Evidence-Aware Entropy Decomposition For  Active Deep Learning

1 INTRODUCTION . Deep learning ( DL ) models establish dominating status among other types of supervised learning models by achieving the state-of-the-art performance in various application domains . However , such an advantage only emerges when a huge amount of labeled training data is available . This limitation slows down the pace of DL , especially when being applied to knowledge-rich domains , such as medicine , biology , and military operations , where large-scale labeled samples are too expensive to obtain from well-trained experts . Meanwhile , active learning ( AL ) has demonstrated great success by showing that for many supervised models , training samples are not equally important in terms of improving the model performance ( Settles , 2009 ) . As a result , a carefully selected smaller training set can achieve a model equally well or even better than a randomly selected large training set . An interesting question arises , which is whether DL models can be actively trained using much less labeled data . Recent efforts show promising results in this direction through Bayesian modeling ( Gal et al. , 2017 ) and batch model sampling ( Sener & Savarese , 2018 ) . However , as DL models are commonly applied to high dimensional data such as images and videos , a fundamental challenge still remains , which is how to most effectively explore the exponentially growing sample space to select the most useful data samples for active model training . Existing AL models usually leverage the model provided information , such as estimated decision boundaries or predicted entropy for data sampling . However , the deep structure and the large number of parameters of DL models make model overfitting almost inevitable especially in the early stage of AL when only very limited training data is available . As a result , the model may provide misleading information that makes data sampling from a high-dimensional search space even more difficult . Besides a high dimensionality , complex data may contain a large number of classes and data samples from certain classes may be completely missing . Such situations are quite common for domains , such as scientific discovery ( e.g. , gene function prediction ) and anomaly detection . AL models should be able to effectively discover these out of distribution ( OOD ) samples for labeling in order to achieve an overall good prediction performance . Uncertainty sampling has been one of the most commonly used pool-based AL models . In particular , a model chooses the data sample that it is least certain about . Thus , once the sample is labeled , model uncertainty can be significantly reduced . As an information-theoretic measure , entropy provides a general criterion for uncertainty sampling . Some commonly used sampling methods , including least confident and margin based strategies , are equivalent to entropy-based sampling in binary classification ( Settles , 2009 ) . It is also natural and straightforward to generalize to multi-class problems . A key challenge of entropy-based sampling for AL is that the predicted entropy may be highly inaccurate , especially in the early state of the AL . Such an issue may become more severe when training a neural network ( NN ) /DL active learner due to model overfitting as described above . Figure 1 ( a ) shows the predicted entropy by an NN active learner trained using nine labeled data samples , which are in black color and evenly distributed in three classes . The standard softmax layer is used in the output layer to generate class probabilities over three classes , each of which is a mixture of two Gaussian ’ s . It turns out that all the data samples in the three small clusters located in the top left , top right , and bottom center , are wrongly predicted with high confidence , as indicated by the low entropy . As a result , data samples from these three clusters are less likely to be selected for labeling . In contrast , the data samples that are close to the center of the three major clusters are more likely to be selected . However , labeling these samples will have the effect of fine-tuning a wrongly predicted decision boundary , leading to a much higher ( but less effective ) labeling cost . Figure 1 ( b ) shows the the result from the proposed active deep learning ( ADL ) model . While the samples from the small clusters are still wrongly predicted due to lack of training data , they are predicted with a much lower low confidence as indicated by the high entropy . However , even with a more accurately predicted entropy , the active learner may still sample from the center of the three major clusters as it is still assigned a high entropy along with the areas that cover the three smaller clusters . By performing a fine-grained analysis of uncertainty under the subjective logic ( SL ) framework ( Jøsang , 2016 ) , we formally decompose entropy into two distinct sources of uncertainty : vacuity and dissonance , which are caused by lack of evidence and conflict of strong evidence , respectively . By putting the vacuity and dissonance as shown in Figures 1 ( c ) and ( d ) together , it is interesting to see that we recover the entropy as shown in Figure 1 ( c ) , which empirically verifies our theoretical results . Entropy decomposition provides further insights on the sources on uncertainty , which is instrumental to guide the data sampling process . Intuitively , given the dataset in Figure 1 , an effective sampling strategy will first choose samples from the three small clusters according to vacuity in the early stage of AL to properly establish the shape of the decision boundary . It can then fine-tune the decision boundary by sampling according to dissonance . Such an uncertainty-aware sampling strategy will be critical for a high-dimensional space with multiple competing classes where data samples are scarcely distributed and the decision boundary becomes more complicated . Our major contribution is threefold : ( 1 ) decomposition of entropy through evidence-based theoretical analysis of belief vacuity and belief dissonance under the SL framework ; ( 2 ) a multi-source uncertainty prediction model that accurately quantifies different sources of uncertainty through kernel density regularized evidence prediction ; ( 3 ) an active deep learning model that systematically integrates different types of uncertainty for effective data sampling in a high-dimensional space . Extensive experiments are conducted over both synthetic and real-world data to demonstrate the effectiveness of the proposed ADL model . 2 RELATED WORK . Uncertainty Quantification in Belief/Evidence Theory : In the belief/evidence theory domain , uncertainty reasoning has been substantially explored such as Fuzzy Logic ( De Silva , 2018 ) , DempsterShafer Theory ( DST ) ( Sentz et al. , 2002 ) , or Subjective Logic ( SL ) ( Jøsang , 2016 ) . Unlike the efforts made in ML/DL above , belief theorists focused on reasoning of inherent uncertainty in information resulting from unreliable , incomplete , deceptive , and/or conflicting evidence . SL considered uncertainty in subjective opinions in terms of vacuity ( i.e. , lack of evidence ) and vagueness ( i.e. , failure of discriminating a belief state ) ( Jøsang , 2016 ) . Vacuity has been used as an effective vehicle to detect out-of-distribution queries through evidence learning , achieved under the typical DL setting with ample training samples ( Sensoy et al. , 2018 ) . Recently , other dimensions of uncertainty have been studied , such as dissonance ( due to conflicting evidence ) and consonance ( due to evidence about composite subsets of state values ) ( Jøsang et al. , 2018 ) . Epistemic Uncertainty in Deep Learning : In DL , aleatoric uncertainty ( AU ) and epistemic uncertainty ( EU ) have been studies using Bayesian Neural Networks ( BNNs ) for computer vision . AU consists of homoscedastic uncertainty ( i.e. , constant errors for different inputs ) and heteroscedastic uncertainty ( i.e. , different errors for different inputs ) ( Gal , 2016 ) . A Bayesian DL ( BDL ) framework was presented to estimate both AU and DU simultaneously in regression ( e.g. , depth regression ) and classification settings ( e.g. , semantic segmentation ) ( Kendall & Gal , 2017 ) . A new type of uncertainty , called distributional uncertainty , is defined based on distributional mismatch between the test and training data distributions ( Malinin & Gales , 2018 ) . Active Learning in Deep Learning : The common AL methods other than DL-based ones are surveyed in ( Settles , 2009 ) . There are limited efforts on actively training DL models for highdimensional data with a few exceptions . In ( Wang & Shang , 2014 ) , an AL model was developed for DL using three metrics for data sampling : least confidence , margin sampling , and entropy . A new approach combines recent advances in BDL into the AL framework to achieve label-efficient DL training ( Gal et al. , 2017 ) . Another approach advances the AL development by introducing a cost-effective strategy to automatically select and annotate the high-confidence samples , which improves the traditional samples selection strategies ( Wang et al. , 2016 ) . Data sampling in DL has also been approached as a core-set selection problem ( Sener & Savarese , 2018 ) , which requires a large batch to work well . Different from all existing works , the proposed ADL model decomposes the accurately estimated uncertainty into vacuity and dissonance and dynamically balances multisource uncertainty to achieve active training of DL models with much less labeled data . 3 EVIDENCE-AWARE ENTROPY DECOMPOSITION . As discussed earlier , a high entropy may be contributed by difference sources of uncertainty with distinct characteristics . In this section , we conduct a fine-grained theoretical analysis of different types of uncertainty that arise in the context of multi-class problems . The decomposition is conducted under the SL framework , which provides key building blocks for our theoretical analysis . 3.1 THEORY OF SUBJECTIVE LOGIC . SL is an uncertain probabilistic logic that is built upon probabilistic logic ( PL ) ( Nilsson , 1986 ) and belief theory ( BT ) ( Shafer , 1976 ) while making two unique extensions . First , SL explicitly represents uncertainty by introducing vacuity of evidence ( or uncertainty mass ) in its opinion representation , which addresses the limitation of using PL to model lack of confidence in probabilities . Second , SL extends the traditional belief function of the BT by incorporating base rates , which serve as the prior probability in Bayesian theory . The Bayesian nature of SL allows it to use second-order uncertainty to express and reason the uncertainty mass , where second-order uncertainty is represented in terms of a probability density function ( PDF ) over first-order probabilities ( Jøsang , 2016 ) . In particular , for multi-class problems , we use a multinomial distribution ( first-order uncertainty ) to model class probabilities and use a Dirichlet PDF ( second-order uncertainty ) to model the distribution of class probabilities . Second-order uncertainty enriches uncertainty representation with evidence information , which plays a central role in entropy decomposition as detailed later . Subjective opinions ( or opinions ) are the arguments in SL . In the multi-class setting , the subjective opinion of a multinomial random variable y in domain Y = { 1 , ... , K } is given by a triplet ω = ( b , u , a ) , with K∑ k=1 bk + u = 1 ( 1 ) where b = ( b1 , ... , bK ) T , u , and a = ( a1 , ... , aK ) T denote the belief mass distribution over Y , uncertainty mass representing vacuity of evidence , and base rate distribution over Y , respectively , and ∀k , ak ≥ 0 , bk ≥ 0 , u ≥ 0 . The probability that y is assigned to the k-class is given by P ( y = k ) = bk + aku , ∀k ∈ Y ( 2 ) which combines the belief mass with the uncertain mass using the base rates . In the multi-class setting , ak can be regarded as the prior preference over the k-th class . When no specific preference is given , we assign all the base rates as 1/K . In existing SL literature , there lacks a clear transition between the first order uncertainty given in equation 2 and the second-order uncertainty expressed as a Dirichlet PDF . Here , we make this transition more explicit by introducing a set of random variables p = ( p1 , ... , pK ) T , where p is distributed on a simplex of dimensionality K − 1 . We introduce a conditional distribution P ( y = k|pk ) = pk , which allows us to represent the marginal distribution in equation 2 by P ( y ) = ∫ P ( y|p ) p ( p ) dp . We define p ( p ) as a Dirichlet PDF over p : Dir ( p|α ) , where α = ( α1 , ... , αK ) T is K-dimensional strength vector , with αk ≥ 0 denoting the effective number of observations of the k-th class . SL explicitly introduces the uncertainty evidence through a non-informative weight W and redefine the strength parameter as αk = rk + akW , with rk ≥ 0 , ∀k ∈ Y ( 3 ) where rk is the amount of evidence ( or the number of observations ) to support the k-th class and W is usually set to K , i.e. , the number of classes . Given the new definition of the strength parameter , the expectation of the class probabilities p = ( p1 , ... , pK ) T is given by E [ pk ] = αk∑K j=1 αj = rk + akW∑K j=1 rj +W ( 4 ) where ak = 1/K . By marginalizing out p , we can derive an evidence-based expression of belief mass and uncertainty mass : bk = rk S ∀k ∈ Y , u = W S , with S = K∑ k=1 αk ( 5 ) SL categorizes uncertainty into two primary sources ( Jøsang , 2016 ) : ( 1 ) basic belief uncertainty that results from specific aspects of belief mass in isolation and ( 2 ) intra-belief uncertainty that results from the relationships between belief masses and uncertainty mass . Since we focus on the multiclass setting , no composite values ( i.e. , simultaneously assigned to multiple classes ) are allowed . As a result , these two sources of uncertainty boil down to vacuity and dissonance , respectively , that correspond to vacuous belief and contradicting beliefs . In particular , vacuity of an opinion vac ( ω ) is captured by uncertainty mass u , which is defined in equation 5 and dissonance of an opinion ( Jøsang et al. , 2018 ) is defined as diss ( ω ) = K∑ k=1 ( bk ∑ j 6=k bjBal ( bj , bk ) ∑ j 6=k bj ) , Bal ( bj , bk ) = { 1− |bj−bk|bj+bk if bibj 6= 0 0 if min ( bi , bj ) = 0 ( 6 ) where Bal ( bj , bk ) is the relative mass balance function between two belief masses . The belief dissonance of an opinion is measured based on how much belief supports individual classes . Consider a binary classification example with a binomial opinion given by ( b1 , b2 , u , a ) = ( 0.49 , 0.49 , 0.02 , a ) . Based on equation 6 , it has a dissonance value of 0.98 . In this case , although the vacuity is close to zero , a high dissonance indicates that one can not make a clear decision because both two classes have the same amount of supporting evidence and hence strongly conflict with each other .

Summary: The authors consider active deep learning. They propose decomposing predictive entropy into a) vacuity (lack of evidence) and b) dissonance (contradictory evidence). They frame this in terms of "subjective logic". In practice this is achieved by having the NN output the parameters of a Dirichlet, which allows an additional degree of freedom describing variance/vacuity. Dissonance is defined in terms of the support of contradictory classes. To get improved estimates of vacuity they augment the loss with a term regularizing the Dirichlet parameters to be small (low precision) at unlabelled points with higher KDE(unlablled points) than KDE(labelled points). They propose initially weighting vacuity and later dissonance as AL proceeds. Encouraging results are presented on simulated 2D data, MNIST and CIFAR10.