Weighted Empirical Risk Minimization: Transfer Learning based on Importance Sampling

′ n differs from the distribution P involved in the risk one seeks to minimize ( referred to as the test distribution ) but is still defined on the same measurable space as P and dominates it . In the unrealistic case where the likelihood ratio Φ ( z ) = dP/dP′ ( z ) is known , one may straightforwardly extends the Empirical Risk Minimization ( ERM ) approach to this specific transfer learning setup using the same idea as that behind Importance Sampling , by minimizing a weighted version of the empirical risk functional computed from the ’ biased ’ training data Z′i with weights Φ ( Z′i ) . Although the importance function Φ ( z ) is generally unknown in practice , we show that , in various situations frequently encountered in practice , it takes a simple form and can be directly estimated from the Z′i ’ s and some auxiliary information on the statistical population P. By means of linearization techniques , we then prove that the generalization capacity of the approach aforementioned is preserved when plugging the resulting estimates of the Φ ( Z′i ) ’ s into the weighted empirical risk . Beyond these theoretical guarantees , numerical results provide strong empirical evidence of the relevance of the approach promoted in this article . 1 Introduction Prediction problems are of major importance in statistical learning . The main paradigm of predictive learning is Empirical Risk Minimization ( ERM in abbreviated form ) , see e.g . Devroye et al . ( 1996 ) . In the standard setup , Z is a random variable ( r.v . in short ) that takes its values in a feature spaceZ with distribution P , Θ is a parameter space and ` : Θ ×Z → R+ is a ( measurable ) loss function . The risk is then defined by : ∀θ ∈ Θ , RP ( θ ) = EP [ ` ( θ , Z ) ] , ( 1 ) and more generally for any measure Q onZ : RQ ( θ ) = ∫ Z ` ( θ , z ) dQ ( z ) . In most practical situations , the distribution P involved in the definition of the risk is unknown and learning is based on the sole observation of an independent and identically distributed ( i.i.d . ) sample Z1 , . . . , Zn drawn from P and the risk ( 1 ) must be replaced by an empirical counterpart ( or a possibly smoothed/penalized version of it ) , typically : R̂P ( θ ) = 1 n n∑ i=1 ` ( θ , Zi ) = RP̂n ( θ ) , ( 2 ) where P̂n = ( 1/n ) ∑n i=1 δZi is the empirical measure of P and δz denotes the Dirac measure at any point z . With the design of successful algorithms such as neural networks , support vector machines or boosting methods to perform ERM , the practice of predictive learning has recently received a significant attention and is now supported by a sound theory based on results in empirical process theory . The performance of minimizers of ( 2 ) can be indeed studied by means of concentration inequalities , quantifying the fluctuations of the maximal deviations supθ∈Θ |R̂P ( θ ) − RP ( θ ) | under various complexity assumptions for the functional class F = { ` ( θ , · ) : θ ∈ Θ } ( e.g . VC dimension , metric entropies , Rademacher averages ) , see Boucheron et al . ( 2013 ) for instance . Although , in the Big Data era , the availability of massive digitized information to train predictive rules is an undeniable opportunity for the widespread deployment of machine-learning solutions , the poor control of the data acquisition process one is confronted with in many applications puts practicioners at risk of jeopardizing the generalization ability of the rules produced by the algorithms implemented . Bias selection issues in machine-learning are now the subject of much attention in the literature , see Bolukbasi et al . ( 2016 ) , Zhao et al . ( 2017 ) , Burns et al . ( 2019 ) , Liu et al . ( 2016 ) or Huang et al . ( 2007 ) . In the context of face analysis , a research area including a broad range of applications such as face detection , face recognition or face attribute detection , machine learning algorithms trained with baised training data , e.g . in terms of gender or ethnicity , raise concerns about fairness in machine learning . Unfair algorithms may induce systemic undesired disadvantages for specific social groups , see Das et al . ( 2018 ) for further details . Several examples of bias in deep learning based face recognition systems are discussed in Nagpal et al . ( 2019 ) . Throughout the present article , we consider the case where the i.i.d . sample Z′1 , . . . , Z ′ n available for training is not drawn from P but from another distribution P′ , with respect to which P is absolutely continuous , and the goal pursued is to set theoretical grounds for the application of ideas behind Importance Sampling ( IS in short ) methodology to extend the ERM approach to this learning setup . We highlight that the problem under study is a very particular case of Transfer Learning ( see e.g . Pan & Yang ( 2010 ) , Ben-David et al . ( 2010 ) and Storkey ( 2009 ) ) , a research area currently receiving much attention in the literature and encompassing general situations where the information/knowledge one would like to transfer may take a form in the target space very different from that in the source space ( referred to as domain adaptation ) . Weighted ERM ( WERM ) . In this paper , we investigate conditions guaranteeing that values for the parameter θ that nearly minimize ( 1 ) can be obtained through minimization of a weighted version of the empirical risk based on the Z′i ’ s , namely R̃w , n ( θ ) = RP̃w , n ( θ ) , ( 3 ) where P̃w , n = ( 1/n ) ∑n i=1 wiδZ′i and w = ( w1 , . . . , wn ) ∈ R n + is a certain weight vector . Of course , ideal weights w∗ are given by the likelihood function Φ ( z ) = ( dP/dP′ ) ( z ) : w∗i = Φ ( Z ′ i ) for i ∈ { 1 , . . . , n } . In this case , the quantity ( 3 ) is obviously an unbiased estimate of the true risk ( 1 ) : EP′ [ RP̃w∗ , n ( θ ) ] = RP ( θ ) , ( 4 ) and generalization bounds for the RP-risk excess of minimizers of R̃w∗ , n can be directly established by studying the concentration properties of the empirical process related to the Z′i ’ s and the class of functions { Φ ( · ) ` ( θ , · ) : θ ∈ Θ } ( see section 2 below ) . However , the importance function Φ is unknown in general , just like distribution P. It is the major purpose of this article to show that , in far from uncommon situations , the ( ideal ) weights w∗i can be estimated from the Z ′ i s combined with auxiliary information on the target population P. As shall be seen below , such favorable cases include in particular classification problems where class probabilities in the test stage differ from those in the training step , risk minimization in stratified populations ( see Bekker & Davis ( 2018 ) ) , with strata statistically represented in a different manner in the test and training populations , positive-unlabeled learning ( PU-learning , see e.g . du Plessis et al . ( 2014 ) ) . In each of these cases , we show that the stochastic process obtained by plugging the weight estimates in the weighted empirical risk functional ( 3 ) is much more complex than a simple empirical process ( i.e . a collection of i.i.d . averages ) but can be however studied by means of linearization techniques , in the spirit of the ERM extensions established in Clémençon et al . ( 2008 ) or Clémençon & Vayatis ( 2009 ) . Learning rate bounds for minimizers of the corresponding risk estimate are proved and , beyond these theoretical guarantees , the performance of the weighted ERM approach is supported by convincing numerical results . The article is structured as follows . In section 2 , the ideal case where the importance function Φ is known is preliminarily considered and a first basic example where the optimal weights can be easily inferred and plugged into the risk without deteriorating the learning rate is discussed . The main results of the paper are stated in section 3 , which shows that the methodology promoted can be applied to two important problems in practice , risk minimization in stratified populations and PU-learning , with generalization guarantees . Illustrative numerical experiments are displayed in section 4 , while some concluding remarks are collected in section 5 . Proofs and additional results are deferred to the Supplementary Material . 2 Importance Sampling - RiskMinimization with Biased Data Here and throughout , the indicator function of any event E is denoted by I { E } , the sup norm of any bounded function h : Z → R by ||h||∞ . We place ourselves in the framework of statistical learning based on biased training data previously introduced . As a first go , we consider the unrealistic situation where the importance function Φ is known , insofar as we shall subsequently develop techniques aiming at mimicking the minimization of the ideally weighted empirical risk R̃w∗ , n ( θ ) = 1 n n∑ i=1 w∗i ` ( θ , Z ′ i ) , ( 5 ) namely the ( unbiased ) Importance Sampling estimator of ( 1 ) based on the instrumental data Z′1 , . . . , Z ′ n. The following result describes the performance of minimizers θ̃ ∗ n of ( 5 ) . Since the goal of this paper is to promote the main ideas of the approach rather than to state results with the highest level of generality due to space limitations , we assume throughout the article for simplicity that ` and Φ are both bounded functions . For σ1 , . . . , σn independent Rademacher random variables ( i.e . symmetric { −1 , 1 } -valued r.v. ’ s ) , independent from the Z′i ’ s , we define the Rademacher average associated to the class of function F as R′n ( F ) : = Eσ [ supθ∈Θ 1 n ∣∣∣∑ni=1 σi ` ( θ , Z′i ) ∣∣∣ ] . This quantity can be bounded by metric entropy methods under appropriate complexity assumptions on the class F , it is for instance of order OP ( 1/ √ n ) when F is a VC major class with finite VC dimension , see e.g . Boucheron et al . ( 2005 ) . Lemma 1 . With probability at least 1 − δ , we have : ∀n ≥ 1 , RP ( ̃θ∗n ) −min θ∈Θ RP ( θ ) ≤ 4||Φ||∞E [ R′n ( F ) ] + 2||Φ||∞ sup ( θ , z ) ∈Θ×Z ` ( θ , z ) √ 2 log ( 1/δ ) n . Of course , when P′ = P , we have Φ ≡ 1 and the bound stated above simply describes the performance of standard empirical risk minimizers . The proof is based on the standard bound RP ( ̃θ∗n ) −min θ∈Θ RP ( θ ) ≤ 2 sup θ∈Θ ∣∣∣∣R̃w∗ , n ( θ ) − E [ R̃w∗ , n ( θ ) ] ∣∣∣∣ , combined with basic concentration results for empirical processes , see the Supplementary Material for further details . Of course , the importance function Φ is generally unknown and must be estimated in practice . As illustrated by the elementary example below ( related to binary classification , in the situation where the probability of occurence of a positive instance significantly differs in the training and test stages ) , in certain statistical learning problems with biased training distribution , Φ takes a simplistic form and can be easily estimated from the Z′i ’ s combined with auxiliary information on P. Binary classification with varying class probabilities . The flagship problem in supervised learning corresponds to the simplest situation , where Z = ( X , Y ) , Y being a binary variable valued in { −1 , +1 } say , and the r.v . X takes its values in a measurable space X and models some information hopefully useful to predict Y . The parameter space Θ is a set G of measurable mappings ( i.e . classifiers ) g : X → { −1 , +1 } and the loss function is given by ` ( g , ( x , y ) ) = I { g ( x ) , y } for all g in G and any ( x , y ) ∈ X × { −1 , +1 } . The distribution P of the random pair ( X , Y ) can be either described by X ’ s marginal distribution µ ( dx ) and the posterior probability η ( x ) = P { Y = +1 | X = x } or else by the triplet ( p , F+ , F− ) where p = P { Y = +1 } and Fσ ( dx ) is X ’ s conditional distribution given Y = σ1 with σ ∈ { − , + } . It is very common that the fraction of positive instances in the training dataset is significantly lower than the rate p expected in the test stage , supposed to be known here ( see the Supplementary Material for the case where the rate p is only approximately known ) . We thus consider the case where the distribution P′ of the training data ( X′1 , Y ′ 1 ) , . . . , ( X ′ n , Y ′ n ) is described by the triplet ( p′ , F+ , F− ) with p′ < p. The likelihood function takes the simple following form Φ ( x , y ) = I { y = +1 } p p′ + I { y = −1 } 1 − p 1 − p′ de f = φ ( y ) , which reveals that it depends on the label y solely , and the ideally weighted empirical risk process is R̃w∗ , n ( g ) = p p′ 1 n ∑ i : Y ′i =1 I { g ( X′i ) = −1 } + 1 − p 1 − p′ 1 n ∑ i : Y ′i =−1 I { g ( X′i ) = +1 } . ( 6 ) In general the theoretical rate p′ is unknown and one replaces ( 6 ) with R̃ŵ∗ , n ( g ) = p n′+ ∑ i : Y ′i =1 I { g ( X′i ) = −1 } + 1 − p n′− ∑ i : Y ′i =−1 I { g ( X′i ) = +1 } , ( 7 ) where n′+ = ∑n i=1 I { Y ′i = +1 } = n− n′− , ŵ∗i = φ̂ ( Y ′i ) and φ̂ ( y ) = I { y = +1 } np/n′+ + I { y = −1 } n ( 1− p ) /n′− . The stochastic process above is not a standard empirical process but a collection of sums of two ratios of basic averages . However , the following result provides a uniform control of the deviations between the ideally weighted empirical risk and that obtained by plugging the empirical weights into the latter . Lemma 2 . Let ε ∈ ( 0 , 1/2 ) . Suppose that p′ ∈ ( ε , 1−ε ) . For any δ ∈ ( 0 , 1 ) , we have with probability larger than 1 − δ : sup g∈G ∣∣∣∣R̃ŵ∗ , n ( g ) − R̃w∗ , n ( g ) ∣∣∣∣ ≤ 2 ε2 √ log ( 2/δ ) 2n , as soon as n ≥ 2 log ( 2/δ ) /ε2 . See the Appendix for the technical proof . Consequently , minimizing ( 7 ) nearly boils down to minimizing ( 6 ) . Combining Lemmas 2 and 1 , we immediately get the generalization bound stated in the result below . Corollary 1 . Suppose that the hypotheses of Lemma 2 are fulfilled . Let g̃n be any minimizer of R̃ŵ∗ , n over class G. We have with probability at least 1 − δ : RP ( ̃gn ) − inf g∈G RP ( g ) ≤ 2 max ( p , 1 − p ) ε 2E [ R′n ( G ) ] + √ 2 log ( 2/δ ) n + 4ε2 √ log ( 4/δ ) 2n , as soon as n ≥ 2 log ( 4/δ ) /ε2 ; where R′n ( G ) = ( 1/n ) Eσ [ supg∈G | ∑n i=1 σiI { g ( X′i ) , Y ′i } | ] . Hence , some side information ( i.e . knowledge of parameter p ) has permitted to weight the training data in order to build an empirical risk functional that approximates the target risk and to show that minimization of this risk estimate yields prediction rules with optimal ( in the minimax sense ) learning rates . The purpose of the subsequent analysis is to show that this remains true for more general problems . Observe in addition that the bound in Corollary 1 deteriorates as ε decays to zero : the method used here is not intended to solve the few shot learning problem , where almost no training data with positive labels is available ( i.e . p′ ≈ 0 ) . As shall be seen in subsection 3.2 , alternative estimators of the importance function must be considered in this situation . Remark 1 . Although the quantity ( 7 ) can be viewed as a cost-sensitive version of the empirical classification risk based on the ( X′i , Y ′ i ) ’ s ( see e.g . Bach et al . ( 2006 ) ) , we point out that the goal pursued here is not to achieve an appropriate trade-off between type I and type II errors in the P′ classification problem as in biometric applications for instance ( i.e . optimization of the ( F+ , F− ) -ROC curve at a specific point ) but to transfer knowledge gained in analyzing the biased data drawn from P′ to the classification problem related to distribution P. Related work . We point out that the natural idea of using weights in ERM problems that mimic those induced by the importance function has already been used in Sugiyama et al . ( 2008 ) for covariate shift adaptation problems ( i.e . supervised situations , where the conditional distribution of the output given the input information is the same in the training and test domains ) , when , in contrast to the framework considered here , a test sample is additionally available ( a method for estimating directly the importance function based on Kullback-Leibler divergence minimization is proposed , avoiding estimation of the test density ) . Importance sampling estimators have been also considered in Garcke & Vanck ( 2014 ) in the setup of inductive transfer learning ( the tasks between source and target are different , regardless of the similarities between source and target domains ) , where the authors have proposed two methods to approximate the importance function , among which one is again based on minimizing the Kullback-Leibler divergence between the two distributions . In Cortes et al . ( 2008 ) , the sample selection bias is assumed to be independent from the label , which is not true under our stratum-shift assumption or for the PU learning problem ( see section 3 ) . Lemma 1 assumes that the exact importance function is known , as does Cortes et al . ( 2010 ) . The next section introduces new results for more realistic settings where it has to be learned from the data . 3 Weighted Empirical RiskMinimization - Generalization Guarantees Through two important and generic examples , relevant for many applications , we show that the approach sketched above can be applied to general situations , where appropriate auxiliary information on the target distribution is available , with generalization guarantees . 3.1 Statistical Learning from Biased Data in a Stratified Population A natural extension of the simplistic problem considered in section 2 is multiclass classification in a stratified population . The random labels Y and Y ′ are supposed to take their values in { 1 , . . . , J } say , with J ≥ 1 , and each labeled observation ( X , Y ) belongs to a certain random stratum S in { 1 , . . . , K } with K ≥ 1 . Again , the distribution P of a random element Z = ( X , Y , S ) may be described by the parameters { ( p j , k , F j , k ) : 1 ≤ j ≤ J , 1 ≤ k ≤ K } where F j , k is the conditional distribution of X given ( Y , S ) = ( j , k ) and p j , k = P ( X , Y , S ) ∼P { Y = j , S = k } . Then , we have dP ( x , y , s ) = J∑ j=1 K∑ k=1 I { y = j , s = k } p j , kdF j , k ( x ) , and considering a distribution P′ with F j , k ≡ F′j , k but possibly different class-stratum probabilities p′j , k , the likelihood function becomes dP dP′ ( x , y , s ) = J∑ j=1 K∑ k=1 p j , k p′j , k I { y = j , s = k } de f= φ ( y , s ) . A more general framework can actually encompass this specific setup by defining ’ meta-strata ’ in { 1 , . . . , J } × { 1 , . . . , K } . Strata may often correspond to categorical input features in practice . The formalism introduced below is more general and includes the example considered in the preceding section , where strata are defined by labels . Learning from biased stratified data . Consider a general mixture model , where distributions P and P′ are stratified over K ≥ 1 strata . Namely , Z = ( X , S ) and Z′ = ( X′ , S ′ ) with auxiliary random variables S and S ′ ( the strata ) valued in { 1 , . . . , K } . We place ourselves in a stratum-shift context , assuming that the conditional distribution of X given S = k is the same as that of X′ given S ′ = k , denoted by Fk ( dx ) , for any k ∈ { 1 , . . . , K } . However , stratum probabilities pk = P ( S = k ) and p′k = P ( S ′ = k ) may possibly be different . In this setup , the likelihood function depends only on the strata and can be expressed in a very simple form , as follows : dP dP′ ( x , s ) = K∑ k=1 I { s = k } pk p′k de f = φ ( s ) . In this case , the ideally weighted empirical risk writes R̃w∗ , n ( θ ) = 1 n n∑ i=1 ` ( θ , Z′i ) K∑ k=1 I { S ′i = k } pk p′k . If the strata probabilities pk ’ s for the test distribution are known , an empirical counterpart of the ideal empirical risk above is obtained by simply plugging estimates of the p′k ’ s computed from the training data : R̃ŵ∗ , n ( θ ) = n∑ i=1 ` ( θ , Z′i ) K∑ k=1 I { S ′i = k } pk n′k , ( 8 ) with n′k = ∑n i=1 I { S ′i = k } , ŵ∗i = φ̂ ( S ′i ) and φ̂ ( s ) = ∑K k=1 I { s = k } npk/n′k . A bound for the excess of risk is given in Theorem 1 , that can be viewed as a generalization of Corollary 1 . Theorem 1 . Let ε ∈ ( 0 , 1/2 ) and assume that p′k ∈ ( ε , 1−ε ) for k = 1 , . . . , K. Let θ̃∗n be any minimizer of R̃ŵ∗ , n as defined in ( 8 ) over class Θ . We have with probability at least 1 − δ : RP ( ̃θ∗n ) − inf θ∈Θ RP ( θ ) ≤ 2 maxk pk ε 2E [ R′n ( F ) ] + L √ 2 log ( 2/δ ) n + 4Lε2 √ log ( 4K/δ ) 2n , as soon as n ≥ 2 log ( 4K/δ ) / 2 ; where R′n ( F ) = ( 1/n ) Eσ [ supθ∈Θ | ∑n i=1 σi ` ( θ , Z ′ i ) | ] , and the loss is bounded by L = sup ( θ , z ) ∈Θ×Z ` ( θ , z ) . Just like in Corollary 1 , the bound in Theorem 1 explodes when ε vanishes , which corresponds to the situation where a stratum k ∈ { 1 , . . . , K } is very poorly represented in the training data , i.e . when p′k < < pk . Again , as highlighted by the experiments carried out , reweighting the losses in a frequentist ( ERM ) approach guarantees good generalization properties in a specific setup only , where the training information , though biased , is sufficiently informative . 3.2 Positive-Unlabeled Learning Relaxing the stratum-shift assumption made in the previous subsection , the importance function becomes more complex and writes : Φ ( x , s ) = dP dP′ ( x , s ) = K∑ k=1 I { s = k } pk p′k dFk dF′k ( x ) , where Fk and F′k are respectively the conditional distributions of X given S = k and of X ′ given S ′ = k. The Positive-Unlabeled ( PU ) learning problem , which has recently been the subject of much attention ( see e.g . du Plessis et al . ( 2014 ) , Du Plessis et al . ( 2015 ) , Kiryo et al . ( 2017 ) ) , provides a typical example of this situation . Re-using the notations introduced in section 2 , in the PU problem , the testing and training distributions P and P′ are respectively described by the triplets ( p , F+ , F− ) and ( q , F+ , F ) , where F = pF+ + ( 1 − p ) F− is the marginal distribution of X . Hence , the objective pursued is to solve a binary classification task , based on the sole observation of a training sample pooling data with positive labels and unlabeled data , q denoting the theoretical fraction of positive data among the dataset . As noticed in du Plessis et al . ( 2014 ) ( see also Du Plessis et al . ( 2015 ) , Kiryo et al . ( 2017 ) ) , the likelihood/importance function can be expressed in a simple manner , as follows : ∀ ( x , y ) ∈ X × { −1 , +1 } , Φ ( x , y ) = p q I { y = +1 } + 1 1 − q I { y = −1 } − p 1 − q dF+ dF ( x ) I { y = −1 } . ( 9 ) Based on an i.i.d . sample ( X′1 , Y ′ 1 ) , . . . , ( X ′ n , Y ′ n ) drawn from P ′ combined with the knowledge of p ( which can also be estimated from PU data , see e.g . Du Plessis & Sugiyama ( 2014 ) ) and using that F− = ( 1/ ( 1 − p ) ) ( F − pF+ ) , one may obtain estimators of q , F+ and F by computing n′+/n = ( 1/n ) ∑n i=1 I { Y ′i = +1 } , F̂+ = ( 1/n′+ ) ∑n i=1 I { Y ′i = +1 } δX′i and F̂ = ( 1/n ′ − ) ∑n i=1 I { Y ′i = −1 } δX′i . However , plugging these quantities into ( 9 ) do not permit to get a statistical version of the importance function , insofar as the probability measures F̂+ and F̂ are mutually singular with probability one , as soon as F+ is continuous . Of course , as proposed in du Plessis et al . ( 2014 ) , one may use statistical methods ( e.g . kernel smoothing ) to build distribution estimators , that ensures absolute continuity but are subject to the curse of dimensionality . However , WERM can still be applied in this case , by observing that : ∀g ∈ G , RP ( g ) = −p + EP′ [ 2p q I { g ( X′ ) = −1 , Y ′ = +1 } + 1 1 − q I { g ( X ′ ) = +1 , Y ′ = −1 } ] , ( 10 ) which leads to the weighted empirical risk 2p n′+ ∑ i : Y ′i =+1 I { g ( X′i ) = −1 } + 1 n′− ∑ i : Y ′i =−1 I { g ( X′i ) = +1 } . ( 11 ) Minimization of ( 11 ) yields rules g̃n whose generalization ability regarding the binary problem related to ( p , F+ , F− ) can be guaranteed , as shown by the following result , the form of the weighted empirical risk in this case being quite similar to ( 7 ) . Theorem 2 . Let ε ∈ ( 0 , 1/2 ) . Suppose that q ∈ ( ε , 1 − ε ) . Let g̃n be any minimizer of the weighted empirical risk ( 11 ) over class G. We have with probability at least 1 − δ : RP ( ̃gn ) − inf g∈G RP ( g ) ≤ 2 max ( 2p , 1 ) ε 2E [ R′n ( G ) ] + √ 2 log ( 2/δ ) n + 4 ( 2p + 1 ) ε2 √ log ( 4/δ ) 2n , as soon as n ≥ 2 log ( 4/δ ) /ε2 ; where R′n ( G ) = ( 1/n ) Eσ [ supg∈G | ∑n i=1 σiI { g ( X′i ) , Y ′i } | ] . Remark 2 . Let η ( x ) = P { Y = +1 | X = x } denote the posterior probability and recall that ( dF+/dF− ) ( x ) = ( ( 1 − p ) /p ) ( η ( x ) / ( 1 − η ( x ) ) . Observing that Φ ( x , y ) = p q I { y = +1 } + 1 − η ( x ) 1 − q I { y = −1 } , ( 12 ) in the case when an estimate η̂ ( x ) of η ( x ) is available , one can perform WERM using the empirical weight function Φ̂ ( x , y ) = np n′+ I { y = +1 } + 1 − η̂ ( x ) 1 − n′+/n I { y = −1 } . ( 13 )

Summary: This paper targets the transfer learning problem. It wants to construct the unbiased estimator of the true risk for the target domain based on data from the source domain. To have the unbiased estimator, samples in the source domain are weighted based on some auxiliary information of both the source domain data distribution and the target domain data distribution. Especially, similar to previous works, the paper first assumes P(Y=1) is known for the target domain, and give a generalization bound for learning on the target domain. Then they consider two more concrete problems, one is learning with stratified information when the conditional probability given the stratified information of the source domain is equal to that of the target domain. Then the paper considers PU learning. Generalization bounds are also given for these two problems. Finally, the paper shows some empirical results showing the reweighting effect of its proposal.