(xi) log (P(xi)/Q(xi))

= i P(xi) log Q(xi) – i P(xi) log P(xi)

= -H(P,Q) + H(P)

= -Cross-entropy + entropy

93

Learning Bayesian Networks

E

R

B

A

C

.9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

Data+

Prior Knowledge

• Model Building• Parameter estimation• Causal structure discovery

• Passive Learning vs Active Learning

Learner

94

Active Learning

• Selective Active Learning

• Interventional Active Learning

• Obtain measure of quality of current model • Choose query that most improves quality • Update model

95

Active Learning: Parameter Estimation

[Tong & Koller, NIPS-2000]• Given a BN structure G• A prior distribution p(θ)• Learner request a particular instantiation q (Query)

Training data

Active Learner

Response (x)

+

E B

A

Initial Network G, p(θ)

E B

A

Updated distribution

p´(θ)

How to update parameter densityHow to select next query based on p

Query (q) E B

A

Updated distribution

p´(θ)

96

•Do not update A since we are fixing it•If we select A then do not update B

•Sampling from P(B|A=a) P(B)

•If we force A then we can update B

•Sampling from P(B|A:=a) = P(B)*

•Update all other nodes as usual

•Obtain new density

*Pearl 2000

A

B

J M

Updating parameter density

) a,A| (p xX θ

97

• Goal: a single estimate – instead of a distribution p over

• If we choose and the true model is ’ then we incur some loss, L(’ || )

Bayesian point estimation

’

p()

~

98

• We do not know the true ’ • Density p represents optimal beliefs over ’ • Choose that minimizes the expected loss

= argmin ∫p(’) L(’ || ) d’

• Call the Bayesian point estimate• Use the expected loss of the Bayesian point

estimate as a measure of quality of p(): – Risk(p) = ∫p(’) L(’ || ) d’

Bayesian point estimation

~

~

99

• Set the controllable variables so as to minimize the expected posterior risk:

• KL divergence will be used for loss

– KL( || ’)=∑KL(Pθ(Xi|Ui)|| Pθ’(Xi|Ui))

The Querying component

θθθθ d )~

||(K )|(p)|(P LxqQxXx

ExPRisk(p | Q=q)

Conditional KL-divergence

100

Algorithm Summary

• For each potential query q • Compute Risk(X|q) • Choose q for which Risk(X|q) is greatest

– Cost of computing Risk(X|q):

• Cost of Bayesian network inference • Complexity: O (|Q|. Cost of inference)

101

Uncertainty samplingMaintain a single hypothesis, based on labels seen so far.Query the point about which this hypothesis is most “uncertain”.

Problem: confidence of a single hypothesis may not accurately represent the true diversity of opinion in the hypothesis class.

X

-

-

--

-

-

-

++

+

+

+ --

102

103

Region of uncertainty

current version spaceSuppose data lies on circle in R2; hypotheses are linear separators.

(spaces X, H superimposed) region of

uncertainty in data space

Current version space: portion of H consistent with labels so far.“Region of uncertainty” = part of data space about which there is still some uncertainty (i.e. disagreement within version space)

++

104

Region of uncertainty

current version spaceData and hypothesis spaces, superimposed:

(both are the surface of the unit sphere in Rd)

region of uncertainty in data space

Algorithm [CAL92]:of the unlabeled points which lie in the region of uncertainty, pick one at random to query.

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Region of uncertainty

Number of labels needed depends on H and also on P.

Special case: H = {linear separators in Rd}, P = uniform distribution over unit sphere.

Then: just d log 1/ labels are needed to reach a hypothesis with error rate < .

[1] Supervised learning: d/ labels.[2] Best we can hope for.

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Region of uncertaintyAlgorithm [CAL92]:of the unlabeled points which lie in the region of uncertainty, pick one at random to query.

For more general distributions: suboptimal…

Need to measure quality of a query – or alternatively, size of version space.

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Uncertainty sampling!

ExpectedInfogainof sample

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