Complementary-label learning for arbitrary losses and models

1 INTRODUCTION . Modern classification methods usually require massive data with high-quality labels , but preparing such datasets is unrealistic in many practical domains . To mitigate the problem , many previous works have investigated ways to learn from weak supervision : semi-supervised learning ( Chapelle et al. , 2006 ; Miyato et al. , 2016 ; Kipf & Welling , 2017 ; Sakai et al. , 2017 ; Tarvainen & Valpola , 2017 ; Oliver et al. , 2018 ) , learning from noisily-labeled data ( Natarajan et al. , 2013 ; Patrini et al. , 2017 ; Ma et al. , 2018 ) , learning from positive-unlabeled data ( Elkan & Noto , 2008 ; du Plessis et al. , 2014 ; 2015 ; Kiryo et al. , 2017 ) , learning from similar-unlabeled data ( Bao et al. , 2018 ) , learning from positive-confidence data ( Ishida et al. , 2018 ) , and others . In this paper , we consider learning from another type of weak but natural supervision called complementary-label learning ( Ishida et al. , 2017 ; Yu et al. , 2018 ) , where the label only specifies one of the classes that the pattern does not belong to . In contrast to the ordinary case where the true class is given to each pattern ( which often needs to be chosen out of many candidate classes precisely ) , collecting these complementary labels is obviously much easier and less costly . A natural question is , however , is it possible to learn from such complementary labels ( without any true labels ) ? The problem has previously been tackled by Ishida et al . ( 2017 ) , showing that the classification risk can be recovered only from complementarily labeled data . They also gave consistency gaurantee in theoretical analysis . However , they required strong restrictions on the loss functions , allowing only one-versus-all and pairwise comparison multi-class loss functions ( Zhang , 2004 ) with certain non-convex binary losses . This is a severe limitation when we use deep learning since the softmax cross-entropy loss is often used to boost the classification performance . Later , Yu et al . ( 2018 ) proposed a different formulation for complementary labels by employing the forward loss correction technique ( Patrini et al. , 2017 ) to adjust the learning objective . Their proposed risk estimator is not necessarily unbiased but the minimizer is theoretically guaranteed to be consistent with the minimizer of the risk for ordinary labels ( under an implicit assumption on the model for convergence analysis ) . They also extended the problem setting to where complementary labels are chosen in an uneven ( biased ) way . This is a realistic problem setting because labelers are more likely to complementarily label a pattern when they feel it is not a certain class which they have more knowledge or experience about . In this paper , we first derive an unbiased risk estimator with a general loss function , making any loss functions available for use : not only the softmax cross-entropy loss function but other convex/nonconvex loss functions can also be applied . We also do not have implicit assumptions on the classifier , allowing both linear and non-linear models . Yu et al . ( 2018 ) does not have an unbiased risk estimator , which means users will need clean data with true labels to calculate the error rate during the validation process . On the other hand , our proposed unbiased risk estimator can handle complementarily labeled validation data not only for our learning objective , but also for Yu et al . ( 2018 ) . This is helpful since collecting clean data is usually much more expensive . Finally , our proposed unbiased risk estimator has an issue that it is unbounded from below and suffers from the classification risk going to negative after learning , leading to overfitting . We further propose a non-negative correction to the original unbiased risk estimator to improve our estimator . We experimentally show that our proposed method is comparable to or better than previous methods ( Ishida et al. , 2017 ; Yu et al. , 2018 ) in terms of classification accuracy . 2 REVIEW OF PREVIOUS WORKS . In this section , we explain the notations and review the formulations of learning from ordinary labels , learning from complementary labels , and learning from both ordinary and complementary labels . Learning from ordinary labels Let X be an instance space and D be the joint distribution over X × [ K ] for class label set [ K ] : = { 1 , 2 , . . . , K } , with random variables ( X , Y ) ∼ D. The data at hand is sampled independently and identically from the joint distribution : { ( xi , yi ) } ni=1 i.i.d.∼ D. The joint distribution D can be either decomposed into class-conditionals { Pk } Kk=1 and base rate { πk } Kk=1 , where Pk : = P ( X|Y = k ) and πk : = P ( Y = k ) , or the marginal M and class-probability function η : X → ∆k , where M : = P ( X ) and ηk ( x ) : = P ( Y = k|X = x ) . A loss is any ℓ : [ K ] × RK → R+ and the decision function is any g : X → RK . The risk for the decision function g with respect to loss ℓ and implicit distribution D is : R ( g ; ℓ ) : = E ( X , Y ) ∼D [ ℓ ( Y , g ( X ) ) ] , ( 1 ) where E denotes the expectation . Two useful equivalent expressions of classification risk ( 1 ) used in later sections are R ( g ; ℓ ) : = EX [ η ( x ) T ℓ ( g ( X ) ) ] = K∑ k=1 πkEPk [ ℓ ( k , g ( X ) ) ] , ( 2 ) where ℓ : = [ ℓ ( 1 , g ) , ℓ ( 2 , g ) , . . . , ℓ ( K , g ) ] T . The goal of classification is to learn the decision function g that minimizes the risk . In the usual classification case with ordinarily labeled data at hand , approximating the risk empirically is straightforward : R̂ ( g ; ℓ ) : = 1n ∑n i=1 ℓ ( yi , g ( xi ) ) . Learning from complementary labels Next we consider the problem of learning from complementary labels ( Ishida et al. , 2017 ) . We observe patterns each equipped with a complementary label { ( xi′ , yi′ ) } n ′ i′=1 sampled independently and identically from a different joint distribution D ̸= D. We denote random variables as ( X , Y ) ∼ D. As before , we assume this distribution can be decomposed into either class-conditionals { P k } Kk=1 and base rate { π } Kk=1 , or marginal M and classprobability function η : X → ∆K , where P k : = P ( X|Y = k ) , πk : = P ( Y = k ) , M : = P ( X ) , ηk ( x ) : = P ( Y = k|X = x ) , Y is the complementary label , and ∆K is the conditional probability simplex for K classes . Without any assumptions on D , it is impossible to design a suitable learning procedure . The assumption for unbiased complementary learning used in Ishida et al . ( 2017 ) was η ( x ) = Tη ( x ) , ( 3 ) where T ∈ RK×K is a matrix that takes 0 on diagonals and 1K−1 on non-diagonals . Under this assumption , Ishida et al . ( 2017 ) proved that they can recover the classification risk ( 1 ) from an alternative formulation using only complementarily labeled data when he loss function satisfies certain conditions . More specifically , usable loss functions are pairwise comparison or one-versus-all multi-class loss functions ( Zhang , 2004 ) each with binary loss function ℓ′ ( z ) : R → R+ that satisfies ℓ′ ( z ) + ℓ′ ( −z ) = 1 , such as ramp loss ℓ′R ( z ) = 12 max ( 0 , min ( 2 , 1 − z ) ) or sigmoid loss ℓ′S ( z ) = 1 1+ez . Having an unbiased risk estimator is also helpful for the validation process . Since we do not have ordinary labels in our validation set in the complementary-label learning setting , we can not follow the usual validation procedure that uses zero-one error or accuracy . If we have an unbiased estimator of the original classification risk ( which can be interpreted as zero-one error ) , we can use the empirical risk for ( cross ) -validated complementary data to select the best hyper-parameter or deploy early stopping . An extension of the above method was considered in Yu et al . ( 2018 ) by using a different assumption than the unbiased complementary learning of Ishida et al . ( 2017 ) : there is some bias amongst the possible complementary labels that can be chosen , thus the non-diagonals of T is not restricted to 1 K−1 . However , one will need to prepare a separate dataset with ordinary labels in order to estimate T beforehand . Unlike Ishida et al . ( 2017 ) , Yu et al . ( 2018 ) did not directly provide a risk estimator , but they showed that the minimizer of their learning objective agrees with the minimizer of the original classification risk ( 1 ) . Note that , in their formulation , the loss function is restricted to the softmax cross-entropy loss . Furthermore , the use of a highly non-linear model is supposed for consistency guarantee in their theoretical analysis . Since the learning objective of Yu et al . ( 2018 ) does not correspond to the classification risk , one will need clean data with true labels to calculate the error rate during the validation process . On the other hand , our proposed risk estimator can cope with complementarily labeled validation data not only for our own learning objective , but can be used to select hyperparameters for others such as Yu et al . ( 2018 ) . Learning from both ordinary and complementary labels In many practical situations , we may also have ordinarily labeled data in addition to complementarily labeled data . Ishida et al . ( 2017 ) touched on the idea of crowdsourcing for an application with both types of data . For example , we may choose one of the classes randomly by following the uniform distribution , with probability 1K−1 for each class , and ask crowdworkers whether a pattern belongs to the chosen class or not . Then the pattern is treated as ordinarily labeled if the answer is yes ; otherwise , the pattern is regarded as complementarily labeled . If the true label was y for a pattern , we can naturally assume that the crowdworker will answer yes by P ( Y = y|X = x ) and no by 1 − P ( Y = y|X = x ) . This way , ordinarily labeled data can be regarded as samples from D , and complementarily labeled data from D , justifying the assumption of unbiased complementary learning ( 3 ) . In Ishida et al . ( 2017 ) , they considered a convex combination of the classification risks derived from ordinarily labeled data and complementarily labeled data : αR ( g ; ℓ ) + ( 1 − α ) R ( g ; ℓ ) , where α ∈ [ 0 , 1 ] is a hyper-parameter that interpolates between the two risks . The combined ( also unbiased ) risk estimator can utilize both kinds of data in order to obtain better classifiers , which was demonstrated to perform well in experiments .

Summary: This paper proposes an improved approach to the "complementary-label" form of weak supervision, in which a label that is *not* the true label is marked. Specifically, this paper proposes an unbiased estimator that accepts arbitrary loss functions and models. Noting that this proposed estimator can suffer from overfitting due to unbounded negative loss, a lower-bounded estimator is proposed. Experiments are then performed on several image classification datasets.