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Sample Selection Bias – Covariate Shift: Problems, Solutions, and Applications. Wei Fan, IBM T.J.Watson Research Masashi Sugiyama, Tokyo Institute of Technology Updated PPT is available: http// Overview of Sample Selection Bias Problem. A Toy Example.

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sample selection bias covariate shift problems solutions and applications

Sample Selection Bias – Covariate Shift: Problems, Solutions, and Applications

Wei Fan, IBM T.J.Watson Research

Masashi Sugiyama, Tokyo Institute of Technology

Updated PPT is available:


a toy example
A Toy Example

Two classes:

red and green

red: f2>f1

green: f2<=f1

unbiased and biased samples
Unbiased and Biased Samples

Not so-biased sampling

Biased sampling

effect on learning

Unbiased 96.9%

Unbiased 97.1%

Unbiased 96.405%

Biased 95.9%

Biased 92.7%

Biased 92.1%

Effect on Learning
  • Some techniques are more sensitive to bias than others.
  • One important question:
    • How to reduce the effect of sample selection bias?

Normally, banks only have data of their own customers

  • “Late payment, default” models are computed using their own data
  • New customers may not completely follow the same distribution.
  • Loan Approval
  • Drug screening
  • Weather forecasting
  • Ad Campaign
  • Fraud Detection
  • User Profiling
  • Biomedical Informatics
  • Intrusion Detection
  • Insurance
  • etc
face recognition
Face Recognition
  • Sample selection bias:
    • Training samples are taken inside research lab, where there are a few women.
    • Test samples: in real-world, men-women ratio is almost 50-50.

The Yale Face Database B

brain computer interface bci
Brain-Computer Interface (BCI)
  • Control computers by EEG signals:
    • Input: EEG signals
    • Output: Left or Right

Figure provided by Fraunhofer FIRST, Berlin, Germany

  • Imagine left/right-hand movement following the letter on the screen

Movie provided by Fraunhofer FIRST, Berlin, Germany

testing playing games
Testing: Playing Games
  • “Brain-Pong”

Movie provided by Fraunhofer FIRST, Berlin, Germany

non stationarity in eeg features
Non-Stationarity in EEG Features
  • Different mental conditions (attention, sleepiness etc.) between training and test phases may change the EEG signals.

Bandpower differences between

training and test phases

Features extracted from brain activity

during training and test phases

Figures provided by Fraunhofer FIRST, Berlin, Germany

robot control by reinforcement learning
Robot Controlby Reinforcement Learning
  • Let the robot learn how to autonomously move without explicit supervision.

Khepera Robot


Robot moves autonomously

= goes forward without hitting wall

  • Give robot rewards:
    • Go forward: Positive reward
    • Hit wall: Negative reward
  • Goal: Learn the control policy that maximizes future rewards
  • After learning:
policy iteration and covariate shift
Policy Iteration and Covariate Shift
  • Policy iteration:
  • Updating the policy correspond to changing the input distributions!


control policy


control policy

bias as distribution
Bias as Distribution
  • Think of “sampling an example (x,y) into the training data” as an event denoted by random variable s
    • s=1: example (x,y) is sampled into the training data
    • s=0: example (x,y) is not sampled.
  • Think of bias as a conditional probability of “s=1” dependent on x and y
  • P(s=1|x,y) : the probability for (x,y) to be sampled into the training data, conditional on the example’s feature vector x and class label y.
categorization zadrozy 04 fan et al 05 fan and davidson 07
Categorization(Zadrozy’04, Fan et al’05, Fan and Davidson’07)
  • No Sample Selection Bias
    • P(s=1|x,y) = P(s=1)
  • Feature Bias/Covariate Shift
    • P(s=1|x,y) = P(s=1|x)
  • Class Bias
    • P(s=1|x,y) = P(s=1|y)
  • Complete Bias
    • No more reduction
bias for a training set
Bias for a Training Set
  • How P(s=1|x,y) is computed
  • Practically, for a given training set D
    • P(s=1|x,y) = 1: if (x,y) is sampled into D
    • P(s=1|x,y) = 0: otherwise
  • Alternatively, consider D of the size can be sampled “exhaustively” from the universe of examples.
realistic datasets are biased
Realistic Datasets are biased?
  • Most datasets are biased.
  • Unlikely to sample each and every feature vector.
  • For most problems, it is at least feature bias.
    • P(s=1|x,y) = P(s=1|x)
effect on learning1
Effect on Learning
  • Learning algorithms estimate the “true conditional probability”
    • True probability P(y|x), such as P(fraud|x)?
    • Estimated probabilty P(y|x,M): M is the model built.
  • Conditional probability in the biased data.
    • P(y|x,s=1)
  • Key Issue:
    • P(y|x,s=1) = P(y|x) ?
heckman s two step approach
Heckman’s Two-Step Approach
  • Estimate one’s donation amount if one does donate.
  • Accurate estimate cannot be obtained by a regression using only data from donors.
  • First Step: Probit model to estimate probability to donate:
  • Second Step: regression model to estimate donation:
  • Expected error
  • Gaussian assumption
covariate shift or feature bias
Covariate Shift or Feature Bias
  • However, no chance for generalization if training and test samples have nothing in common.
  • Covariate shift:
    • Input distribution changes
    • Functional relation remains unchanged
example of covariate shift
Example of Covariate Shift

(Weak) extrapolation:

Predict output values outside training region

Training samples

Test samples

covariate shift adaptation
Covariate Shift Adaptation
  • To illustrate the effect of covariate shift, let’s focus on linear extrapolation

Training samples

Test samples

True function

Learned function

model specification
Model Specification
  • Model is said to be correctly specified if
  • In practice, our model may not be correct.
  • Therefore, we need a theory for misspecified models!
ordinary least squares ols
If model is correct:

OLS minimizes bias asymptotically

If model is misspecified:

OLS does not minimize bias even asymptotically.

We want to reduce bias!

Ordinary Least-Squares (OLS)
law of large numbers
Law of Large Numbers
  • Sample average converges to the population mean:
  • We want to estimate the expectation overtest input points only using training input points .
key trick importance weighted average
Key Trick:Importance-Weighted Average
  • Importance: Ratio of test and training input densities
  • Importance-weighted average:

(cf. importance sampling)

importance weighted ls
Even for misspedified models, IWLS minimizes bias asymptotically.

We need to estimate importance in practice.

Importance-Weighted LS

(Shimodaira, JSPI2000)

:Assumed strictly positive

use of unlabeled samples importance estimation
Use of Unlabeled Samples: Importance Estimation
  • Assumption: We have training inputs and test inputs .
  • Naïve approach: Estimate and separately, and take the ratio of the density estimates
  • This does not work well since density estimation is hard in high dimensions.
vapnik s principle
Vapnik’s Principle

When solving a problem,

more difficult problems shouldn’t be solved.

  • Directly estimating the ratio is easier than estimating the densities!

(e.g., support vector machines)

Knowing densities

Knowing ratio

modeling importance function
Modeling Importance Function
  • Use a linear importance model:
  • Test density is approximated by
  • Idea: Learn so that well approximates .
kullback leibler divergence
Kullback-Leibler Divergence



learning importance function
Learning Importance Function
  • Thus
  • Since is density,

(objective function)


kliep kullback leibler importance estimation procedure
KLIEP (Kullback-LeiblerImportance Estimation Procedure)

(Sugiyama et al., NIPS2007)

  • Convexity: unique global solution is available
  • Sparse solution: prediction is fast!
experiments setup
Experiments: Setup
  • Input distributions: standard Gaussian with
    • Training: mean (0,0,…,0)
    • Test: mean (1,0,…,0)
  • Kernel density estimation (KDE):
    • Separately estimate training and test input densities.
    • Gaussian kernel width is chosen by likelihood cross-validation.
    • Gaussian kernel width is chosen by likelihood cross-validation
experimental results
Experimental Results
  • KDE:Error increases as dim grows
  • KLIEP: Error remains small for large dim


Normalized MSE



ensemble methods fan and davidson 07
Ensemble Methods (Fan and Davidson’07)

Averaging of estimated class probabilities weighted by posterior



Integration Over

Model Space



Removes model uncertainty by averaging

how to use them
How to Use Them
  • Estimate “joint probability” P(x,y) instead of just conditional probability, i.e.,
    • P(x,y) = P(y|x)P(x)
    • Makes no difference use 1 model, but

Multiple models

examples of how this works
Examples of How This Works
  • P1(+|x) = 0.8 and P2(+|x) = 0.4
  • P1(-|x) = 0.2 and P2(-|x) = 0.6
  • model averaging,
    • P(+|x) = (0.8 + 0.4) / 2 = 0.6
    • P(-|x) = (0.2 + 0.6)/2 = 0.4
    • Prediction will be –
But if there are two P(x) models, with probability 0.05 and 0.4
  • Then
    • P(+,x) = 0.05 * 0.8 + 0.4 * 0.4 = 0.2
    • P(-,x) = 0.05 * 0.2 + 0.4 * 0.6 = 0.25
  • Recall with model averaging:
    • P(+|x) = 0.6 and P(-|x)=0.4
    • Prediction is +
  • But, now the prediction will be – instead of +
  • Key Idea:
    • Unlabeled examples can be used as “weights” to re-weight the models.
structure discovery ren et al 08
Structure Discovery (Ren et al’08)

Structural Discovery

Original Dataset

Structural Re-balancing

Corrected Dataset

active learning
Active Learning
  • Quality of learned functions depends on training input location .
  • Goal: optimize training input location

Good input location

Poor input location



  • Generalization error is unknown and needs to be estimated.
  • In experiment design, we do not have training output valuesyet.
  • Thus we cannot use, e.g., cross-validationwhich requires .
  • Only training input positions can be used in generalization error estimation!
agnostic setup
Agnostic Setup
  • The model is not correctin practice.
  • Then OLS is not consistent.
  • Standard “experiment design” method does not work!

(Fedorov 1972; Cohn et al., JAIR1996)

bias reduction by importance weighted ls iwls
Bias Reduction byImportance-Weighted LS (IWLS)

(Wiens JSPI2001; Kanamori & Shimodaira JSPI2003; Sugiyama JMLR2006)

  • The use of IWLS mitigates the problem of in consistency under agnostic setup.
  • Importance is known in active learning setup since is designed by us!


model selection
Model Selection

Polynomial of order 1

Polynomial of order 2

Polynomial of order 3

  • Choice of models is crucial:
  • We want to determine the model so that generalization error is minimized:
generalization error estimation
Generalization Error Estimation
  • Generalization error is not accessible since the target function is unknown.
  • Instead, we use a generalization error estimate.

Model complexity

Model complexity

cross validation
  • Divide training samples into groups.
  • Train a learning machine with groups.
  • Validate the trained machine using the rest.
  • Repeat this for all combinations and output the mean validation error.
  • CV is almost unbiased without covariate shift.
  • But, itis heavily biased under covariate shift!

Group 1

Group 2

Group k-1

Group k



importance weighted cv iwcv
Importance-Weighted CV (IWCV)

(Zadrozny ICML2004; Sugiyama et al., JMLR2007)

  • When testing the classifier in CV process, we also importance-weight the test error.
  • IWCV gives almost unbiased estimates of generalization error even under covariate shift

Set 1

Set 2

Set k-1

Set k



example of iwcv
Example of IWCV
  • IWCV gives better estimates of generalization error.
  • Model selection by IWCV outperforms CV!
reserve testing fan and davidson 06





test data










Reserve Testing (Fan and Davidson’06)




Estimate the performance of MA and MB based on the order of MAA, MAB, MBA and MBB

  • If “A’s labeled test data” can construct “more accurate models” for both algorithm A and B evaluated on labeled training data, then A is expected to be more accurate.
    • If MAA > MAB and MBA > MBB then choose A
  • Similarly,
    • If MAA < MAB and MBA < MBB then choose B
  • Otherwise, undecided.
ozone day prediction zhang et al 06
Ozone Day Prediction (Zhang et al’06)
  • Daily summary maps of two datasets from Texas Commission on Environmental Quality (TCEQ)

Challenges as a Data Mining Problem

  • Rather skewed and relatively sparse distribution
    • 2500+ examples over 7 years (1998-2004)
    • 72 continuous features with missing values
    • Large instance space
      • If binary and uncorrelated, 272 is an astronomical number
    • 2% and 5% true positive ozone days for 1-hour and 8-hour peak respectively
A large number of irrelevant features
    • Only about 10 out of 72 features verified to be relevant,
    • No information on the relevancy of the other 62 features
    • For stochastic problem, given irrelevant features Xir , where X=(Xr, Xir),

P(Y|X) = P(Y|Xr) only if the data is exhaustive.

    • May introduce overfitting problem, and change the probability distribution represented in the data.
      • P(Y = “ozone day”| Xr, Xir) 1
      • P(Y = “normal day”|Xr, Xir) 0















Testing Distribution

Training Distribution

  • “Feature sample selection bias”.
    • Given 7 years of data and 72 continuous features, hard to find many days in the training data that is very similar to a day in the future
    • Given these, 2 closely-related challenges
        • How to train an accurate model
        • How to effectively use a model to predict the future with a different and yet unknown distribution
reliable probability estimation under irrelevant features
Reliable probability estimation under irrelevant features
  • Recall that due to irrelevant features:
    • P(Y = “ozone day”| Xr, Xir) 1
    • P(Y = “normal day”|Xr, Xir) 0
  • Construct multiple models
  • Average their predictions
    • P(“ozone”|xr): true probability
    • P(“ozone”|Xr, Xir, θ): estimated probability by model θ
    • MSEsinglemodel:
      • Difference between “true” and “estimated”.
    • MSEAverage
      • Difference between “true” and “average of many models”
    • Formally show that MSEAverage ≤ MSESingleModel








1 fold




10 fold




2 fold












TrainingSet Algorithm












P(y=“ozoneday”|x,θ) Lable

7/1/98 0.1316 Normal

7/3/98 0.5944 Ozone

7/2/98 0.6245 Ozone


P(y=“ozoneday”|x,θ) Lable

7/1/98 0.1316 Normal

7/2/98 0.6245 Ozone

7/3/98 0.5944 Ozone








Testing Distribution

Training Distribution

  • A CV based procedure for decision threshold selection
  • Prediction with feature sample selection bias
addressing data mining challenges

Classification on future days

Whole TrainingSet

if P(Y = “ozonedays”|X,θ ) ≥ VE


Predict “ozonedays”

Addressing Data Mining Challenges
  • Prediction with feature sample selection bias
    • Future prediction based on decision threshold selected
task 1
Task 1
  • Task 1: Who rated what in 2006
    • Given a list of 100,000 pairs of users and movies, predict for each pair the probability that the user rated the movie in 2006
    • Result: They are the close runner-up, No 3 out of 39 teams
  • Challenges:
    • Huge amount of data how to sample the data so that any learning algorithms can be applied is critical
    • Complex affecting factors: decrease of interest in old movies, growing tendency of watching (reviewing) more movies by Netflix users
netflix data generation process
NETFLIX data generation process

NO User

or Movie


User Arrival

Movie Arrival

Task 1

17K movies

Task 2

Training Data

1998 Time 2005 2006




task 1 effective sampling strategies


Movie5 .0011


Movie3 .001


Movie4 .0007








Movie5 User 7


Movie3 User 7


Movie4 .User 8


User7 .0007


User6 .00012


User8 .00003


Task 1: Effective Sampling Strategies
  • Sampling the movie-user pairs for “existing” users and “existing” movies from 2004, 2005 as training set and 4Q 2005 as developing set
    • The probability of picking a movie was proportional to the number of ratings that movie received; the same strategy for users






Learning Algorithm:

    • Single classifiers: logistic regression, Ridge regression, decision tree, support vector machines
    • Naïve Ensemble: combining sub-classifiers built on different types of features with pre-set weights
    • Ensemble classifiers: combining sub-classifiers with weights learned from the development set
brain computer interface bci1
Brain-Computer Interface (BCI)
  • Control computers by brain signals:
    • Input: EEG signals
    • Output: Left or Right
bci results
BCI Results

KL divergence from training

to test input distributions

  • When KL is large, covariate shift adaptation tends to improve accuracy.
  • When KL is small, no difference.
robot control by reinforcement learning1
Robot Control byReinforcement Learning
  • Swing-up inverted pendulum:
    • Swing-up the pole by controlling the car.
    • Reward:

Covariate shift adaptation

Existing method (b)

Existing method (a)

wafer alignment in semiconductor exposure apparatus
Wafer Alignment inSemiconductor Exposure Apparatus
  • Recent silicon wafers have layer structure.
  • Circuit patterns are exposed multiple times.
  • Exact alignment of wafers is very important.
markers on wafer
Markers on Wafer
  • Wafer alignment process:
    • Measure marker location printed on wafers.
    • Shift and rotate the wafer to minimize the gap.
  • For speeding up, reducing the number of markers to measure is very important.

Active learning problem!

non linear alignment model
Non-linear Alignment Model
  • When gap is only shift and rotation, linear model is exact:
  • However, non-linear factors exist, e.g.,
    • Warp
    • Biased characteristic of measurement apparatus
    • Different temperature conditions
  • Exactly modeling non-linear factors is very difficult in practice!

Agnostic setup!

experimental results1
Experimental Results

(Sugiyama & Nakajima ECML-PKDD2008)

Mean squared error of wafer position estimation

  • IWLS-based active learning works very well!
  • 20 markers (out of 38) are chosen by experiment design methods.
  • Gaps of all markers are predicted.
  • Repeated for 220 different wafers.
  • Mean (standard deviation) of the gap prediction error
  • Red: Significantly better by 5% Wilcoxon test
  • Blue: Worse than the baseline passive method
book on dataset shift
Book on Dataset Shift
  • Quiñonero-Candela, Sugiyama, Schwaighofer & Lawrence (Eds.), Dataset Shift in Machine Learning, MIT Press, Cambridge, 2008.