Multi-Label Learning with Pairwise Relevance Ordering

1 Introduction . Multi-label learning ( MLL ) solves problems where each object is assigned with multiple class labels simultaneously [ Zhang and Zhou , 2013 ] . For example , an image may be annotated with labels building , street and person . The goal of multi-label learning is to train a classification model that can predict all the relevant labels for unseen instances . A large number of recent works have witnessed the great successes that MLL has achieved in many real-world applications , e.g. , image annotation [ Chen et al. , 2019 ] , human attribute recognition [ Li et al. , 2016 ] , user profiling [ Liu et al. , 2021 ] , and protein function prediction [ Elisseeff and Weston , 2002 ] . Traditional multi-label learning studies assume that each instance has been precisely annotated with all of its relevant labels . However , in many real-world scenarios , it is difficult and costly to collect the precise annotations . Instead , each instance may be provided with the relative order of label pairs , where each label pair y y′ ( or y ≺ y′ ) indicates that label y is more relevant ( or irrelevant ) than label y′ to instance x , i.e. , p ( y = 1|x ) > p ( y′ = 1|x ) ( or p ( y = 1|x ) < p ( y′ = 1|x ) ) . Generally , deciding the relative order of label pairs would be much easier than collecting the precise annotations and thus less costly . For example , in medical image analysis , only experts with rich experiences can accurately identify the disease for a patient based on the medical image . In contrast , if the question is to decide which of two given diseases is more likely suffered by the patient , then even medical students with basic knowledge may easily provide the answer . While the annotation cost is significantly reduced with pairwise relevance ordering , the learning task becomes more challenging , since the supervised information of pairwise relevance ordering is much less than exact labels . We formalize this learning problem as a new framework called multi-label learning with pairwise relevance ordering ( PRO ) . Specifically , PRO attempts to learn a classification model from multi-label ∗Correspondence to : Sheng-Jun Huang ( huangsj @ nuaa.edu.cn ) . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . examples with the relative order of label pairs , where each label pair y y′ indicates three possible cases : 1 ) both y and y′ are relevant to x , i.e. , y = 1 , y′ = 1 ; 2 ) y is relevant to x while y′ is not , i.e. , y = 1 , y′ = 0 ; 3 ) both y and y′ are irrelevant to x , i.e. , y = 0 , y′ = 0 . PRO is a novel learning framework with significant difference from exiting settings . For example , semi-supervised multi-label learning ( SSMLL ) learns a classifier by exploiting a few of labeled examples as well as a large number of unlabeled examples [ Liu et al. , 2006 ] ; multi-label learning with missing labels ( MLML ) assumes that only a subset of labels are available for each instance [ Sun et al. , 2010 , Yu et al. , 2014 ] ; partial multi-label learning ( PML ) assigns each instance with a candidate label set [ Xie and Huang , 2018 , Zhang and Fang , 2020 ] ; multi-label learning with noisy labels assumes that multiple class labels may be flipped simultaneously with their respective probabilities [ Xie and Huang , 2021a ] . However , these frameworks do not consider multi-label examples with pairwise relevance ordering , and can not be employed to solve PRO problems . To deal with multi-label data with pairwise relevance ordering , we propose a cost-sensitive loss function for learning a multi-label classifier with empirical risk minimization . Theoretically , we show that the unbiased estimator of classification risk can be derived from only PRO examples if the surrogate loss function satisfies a mild condition , i.e. , the symmetric condition . The estimation error bound is established for the unbiased estimator , showing that learning with PRO examples can be multi-label consistent to the commonly used ranking loss . Extensive experimental results on multiple datasets and evaluation metrics demonstrate the practical usefulness of the proposed method . 2 Related Works . There are plenty of literature on multi-label learning . As one of the earliest representative methods , Binary Relevance simply decomposes the multi-label learning task into a set of binary classification problems [ Zhang and Zhou , 2013 ] . Nevertheless , such method neglects the label correlations , which are regarded as an essential information for multi-label classification . Therefore , there are many studies trying to learn a multi-label classifier by exploiting the label correlations [ Read et al. , 2011 ] . Some of these works focus on the pairwise correlation [ Elisseeff and Weston , 2002 , Li et al. , 2017 ] , while some others consider the high order correlation among all labels [ Chen et al. , 2019 ] . To solve SSMLL problems , some works attempt to learn a multi-label classifier based on the graph models [ Kong et al. , 2011 ] while some others utilize the low-rank assumption [ Jing et al. , 2015 ] . In addition to these methods , the co-training strategy [ Zhan and Zhang , 2017 ] and matrix factorization [ Liu et al. , 2006 ] are also employed to solve SSMLL problems . The pioneering MLML study [ Sun et al. , 2010 ] tries to construct a similarity graph for each label and the manifold regularization term is added to recover the missing labels . A linear classifier with the low-rank constraint is proposed to deal with large scale data with missing labels [ Yu et al. , 2014 ] . In [ Kanehira and Harada , 2016 ] , authors solve the MLML problem by viewing it as a positive-unlabeled learning task . Some other techniques also employ the robust loss [ Xu et al. , 2019 ] or the group lasso regularizer [ Bucak et al. , 2011 ] to solve MLML problems . In order to deal with partial-labeled data , the most commonly used strategy is disambiguation [ Cour et al. , 2011 ] , which recovers ground-truth labeling information for candidate labels . Some methods perform the disambiguation strategy by estimating a confidence for each candidate label [ Xie and Huang , 2018 , Yu et al. , 2018 , 2020 ] . Other methods utilize the decomposition scheme [ Sun et al. , 2019 ] or adversarial training [ Yan and Guo , 2021 ] . In [ Xie and Huang , 2021b ] , authors first solve PML problems by considering the generation process of noisy labels . A recent work utilizes the meta disambiguation strategy to deal with partial-labeled data [ Xie et al. , 2021 ] . Although it allows noisy labels hidden in the candidate set ( e.g. , PML ) or some labels are missed ( e.g. , SSMLL and MLML ) , the aforementioned frameworks consider the supervised information of the label-level , i.e. , each of labels is relevant or not , which can be still costly . Instead our proposed PRO framework considers pairwise relevance ordering , which can be much easier for annotators and thus less costly . In [ Huang et al. , 2015 ] , authors propose a multi-label active learning framework called AURO which queries the relevance ordering between two labels in every iteration . However , different from the proposed PRO framework , AURO directly asks annotators to provide each label pair with one of three possible cases that have been discussed in Section 1 , which can be regarded as a stronger supervised information with higher cost . AURO can not be used to solve the PRO problem . Furthermore , authors consider the pairwise supervision of instances and performing binary classification based on similar paired data and unlabeled examples [ Bao et al. , 2018 , Shimada et al. , 2021 , Bao et al. , 2020 ] . Another recent work trains the binary classifier under the supervision of pairwise confidence comparisons [ Feng et al. , 2021 ] . These methods focus on binary classification and can not be directly applied to solve the PRO problem . 3 Preliminaries . In this section , before deriving our main results for solving the PRO problem , we introduce some notations and provide the necessary preliminaries . In traditional multi-label learning task , let x ∈ X be a feature vector and y ⊆ Y its corresponding labels , where X = Rd is the feature space and Y = { 0 , 1 } q is the target space with q possible class labels . Here , yj = 1 indicates the j-th label is relevant to the instance ; yj = 0 , otherwise . Let D = { xi , yi } ni=1 be the given training examples , where each example is drawn i.i.d . according to the joint distribution p ( x , y ) . In multi-label learning , many loss functions have been proposed to measure the performance of learning algorithms , such as ranking loss , hamming loss , coverage and average precision [ Zhang and Zhou , 2013 ] . Among them , the ranking loss concerns about label pairs that are ordered reversely for an instance , which naturally considers the pairwise label correlation . Given the decision function f : X → Rd , the ranking loss can be defined as follows : L ( f ( x ) , y ) = ∑ 1≤j < k≤q I ( yj = 1 , yk = 0 ) ` ( fk , fj ) + I ( yj = 0 , yk = 1 ) ` ( fj , fk ) , ( 1 ) where ` ( fj , fk ) = I ( fj > fk ) + 1 2 I ( fj = fk ) . ( 2 ) Here , fj is the j-th component of f ( x ) and I ( · ) is the indicator function , which outputs 1 if the condition holds while outputs 0 otherwise . The goal of multi-label learning tasks is to learn the optimal classifier f by minimizing the following expected classification risk : R ( f ) = Ep ( x , y ) [ L ( f ( x ) , y ) ) ] = ∑ y∈Y p ( y ) Ep ( x|y ) [ L ( f ( x ) , y ) ) ] , ( 3 ) = ∑ y∈Y p ( y ) ∑ 1≤j < k≤q Ep ( x|yj=1 , yk=0 ) [ ` ( fk , fj ) ] + Ep ( x|yj=0 , yk=1 ) [ ` ( fj , fk ) ] , = ∑ 1≤j < k≤q π10jkEp10jk ( x ) [ ` ( fk , fj ) ] + π 01 jkEp01jk ( x ) [ ` ( fj , fk ) ] , where π10jk = p ( yj = 1 , yk = 0 ) ( or π 01 jk = p ( yj = 0 , yk = 1 ) ) denotes the positive-negative ( or negative-positive ) label pair prior probability , and p10jk ( x ) = p ( x|yj = 1 , yk = 0 ) ( or p01jk ( x ) = p ( x|yj = 0 , yk = 1 ) ) denotes the class-conditional probability density of data given the positivenegative ( negative-positive ) label pair . Accordingly , we define the minimal risk ( also called the Bayes risk ) as R∗ = inff R ( f ) . However , the loss function L is highly discontinuous and computationally NP-hard , which often makes the corresponding optimization problem hard to solve [ Gao and Zhou , 2013 ] . In practice , a feasible solution is to consider alternatively a surrogate loss function L which can be solved efficiently . Accordingly , the L-risk with respect to p ( x , y ) can be defined as : RL ( f ) = Ep ( x , y ) [ L ( f ( x ) , y ) ) ] = ∑ 1≤j < k≤q π10jkEp10jk ( x ) [ φ ( fj−fk ) ] +π 01 jkEp01jk ( x ) [ φ ( fk−fj ) ] , ( 4 ) where φ is a surrogate loss function . A common choice is hinge loss φ ( t ) = max ( 0 , 1 − t ) in [ Elisseeff and Weston , 2002 ] . Accordingly , we define the minimal L-risk ( also called the Bayes L-risk ) as R∗L = inff RL ( f ) .

Summary: This paper studies the multi-label classification problem in a weakly supervised setting, where each instance is assigned with the relative order of label pairs. Authors formalize this task as a new framework called pairwise relevance ordering. An empirical estimator of the classification risk based on a cost-sensitive loss is proposed. Then authors theoretically prove that the proposed estimator is unbiased with the symmetric condition of the loss function. The error bound and consistency of the estimator is also proved. In addition to the theoretical results, the paper also validate the effectiveness of the proposed method empirically on multiple datasets.