Revisiting Explicit Regularization in Neural Networks for Reliable Predictive Probability

1 INTRODUCTION . As deep learning models have become pervasive in real-world decision-systems , the importance of producing a reliable predictive probability is increasing . In this paper , we call predictive probability reliable if it is well-calibrated and precisely represents uncertainty about its predictions . The calibrated behavior refers to the ability to match its predictive probability of an event to the longterm frequency of the event occurrence ( Dawid , 1982 ) . The reliable predictive probability benefits many downstream tasks such as anomaly detection ( Malinin & Gales , 2019 ) , classification with rejection ( Lakshminarayanan et al. , 2017 ) , and exploration in reinforcement learning ( Gal & Ghahramani , 2016 ) . More importantly , deep learning systems with more reliable predictive probability can provide better feedback for explaining what is going on , situations when its prediction becomes uncertain , and unexpected anomalies to users . Unfortunately , neural networks are prone to be overconfident and lack uncertainty representation ability , and this problem has become a fundamental concern in the deep learning community . Bayesian methods have innate abilities to produce reliable predictive probability . Specifically , they express the probability distribution over parameters , in which uncertainty in the parameter space is automatically determined by data ( MacKay , 1992 ; Neal , 1993 ) . Then , uncertainty in prediction can be represented by means of providing rich information about aggregated predictions from different parameter configurations such as entropy and mutual information . From this perspective , deterministic neural networks selecting a single parameter configuration that can not provide such rich information naturally lack the uncertainty representation ability . However , the automatic determination of parameter uncertainty in the light of data , i.e. , posterior inference , comes with prohibitive computational costs . Therefore , the mainstream approach for improving the predictive probability quality has been an efficient adoption of the Bayesian principle into neural networks ( Gal & Ghahramani , 2016 ; Ritter et al. , 2018 ; Teye et al. , 2018 ; Joo et al. , 2020a ) . Recent works ( Lakshminarayanan et al. , 2017 ; Müller et al. , 2019 ; Thulasidasan et al. , 2019 ) has discovered the hidden gems of label smoothing ( Szegedy et al. , 2016 ) , mixup ( Zhang et al. , 2018 ) , and adversarial training ( Goodfellow et al. , 2015 ) , which improve the calibration performance and the uncertainty representation ability . These findings present a new possibility of improving the reliability of the predictive probability without changing the deterministic nature of neural networks . This direction is appealing because it can be applied in a plug-and-play fashion to the existing building blocks . This means that they can inherit the scalability , computational efficiency , and surprising generalization performance of the deterministic neural networks , for which Bayesian neural networks often struggle ( Wu et al. , 2019 ; Osawa et al. , 2019 ; Joo et al. , 2020a ) . Motivated by these observations , we investigate a general direction from the regularization perspective to mitigate the unreliable predictive probability problem , rather than proposing new constructive heuristics or discovering hidden properties of specific methods . Our main contribution is twofold . First , we present a new direction for alleviating the unreliable predictive behavior , which is readily applicable , computationally efficient , and scalable to large-scale models compared to Bayesian neural networks or ensemble methods . Second , our findings provide a novel view of the role of explicit regularization in deep learning , which improves the reliability of the predictive probability . 2 ANALYZING THE CAUSE OF UNRELIABLE PREDICTIVE PROBABILITY . 2.1 BACKGROUND . We consider a classification problem with i.i.d . training samples D = { ( x ( i ) , y ( i ) ) } N i=1 drawn from unknown distributions Px , y whose corresponding tuple of random variables is ( x , y ) . We denote X as an input space and Y as a set of categories { 1 , 2 , · · · , K } . Let fW : X → Z be a neural network with parameters W where Z = RK is a logit space . On top of the logit space , the softmax σ : RK →4K−1 normalizes the exponential of logits : φWk ( x ) = exp ( fWk ( x ) ) ∑ i exp ( f W i ( x ) ) ( 1 ) where we let φWk ( x ) = σk ( f W ( x ) ) for brevity . σk ( fW ( x ) ) is often interpreted as the predictive probability that the label of x belongs to class k ( Bridle , 1990 ) The probabilistic interpretation of neural network outputs gives the natural minimization objective for classification–the cross-entropy between the predictive probability and the one-hot encoded label : lCE ( y , φW ( x ) ) = − ∑ k 1y ( k ) log φ W k ( x ) , where 1A ( ω ) is an indicator function taking one if ω ∈ A and zero otherwise . By minimizing the cross-entropy ( or equivalently maximizing the log-likelihood ) with stochastic gradient descent ( SGD ) ( Robbins & Monro , 1951 ) or its variants , modern neural networks achieve the surprising generalization performance . As the demand for neural networks in real-world decision-making is emerging , reliable predictive probability has been of interest in the machine learning community . One important quality of predictive probability is calibrated behavior . Specifically , based on the notion of calibration in classical forecasting problem ( Dawid , 1982 ) , the perfectly calibrated model can be defined as follows : p ( y = k|φW ( x ) = p ) = pk , ∀p ∈ 4K−1 , k ∈ { 1 , 2 , · · · , K } ( 2 ) Here note that the calibrated model does not necessarily be ones producing φWk ( x ) = p ( y = k|x ) . In practice , expected calibration error ( ECE ) ( Naeini et al. , 2015 ) is widely used for calibration performance measure . ECE on dataset DT can be computed by binning predictions into M -groups based on their confidences1 and then averaging their calibration scores by : M∑ i=1 |Gi| |DT | |acc ( Gi ) − conf ( Gi ) | ( 3 ) where Gi = { x : i/M < maxk φ W k ( x ) ≤ ( 1 + i ) /M , x ∈ DT } ; acc ( Gi ) and conf ( Gi ) are average accuracy and confidence of predictions in group Gi , respectively . 1Throughout this paper , the confidence ( or predictive confidence ) at x refers to maxk φWk ( x ) , which is different from the confidence in statistics literature . Another metric for evaluating the reliability of predictive probability is based on predictive entropy , which evaluates how well is the model aware of its ignorance . To this end , predictive entropy is measured on samples that the model is ignorant of , such as misclassified or out-of-distribution ( OOD ) samples . For the classifier having reliable predictive probability , we expect the high uncertainty of φW ( x ) on such samples , i.e. , the answer “ I don ’ t know. ” However , several recent findings show that the resulting neural network produces unreliable predictive probability , which interprets the softmax output as the “ predictive probability ” implausible ( Gal & Ghahramani , 2016 ) . For example , Figure 1 illustrates the unreliable predictive behavior of ResNet ( He et al. , 2016 ) : the network produces outputs with high confidence in misclassified examples ( Figure 1 ( upper ) ) and provides low predictive entropy on out-of-distribution samples , albeit the samples belong to none of the classes it learned ( Figure 1 ( lower ) ) . One simple yet effective solution for improving the predictive probability ’ s quality is temperature scaling ( Guo et al. , 2017 ) , which adjusts the smoothness of the softmax so that the resulting predictive probability maximizes the log-likelihood on unseen dataset D′ : max τ ∑ ( x , y ) ∈D′ log exp ( fWy ( x ) /τ ) ∑ j exp ( f W j ( x ) /τ ) ( 4 ) where W is a fixed pretrained weight and τ is a temperature controlling the smoothness of the softmax output . This simple method makes the softmax output more reliable predictive probability . For instance , the predictive confidence matches its actual accuracy well , and the predictive entropy on out-of-distribution samples significantly increases ( Figure 1 ) . 2.2 A CLOSER LOOK AT THE LOG-LIKELIHOOD ON UNSEEN SAMPLES .. Motivated by the success of the temperature scaling , we aim to find the relationship between the loglikelihood and the calibration performance . To this end , let s be a binary random variable indicating whether a model correctly classifies a sample . Then , we can derive the following upper bound by using the law of total expectation : Ex , y [ log φWy ( x ) ] = Ex [ ps|x ( s = 1 ) Ey|s=1 , x [ log φWy ( x ) ] + ps|x ( s = 0 ) Ey|s=0 , x [ log φWy ( x ) ] ] ≤ Ex [ ps|x ( s = 1 ) log φ W mx ( x ) + ps|x ( s = 0 ) log ( 1− φ W mx ( x ) ) ] ( 5 ) where mx is the predictive class such that mx = argmaxk f W k ( x ) , ps|x ( s = 1 ) = p ( y = argmaxk φ W mk ( x ) |x ) , and the inequality comes from the fact that Ey|s=0 , x [ log φWy ( x ) ] ≤ maxk 6=mx log φ W k ( x ) ≤ log ( 1− φWmx ( x ) ) . The upper bound can be thought of as the probabilistic measure of the calibration performance . Specifically , suppose a neural network produces an answer with probability φWmx ( x ) and refuses to answer otherwise . Then , the upper bound becomes the divergence between the model ’ s ability to correctly predict the sample and the model ’ s willingness to answer . When we consider the inequality in equation 5 is applied to the empirical mean on the test dataset , this inequality clearly explains why the temperature scaling is helpful ; it increases a lower bound of the calibration error with modified confidence by τ closer to its accuracy . More importantly , the inequality in equation 5 explains the impacts of the cross-entropy minimization on the behavior of the neural network . Specifically , suppose E ( x , y ) ∼D [ log φWy ( x ) ] → 0 for some dataset D. Then , it can be shown that ps|x ( s = 1 ) → 1 and φWmx ( x ) → 1 for all ( x , y ) ∈ D. This holds because we have mink φWk ( x ) ≥ 1/K for any x and the minimum of p log q + ( 1 − p ) log ( 1 − q ) are ( p , q ) → ( 1 , 1 ) or ( p , q ) → ( 0 , 0 ) . Therefore , for high-capacity models that can make the log-likelihood on the training dataset Dtr close to zero , e.g. , deep neural networks , its behavior on the training set will converge to a configuration that corrects all samples with perfect confidence . Figure 2 illustrates this phenomenon in ResNet trained on CIFAR-100 : Ex∼Dtr [ log φWmx ] → 0 ( a ) and E ( x , y ) ∼Dtr [ ps|x ( s = 1 ) ] → 1 ( b ) as the training continues . Why are these convergences problematic ? We observe that function ’ s properties , which depends only on x , evaluated on two different datasets are very close to each other if they are drawn from the same distribution , unlike values that depend on the external randomness y|x ; that is , |Ex∼Dtr [ g ( x ) ] − Ex′∼Dval [ g ( x′ ) ] | |E ( x , y ) ∼Dtr [ h ( x , y ) ] − E ( x′ , y′ ) ∼Dval [ h ( x′ , y′ ) ] | for some functions g ( · ) and h ( · ) . For example , the empirical mean of the maximum log-probability log φWmx ( x ) ( Figure 2 ( a ) ) and L2 norm of fW ( Figure 2 ( c ) ) 2 on training samples are significantly similar to those values on unseen samples compared to the log-likelihood ( Figure 2 ( a ) ) and the accuracy ( Figure 2 ( b ) ) . We conjecture that the log-likelihood maximization with a high-capacity neural network naturally results in a high calibration error on unseen samples due to this discrepancy ; that is , it produces perfect confidence on unseen samples as it does on the training samples where it can not produce the perfect accuracy . Fortunately , this result also indicates that restricting the predictive confidence φWmx ( x ) on training samples will directly impact the predictive confidence on unseen samples and therefore can reduce the calibration error . For this reason , we explore various ways to restrict confidence and show their efficacy on a wide range of tasks for the rest of the paper .

Summary: The paper studies how explicit regularization affects the reliable predictive probability (calibration) of classification tasks. An analysis of the log-likelihood is presented which motivates the use of explicit regularization. Next, two regularization terms are proposed to improve the predictive probability, and experiments on CIFAR – 10/100 show that these regularization terms improve both the accuracy and the calibration error.