Learning to Predict Trustworthiness with Steep Slope Loss

1 Introduction . Classification is a ubiquitous learning problem that categorizes objects according to input features . It is widely used in a range of applications , such as robotics [ 1 ] , environment exploration [ 2 ] , medical diagnosis [ 3 ] , etc . In spite of the successful development of deep learning methods in recent decades , high-performance classifiers would still have a chance to make mistakes due to the improvability of models and the complexity of real-world data [ 4 , 5 , 6 , 7 ] . To assess whether the prediction yielded by a classifier can be trusted or not , there are growing efforts towards learning to predict trustworthiness [ 8 , 9 ] . These methods are evaluated on small-scale datasets , e.g. , MNIST [ 10 ] , where the data is relatively simple and existing classifiers have achieved high accuracy ( > 99 % ) . As a result , there are a dominant proportion of correct predictions and the trustworthiness predictors are prone to classify incorrect predictions as trustworthy predictions . The characteristics that the simple data is easy-to-classify aggravate the situation . To further understand the prowess of predicting trustworthiness , we study this problem on the real-world large-scale datasets , i.e. , ImageNet [ 11 ] . This is a challenging theme for classification in terms of boundary complexity , class ambiguity , and feature dimensionality [ 12 ] . As a result , failed predictions are inevitable . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . A general illustration of predicting trustworthiness [ 9 , 13 ] is shown in Fig . 1a . The trustworthiness predictor [ 13 ] that is based on the maximum confidence have been proven to be unreliable [ 8 , 14 , 15 , 16 ] . Instead , Corbiere et al . [ 9 ] propose the true class probability ( TCP ) that uses the confidence w.r.t . the ground-truth class to determine whether to trust the classifier ’ s prediction or not . Nevertheless , the classification confidence is sensitive to the data . As shown in Fig . 1b , TCP predicts that all the incorrect predictions ( 0.9 % in predictions ) are trustworthy on MNIST [ 10 ] and predicts that all the incorrect predictions ( ∼16 % in predictions ) are trustworthy on ImageNet [ 11 ] . To comprehensively understand this problem , we follow the learning scheme used in [ 9 ] and use two state-of-the-art backbones , i.e. , ViT [ 7 ] and ResNet [ 5 ] , as the trustworthiness predictors . For simplicity , we call the “ trustworthiness predictor '' an oracle . We find that the oracles trained with cross entropy loss [ 17 ] , focal loss [ 18 ] , and TCP confidence loss [ 9 ] on ImageNet are prone to overfit the training samples , i.e. , the true positive rate ( TPR ) is close to 100 % while the true negative rate ( TNR ) is close to 0 % . To improve the generalizability of oracles , we propose a novel loss function named the steep slope loss . The proposed steep slope loss consists of two slide-like curves that cross with each other and face in the opposite direction to separate the features w.r.t . trustworthy and untrustworthy predictions . It is tractable to control the slopes by indicating the heights of slides . In this way , the proposed loss is able to be flexible and effective to push the features w.r.t . correct and incorrect predictions to the well-classified regions . Predicting trustworthiness is similar to as well as different from conventional classification tasks . On one hand , predicting trustworthiness can be formulated as a binary classification problem . On the other hand , task-specific semantics are different between the classification task and predicting trustworthiness . The classes are referred to visual concepts , such as dog , cat , etc. , in the classification task , while the ones in predicting trustworthiness are abstract concepts . The trustworthiness could work on top of the classes in the classification task . In other words , the classes in the classification task are specific and closed-form , while trustworthiness is open-form and is related to the classes in the classification task . The contribution of this work can be summarized as follows . • We study the problem of predicting trustworthiness with widely-used classifiers on ImageNet . Specifically , we observe that a major challenge of this learning task is that the cross entropy loss , focal loss , and TCP loss are prone to overfit the training samples , where correct predictions are dominant over incorrect predictions . • We propose the steep slope loss function that improves the generalizability of trustworthiness predictors . We conduct comprehensive experiments and analyses , such as performance on both small-scale and large-scale datasets , analysis of distributions separability , comparison to the class-balanced loss , etc. , which verify the efficacy of the proposed loss . • To further explore the practicality of the proposed loss , we train the oracle on the ImageNet training set and evaluate it on two variants of ImageNet validation set , i.e. , the stylized validation set and the adversarial validation set . The two variants ’ domains are quite different from the domain of the training set . We find that the learned oracle is able to consistently differentiate the trustworthy predictions from the untrustworthy predictions . 2 Preliminaries . In this section , we first recap how a deep learning model learns in the image classification task . Then , we show how the task of predicting trustworthiness connects to the classification task . Supervised Learning for Classification . In classification tasks , given a training sample , i.e. , image x ∈ Rm and corresponding ground-truth label y ∈ Y = { 1 , . . . , K } , we assume that samples are drawn i.i.d . from an underlying distribution . The goal of the learning task is to learn to find a classifier f ( cls ) ( · ; θ′ ) with training samples for classification . θ′ is the set of parameters of the classifier . Let f ( cls ) θ′ ( · ) = f ( cls ) ( · ; θ′ ) . The optimization problem is defined as f ∗ ( cls ) θ′ = argmin f ( cls ) θ′ R̂ ( f ( cls ) θ′ , ` ( cls ) , Dtr ) , ( 1 ) where f∗ ( cls ) θ′ is the learned classifier , R̂ is the empirical risk , ` ( cls ) is a loss function for classification , and Dtr is the set of training samples . Supervised Learning for Predicting Trustworthiness . In contrast to the learning task for classification , which is usually a multi-class single-label classification task [ 4 , 5 , 6 , 7 ] , learning to predict trustworthiness is a binary classification problem , where the two classes are positive ( i.e. , trustworthy ) or negative ( i.e. , untrustworthy ) . Similar to [ 9 ] , given a pair ( x , y ) and a classifier f ( cls ) θ′ , we define the ground-truth label o for predicting trustworthiness as o = { 1 , if argmax f ( cls ) θ′ ( x ) = y 0 , otherwise ( 2 ) In other words , the classifier correctly predicts the image ’ s label so the prediction is trustworthy in hindsight , otherwise the prediction is untrustworthy . The learning task for predicting trustworthiness follows a similar learning framework in the classification task . Let fθ ( · ) be an oracle ( i.e. , a trustworthiness predictor ) . A generic loss function ` : Rm×R → R≥0 , where R≥0 is a non-negative space and m is the number of classes . Given training samples ( x , y ) ∈ Dtr , the optimization problem for predicting trustworthiness is defined as f∗θ = argmin fθ 1 |Dtr| |Dtr|∑ i=1 ` ( fθ ( xi ) , oi ) , ( 3 ) where |Dtr| is the cardinality of Dtr . Particularly , we consider two widely-used loss functions for classification and the loss function used for training trustworthiness predictors as baselines . They are the cross entropy loss [ 17 ] , focal loss [ 18 ] , and TCP confidence loss [ 9 ] . Let p ( o = 1|θ , x ) = 1/ ( 1 + exp ( −z ) ) be the trustworthiness confidence , where z ∈ R is the descriminative feature produced by the oracle , i.e. , z = fθ ( x ) . The three loss functions can be written as ` CE ( fθ ( x ) , o ) = −o · log p ( o = 1|θ , x ) − ( 1− o ) · log ( 1− p ( o = 1|θ , x ) ) , ( 4 ) ` Focal ( fθ ( x ) , o ) = −o · ( 1− p ( o = 1|θ , x ) ) γ log p ( o = 1|θ , x ) − ( 1− o ) · ( p ( o = 1|θ , x ) ) γ log ( 1− p ( o = 1|θ , x ) ) , ( 5 ) ` TCP ( fθ ( x ) , y ) = ( fθ ( x ) − p ( ŷ = y|θ′ , x ) ) 2 . ( 6 ) In the focal loss , γ is a hyperparameter . In the TCP confidence loss , ŷ is the predicted label and p ( ŷ = y|θ′ , x ) is the classification probability w.r.t . the ground-truth class . Consequently , the learned oracle would yield z to generate the trustworthiness confidence . In the cases of ` CE and ` Focal , the oracle considers a prediction is trustworthy if the corresponding trustworthiness confidence is greater than the positive threshold 0.5 , i.e. , p ( o = 1|θ , x ) > 0.5 . The predictions whose trustworthiness confidences are equal to or lower than the negative threshold 0.5 are viewed to be untrustworthy . In the case of ` TCP , the positive threshold is also 0.5 , but the negative threshold correlates to the number of classes in the classification task . It is defined as 1/K in [ 9 ] . 3 Methodology . In this section , we first introduce the overall learning framework for predicting trustworthiness . Then , we narrow down to the proposed steep slope loss function . At last , we provide the generalization bound that is related to the proposed steep slope loss function . 3.1 Overall Design . Corbière et al . [ 9 ] provide a good learning scheme for predicting trustworthiness . Briefly , it first trains a classifier with the training samples . Then , the classifier is frozen and the confidence network ( i.e. , trustworthiness predictor ) is trained ( or fine-tuned ) with the training samples . In general , we follows this learning scheme . This work focuses on the trustworthiness on the predictions yielded by the publicly available pretrained classifiers , i.e. , ViT [ 7 ] and ResNet [ 5 ] . We use the pre-trained backbones as the backbones of the oracles for general purposes . In this sense , the set of the oracle ’ s parameters can be split into two parts , one is related to the backbone and the other one is related to the head , i.e. , θ = { θbackbone , θhead } . θbackbone is used to generated the intermediate feature xout and θhead = { w , b } are usually the weights of a linear function to generate the discriminative feature z = w > xout + b . With the classifier , the oracle , a pre-defined loss , and the training samples , we can optimize problem ( 3 ) to find the optimal parameters for the oracle .

Summary: The paper proposes the steep slope loss for learning trustworthiness under a supervised training setup. A pre-trained neural network is further trained on the steep slope loss to learn trustworthiness as a binary classification problem. The approach is evaluated on ImageNet for a ResNet and a ViT model. In addition, evaluations on MNIST and CIFAR-10 are provided for comparisons with the related work. In all experiments, the proposed approach shows good performance.