Instance-Dependent Partial Label Learning

1 Introduction . Partial label learning ( PLL ) deals with the problem where each training example is associated with a set of candidate labels , among which only one label is valid [ 7 , 5 , 37 ] . Due to the difficulty in collecting exactly labeled data in many real-world scenarios , PLL leverages inexact supervision instead of exact labels . The need to learn from the inexact supervision leads to a wide range of applications for PLL techniques , such as web mining [ 25 ] , multimedia content analysis [ 38 , 4 ] , ecoinformatics [ 24 , 31 ] , etc . To accomplish the task of learning from partial label data , many approaches have been proposed . Identification-based PLL approaches [ 17 , 28 , 24 , 5 , 37 ] regard the ground-truth label as a latent variable and try to identify it . Average-based approaches [ 15 , 7 , 40 ] treat all the candidate labels equally and average the modeling outputs as the prediction . For confidence-based approaches [ 8 , 34 , 42 ] , the confidence of each label is estimated instead of identifying the ground-truth label . These approaches always adopt the randomly picked candidate labels to corrupt benchmark data into partially labeled version despite having no explicit generation process of candidate label sets . To depict the instance-independent generation process of candidate label sets , Feng [ 9 ] proposes a statistical model and deduces a risk-consistent method and a classifier-consistent method . Under the ∗Corresponding author 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . same generation process , another classifier-consistent risk estimator is proposed for deep model and stochastic optimizers [ 26 ] . The previous methods assume that the candidate labels are randomly sampled with the uniform generating procedure [ 26 , 9 ] , which is commonly adopted to corrupt benchmark datasets into partially labeled versions in their experiments . However , the candidate labels are always instance-dependent ( feature-dependent ) in practice as the incorrect labels related to the feature are more likely to be picked as candidate label set for each instance . These methods usually do not perform as well as expected due to the unrealistic assumption on the generating procedure of candidate label sets . In this paper , we consider instance-dependent PLL and assume that each instance in PLL is associated with a latent label distribution [ 33 , 35 , 11 ] constituted by the real number of each label , representing the degree to each label describing the feature . Then , the incorrect label with a high degree in the latent label distribution is more likely to be annotated as the candidate label . For example , the candidate label set of the handwritten digits in Figure 1 ( a ) contains “ 1 ” , “ 3 ” and “ 5 ” , where “ 1 ” and “ 3 ” are not ground-truth but selected as candidate labels due to their high degrees in the latent label distribution of the instance . The object in Figure 1 ( b ) is annotated with “ bird ” and “ airplane ” as the degrees of these two labels are much higher than others in the label distribution . The intrinsical ambiguity increases the difficulty of annotating , which leads to the result that annotators pick the candidate labels with high degrees in the latent label distribution of each instance instead of annotating the ground-truth label directly in PLL . Therefore , the latent label distribution is the essential labeling information in partially labeled examples and worth being leveraged for predictive model training . Motivated by the above consideration , we deal with the PLL problem from two aspects . First , we enhance the labeling information by recovering the latent label distribution for each training example as a label enhancement process [ 33 , 35 ] . Second , we run label enhancement and train the predictive model with recovered label distributions iteratively . The proposed method named VALEN , i.e. , VAriational Label ENhancement for instance-dependent partial label learning , uses the candidate labels to initialize the predictive model in the warm-up training stage , then recovers the latent label distributions via inferring the variational posterior density parameterized by an inference model with the deduced evidence lower bound , and trains the predictive model with a risk estimator by leveraging the candidate labels as well as the label distributions . Our contributions can be summarized as follows : • We for the first time consider the instance-dependent PLL and assume that each partially labeled example is associated with a latent label distribution , which is the essential labeling information and worth being recovered for predictive model training . • We infer the posterior density of the latent label distribution via taking on the approximate Dirichlet density parameterized by an inference model and deduce the evidence lower bound for optimization , in which the topological information and the features extracted from the predictive model are leveraged . • We train predictive model with a proposed empirical risk estimator by leveraging the candidate labels as well as the label distributions . We iteratively recover the latent label distributions and train the predictive model in every epoch . After the network has been fully trained , the predictive model can perform predictions for future test examples alone . Experiments on the corrupted benchmark datasets and real-world PLL datasets validate the effectiveness of the proposed method . 2 Proposed Method . First of all , we briefly introduce some necessary notations . Let X = Rq be the q-dimensional instance space and Y = { y1 , y2 , ... , yc } be the label space with c class labels . Given the PLL training set D = { ( xi , Si ) |1 ≤ i ≤ n } where xi denotes the q-dimensional instance and Si ⊆ Y denotes the candidate label set associated with xi . Note that Si contains the correct label of xi and the task of PLL is to induce a multi-class classifier f : X 7→ Y from D. For each PLL training example ( xi , Si ) , we use the logical label vector li = [ l y1 i , l y2 i , . . . , l yc i ] > ∈ { 0 , 1 } c to represent whether yj is the candidate label , i.e. , lyji = 1 if yj ∈ Si , otherwise l yj i = 0 . The label distribution of xi is denoted by di = [ d y1 i , d y2 i , . . . , d yc i ] > ∈ [ 0 , 1 ] c where ∑c j=1 d yj i = 1 . Then L = [ l1 , l2 , . . . , ln ] and D = [ d1 , d2 , . . . , dn ] represent the logical label matrix and label distribution matrix , respectively . 2.1 Overview . To deal with PLL problem , we iteratively recover the latent label distribution for each example x and train the predictive model by leveraging the recovered label distribution . We start with a warm-up period , in which we train the predictive model with the PLL minimal loss [ 26 ] . This allows us to attain a reasonable predictive model before it starts fitting incorrect labels . After the warm-up period , the features extracted from the predictive model can help for recovering the latent label distribution . Benefited from the essential labeling information in the recovered label distribution , the performance of the predictive model could be further improved . VALEN implements label enhancement and classifier training iteratively in every epoch . In label enhancement , we assume the true posterior density of the latent label distribution takes on the variational approximate Dirichlet density parameterized by an inference model . Then the evidence lower bound is deduced for optimizing the inference model and the label distributions can be generated from the variational posterior . In classifier training , the predictive model is trained by leveraging the recovered label distributions and candidate labels with an empirical risk estimator . After the models has been fully trained , the predictive model can perform prediction for future test instances alone . 2.2 Warm-up Training . The predictive model θ is trained on partially labeled examples by minimizing the following PLL minimal loss function [ 26 ] : Lmin = n∑ i=1 min yj∈Si ` ( f ( xi ) , e yj ) , ( 1 ) where ` is cross-entropy loss and eY = { eyj : yj ∈ Y } denotes the standard canonical vector inRc , i.e. , the j-element in eyj equals 1 and others equal 0 . Similar to [ 26 ] , the min operator in Eq . ( 1 ) is replaced by using the current predictions for slightly weighting on the possible labels in warm-up training . Then we could extract the feature φ of each x via using the predictive model . 2.3 Label Enhancement . We assume that the prior density p ( d ) is a Dirichlet with α̂ , i.e. , p ( d ) = Dir ( d | α̂ ) where α̂ = [ ε , ε , . . . , ε ] > is a c-dimensional vector with a minor value ε . Then we let the prior density p ( D ) be the product of each Dirichlet p ( D ) = n∏ i=1 Dir ( di|α̂ ) . ( 2 ) We consider the topological information of the feature space , which is represented by the affinity graph G = ( V , E , A ) . Here , the feature vector φi of each example could be extracted from the predictive model θ in current epoch , V = { φi | 1 ≤ i ≤ n } corresponds to the vertex set consisting of feature vectors , E = { ( φi , φj ) | 1 ≤ i 6= j ≤ n } corresponds to the edge set , and a sparse adjacency matrix A = [ aij ] n×n can be obtained by aij = { 1 if φi ∈ N ( φj ) 0 otherwise , ( 3 ) where N ( φj ) is the set for k-nearest neighbors of φj and the diagonal elements of A are set to 1 . Let features matrix Φ = [ φ1 , φ2 , . . . , φn ] , adjacency matrix A and logical labels L be observed matrix , VALEN aims to infer the posterior density p ( D|L , Φ , A ) . As the computation of the exact posterior density p ( D|L , Φ , A ) is intractable , a fixed-form density q ( D|L , Φ , A ) is employed to approximate the true posterior . We let the approximate posterior be the product of each Dirichlet parameterized by a vector αi = [ α1i , α 2 i , . . . , α c i ] > : qw ( D | L , Φ , A ) = n∏ i=1 Dir ( di|αi ) . ( 4 ) Here , the parameters ∆ = [ α1 , α2 , . . . , αn ] are outputs of the inference model parameterized by w , which is defined as a two-layer GCN [ 20 ] by GCN ( L , Φ , A ) = Ã ReLU ( ÃZW0 ) W1 , with Z = [ Φ ; L ] and weight W0 , W1 . Here Ã = Â− 1 2 AÂ− 1 2 is the symmetrically normalized weight matrix where Â is the degree matrix of A . By following the Variational Bayes techniques , a lower bound on the marginal likelihood of the model is derived which ensures that qw ( D|L , Φ , A ) is as close as possible to p ( D|L , Φ , A ) . For logical label matrix L , feature matrix Φ , and the corresponding A , the log marginal probability can be decomposed as follows 1 : log p ( L , Φ , A ) = LELBO + KL [ qw ( D|L , Φ , A ) ||p ( D|L , Φ , A ) ] . ( 5 ) where LELBO = Eqw ( D|L , Φ , A ) [ log p ( L , Φ , A|D ) ] − KL [ qw ( D|L , Φ , A ) ||p ( D ) ] . ( 6 ) Due to the non-negative property of KL divergence , the first term LELBO constitutes a lower bound of log p ( L , Φ , A ) , which is often called as evidence lower bound ( ELBO ) , i.e. , log p ( L , Φ , A ) ≥ LELBO . According to Eq . ( 2 ) and Eq . ( 4 ) , the KL divergence in Eq . ( 6 ) can be analytically calculated as follows : KL ( qw ( D|L , Φ , A ) ‖p ( D ) ) = ∑n i=1 ( log Γ ( ∑c j=1 α j i ) − ∑c j=1 log Γ ( αji ) − log Γ ( c · ε ) + c log Γ ( ε ) + ∑c j=1 ( αji − ε ) ( ψ ( αji ) − ψ ( ∑c j=1 α j i ) ) ) . ( 7 ) where Γ ( · ) and ψ ( · ) are Gamma function and Digamma function , respectively . As the first part of Eq . ( 6 ) is intractable , we employ the implicit reparameterization trick [ 10 ] to approximate it by Monte Carlo ( MC ) estimation . Inspired by [ 20 ] , we simply drop the dependence on Φ : p ( L | A , D ) = n∏ i=1 p ( li | A , D ) , p ( A | D ) = n∏ i=1 n∏ j=1 p ( aij | di , dj ) , with p ( aij = 1 | di , dj ) = s ( d > i dj ) . ( 8 ) Here , s ( · ) is the logistic sigmoid function . We further assume that p ( li|A , D ) is a multivariate Bernoulli with probabilities τi . In order to simplify the observation model , T ( m ) = [ τ ( m ) 1 , τ ( m ) 2 , . . . , τ ( m ) n ] is computed from m-th sampling D ( m ) with a three-layer MLP parame- terized by η . Then the first part of Eq . ( 6 ) can be tractable : Eqw ( D|L , Φ , A ) [ log pη ( L , Φ , A|D ) ] = 1 M M∑ m=1 ( tr ( ( I− L ) > log ( I−T ( m ) ) ) + tr ( L > log T ( m ) ) − ‖A− S ( D ( m ) D ( m ) > ) ‖2F ) . ( 9 ) 1More detailed calculations can be seen in Appendix A.1 . Algorithm 1 VALEN Algorithm Input : The PLL training set D = { ( xi , Si ) } ni=1 , epoch T and iteration I ; 1 : Initialize the predictive model θ by warm-up training , the reference model w and observation model η ; 2 : Extract the features Φ from predictive model θ and calculate the adjacency matrix A ; 3 : for t = 1 , . . . , T do 4 : Shuffle training set D = { ( xi , Si ) } ni=1 into I mini-batches ; 5 : for k = 1 , . . . , I do 6 : Obtain label distribution di for each example xi by Eq . ( 4 ) ; 7 : Update θ , w and η by forward computation and back-propagation by fusing Eq . ( 12 ) and Eq . ( 13 ) ; 8 : end for 9 : end for Output : The predictive model θ . Note that we can use only one MC sample in Eq . ( 9 ) during the training process as suggested in [ 19 , 35 ] . In addition , VALEN improves the label enhancement by employing the compatibility loss , which enforces that the recovered label distributions should not be completely different from the confidence ζ ( xi ) [ 9 , 26 ] estimated by current prediction f ( xi ; θ ) : Lo = − 1 n n∑ i=1 c∑ j=1 ζj ( xi ) log d yj i ( 10 ) where ζj ( xi ) = { fj ( xi ; θ ) / ∑ yk∈Si fk ( xi ; θ ) if yj ∈ Si 0 otherwise ( 11 ) Now we can easily get the objective of label enhancement LLE as follows : LLE = λLo − LELBO ( 12 ) where λ is a hyper-parameter . The label distribution matrix D is sampled from q ( D|L , Φ , A ) , i.e. , di ∼ Dir ( αi ) . Note that the implicit reparameterization gradient [ 10 ] is employed , which avoids the inversion of the standardization function , which makes the gradients can be computed analytically in backward pass .

Summary: This paper studies the problem of feature-dependent partial label learning, which is an interesting weakly supervised learning problem where the candidate labels of each instance are feature-dependent. Accordingly, the first attempt to feature-dependent partial label learning with latent label distribution is proposed.  The performance advantage of the proposed approach over state-of-the-art approaches is clearly validated via extensive experimental studies.