Sample Selection Bias – Covariate Shift: Problems, Solutions, and Applications Wei Fan, IBM T.J.Watson Research Masashi Sugiyama, Tokyo Institute of Technology.
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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//www.weifan.info/tutorial.htm
A Toy Example Two classes: red and green red: f2>f1 green: f2<=f1
Unbiased and Biased Samples Not so-biased sampling Biased sampling
Effect on Learning Unbiased 97.1%Biased 92.1%Unbiased 96.9%Biased 95.9%Unbiased 96.405%Biased 92.7% Some techniques are more sensitive to bias than others. One important question: –How to reduce the effect of sample selection bias?
Ubiquitous Loan Approval Drug screening Weather forecasting Ad Campaign Fraud Detection User Profiling Biomedical Informatics Intrusion Detection Insurance etc 1.Normally, banks only have data of their own customers 2.“Late payment, default” models are computed using their own data 3.New customers may not completely follow the same distribution.
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) Control computers by EEG signals: –Input: EEG signals –Output: Left or Right Figure provided by Fraunhofer FIRST, Berlin, Germany
Training Imagine left/right-hand movement following the letter on the screen Movie provided by Fraunhofer FIRST, Berlin, Germany
Testing: Playing Games “Brain-Pong” Movie provided by Fraunhofer FIRST, Berlin, Germany
Non-Stationarity in EEG Features Bandpower differences between training and test phases Different mental conditions (attention, sleepiness etc.) between training and test phases may change the EEG signals. Features extracted from brain activity during training and test phases Figures provided by Fraunhofer FIRST, Berlin, Germany
Robot Control by Reinforcement Learning Let the robot learn how to autonomously move without explicit supervision. Khepera Robot
Rewards Give robot rewards: –Go forward: Positive reward –Hit wall: Negative reward Goal: Learn the control policy that maximizes future rewards Robot moves autonomously = goes forward without hitting wall
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) –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
Alternatively, consider D of the size can be sampled “exhaustively” from the universe of examples. 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
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 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 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 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 (Weak) extrapolation: Predict output values outside training region Training samples Test samples
Covariate Shift Adaptation Training samples Test samples To illustrate the effect of covariate shift, let’s focus on linear extrapolation True function Learned function
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!
Law of Large Numbers Sample average converges to the population mean: We want to estimate the expectation over test input points only using training input points.
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. :Assumed strictly positive (Shimodaira, JSPI2000)
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 Directly estimating the ratio is easier than estimating the densities! When solving a problem, more difficult problems shouldn’t be solved. Knowing densitiesKnowing ratio (e.g., support vector machines)
Modeling Importance Function Use a linear importance model: Test density is approximated by Idea: Learn so that well approximates.
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. KLIEP –Gaussian kernel width is chosen by likelihood cross-validation
KDE: Error increases as dim grows KLIEP: Error remains small for large dim Experimental Results KDE KLIEP Normalized MSE dim
Ensemble Methods (Fan and Davidson’07) Posterior weighting Class Probability Integration Over Model Space Averaging of estimated class probabilities weighted by posterior Removes model uncertainty by averaging
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 P 1 (+|x) = 0.8 and P 2 (+|x) = 0.4 P 1 (-|x) = 0.2 and P 2 (-|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) Original Dataset Structural Discovery Structural Re-balancing Corrected Dataset
Active Learning Quality of learned functions depends on training input location. Goal: optimize training input location Good input location Poor input location Target Learned
Challenges Generalization error is unknown and needs to be estimated. In experiment design, we do not have training output values yet. Thus we cannot use, e.g., cross-validation which requires. Only training input positions can be used in generalization error estimation!
Agnostic Setup The model is not correct in 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) 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! Importance (Wiens JSPI2001; Kanamori & Shimodaira JSPI2003; Sugiyama JMLR2006)
Model Selection Choice of models is crucial: We want to determine the model so that generalization error is minimized: Polynomial of order 1Polynomial of order 2Polynomial of order 3
Generalization Error Estimation Generalization error is not accessible since the target function is unknown. Instead, we use a generalization error estimate. 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, it is heavily biased under covariate shift! Group 1Group 2Group kGroup k-1 … TrainingValidation
Importance-Weighted CV (IWCV) 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 1Set 2Set kSet k-1 … TrainingTesting (Zadrozny ICML2004; Sugiyama et al., JMLR2007)
Example of IWCV IWCV gives better estimates of generalization error. Model selection by IWCV outperforms CV!
Reserve Testing (Fan and Davidson’06) Train A B MA MB Test A B MAA MAB MBA MBB Train Estimate the performance of MA and MB based on the order of MAA, MAB, MBA and MBB DA DB Labeled test data
Rule 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) – Daily summary maps of two datasets from Texas Commission on Environmental Quality (TCEQ)
1. 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, 2 72 is an astronomical number – 2% and 5% true positive ozone days for 1- hour and 8-hour peak respectively Challenges as a Data Mining Problem
3.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 X ir, where X=(X r, X ir ), P(Y|X) = P(Y|X r ) only if the data is exhaustive. – May introduce overfitting problem, and change the probability distribution represented in the data. P(Y = “ozone day”| X r, X ir ) 1 P(Y = “normal day”| X r, X ir ) 0
4.“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 1.How to train an accurate model 2.How to effectively use a model to predict the future with a different and yet unknown distribution Training Distribution Testing Distribution 1 2 3 1 2 3 + + + + + + - -
Reliable probability estimation under irrelevant features – Recall that due to irrelevant features: P(Y = “ozone day”| X r, X ir ) 1 P(Y = “normal day”| X r, X ir ) 0 – Construct multiple models – Average their predictions P(“ozone”|x r ): true probability P(“ozone”| X r, X ir, θ ): estimated probability by model θ MSE singlemodel: – Difference between “true” and “estimated”. MSE Average – Difference between “true” and “average of many models” Formally show that MSE Average ≤ MSE SingleModel
Prediction with feature sample selection bias TrainingSet Algorithm ….. Estimated probability values 1 fold Estimated probability values 10 fold 10CV Estimated probability values 2 fold Decision threshold V E VEVE “Probability- TrueLabel” file Concatenate P(y=“ozoneday”|x,θ) Lable 7/1/98 0.1316 Normal 7/2/98 0.6245 Ozone 7/3/98 0.5944 Ozone ……… PrecRec plot Recall Precision Ma Mb – A CV based procedure for decision threshold selection Training Distribution Testing Distribution 1 2 3 1 2 3 + + + + + + - - P(y=“ozoneday”|x,θ) Lable 7/1/98 0.1316 Normal 7/3/98 0.5944 Ozone 7/2/98 0.6245 Ozone ………
Addressing Data Mining Challenges Prediction with feature sample selection bias – Future prediction based on decision threshold selected Whole Training Set θ Classification on future days if P(Y = “ozonedays”|X,θ ) ≥ V E Predict “ozonedays”
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
17K movies Training Data Task 2 Task 1 Movie Arrival 1998 Time 2005 2006 User Arrival 45? 3 2 ? Qualifier Dataset 3M NO User or Movie Arrival NETFLIX data generation process
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 Movies Users History Samples …… Movie5.0011 …… Movie3.001 …… Movie4.0007 …… User7.0007 …… User6.00012 …… User8.00003 …… Movie5 User 7 …… Movie3 User 7 …… Movie4.User 8 …. 1488844,3,2005-09-06 822109,5,2005-05-13 885013,4,2005-10-19 30878,4,2005-12-26 823519,3,2004-05-03 …
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 (BCI) Control computers by brain signals: –Input: EEG signals –Output: Left or Right
BCI Results When KL is large, covariate shift adaptation tends to improve accuracy. When KL is small, no difference. SubjectTrial No adaptation With adaptation KL 1 19.3 %10.0 %0.76 28.8 % 1.11 34.3 % 0.69 2 140.0 % 0.97 239.3 %38.7 %1.05 325.5 % 0.43 3 136.9 %34.4 %2.63 221.3 %19.3 %2.88 322.5 %17.5 %1.25 4 121.3 % 9.23 22.4 % 5.58 36.4 % 1.83 5 121.3 % 0.79 215.3 %14.0 %2.01 KL divergence from training to test input distributions
Robot Control by Reinforcement Learning Swing-up inverted pendulum: –Swing-up the pole by controlling the car. –Reward:
Wafer Alignment in Semiconductor Exposure Apparatus Recent silicon wafers have layer structure. Circuit patterns are exposed multiple times. Exact alignment of wafers is very important.
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 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 Results 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 IWLS-basedOLS-based“Outer” heuristicPassive 2.27(1.08)2.37(1.15)2.36(1.15)2.32(1.11) (Sugiyama & Nakajima ECML-PKDD2008) Mean squared error of wafer position estimation