Original: A NGULAR V ISUAL H ARDNESS
Revised: A NGULAR V ISUAL H ARDNESS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Although convolutional neural networks (CNNs) are inspired by the mechanisms behind human visual systems, they diverge on many measures such as ambiguity or hardness.
Revised: Although convolutional neural networks (CNNs) are inspired by the mechanisms behind human visual systems, they diverge on many measures such as ambiguity or hardness.

Original: In this paper, we make a surprising discovery: there exists a (nearly) universal score function for CNNs whose correlation is statistically signiﬁcant than the widely used model conﬁdence with human visual hardness.
Revised: In this paper, we make a surprising discovery: there exists a (nearly) universal score function for CNNs whose correlation is statistically signiﬁcant than the widely used model conﬁdence with human visual hardness.

Original: We term this function as angular visual hardness (AVH) which is given by the normalized angular distance between a feature embedding and the classiﬁer weights of the corresponding target category in a CNN.
Revised: We term this function as angular visual hardness (AVH) which is given by the normalized angular distance between a feature embedding and the classiﬁer weights of the corresponding target category in a CNN.

Original: We conduct an in-depth scientiﬁc study.
Revised: We conduct an in-depth scientiﬁc study.

Original: We observe that CNN models with the highest accuracy also have the best AVH scores.
Revised: We observe that CNN models with the highest accuracy also have the best AVH scores.

Original: This agrees with an earlier ﬁnding that state-of-art models tend to improve on classiﬁcation of harder training examples.
Revised: This agrees with an earlier ﬁnding that state-of-art models tend to improve on classiﬁcation of harder training examples.

Original: We ﬁnd that AVH displays interesting dynamics during training: it quickly reaches a plateau even though the training loss keeps improving.
Revised: We ﬁnd that AVH displays interesting dynamics during training: it quickly reaches a plateau even though the training loss keeps improving.

Original: This suggests the need for designing better loss functions that can target harder examples more effectively.
Revised: This suggests the need for designing better loss functions that can target harder examples more effectively.

Original: Finally, we empirically show signiﬁcant improvement in performance by using AVH as a measure of hardness in self-training methods for domain adaptation.
Revised: Finally, we empirically show signiﬁcant improvement in performance by using AVH as a measure of hardness in self-training methods for domain adaptation.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: The invention and development of Convolutional Neural Networks (CNNs) were inspired by natural visual processing systems.
Revised: The invention and development of Convolutional Neural Networks (CNNs) were inspired by natural visual processing systems.

Original: For example, artiﬁcial neurons were designed to mimic neurons taking and transforming information [48], and neocognitron, the origin of the CNN architecture, was enlightened by early ﬁndings of receptive ﬁelds in the visual cortex [15].
Revised: For example, artiﬁcial neurons were designed to mimic neurons taking and transforming information [48], and neocognitron, the origin of the CNN architecture, was enlightened by early ﬁndings of receptive ﬁelds in the visual cortex [15].

Original: CNNs have achieved a great success in pushing the boundaries in a wide range of computer vision tasks such as image classiﬁcation [24, 30, 59], face recognition [39, 63, 64], and scene analysis [19, 43, 70].
Revised: CNNs have achieved a great success in pushing the boundaries in a wide range of computer vision tasks such as image classiﬁcation [24, 30, 59], face recognition [39, 63, 64], and scene analysis [19, 43, 70].

Original: Speciﬁcally, on certain large-scale benchmarks such as ImageNet [10], CNNs have even surpassed human-level accuracy.
Revised: Speciﬁcally, on certain large-scale benchmarks such as ImageNet [10], CNNs have even surpassed human-level accuracy.

Original: Despite such notable progress, CNNs are still far from matching human vision on many measures such as robustness, adaptability and few-shot learning [23], and could suffer from various biases.
Revised: Despite such notable progress, CNNs are still far from matching human vision on many measures such as robustness, adaptability and few-shot learning [23], and could suffer from various biases.

Original: For example, CNNs pre-trained on Imagenet are biased towards textures [18].
Revised: For example, CNNs pre-trained on Imagenet are biased towards textures [18].

Original: These biases can result in CNNs being overconﬁdent, or prone to domain gaps and adversarial attacks.
Revised: These biases can result in CNNs being overconﬁdent, or prone to domain gaps and adversarial attacks.

Original: Therefore, to fundamentally solve the above problems, the efforts should be made to improve CNN’s capabilities to model human visual system [47, 62].
Revised: Therefore, to fundamentally solve the above problems, the efforts should be made to improve CNN’s capabilities to model human visual system [47, 62].

Original: The popular measure of CNN conﬁdence, softmax score, has been widely used in many applications, yet causing calibration problems and tending to make CNNs overconﬁdent even if they are wrong [21, 34].
Revised: The popular measure of CNN conﬁdence, softmax score, has been widely used in many applications, yet causing calibration problems and tending to make CNNs overconﬁdent even if they are wrong [21, 34].

Original: However, this is not the case with human vision.
Revised: However, this is not the case with human vision.

Original: Thus, there is a gap between the current measure of hard examples that appear to be ambiguous or uncertain in these two systems.
Revised: Thus, there is a gap between the current measure of hard examples that appear to be ambiguous or uncertain in these two systems.

Original: We denote human visual hardness as the measure of how hard an example is to human visual system.
Revised: We denote human visual hardness as the measure of how hard an example is to human visual system.

Original: In this paper, we bridge the gap by proposing a novel score function on CNNs that correlates closely with human visual hardness.
Revised: In this paper, we bridge the gap by proposing a novel score function on CNNs that correlates closely with human visual hardness.

Original: The ﬁrst piece of this puzzle starts with the question of what is a good measure of human visual hardness.
Revised: The ﬁrst piece of this puzzle starts with the question of what is a good measure of human visual hardness.

Original: Recently, [52] argued that human selection frequency is a good measure.
Revised: Recently, [52] argued that human selection frequency is a good measure.

Original: This is the average number of times an image gets picked by a crowd of annotators, when they are asked to pick an image from a pool that belongs to a certain speciﬁed category.
Revised: This is the average number of times an image gets picked by a crowd of annotators, when they are asked to pick an image from a pool that belongs to a certain speciﬁed category.

Original: Intuitively, human selection frequency depends on various factors like object sizes, poses, special ﬁlters applied to images, etc.
Revised: Intuitively, human selection frequency depends on various factors like object sizes, poses, special ﬁlters applied to images, etc.

Original: [52] collected human selection frequency scores on ImageNet validation set using the MTurk platform.
Revised: [52] collected human selection frequency scores on ImageNet validation set using the MTurk platform.

Original: In this paper, we use this dataset to verify several hypotheses on correlations between CNNs and human visual systems in section 3.2.
Revised: In this paper, we use this dataset to verify several hypotheses on correlations between CNNs and human visual systems in section 3.2.

Original: Moreover, an automatic detection of examples that are hard for human vision has numerous applications.
Revised: Moreover, an automatic detection of examples that are hard for human vision has numerous applications.

Original: [52] showed that state-of-the-art models perform better on hard examples ( i.e. , hard for humans).
Revised: [52] showed that state-of-the-art models perform better on hard examples ( i.e. , hard for humans).

Original: This implies that in order to improve generalization, the models need to improve accuracy on hard examples.
Revised: This implies that in order to improve generalization, the models need to improve accuracy on hard examples.

Original: This can be achieved through various learning algorithms such as curriculum learning [2] and self-paced learning [32] where being able to detect hard examples is crucial.
Revised: This can be achieved through various learning algorithms such as curriculum learning [2] and self-paced learning [32] where being able to detect hard examples is crucial.

Original: Measuring sample conﬁdence is also important in partially-supervised problems such as semi-supervised learning [71, 72], unsupervised domain adaptation [8] and weakly-supervised learning [65] due to their under-constrained nature.
Revised: Measuring sample conﬁdence is also important in partially-supervised problems such as semi-supervised learning [71, 72], unsupervised domain adaptation [8] and weakly-supervised learning [65] due to their under-constrained nature.

Original: For instance, self-training [73] can easily reach to trivial solutions if one does not select pseudo-labels carefully based on correct measure of hardness.
Revised: For instance, self-training [73] can easily reach to trivial solutions if one does not select pseudo-labels carefully based on correct measure of hardness.

Original: Furthermore, by identifying hard examples, one can detect various biases in current CNN models.
Revised: Furthermore, by identifying hard examples, one can detect various biases in current CNN models.

Original: Sample hardness can also be used to identify implicit distribution imbalance in datasets to ensure fairness and remove societal biases [4].
Revised: Sample hardness can also be used to identify implicit distribution imbalance in datasets to ensure fairness and remove societal biases [4].

Original: [Figure]
Revised: [Figure]

Original: Our contributions are summarized as follows:
Revised: Our contributions are summarized as follows:

Original: Angular visual hardness (AVH): Given a CNN, we propose a novel score function that has stronger correlation with human visual hardness than softmax score.
Revised: Angular visual hardness (AVH): Given a CNN, we propose a novel score function that has stronger correlation with human visual hardness than softmax score.

Original: It is the normalized angular distance between the image feature embedding and the weights of the target category (See Figure 1).
Revised: It is the normalized angular distance between the image feature embedding and the weights of the target category (See Figure 1).

Original: The normalization takes into account the angular distances to other categories.
Revised: The normalization takes into account the angular distances to other categories.

Original: We argue that the semantic ambiguity that affects human visual hardness has stronger correlation with this score and we ﬁnd empirical evidence to support this claim.
Revised: We argue that the semantic ambiguity that affects human visual hardness has stronger correlation with this score and we ﬁnd empirical evidence to support this claim.

Original: The AVH score is inspired by the observation from Figure 1 and also [42] that samples from each class concentrate in a convex cone in the embedding space.
Revised: The AVH score is inspired by the observation from Figure 1 and also [42] that samples from each class concentrate in a convex cone in the embedding space.

Original: In addition, some existing theoretical results [61] show that for minimization of logistic loss or crossentropy loss which is often used in CNNs, gradient descent converges to the same direction as maximum margin solutions irrelevant to the (cid:96) 2 norm of classiﬁer weights or feature embeddings.
Revised: In addition, some existing theoretical results [61] show that for minimization of logistic loss or crossentropy loss which is often used in CNNs, gradient descent converges to the same direction as maximum margin solutions irrelevant to the (cid:96) 2 norm of classiﬁer weights or feature embeddings.

Original: This also provides the intuition behind why we can show AVH score rather than current model conﬁdence, softmax score, correlates better with human visual hardness.
Revised: This also provides the intuition behind why we can show AVH score rather than current model conﬁdence, softmax score, correlates better with human visual hardness.

Original: We validate that there is a statistically-signiﬁcant stronger correlation between AVH and human selection frequency across a wide range of CNN models.
Revised: We validate that there is a statistically-signiﬁcant stronger correlation between AVH and human selection frequency across a wide range of CNN models.

Original: Hence, it serves as its proxy on datasets where such information is not available and is beneﬁcial to a number of downstream tasks.
Revised: Hence, it serves as its proxy on datasets where such information is not available and is beneﬁcial to a number of downstream tasks.

Original: We observed the evolution of AVH score during training of CNN models.
Revised: We observed the evolution of AVH score during training of CNN models.

Original: It plateaus early in training even if the training (cross-entropy) loss keeps improving.
Revised: It plateaus early in training even if the training (cross-entropy) loss keeps improving.

Original: This suggests the need to design better loss functions that can improve performance on hard examples.
Revised: This suggests the need to design better loss functions that can improve performance on hard examples.

Original: We also validate the argument in [52] that improving on hard examples is the key to improve the generalization by verifying that the state-of-the-art models have the best average AVH scores.
Revised: We also validate the argument in [52] that improving on hard examples is the key to improve the generalization by verifying that the state-of-the-art models have the best average AVH scores.

Original: Finally, we empirically show the superiority of AVH with its application to self-training for unsupervised domain adaptation.
Revised: Finally, we empirically show the superiority of AVH with its application to self-training for unsupervised domain adaptation.

Original: With AVH being an improved conﬁdence measure, our proposed self-training framework renders considerably improved pseudo-label selection and category estimation, leading to state-of-the-art results with signiﬁcant performance gain over baselines.
Revised: With AVH being an improved conﬁdence measure, our proposed self-training framework renders considerably improved pseudo-label selection and category estimation, leading to state-of-the-art results with signiﬁcant performance gain over baselines.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Example hardness measures: Recently, measuring sample hardness for deep learning models has been widely studied with loss value [58], relative Euclidean distance [56, 66] and gradient norm [28].
Revised: Example hardness measures: Recently, measuring sample hardness for deep learning models has been widely studied with loss value [58], relative Euclidean distance [56, 66] and gradient norm [28].

Original: On the other hand, there is a rich history in cognitive and neuroscience communities to understand human visual perception [6, 7, 14, 45], where many of them focus on mechanisms used by the human brain to translate visual information to mental representations.
Revised: On the other hand, there is a rich history in cognitive and neuroscience communities to understand human visual perception [6, 7, 14, 45], where many of them focus on mechanisms used by the human brain to translate visual information to mental representations.

Original: These representations are subject to many correspondence differences and errors and thereby are not isomorphic to the real world [37].
Revised: These representations are subject to many correspondence differences and errors and thereby are not isomorphic to the real world [37].

Original: They can be affected by the ambiguity of different semantics [27] such as occlusion, distortion, motion blur, and inherent similarity among objects.
Revised: They can be affected by the ambiguity of different semantics [27] such as occlusion, distortion, motion blur, and inherent similarity among objects.

Original: Due to the expensive human labeling process, such detailed semantic information is typically not present in large-scale image benchmarks used to train the CNN models.
Revised: Due to the expensive human labeling process, such detailed semantic information is typically not present in large-scale image benchmarks used to train the CNN models.

Original: Angular distance in CNNs: [69] uses the deep features to quantify the semantic difference between images, indicating that deep features contain the most crucial semantic information.
Revised: Angular distance in CNNs: [69] uses the deep features to quantify the semantic difference between images, indicating that deep features contain the most crucial semantic information.

Original: It empirically shows that the angular distance between feature maps in deep neural networks is very consistent with the human in distinguishing the semantic difference.
Revised: It empirically shows that the angular distance between feature maps in deep neural networks is very consistent with the human in distinguishing the semantic difference.

Original: However, because of the different goal mentioned above, they have not studied or shown any strong correlation of human visual hardness and the angular distance on natural images.
Revised: However, because of the different goal mentioned above, they have not studied or shown any strong correlation of human visual hardness and the angular distance on natural images.

Original: [40] proposes a hyperspherical neural network that constrains the parameters of neurons on a unit hypersphere and uses angular similarity to replace the inner product similarity.
Revised: [40] proposes a hyperspherical neural network that constrains the parameters of neurons on a unit hypersphere and uses angular similarity to replace the inner product similarity.

Original: [42] decouples the inner product as the norm and the angle and argues that the norm corresponds to intra-class variation, and the angle corresponds to inter-class semantic difference.
Revised: [42] decouples the inner product as the norm and the angle and argues that the norm corresponds to intra-class variation, and the angle corresponds to inter-class semantic difference.

Original: However, this work does not consider any human factors, while our goal is to bridge the gap between CNNs and human perception.
Revised: However, this work does not consider any human factors, while our goal is to bridge the gap between CNNs and human perception.

Original: [35, 41] propose well-performing regularizations based on angular diversity to improve the network generalization.
Revised: [35, 41] propose well-performing regularizations based on angular diversity to improve the network generalization.

Original: Image degradation: Because CNNs and humans achieve similar accuracy on a wide range of tasks on benchmark datasets, a number of works have investigated similarities and differences between
Revised: Image degradation: Because CNNs and humans achieve similar accuracy on a wide range of tasks on benchmark datasets, a number of works have investigated similarities and differences between

Original: CNNs and human vision [3, 5, 9, 11, 12, 29, 46, 51, 67].
Revised: CNNs and human vision [3, 5, 9, 11, 12, 29, 46, 51, 67].

Original: Since human annotation data is hard to come by, researchers have proposed an alternative measure of visual hardness on images based on image degradation [37].
Revised: Since human annotation data is hard to come by, researchers have proposed an alternative measure of visual hardness on images based on image degradation [37].

Original: It involves adding noise or changing image properties such as contrast, blurriness, and brightness.
Revised: It involves adding noise or changing image properties such as contrast, blurriness, and brightness.

Original: [17] employed psychological studies to validate the degradation method as a way to measure human visual hardness.
Revised: [17] employed psychological studies to validate the degradation method as a way to measure human visual hardness.

Original: It should be noted that the artiﬁcial visual hardness introduced by degradation is a different concept from the natural visual hardness.
Revised: It should be noted that the artiﬁcial visual hardness introduced by degradation is a different concept from the natural visual hardness.

Original: The hardness based on degradation only reﬂects the hardness of a single original image with various of transformations, while natural visual hardness based on the ambiguity of human perception across a distribution of natural images.
Revised: The hardness based on degradation only reﬂects the hardness of a single original image with various of transformations, while natural visual hardness based on the ambiguity of human perception across a distribution of natural images.

Original: In this paper, we consider both as the surrogates of human visual hardness.
Revised: In this paper, we consider both as the surrogates of human visual hardness.

Original: Deep model calibration .
Revised: Deep model calibration .

Original: Conﬁdence calibration is the problem of predicting probability estimates representative of the true correctness likelihood [21].
Revised: Conﬁdence calibration is the problem of predicting probability estimates representative of the true correctness likelihood [21].

Original: It is well-known that the deep neural networks are mis-calibrated and there has been a rich literature trying to solve this problem [21, 31].
Revised: It is well-known that the deep neural networks are mis-calibrated and there has been a rich literature trying to solve this problem [21, 31].

Original: However, this is a somewhat different issue because the conﬁdence calibration is a problem introduced by two measurements of the model, which does not have any involvement of human visual hardness.
Revised: However, this is a somewhat different issue because the conﬁdence calibration is a problem introduced by two measurements of the model, which does not have any involvement of human visual hardness.

Original:  A D ISCOVERY FROM S CIENTIFIC T ESTING : A NGULAR V ISUAL H ARDNESS 3. N OTATIONS AND S ETUP
Revised:  A D ISCOVERY FROM S CIENTIFIC T ESTING : A NGULAR V ISUAL H ARDNESS 3. N OTATIONS AND S ETUP

Original: In order to quantify Human Visual Hardness and Model Predictions for convenience purposes in experiments, we use corresponding surrogates which are formally deﬁned as the following throughout the paper.
Revised: In order to quantify Human Visual Hardness and Model Predictions for convenience purposes in experiments, we use corresponding surrogates which are formally deﬁned as the following throughout the paper.

Original: We use the ImageNet [10] benchmark in all following experiments.
Revised: We use the ImageNet [10] benchmark in all following experiments.

Original: Particularly, we take advantage of the Human Selection Frequency information for validation images provided by the recent paper [52].
Revised: Particularly, we take advantage of the Human Selection Frequency information for validation images provided by the recent paper [52].

Original: Recall that such information can serve as one of the proxy for Human Visual Hardness.
Revised: Recall that such information can serve as one of the proxy for Human Visual Hardness.

Original: To test if our ﬁndings with Human Selection Frequency hold on another proxy, image degradation, we create an augmented validation set based on two image degradation methods, decreasing contrast and adding noise.
Revised: To test if our ﬁndings with Human Selection Frequency hold on another proxy, image degradation, we create an augmented validation set based on two image degradation methods, decreasing contrast and adding noise.

Original: We label them with corresponding degradation level (results shown in the appendix).
Revised: We label them with corresponding degradation level (results shown in Appendix A. and A.5).

Original: Besides, in order to verify that the our experimental results hold consistently across models instead of a particular model, we use four popular ImageNet pre-trained models AlexNet [30],VGG19 [59],
Revised: Besides, in order to verify that the our experimental results hold consistently across models instead of a particular model, we use four popular ImageNet pre-trained models AlexNet [30],

Original:  DenseNet121 [26], ResNet50 [24].
Revised: VGG19 [59], DenseNet121 [26], ResNet50 [24].

Original: We select ResNet50 as the representative model for some experiments.
Revised: We select ResNet50 as the representative model for some experiments.

Revised: Most importantly, we also provide experimental results on different datasets,

Revised: MNIST and CIFAR10/100, in Appendix A.3 and A.4 to better support our proposal.

Original: Denote S n as the unit n -sphere, formally, S n = { x ∈ R n +1 |(cid:107) x (cid:107) 2 = 1 } .
Revised: Denote S n as the unit n -sphere, formally, S n = { x ∈ R n +1 |(cid:107) x (cid:107) 2 = 1 } .

Original: Below by A ( · , · ) , we denote the angular distance between two points on S n , i.e. , A ( u , v ) =
Revised: Below by A ( · , · ) , we denote the angular distance between two points on S n , i.e. , A ( u , v ) =

Original: arccos( (cid:104) u , v (cid:105) (cid:107) u (cid:107)(cid:107) v (cid:107) ) .
Revised: arccos( (cid:104) u , v (cid:105) (cid:107) u (cid:107)(cid:107) v (cid:107) ) .

Original: Let x be the feature embeddings input for the layer before the last one of the classiﬁer of the pretrained CNN models, eg.
Revised: Let x be the feature embeddings input for the layer before the last one of the classiﬁer of the pretrained CNN models, eg.

Original: FC2 for VGG19.
Revised: FC2 for VGG19.

Original: Let C be the number of classes for a classiﬁcation task.
Revised: Let C be the number of classes for a classiﬁcation task.

Original: Denote
Revised: Denote

Original: W = { w i | 0 < i ≤ C} as the set of weights for all C classes in the ﬁnal layer of the classiﬁer.
Revised: W = { w i | 0 < i ≤ C} as the set of weights for all C classes in the ﬁnal layer of the classiﬁer.

Original: Deﬁnition 1 (Angular Visual Hardness (AVH)) .
Revised: Deﬁnition 1 (Angular Visual Hardness (AVH)) .

Original: AVH, for any ( x, y ) , is deﬁned as:
Revised: AVH, for any ( x, y ) , is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: which w y represents the weights of the target class.
Revised: which w y represents the weights of the target class.

Original: [Figure]
Revised: [Figure]

Original: AV H ( x )       # o f S a m p l e s 0.
Revised: AV H ( x )       # o f S a m p l e s 0.

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: M
Revised: M

Original: [Figure]
Revised: [Figure]

Original: d
Revised: d

Original: [Figure]
Revised: [Figure]

Original: l
Revised: l

Original: [Figure]
Revised: [Figure]

Original: n f i
Revised: n f i

Original: [Figure]
Revised: [Figure]

Original: [Table]
Revised: [Table]

Original: Deﬁnition 2 (Model Conﬁdence) .
Revised: Deﬁnition 2 (Model Conﬁdence) .

Original: We deﬁne model conﬁdence on a single sample as the probability score of the true objective class output by the CNN models, formally,
Revised: We deﬁne model conﬁdence on a single sample as the probability score of the true objective class output by the CNN models, formally,

Original: [Table]
Revised: [Table]

Original: Deﬁnition 3 (Human Selection Frequency) .
Revised: Deﬁnition 3 (Human Selection Frequency) .

Original: We deﬁne one way to measure human visual hardness on pictures as Human Selection Frequency.
Revised: We deﬁne one way to measure human visual hardness on pictures as Human Selection Frequency.

Original: Quantitatively, given m number of human workers in a labeling process described in [52], if b out of m label a picture as a particular class and that class is the target class of that picture in the ﬁnal dataset, then Human Selection Frequency is deﬁned as bm .
Revised: Quantitatively, given m number of human workers in a labeling process described in [52], if b out of m label a picture as a particular class and that class is the target class of that picture in the ﬁnal dataset, then Human Selection Frequency is deﬁned as bm .

Original: 3. C ONNECTIONS AND G APS BETWEEN H UMAN V ISUAL S YSTEM AND CNN
Revised: 3. C ONNECTIONS AND G APS BETWEEN H UMAN V ISUAL S YSTEM AND CNN

Original: Studying the precise connection or gap between human visual hardness and model predictions is not feasible because data collection involving human labelling or annotation requires large amount of work.
Revised: Studying the precise connection or gap between human visual hardness and model predictions is not feasible because data collection involving human labelling or annotation requires large amount of work.

Original: In addition, usually those human data is application or dataset speciﬁc, which makes the scalability of this study even worse.
Revised: In addition, usually those human data is application or dataset speciﬁc, which makes the scalability of this study even worse.

Original: Therefore, all the testing and experiments we design are at best effort given the limited resources.
Revised: Therefore, all the testing and experiments we design are at best effort given the limited resources.

Original: That is exactly another motivation for us to bridge the gap between
Revised: That is exactly another motivation for us to bridge the gap between

Original: Human and models because models predictions require minimum costs compared to human efforts.
Revised: Human and models because models predictions require minimum costs compared to human efforts.

Original: In this section, We ﬁrst provide four hypothesis and test them accordingly.
Revised: In this section, We ﬁrst provide four hypothesis and test them accordingly.

Original: Hypothesis 1.
Revised: Hypothesis 1.

Original: AVH has a correlation with Human Selection Frequency.
Revised: AVH has a correlation with Human Selection Frequency.

Original: Outcome: Null Hypothesis Rejected
Revised: Outcome: Null Hypothesis Rejected

Original: Correspondingly, after evaluating each validation sample on pre-trained models, we extract feature embeddings x and also the class weights W to compute AVH ( x ) .
Revised: Correspondingly, after evaluating each validation sample on pre-trained models, we extract feature embeddings x and also the class weights W to compute AVH ( x ) .

Original: Noted that we linear scale the range of AVH ( x ) to [0 , 1] .
Revised: Noted that we linear scale the range of AVH ( x ) to [0 , 1] .

Original: Table 1 shows the overall strong correlation of AVH ( x ) and Human Selection
Revised: Table 1 shows the overall strong correlation of AVH ( x ) and Human Selection

Original: Frequency consistently (p-value is < 0 .
Revised: Frequency consistently (p-value is < 0 .

Original: 001 rejecting the null hypothesis).
Revised: 001 rejecting the null hypothesis).

Original: From the coefﬁcients represented for different bins of example hardness, we can see that the harder the examples, the weaker the correlation.
Revised: From the coefﬁcients represented for different bins of example hardness, we can see that the harder the examples, the weaker the correlation.

Original: Noted that we also check the results across four different CNN architectures and we found that better model has higher coefﬁcient.
Revised: Noted that we also check the results across four different CNN architectures and we found that better model has higher coefﬁcient.

Original: The plot on the left in Figure 2 help visualize the strong correlation between AVH ( x ) and Human
Revised: The plot on the left in Figure 2 help visualize the strong correlation between AVH ( x ) and Human

Original: Selection Frequency for validation images.
Revised: Selection Frequency for validation images.

Original: One intuition behind this correlation is that the class weights W might corresponds to human semantic for each category and thereby AVH ( x ) corresponds to human semantic categorization of an image.
Revised: One intuition behind this correlation is that the class weights W might corresponds to human semantic for each category and thereby AVH ( x ) corresponds to human semantic categorization of an image.

Original: In order to test if the strong correlation holds for all models, we perform the same experiments on AlexNet, VGG19 and DenseNet121.
Revised: In order to test if the strong correlation holds for all models, we perform the same experiments on AlexNet, VGG19 and DenseNet121.

Original: Hypothesis 2.
Revised: Hypothesis 2.

Original: Model Conﬁdence has a correlation with Human Selection Frequency
Revised: Model Conﬁdence has a correlation with Human Selection Frequency

Original: . Outcome: Null Hypothesis Rejected
Revised: . Outcome: Null Hypothesis Rejected

Original: An interesting observation in [52] shows that Human Selection Frequency has strong inﬂuence on the Model Conﬁdence.
Revised: An interesting observation in [52] shows that Human Selection Frequency has strong inﬂuence on the Model Conﬁdence.

Original: Speciﬁcally, examples with low Human Selection Frequency tends to have relatively low Model Conﬁdence.
Revised: Speciﬁcally, examples with low Human Selection Frequency tends to have relatively low Model Conﬁdence.

Original: Naturally we examine if the correlation between Model Conﬁdence and Human Selection Frequency is strong.
Revised: Naturally we examine if the correlation between Model Conﬁdence and Human Selection Frequency is strong.

Original: Speciﬁcally, all ImageNet validation images are evaluated by the pre-trained models.
Revised: Speciﬁcally, all ImageNet validation images are evaluated by the pre-trained models.

Original: The corresponding output is simply the Model Conﬁdence on each image.
Revised: The corresponding output is simply the Model Conﬁdence on each image.

Original: From table 1, we can ﬁrst see that it is clear that because p-value is < 0 .
Revised: From table 1, we can ﬁrst see that it is clear that because p-value is < 0 .

Original: 001 , Model Conﬁdence does have a strong correlation with Human Selection Frequency.
Revised: 001 , Model Conﬁdence does have a strong correlation with Human Selection Frequency.

Original: However, the correlation coefﬁcient for Model Conﬁdence and Human Selection Frequency is consistently lower than that of AVH and
Revised: However, the correlation coefﬁcient for Model Conﬁdence and Human Selection Frequency is consistently lower than that of AVH and

Original: Human Selection Frequency.
Revised: Human Selection Frequency.

Original: The middle plot in ﬁgure 2 presents a two-dimensional histogram for the correlation visualization.
Revised: The middle plot in ﬁgure 2 presents a two-dimensional histogram for the correlation visualization.

Original: The x-axis represents Human Selection Frequency, and the y-axis represents Model Conﬁdence.
Revised: The x-axis represents Human Selection Frequency, and the y-axis represents Model Conﬁdence.

Original: Each bin exhibits the number of images which lie in the corresponding range.
Revised: Each bin exhibits the number of images which lie in the corresponding range.

Original: We can observe the high density at the right corner, which means the majority of the images have both high human and model accuracy.
Revised: We can observe the high density at the right corner, which means the majority of the images have both high human and model accuracy.

Original: However, there is a considerable amount of density on the range of medium human accuracy but either extremely low or high model accuracy.
Revised: However, there is a considerable amount of density on the range of medium human accuracy but either extremely low or high model accuracy.

Original: One may question that the difference of the correlation coefﬁcient is not large, thereby we also run statistical testing on the signiﬁcance of the gap, naturally our next step is to test if the difference is signiﬁcant.
Revised: One may question that the difference of the correlation coefﬁcient is not large, thereby we also run statistical testing on the signiﬁcance of the gap, naturally our next step is to test if the difference is signiﬁcant.

Original: Hypothesis 3.
Revised: Hypothesis 3.

Original: AVH has a stronger correlation to Human Selection Frequency than Model Conﬁdence.
Revised: AVH has a stronger correlation to Human Selection Frequency than Model Conﬁdence.

Original: Outcome: Null Hypothesis Rejected
Revised: Outcome: Null Hypothesis Rejected

Revised: There are three steps for testing if two correlation coefﬁcient are signiﬁcantly different.

Revised: First step is applying Fisher Z-Transformation to both coefﬁcient.

Revised: The Fisher Z-Transformation is a way to transform the sampling distribution of the correlation coefﬁcient so that it becomes normally distributed.

Original: We ﬁrst apply ﬁsher transformation for each correlation coefﬁcient: Z score for coefﬁcient of AVH becomes 0 .
Revised: Therefore, we apply ﬁsher transformation for each correlation coefﬁcient: Z score for coefﬁcient of AVH becomes 0 .

Original: 377 and that of Model Conﬁdence becomes 0 .
Revised: 377 and that of Model Conﬁdence becomes 0 .

Revised: The second step is to compute the Z value of two Z scores.

Original: Then we determined the Z value to be 4 . 85 from the two above-mentioned Z scores and sample sizes.
Revised: Then we determined the Z value to be 4 . 85 from the two above-mentioned Z scores and sample sizes.

Revised: The last step is that ﬁnd out the p-value according to the Z table.

Original: According to Z table, pvalue is 0.00001.
Revised: According to Z table, p-value is 0.00001.

Original: Therefore, we reject the null hypothesis and conclude that AVH has statistically signiﬁcant stronger correlation with Human Selection Frequency than Model Conﬁdence.
Revised: Therefore, we reject the null hypothesis and conclude that AVH has statistically signiﬁcant stronger correlation with Human Selection Frequency than Model Conﬁdence.

Original: In later section 5, we also empirically show that such stronger correlation brings cumulative advantages in some applications.
Revised: In later section 5, we also empirically show that such stronger correlation brings cumulative advantages in some applications.

Revised: In appendix A.1, besides Spearman correlation coefﬁcient, we have also shown Pearson and Kendall Tau ones.

Revised: Further more, to test if the conclusion holds for different models, we run the same tests on all four different architectures.

Revised: The conclusion is for all the models and under different testings, AVH correlate signiﬁcantly stronger than model conﬁdence, but the correlation is even stronger for better models.

Original: Hypothesis 4.
Revised: Hypothesis 4.

Original: (cid:107) x (cid:107) 2 has a correlation with Human Selection Frequency.
Revised: (cid:107) x (cid:107) 2 has a correlation with Human Selection Frequency.

Original: Outcome: Failure to Reject Null Hypothesis
Revised: Outcome: Failure to Reject Null Hypothesis

Original: [42] conjectures that (cid:107) x (cid:107) 2 accounts for intra-class Human/Model Conﬁdence.
Revised: [42] conjectures that (cid:107) x (cid:107) 2 accounts for intra-class Human/Model Conﬁdence.

Original: Particularly, if the norm is larger, the prediction from the model is also more conﬁdent, to some extent.
Revised: Particularly, if the norm is larger, the prediction from the model is also more conﬁdent, to some extent.

Original: Therefore, we conduct similar experiments like previous section to demonstrate the correlation between (cid:107) x (cid:107)and Human Selection Frequency.
Revised: Therefore, we conduct similar experiments like previous section to demonstrate the correlation between (cid:107) x (cid:107)and Human Selection Frequency.

Original: Initially, we compute the (cid:107) x (cid:107) 2 for every validation sample for all models.
Revised: Initially, we compute the (cid:107) x (cid:107) 2 for every validation sample for all models.

Original: Then we normalize (cid:107) x (cid:107) 2 within each class.
Revised: Then we normalize (cid:107) x (cid:107) 2 within each class.

Original: Table 1 presents the results for the correlation test.
Revised: Table 1 presents the results for the correlation test.

Original: We omit the results for p-value in the table and report here that they are all much higher than 0 . 05 , indicating there is no correlation between (cid:107) x (cid:107) 2 and Human Selection Frequency.
Revised: We omit the results for p-value in the table and report here that they are all much higher than 0 . 05 , indicating there is no correlation between (cid:107) x (cid:107) 2 and Human Selection Frequency.

Original: The right plot in ﬁgure 2 uses a two-dimensional histogram to show the correlation for all the validation images.
Revised: The right plot in ﬁgure 2 uses a two-dimensional histogram to show the correlation for all the validation images.

Original: Given that the norm has been normalized with each class, naturally, there is notable density when the norm is 0 or 1 .
Revised: Given that the norm has been normalized with each class, naturally, there is notable density when the norm is 0 or 1 .

Original: Except for that, there is no obvious correlation between (cid:107) x (cid:107)and Human Selection Frequency.
Revised: Except for that, there is no obvious correlation between (cid:107) x (cid:107)and Human Selection Frequency.

Original: We further verify if presenting all samples across 1000 different classes affects the visualization of the correlation.
Revised: We further verify if presenting all samples across 1000 different classes affects the visualization of the correlation.

Original: According to WordNet [13] hierarchy, we map the original 1000 ﬁne-grained classes to 45 higher hierarchical classes.
Revised: According to WordNet [13] hierarchy, we map the original 1000 ﬁne-grained classes to 45 higher hierarchical classes.

Original: A ﬁgure in appendix exhibits the relationship between Human Selection Frequency and (cid:107) x (cid:107) 2 for three representative higher classes containing 58 , 7 , 1 ﬁne-grained classes respectively.
Revised: A ﬁgure in appendix exhibits the relationship between Human Selection Frequency and (cid:107) x (cid:107) 2 for three representative higher classes containing 58 , 7 , 1 ﬁne-grained classes respectively.

Original: Noted that there is still not any visible direct proportion between these two variables across all plots. 
Revised: Noted that there is still not any visible direct proportion between these two variables across all plots. [Figure] [Figure]

Original: D YNAMICS OF AVH DURING T RAINING
Revised: AV H ( x ) 1e Freq=[0.0 ,0.2] Freq=[0.2 ,0.4] Freq=[0. [Figure] 0.6] Freq

Revised: [Figure]

Revised: [Figure]

Revised: Freq

Revised: [Figure]

Revised: [Figure]

Revised: Besides the hypothesis testings, we provide a detailed discussion on the difference between AVH and

Revised: Model Conﬁdence in Appendix C.

Original: After discovering the strong correlation of human visual hardness and AVH score, a natural question would be: What role does AVH play during the training process?
Revised: After discovering the strong correlation of human visual hardness and AVH score, a natural question would be: What role does AVH play during the training process?

Original: Optimization algorithms are used to update weights and biases i.e. the internal parameters of a model to improve the training loss.
Revised: Optimization algorithms are used to update weights and biases i.e. the internal parameters of a model to improve the training loss.

Original: Both the angles between the feature embedding and classiﬁers, and the L 2 norm of the embedding can inﬂuence the loss.
Revised: Both the angles between the feature embedding and classiﬁers, and the L 2 norm of the embedding can inﬂuence the loss.

Original: While it is well-known that the training loss or accuracy keeps improving but it

Original: [Figure]

Original: [Figure]

Original: AV H ( x ) 1e

Original: Freq=[0.0 ,0.2] Freq=[0.2 ,0.4] Freq=[0.

Original: [Figure]

Original: 0.6] Freq

Original: [Figure]

Original: [Figure]

Original: Freq [Figure] [Figure] is not obvious what would be the dynamics of the angles and norms separately during training.
Revised: While it is well-known that the training loss or accuracy keeps improving but it is not obvious what would be the dynamics of the angles and norms separately during training.

Original: we design the experiments to observe the training dynamics of various network architectures.
Revised: we design the experiments to observe the training dynamics of various network architectures.

Original: Experiment Settings .
Revised: Experiment Settings .

Original: For datasets and models, we use exactly the same setting as the experiments in 3.1.
Revised: For datasets and models, we use exactly the same setting as the experiments in 3.1.

Original: Nevertheless, observing training dynamics involves training models from scratch on ImageNet training set instead of directly using the pre-trained models.
Revised: Nevertheless, observing training dynamics involves training models from scratch on ImageNet training set instead of directly using the pre-trained models.

Original: Therefore, we follow the standard training process of AlexNet [30], VGG19 [59], ResNet50 [24] and DenseNet121 [26] (DenseNet results are put in Appendix).
Revised: Therefore, we follow the standard training process of AlexNet [30], VGG19 [59], ResNet50 [24] and DenseNet121 [26] (DenseNet results are put in Appendix).

Original: For consistency, we train all four models for 90 epochs and decay the initial learning rate by a factor of 10 every 30 epochs.
Revised: For consistency, we train all four models for 90 epochs and decay the initial learning rate by a factor of 10 every 30 epochs.

Original: The initial learning rate for AlexNet and VGG19 is 0 .
Revised: The initial learning rate for AlexNet and VGG19 is 0 .

Original: 01 and for
Revised: 01 and for

Original: DensetNet121 and ResNet50 is 0 .
Revised: DensetNet121 and ResNet50 is 0 .

Original: For human visual hardness based on Human Selection Frequency, we split all the validation images into 5 bins, [0 . 0 , 0 . 2] , [0 . 2 , 0 . 4] , [0 . 4 , 0 . 6] , [0 . 6 , 0 . 8] , [0 . 8 , 1 . 0] , based on their human selection frequency respectively.
Revised: For human visual hardness based on Human Selection Frequency, we split all the validation images into 5 bins, [0 . 0 , 0 . 2] , [0 . 2 , 0 . 4] , [0 . 4 , 0 . 6] , [0 . 6 , 0 . 8] , [0 . 8 , 1 . 0] , based on their human selection frequency respectively.

Original: For human visual hardness based on Image Degradation Level, we create an augmented validation set based on two image degradation methods, decreasing contrast and adding noise.
Revised: For human visual hardness based on Image Degradation Level, we create an augmented validation set based on two image degradation methods, decreasing contrast and adding noise.

Original: We label them with corresponding degradation level as well.
Revised: We label them with corresponding degradation level as well.

Original: Note that for all the ﬁgures in this section, Epoch starts from 1.
Revised: Note that for all the ﬁgures in this section, Epoch starts from 1.

Original: Observation 1: The norm of feature embeddings keeps increasing during training .
Revised: Observation 1: The norm of feature embeddings keeps increasing during training .

Original: Figure 3,and 11 present the dynamics of the average (cid:107) x (cid:107) 2 and the dynamics of the accuracy for validation samples vary in 90 epochs during the training on three architectures.
Revised: Figure 3,and 11 present the dynamics of the average (cid:107) x (cid:107) 2 and the dynamics of the accuracy for validation samples vary in 90 epochs during the training on three architectures.

Original: Note that we are using the validation data for dynamics observation and therefore have never ﬁt them into the model.
Revised: Note that we are using the validation data for dynamics observation and therefore have never ﬁt them into the model.

Original: The average (cid:107) x (cid:107) 2 increases with a small initial slope but it suddenly climbs after 30 epochs when the ﬁrst learning rate decay happens.
Revised: The average (cid:107) x (cid:107) 2 increases with a small initial slope but it suddenly climbs after 30 epochs when the ﬁrst learning rate decay happens.

Original: The accuracy curve is very similar to that of the average (cid:107) x (cid:107) 2 .
Revised: The accuracy curve is very similar to that of the average (cid:107) x (cid:107) 2 .

Original: The above observations are consistent in all models.
Revised: The above observations are consistent in all models.

Original: More interestingly, we ﬁnd that neural networks with shortcut connections ( e.g. , ResNets and DenseNets) tend to make the norm of the images with different human selection frequency become the same, while the neural networks without shortcuts ( e.g. , AlexNet and VGG) tend to keep the gap of norm among the images with different human visual hardness.
Revised: More interestingly, we ﬁnd that neural networks with shortcut connections ( e.g. , ResNets and DenseNets) tend to make the norm of the images with different human selection frequency become the same, while the neural networks without shortcuts ( e.g. , AlexNet and VGG) tend to keep the gap of norm among the images with different human visual hardness.

Original: Observation 2: AVH hits a plateau very early even when the accuracy or loss is still improving .
Revised: Observation 2: AVH hits a plateau very early even when the accuracy or loss is still improving .

Original: Figure 3, 10 and 11 exhibits the change of average AVH for validation samples in 90 epochs of training on three models.
Revised: Figure 3, 10 and 11 exhibits the change of average AVH for validation samples in 90 epochs of training on three models.

Original: The average AVH for AlexNet and VGG19 decreases sharply at the beginning and then starts to bounce back a little bit before converging.
Revised: The average AVH for AlexNet and VGG19 decreases sharply at the beginning and then starts to bounce back a little bit before converging.

Original: However, the dynamics of the average AVH for DenseNet121 and ResNet50 are different.
Revised: However, the dynamics of the average AVH for DenseNet121 and ResNet50 are different.

Original: They both decrease slightly and then quickly hits a plateau in all three learning rate decay stages.
Revised: They both decrease slightly and then quickly hits a plateau in all three learning rate decay stages.

Original: But the common observation is that they all stop improving even when (cid:107) x (cid:107) 2 and model accuracy are increasing.
Revised: But the common observation is that they all stop improving even when (cid:107) x (cid:107) 2 and model accuracy are increasing.

Original: AVH is more important than (cid:107) x (cid:107) 2 in the sense that it is the key factor deciding which class the input sample is classiﬁed to.
Revised: AVH is more important than (cid:107) x (cid:107) 2 in the sense that it is the key factor deciding which class the input sample is classiﬁed to.

Original: However, optimizing the norm under the current softmax cross-entropy loss would be easier so, which cause the plateau of angles for easy examples.
Revised: However, optimizing the norm under the current softmax cross-entropy loss would be easier so, which cause the plateau of angles for easy examples.

Original: However, the plateau for the hard examples can be caused by the limitation of the model itself.
Revised: However, the plateau for the hard examples can be caused by the limitation of the model itself.

Original: As a result, it shows the necessity and importance of designing loss functions that focus on optimizing angles, such as [35, 39, 41].
Revised: As a result, it shows the necessity and importance of designing loss functions that focus on optimizing angles, such as [35, 39, 41].

Original: Observation 3: AVH’s correlation with human selection frequency consistently holds across models throughout the training process .
Revised: Observation 3: AVH’s correlation with human selection frequency consistently holds across models throughout the training process .

Original: In Figure 3, 10 and 11, we average over validation samples in ﬁve human selection frequency bins or ﬁve degradation level bins separately , and then compute the average embedding norm, AVH and model accuracies.
Revised: In Figure 3, 10 and 11, we average over validation samples in ﬁve human selection frequency bins or ﬁve degradation level bins separately , and then compute the average embedding norm, AVH and model accuracies.

Original: We can observe that for (cid:107) x (cid:107) 2 , the gaps between the samples with different human visual hardness are not obvious in ResNet and
Revised: We can observe that for (cid:107) x (cid:107) 2 , the gaps between the samples with different human visual hardness are not obvious in ResNet and

Original: DenseNet, while they are quite obvious in AlexNet and VGG.
Revised: DenseNet, while they are quite obvious in AlexNet and VGG.

Original: However, for AVH, such AVH gaps are very signiﬁcant and consistent across every network architecture during the entire training process.
Revised: However, for AVH, such AVH gaps are very signiﬁcant and consistent across every network architecture during the entire training process.

Original: Interestingly, even if the network is far from being converged, such AVH gaps are still consistent across different human selection frequency.
Revised: Interestingly, even if the network is far from being converged, such AVH gaps are still consistent across different human selection frequency.

Original: Also the norm gaps are also consistent.
Revised: Also the norm gaps are also consistent.

Original: The intuition behind this could be that the angles for hard examples are much harder to decrease and probably never in the region for correct classiﬁcation.
Revised: The intuition behind this could be that the angles for hard examples are much harder to decrease and probably never in the region for correct classiﬁcation.

Original: Therefore the corresponding norms would not increase otherwise hurting the loss.
Revised: Therefore the corresponding norms would not increase otherwise hurting the loss.

Original: It validates that AVH is a consistent and robust measure for visual hardness (and even generalization).
Revised: It validates that AVH is a consistent and robust measure for visual hardness (and even generalization).

Original: Observation 4: AVH is an indicator of model’s generalization ability .
Revised: Observation 4: AVH is an indicator of model’s generalization ability .

Original: From Figure 3, 18, 10 and 11, we observe that better models ( i.e. , higher accuracy) have lower average AVH throughout the training process and also across samples under different human visual hardness.
Revised: From Figure 3, 18,and 11, we observe that better models ( i.e. , higher accuracy) have lower average AVH throughout the training process and also across samples under different human visual hardness.

Original: For instance, Alexnet is the worst model, and its overall average AVH and average AVH on each of ﬁve bins are worse than those of the other three models.
Revised: For instance, Alexnet is the worst model, and its overall average AVH and average AVH on each of ﬁve bins are worse than those of the other three models.

Revised: In addition, we have found when testing Hypothesisfor better models, its AVH correlation is much more stronger than its Model conﬁdence correlation with Human Selection Frequency.

Original: This observation is aligned with the earlier observations of [52] that better models also generalize better on samples across different human visual hardness.
Revised: The above observations are aligned with the earlier observations of [52] that better models also generalize better on samples across different human visual hardness.

Original: Moreover, we AVH is potentially a better measure for generalization as a pretrained model.
Revised: Moreover, we AVH is potentially a better measure for generalization as a pretrained model.

Original: The norm of feature embeddings is often embedded with training data prior such as data imbalance [39] and class granularity [30].
Revised: The norm of feature embeddings is often embedded with training data prior such as data imbalance [39] and class granularity [30].

Original: But when extracting the features for the classes that do not exist in training set, such training data prior is undesired.
Revised: But when extracting the features for the classes that do not exist in training set, such training data prior is undesired.

Original: Since AVH does not consider the norm of feature embeddings, it may better evaluate the generalization of the deep network.
Revised: Since AVH does not consider the norm of feature embeddings, it may better evaluate the generalization of the deep network.

Original: Conjecture on training dynamics of CNNs .
Revised: Conjecture on training dynamics of CNNs .

Original: From Figure 3 and observations above, we conjecture that the training of CNN has two phases.
Revised: From Figure 3 and observations above, we conjecture that the training of CNN has two phases.

Original: At the beginning of the training, the softmax cross-entropy loss will ﬁrst optimize the angles among different classes while the norm will ﬂuctuate and increase very slowly.
Revised: At the beginning of the training, the softmax cross-entropy loss will ﬁrst optimize the angles among different classes while the norm will ﬂuctuate and increase very slowly.

Original: We argue that it is because changing the norm will not decrease the loss when the angles are not separated enough for correct classiﬁcation.
Revised: We argue that it is because changing the norm will not decrease the loss when the angles are not separated enough for correct classiﬁcation.

Original: As a result, the angles get optimized ﬁrstly.
Revised: As a result, the angles get optimized ﬁrstly.

Original: As the training continues, the angles become more stable and change very slowly while the norm increases rapidly.
Revised: As the training continues, the angles become more stable and change very slowly while the norm increases rapidly.

Original: On the one hand, for easy examples, it is because when the angles get decreased enough for correct classiﬁcation, the softmax cross-entropy loss can be well minimized by purely increasing the norm.
Revised: On the one hand, for easy examples, it is because when the angles get decreased enough for correct classiﬁcation, the softmax cross-entropy loss can be well minimized by purely increasing the norm.

Original: On the other hand, for hard examples, the plateau is caused by that the CNN is unable to decrease the angle to correctly classify examples and thereby also unable to increase the norms (because it may otherwise increase the loss).
Revised: On the other hand, for hard examples, the plateau is caused by that the CNN is unable to decrease the angle to correctly classify examples and thereby also unable to increase the norms (because it may otherwise increase the loss).

Original:  A PPLICATION TO S ELF - TRAINING FOR D OMAIN A DAPTATION
Revised:  A PPLICATION TO S ELF - TRAINING FOR D OMAIN A DAPTATION

Original: Unsupervised domain adaptation [1] presents an important transfer learning problem with wide applications.
Revised: Unsupervised domain adaptation [1] presents an important transfer learning problem with wide applications.

Original: Deep self-training [33] recently emerged as a powerful framework towards addressing this problem [53, 57, 73, 74].
Revised: Deep self-training [33] recently emerged as a powerful framework towards addressing this problem [53, 57, 73, 74].

Original: Here we show the application of AVH as an improved conﬁdence measure in self-training that could signiﬁcantly beneﬁt the domain adaptation task.
Revised: Here we show the application of AVH as an improved conﬁdence measure in self-training that could signiﬁcantly beneﬁt the domain adaptation task.

Original: Dataset: We conduct expeirments on the VisDA-17 [50] dataset which is a widely used major benchmark for domain adaptation in image classiﬁcation.
Revised: Dataset: We conduct expeirments on the VisDA-17 [50] dataset which is a widely used major benchmark for domain adaptation in image classiﬁcation.

Original: The dataset contains a total number of 152 , 409 2D synthetic images from 12 categories in the source training set, and 55 , 400 real images from MS-COCO [36] with the same set of categories as the target domain validation set.
Revised: The dataset contains a total number of 152 , 409 2D synthetic images from 12 categories in the source training set, and 55 , 400 real images from MS-COCO [36] with the same set of categories as the target domain validation set.

Original: We follow the protocol of previous works to train a source model with the synthetic training set, and report the model performance on target validation set upon adaptation.
Revised: We follow the protocol of previous works to train a source model with the synthetic training set, and report the model performance on target validation set upon adaptation.

Original: Baseline: We choose class-balanced self-training (CBST)
Revised: Baseline: We choose class-balanced self-training (CBST)

Original: [73] as our starting self-training baseline considering its good performance on domain adaptaiton.
Revised: [73] as our starting self-training baseline considering its good performance on domain adaptaiton.

Original: We also compare our model with conﬁdence regularized self-training (CRST) 1 [74], a more recent state-of-the-art self-training framework improved over CBST with network prediction/pseudo-label regularized with smoothness.
Revised: We also compare our model with conﬁdence regularized self-training (CRST) 1 [74], a more recent state-of-the-art self-training framework improved over CBST with network prediction/pseudo-label regularized with smoothness.

Original: Speciﬁcally, our work follows the exact implementation of CBST/CRST in [74].
Revised: Speciﬁcally, our work follows the exact implementation of CBST/CRST in [74].

Original: Speciﬁcally, given the labeled source domain training set x s ∈ X S and the unlabeled target domain data x t ∈ X T , with known source labelsy s =
Revised: Speciﬁcally, given the labeled source domain training set x s ∈ X S and the unlabeled target domain data x t ∈ X T , with known source labels

Original: ( y (1) s , ..., y ( K ) s ) ∈ Y S and unknown target labels ˆ y t
Revised: [Equation] and unknown target labels ˆ y t

Original: = (ˆ y (1) t , ..., 
Revised: = (ˆ y (1) t , ..., ˆ y

Revised: [Equation]

Original: ˆ y ( K ) t ) ∈
Revised: K ) [Equation] ) ∈

Original: ˆ Y T from K classes, CBST performs joint network learning and pseudo-label estimation by treating pseudo-labels as discrete learnable latent variables with the following loss:
Revised: ˆ Y T from K classes, CBST performs joint network learning and pseudo-label estimation by treating pseudo-labels as discrete learnable latent variables with the following loss:

Original: [Equation]
Revised: [Equation]

Original: where the feasible set of pseudo-labels is the union of { 0 } and the K dimensional one-hot vector space E K , and w and p ( k | x ; w ) represent the network weights and the classiﬁer’s softmax probability for class k , respectively.
Revised: where the feasible set of pseudo-labels is the union of { 0 } and the K dimensional one-hot vector space E K , and w and p ( k | x ; w ) represent the network weights and the classiﬁer’s softmax probability for class k , respectively.

Original: In addition, λ k serves as a class-balancing parameter controlling the pseudolabel selection of class k , and is determined by the softmax conﬁdence ranked at portion p (in descending order) among samples predicted to class k .
Revised: In addition, λ k serves as a class-balancing parameter controlling the pseudolabel selection of class k , and is determined by the softmax conﬁdence ranked at portion p (in descending order) among samples predicted to class k .

Original: Therefore, only one parameter p is used to determine all λ k ’s. The optimization problem in (1) can be solved via minimizing with respect to w and ˆ Y alternatively, and the solver of ˆ Y can be written as:
Revised: Therefore, only one parameter p is used to determine all λ k ’s. The optimization problem in (1) can be solved via minimizing with respect to w and ˆ Y alternatively, and the solver of ˆ Y can be written as:

Original: [Equation]
Revised: [Equation]

Original: The optimization with respect to w is simply normal network re-training with source labels and estimated pseudo-labels.
Revised: The optimization with respect to w is simply normal network re-training with source labels and estimated pseudo-labels.

Original: And the complete self-training process involves alternatively repeat of network re-training and pseudo-label estimation.
Revised: And the complete self-training process involves alternatively repeat of network re-training and pseudo-label estimation.

Original: CBST+AVH: We seek to improve the pseudo-label solver with better conﬁdence measure from AVH.
Revised: CBST+AVH: We seek to improve the pseudo-label solver with better conﬁdence measure from AVH.

Original: To this end, we propose the following deﬁnition of angular visual conﬁdence (AVC) to represent the predicted probability of class c :
Revised: To this end, we propose the following deﬁnition of angular visual conﬁdence (AVC) to represent the predicted probability of class c :

Original: [Equation]
Revised: [Equation]

Revised: [Table]

Original: and the pseudo-label estimation in CBST+AVH is accordingly deﬁned as:
Revised: and the pseudo-label estimation in CBST+AVH is accordingly deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where λ k is differently determined by referring to AV C ( k | x t ; w ) ranked at portion p among samples predicted to class k by AVH, following a similar deﬁnition of λ k in CBST.
Revised: where λ k is differently determined by referring to AV C ( k | x t ; w ) ranked at portion p among samples predicted to class k by AVH, following a similar deﬁnition of λ k in CBST.

Original: In addition, network re-training in CBST+AVH follows the softmax self-training loss in (1).
Revised: In addition, network re-training in CBST+AVH follows the softmax self-training loss in (1).

Original: One could see that AVH changes the self-training behavior from two ways with the conditions in (4):
Revised: One could see that AVH changes the self-training behavior from two ways with the conditions in (4):

Original: Improved selection: This is determined by AV C ( k | x t ; w ) > λ k .
Revised: Improved selection: This is determined by AV C ( k | x t ; w ) > λ k .

Original: Improved classiﬁcation: This is determined by k = arg max k { AV C ( k | x t ; w ) λ k } .Speciﬁcally, the former determines which samples are not ignored during self-training based on AVC, [Table] [Table]
Revised: Improved classiﬁcation: This is determined by k = arg max k { AV C ( k | x t ; w ) λ k } .

Original: 51.6 CBST [74] 87. 76.4 CRST [74] 88. 78.1 Proposed 93. whereas the latter determines the pseudo-label class by taking argmax over normalized AVC scores.
Revised: Speciﬁcally, the former determines which samples are not ignored during self-training based on AVC, whereas the latter determines the pseudo-label class by taking argmax over normalized AVC scores.

Original: With calibrated conﬁdence that better resembles human visual hardness, both aspects are likely to considerably inﬂuence the performance of self-training.
Revised: With calibrated conﬁdence that better resembles human visual hardness, both aspects are likely to considerably inﬂuence the performance of self-training.

Original: Experimental Results: We present the results of the proposed method in Table 3, and also show its performance with respect to different self-training epochs in Figure 4.
Revised: Experimental Results: We present the results of the proposed method in Table 3, and also show its performance with respect to different self-training epochs in Figure 4.

Original: One could see that CBST+AVH outperforms both CBST and CRST by a very signiﬁcant margin.
Revised: One could see that CBST+AVH outperforms both CBST and CRST by a very signiﬁcant margin.

Original: We would like to emphasize that this is a very compelling result under “apples to apples” comparison with the same source model, implementation and hyper-parameters.
Revised: We would like to emphasize that this is a very compelling result under “apples to apples” comparison with the same source model, implementation and hyper-parameters.

Original: [Figure]
Revised: [Figure]

Original: Analysis: A major challenge of self-training is the ampliﬁcation of error due to misclassiﬁed pseudo-labels.
Revised: Analysis: A major challenge of self-training is the ampliﬁcation of error due to misclassiﬁed pseudo-labels.

Original: Therefore, traditional self-training methods such as CBST often use model conﬁdence as the conﬁdence measure to select conﬁdently labeled examples.
Revised: Therefore, traditional self-training methods such as CBST often use model conﬁdence as the conﬁdence measure to select conﬁdently labeled examples.

Original: The hope is that higher conﬁdence potentially implies lower error rate.
Revised: The hope is that higher conﬁdence potentially implies lower error rate.

Original: While this in generally proves useful, the model tends to focus on the “less informative” samples, whereas ignoring the “more informative”, harder ones near classier boundaries that could be essential for learning a stronger classiﬁer.
Revised: While this in generally proves useful, the model tends to focus on the “less informative” samples, whereas ignoring the “more informative”, harder ones near classier boundaries that could be essential for learning a stronger classiﬁer.

Original: An advantage we observe from AVH is that the improved calibration leads to more frequent sampling of harder samples, whereas the pseudo-label classiﬁcation on these hard samples generally outperforms softmax results.
Revised: An advantage we observe from AVH is that the improved calibration leads to more frequent sampling of harder samples, whereas the pseudo-label classiﬁcation on these hard samples generally outperforms softmax results.

Original: Table 2 shows some statistics of the examples selected with AVH and Model Conﬁdence respectively at the beginning of the training process.
Revised: Table 2 shows some statistics of the examples selected with AVH and Model Conﬁdence respectively at the beginning of the training process.

Original: The true postive rate (TP Rate) for of CBST+AVH remains similar to
Revised: The true postive rate (TP Rate) for of CBST+AVH remains similar to

Original: CBST/CRST, indicating AVH is not overall introducing more noisy examples compare to model conﬁdence.
Revised: CBST/CRST, indicating AVH is not overall introducing more noisy examples compare to model conﬁdence.

Original: On the other hand, it is observed that the average model conﬁdence of AVH selected samples is lower, indicating there are more selected hard samples that are closer to the decision boundary.
Revised: On the other hand, it is observed that the average model conﬁdence of AVH selected samples is lower, indicating there are more selected hard samples that are closer to the decision boundary.

Original: It is also observed that the average sample norm by AVH is also lower, conﬁrming the inﬂuence of sample norm on ultimate model conﬁdence.
Revised: It is also observed that the average sample norm by AVH is also lower, conﬁrming the inﬂuence of sample norm on ultimate model conﬁdence.

Original:  E XTENSIONS AND A PPLICATIONS
Revised:  E XTENSIONS AND A PPLICATIONS

Original: Adversarial Example: A Counter Example?
Revised: Adversarial Example: A Counter Example?

Original: Our claim about the strong correlation between AVH score and human visual hardness does not apply on non-natural images such as adversarial examples.
Revised: Our claim about the strong correlation between AVH score and human visual hardness does not apply on non-natural images such as adversarial examples.

Original: For such examples, the human can not tell the difference visually, but the adversarial example has a worse AVH than the original image, which runs counter to our claim that AVH has strong correlation with human visual hardness.
Revised: For such examples, the human can not tell the difference visually, but the adversarial example has a worse AVH than the original image, which runs counter to our claim that AVH has strong correlation with human visual hardness.

Original: So this claim is limited to distribution of natural images.
Revised: So this claim is limited to distribution of natural images.

Original: However, on a positive note, we do ﬁnd that AVH is slower to change compared to the embedding norm during the dynamics of adversarial training.
Revised: However, on a positive note, we do ﬁnd that AVH is slower to change compared to the embedding norm during the dynamics of adversarial training.

Original: See Appendix for details.
Revised: See Appendix for details.

Revised: [Table]

Revised: [Table]

Original: Connection to deep metric learning:
Revised: Connection to deep metric learning:

Original: Measuring the hardness of samples is also of great importance in the ﬁeld of deep metric learning [49, 60, 66].
Revised: Measuring the hardness of samples is also of great importance in the ﬁeld of deep metric learning [49, 60, 66].

Original: For instance, objective functions in deep metric learning consist of e.g. , triplet loss [56] or contrastive loss [22], which requires data pair/triplet mining in order to perform well in practice.
Revised: For instance, objective functions in deep metric learning consist of e.g. , triplet loss [56] or contrastive loss [22], which requires data pair/triplet mining in order to perform well in practice.

Original: One of the most widely used data sampling strategies is semi-hard negative sample mining [56] and hard negative sample mining.
Revised: One of the most widely used data sampling strategies is semi-hard negative sample mining [56] and hard negative sample mining.

Original: These negative sample mining techniques highly depend on how one deﬁnes the hardness of samples.
Revised: These negative sample mining techniques highly depend on how one deﬁnes the hardness of samples.

Original: AVH can be potentially useful in this setting.
Revised: AVH can be potentially useful in this setting.

Original: Connections to fairness in machine learning: Easy and hard samples can implicitly reﬂect imbalances in latent attributes in the dataset.
Revised: Connections to fairness in machine learning: Easy and hard samples can implicitly reﬂect imbalances in latent attributes in the dataset.

Original: For example, the CASIA-WebFace dataset [68] mostly contains white celebrities, so the neural network trained on CASIA-WebFace is highly biased against the other races.
Revised: For example, the CASIA-WebFace dataset [68] mostly contains white celebrities, so the neural network trained on CASIA-WebFace is highly biased against the other races.

Original: [4] demonstrates a performance drop of faces of darker people due to the biases in the training dataset.
Revised: [4] demonstrates a performance drop of faces of darker people due to the biases in the training dataset.

Original: In order to ensure fairness and remove dataset biases, the ability to identify hard samples automatically can be very useful.
Revised: In order to ensure fairness and remove dataset biases, the ability to identify hard samples automatically can be very useful.

Original: We would like to test if AVH is effective in these settings.
Revised: We would like to test if AVH is effective in these settings.

Original: Connections to knowledge transfer and curriculum learning: The efﬁciency of knowledge transfer [25] is partially determined by the sequence of input training data.
Revised: Connections to knowledge transfer and curriculum learning: The efﬁciency of knowledge transfer [25] is partially determined by the sequence of input training data.

Original: [38] theoretically shows feeding easy samples ﬁrst and hard samples later (known as curriculum learning) can improve the convergence of model.
Revised: [38] theoretically shows feeding easy samples ﬁrst and hard samples later (known as curriculum learning) can improve the convergence of model.

Original: [2] also show that the curriculum of feeding training samples matters in terms of both accuracy and convergence.
Revised: [2] also show that the curriculum of feeding training samples matters in terms of both accuracy and convergence.

Original: We plan to investigate the use of AVH metric in such settings.
Revised: We plan to investigate the use of AVH metric in such settings.

Original:  C ONCLUDING R EMARKS
Revised:  C ONCLUDING R EMARKS

Original: Human perception and deep neural networks in general have different notions of visual hardness.
Revised: Human perception and deep neural networks in general have different notions of visual hardness.

Original: Our paper studies the gap between them, and attempts to bridge this gap by proposing a novel measure for
Revised: Our paper studies the gap between them, and attempts to bridge this gap by proposing a novel measure for

Original: CNN models known as angular visual hardness.
Revised: CNN models known as angular visual hardness.

Original: Our comprehensive empirical studies show that AVH has many nice properties.
Revised: Our comprehensive empirical studies show that AVH has many nice properties.

Original: First, AVH has a strong correlation with human selection frequency and image degradation level.
Revised: First, AVH has a strong correlation with human selection frequency and image degradation level.

Original: Second, this holds across different network architectures and throughout the training process.
Revised: Second, this holds across different network architectures and throughout the training process.

Original: Third, AVH can serve as an indicator of generalization abilities of neural networks, and improving SOTA accuracy entails improving accuracy on hard examples.
Revised: Third, AVH can serve as an indicator of generalization abilities of neural networks, and improving SOTA accuracy entails improving accuracy on hard examples.

Original: Then we empirically show the huge advantage of AVH over Model Conﬁdence in self-training for domain adaptation task.
Revised: Then we empirically show the huge advantage of AVH over Model Conﬁdence in self-training for domain adaptation task.

Original: It is still an open problem of designing an appropriate loss function that can focus on improving AVH during training.
Revised: It is still an open problem of designing an appropriate loss function that can focus on improving AVH during training.

Original: AVH can be very useful in other applications such as deep metric learning, fairness, knowledge transfer, etc.
Revised: AVH can be very useful in other applications such as deep metric learning, fairness, knowledge transfer, etc.

Original: and we plan to investigate them in future
Revised: and we plan to investigate them in future.

Original: Appendix A A DDITIONAL E XPERIMENTS A. A DDITIONAL R ESULTS FOR C ORRELATION T ESTINGS
Revised: Appendix A A DDITIONAL E XPERIMENTS A. A DDITIONAL R ESULTS FOR C ORRELATION T ESTINGS

Original: In order to run rigorous correlation testings, besides computing the Spearman coefﬁcient, we provide additional results on Pearson and Kendall Tau correlation coefﬁcients.
Revised: In order to run rigorous correlation testings, besides computing the Spearman coefﬁcient, we provide additional results on Pearson and Kendall Tau correlation coefﬁcients.

Original: Moreover, we show results for all four architectures, AlexNet, VGG19, ResNet50 and DenseNet121 in Table 4, 5, 6 andrespectively to support our claims in section 3.2.
Revised: Moreover, we show results for all four architectures, AlexNet, VGG19, ResNet50 and DenseNet121 in Table 4, 5, 6 andrespectively to support our claims in section 3.2.

Original: A. A DDITIONAL R ESULTS FOR THE H YPOTHESIS
Revised: A. A DDITIONAL R ESULTS FOR THE H YPOTHESIS

Original: Deﬁnition 4 (Image Degradation Level) .
Revised: Deﬁnition 4 (Image Degradation Level) .

Original: We deﬁne another way to measure human visual hardness on pictures as Image Degradation Level.
Revised: We deﬁne another way to measure human visual hardness on pictures as Image Degradation Level.

Original: We consider two degradation methods in this paper, decreasing contrast and adding noise.
Revised: We consider two degradation methods in this paper, decreasing contrast and adding noise.

Original: Quantitatively, Image Degradation Level for decreasing contrast is directly the contrast level.
Revised: Quantitatively, Image Degradation Level for decreasing contrast is directly the contrast level.

Original: Image Degradation Level for adding noise is the amount of pixel-wise additive uniform noise.
Revised: Image Degradation Level for adding noise is the amount of pixel-wise additive uniform noise.

Original: Hypothesis: AVH has a strong correlation with Image Degradation Level
Revised: Hypothesis: AVH has a strong correlation with Image Degradation Level

Original: In order to test if the results from Prediction 1 hold on another proxy to human visual hardness, Image
Revised: In order to test if the results from Prediction 1 hold on another proxy to human visual hardness, Image

Original: Degradation Level, we perform the similar experiments but on the augmented ImageNet validation set.
Revised: Degradation Level, we perform the similar experiments but on the augmented ImageNet validation set.

Original: The plots in Figure 5 show the strong correlation between AVH ( x ) and Noise Degradation Level while the plots in Figure 6 present the strong correlation between AVH ( x ) and Contrast Degradation
Revised: The plots in Figure 5 show the strong correlation between AVH ( x ) and Noise Degradation Level while the plots in Figure 6 present the strong correlation between AVH ( x ) and Contrast Degradation

Original: Level.
Revised: Level.

Original: They, along with Figure 7, demonstrate that AVH ( x ) strongly correlates with Human Visual
Revised: They, along with Figure 7, demonstrate that AVH ( x ) strongly correlates with Human Visual

Original: Hardness.
Revised: Hardness.

Original: Additional Plots for DenseNet121 is shown in Figure 8.
Revised: Additional Plots for DenseNet121 is shown in Figure 8.

Original: Hypothesis: (cid:107) x (cid:107) 2 has a correlation with Human Selection Frequency
Revised: Hypothesis: (cid:107) x (cid:107) 2 has a correlation with Human Selection Frequency

Original: We further verify if presenting all samples across 1000 different classes affects the visualization of the correlation.
Revised: We further verify if presenting all samples across 1000 different classes affects the visualization of the correlation.

Original: According to WordNet [13] hierarchy, we map the original 1000 ﬁne-grained classes to 45 higher hierarchical classes.
Revised: According to WordNet [13] hierarchy, we map the original 1000 ﬁne-grained classes to 45 higher hierarchical classes.

Original: Figure 9 exhibits the relationship between Human Selection Frequency and (cid:107) x (cid:107) 2 for three representative higher classes containing 58 , 7 , 1 ﬁne-grained classes respectively.
Revised: Figure 9 exhibits the relationship between Human Selection Frequency and (cid:107) x (cid:107) 2 for three representative higher classes containing 58 , 7 , 1 ﬁne-grained classes respectively.

Original: Noted that there is still not any visible direct proportion between these two variables across all plots.
Revised: Noted that there is still not any visible direct proportion between these two variables across all plots.

Original: Hypothesis: AVH has a correlation with Human Selection Frequency Additional results for
Revised: Hypothesis: AVH has a correlation with Human Selection Frequency Additional results for

Original: DenseNet121 are shown in Figure 8.
Revised: DenseNet121 are shown in Figure 8.

Revised: Table 4: presents the spearman rank correlations between human selection frequency and AVH, Model Conﬁdence in AlexNet.

Revised: Noted here we show the absolute value of the coefﬁcient which represents the strength of the correlation.

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: Table 5: presents the spearman rank correlations between human selection frequency and AVH, Model Conﬁdence in VGG19. Noted here we show the absolute value of the coefﬁcient which represents the strength of the correlation.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Figure] [Table] 
Revised: [Table] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Revised: AV H ( x )       # o f S a m p l e s

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure][Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: A. A DDITIONAL E XPERIMENTS FOR O BSERVING D YNAMICS ON MNIST
Revised: A. A DDITIONAL E XPERIMENTS FOR O BSERVING D YNAMICS ON MNIST

Original: Figure 12 illustrates how the average norm of the feature embedding and angles between feature and class embedding for testing samples vary in 60 iterations during the training process.
Revised: Figure 12 illustrates how the average norm of the feature embedding and angles between feature and class embedding for testing samples vary in 60 iterations during the training process.

Original: The average norm increases with a large initial slope but it ﬂattens slightly after 10 iterations.
Revised: The average norm increases with a large initial slope but it ﬂattens slightly after 10 iterations.

Original: On the other hand, the average angle decreases sharply at the beginning and then becomes almost ﬂat after 10 iterations.
Revised: On the other hand, the average angle decreases sharply at the beginning and then becomes almost ﬂat after 10 iterations.

Original: Moreover
Revised: Moreover

Original: [Figure]
Revised: [Figure]

Original: we explore the difference between norm and angle change for easy and hard human examples in more details.
Revised: we explore the difference between norm and angle change for easy and hard human examples in more details.

Original: Figure 13 also plots the angle and norm changes for two examples, which are hard and easy for human visualization, in the training phase.
Revised: Figure 13 also plots the angle and norm changes for two examples, which are hard and easy for human visualization, in the training phase.

Original: Note that both examples are testing data and thereby have never ﬁt into the model.
Revised: Note that both examples are testing data and thereby have never ﬁt into the model.

Original: We can see that for the angle, both of them drop largely initially and then the angle for the easy one converges to a much lower value.
Revised: We can see that for the angle, both of them drop largely initially and then the angle for the easy one converges to a much lower value.

Original: For the norm, both of them are increasing drastically at an early stage but that for the harder example keeps climbing even when that for the easy one saturates
Revised: For the norm, both of them are increasing drastically at an early stage but that for the harder example keeps climbing even when that for the easy one saturates

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: A. A DDITIONAL E XPERIMENTS FOR T RAINING D YNAMICS ON CIFAR10 AND CIFAR 
Revised: A. A DDITIONAL E XPERIMENTS FOR T RAINING D YNAMICS ON CIFAR10 AND CIFAR

Original: Figure 14 and 15 show the dynamics of average (cid:96) 2 norm of the embeddings and average AVH(x) on
Revised: Figure 14 and 15 show the dynamics of average (cid:96) 2 norm of the embeddings and average AVH(x) on

Original: CIFAR10 and CIFAR100 datasets respectively.
Revised: CIFAR10 and CIFAR100 datasets respectively.

Original: We can observe the similar phenomenons we have discussed in section 4.
Revised: We can observe the similar phenomenons we have discussed in section 4.

Original: It further supports our theoretical foundation from [61] that gradient descent converges to the same direction as maximum margin solutions irrelevant to the (cid:96) 2 norm of classiﬁer weights or feature embeddings.
Revised: It further supports our theoretical foundation from [61] that gradient descent converges to the same direction as maximum margin solutions irrelevant to the (cid:96) 2 norm of classiﬁer weights or feature embeddings.

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: A. A DDITIONAL E XPERIMENTS FOR T RAINING D YNAMICS ON I MAGE N ET
Revised: A. A DDITIONAL E XPERIMENTS FOR T RAINING D YNAMICS ON I MAGE N ET

Original: Figure 17 presents the dynamics of the average (cid:107) x (cid:107) 2 and the dynamics of the accuracy for validation samples vary in 90 epochs during the training on AlexNet, VGG19, DenseNet121 and ResNet50.
Revised: Figure 17 presents the dynamics of the average (cid:107) x (cid:107) 2 and the dynamics of the accuracy for validation samples vary in 90 epochs during the training on AlexNet, VGG19, DenseNet121 and ResNet50.

Original: In ﬁgure 16 and 18, we average over validation samples in ﬁve human selection frequency bins separately, and then compute the average embedding norm, AVH and model accuracies.
Revised: In ﬁgure 16 and 18, we average over validation samples in ﬁve human selection frequency bins separately, and then compute the average embedding norm, AVH and model accuracies.

Original: In ﬁgureand 19, we average over validation samples in ﬁve image noise degradation level bins separately, and then compute the average embedding norm, AVH and model accuracies.
Revised: In ﬁgureand 19, we average over validation samples in ﬁve image noise degradation level bins separately, and then compute the average embedding norm, AVH and model accuracies.

Original: In ﬁgure 10 and 20, we average over validation samples in ﬁve image contrast degradation level bins separately, and then compute the average embedding norm, AVH and model accuracies.
Revised: In ﬁgure 10 and 20, we average over validation samples in ﬁve image contrast degradation level bins separately, and then compute the average embedding norm, AVH and model accuracies.

Original: Figure 21 shows the training dynamics of the model conﬁdence on AlexNet, VGG19 and ResNet50.
Revised: Figure 21 shows the training dynamics of the model conﬁdence on AlexNet, VGG19 and ResNet50.

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: B A S PECIAL C ASE : A DVERSARIAL E XAMPLES
Revised: B A S PECIAL C ASE : A DVERSARIAL E XAMPLES

Original: We show a special case in Figure 22 to illustrate how the norm and the angle change when one sample switches from one class to another.
Revised: We show a special case in Figure 22 to illustrate how the norm and the angle change when one sample switches from one class to another.

Original: Speciﬁcally, we change the sample from one class to another using adversarial perturbation.
Revised: Speciﬁcally, we change the sample from one class to another using adversarial perturbation.

Original: It is essentially performing gradient ascent to the ground truth class.
Revised: It is essentially performing gradient ascent to the ground truth class.

Original: In Figure
Revised: In Figure

Original: [Figure]
Revised: [Figure]

Original: the purple line denotes the trajectory of an adversarial sample switching from one class to another.
Revised: the purple line denotes the trajectory of an adversarial sample switching from one class to another.

Original: We can see that the sample will ﬁrst shrink its norm towards origin and then push its angle away from the ground truth class.
Revised: We can see that the sample will ﬁrst shrink its norm towards origin and then push its angle away from the ground truth class.

Original: Such a trajectory indicates that the adversarial sample will ﬁrst approach to the origin in order to become a hard sample for this class.
Revised: Such a trajectory indicates that the adversarial sample will ﬁrst approach to the origin in order to become a hard sample for this class.

Original: Then the sample will change the angle in order to switch its label.
Revised: Then the sample will change the angle in order to switch its label.

Original: This special example fully justiﬁes the importance of both norm and angle in terms of the hardness of samples.
Revised: This special example fully justiﬁes the importance of both norm and angle in terms of the hardness of samples.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: C M ORE DISCUSSIONS ON THE D IFFERENCE BETWEEN AVH AND M ODEL C ONFIDENCE
Revised: C M ORE DISCUSSIONS ON THE D IFFERENCE BETWEEN AVH AND M ODEL C ONFIDENCE

Original: The difference between AVH and model conﬁdence lies in the feature norm and its role during training.
Revised: The difference between AVH and model conﬁdence lies in the feature norm and its role during training.

Original: To illustrate the difference, we consider a simple binary classiﬁcation case where the softmax score (i.e., model conﬁdence) for class 1 is
Revised: To illustrate the difference, we consider a simple binary classiﬁcation case where the softmax score (i.e., model conﬁdence) for class 1 is

Original: [Equation]
Revised: [Equation]

Original: where w i is the classiﬁer weights of class i , x is the input deep feature and θ w i , x is the angle between w i and x .
Revised: where w i is the classiﬁer weights of class i , x is the input deep feature and θ w i , x is the angle between w i and x .

Original: To simplify, we assume the norm of w 1 and w 2 are the same, and then the classiﬁcation result is based on the angle now.
Revised: To simplify, we assume the norm of w 1 and w 2 are the same, and then the classiﬁcation result is based on the angle now.

Original: Once θ w 1 , x is smaller than θ w 2 , x , the network will classify the sample x as class 1.
Revised: Once θ w 1 , x is smaller than θ w 2 , x , the network will classify the sample x as class 1.

Original: However, in order to further minimize the cross-entropy loss after making θ w 1 , x smaller than θ w 2 , x , the network has a trivial solution: increasing the feature norm (cid:107) x (cid:107) instead of further minimizing the θ w 1 , x .
Revised: However, in order to further minimize the cross-entropy loss after making θ w 1 , x smaller than θ w 2 , x , the network has a trivial solution: increasing the feature norm (cid:107) x (cid:107) instead of further minimizing the θ w 1 , x .

Original: It is obviously a much more difﬁcult task to minimize θ w 1 , x rather than increasing (cid:107) x (cid:107) .
Revised: It is obviously a much more difﬁcult task to minimize θ w 1 , x rather than increasing (cid:107) x (cid:107) .

Original: Therefore, the network will tend to increase the feature norm (cid:107) x (cid:107) to minimize the cross-entropy loss, which is equivalent to maximizing the model conﬁdence in class 1.
Revised: Therefore, the network will tend to increase the feature norm (cid:107) x (cid:107) to minimize the cross-entropy loss, which is equivalent to maximizing the model conﬁdence in class 1.

Original: In fact, this also matches our empirical observation that the feature norm keeps increasing during training.
Revised: In fact, this also matches our empirical observation that the feature norm keeps increasing during training.

Original: Most importantly, one can notice that AVH will stay unchanged no matter how large the feature norm (cid:107) x (cid:107) is.
Revised: Most importantly, one can notice that AVH will stay unchanged no matter how large the feature norm (cid:107) x (cid:107) is.

Original: Moreover, this also matches our empirical result that AVH easily gets saturated while model conﬁdence can keep improving.
Revised: Moreover, this also matches our empirical result that AVH easily gets saturated while model conﬁdence can keep improving.

Original: Therefore, AVH is able to better characterize the visual hardness, since it is trivial for the network to increase feature norm.
Revised: Therefore, AVH is able to better characterize the visual hardness, since it is trivial for the network to increase feature norm.

Original: This is the fundamental difference between model conﬁdence and AVH.
Revised: This is the fundamental difference between model conﬁdence and AVH.

Original: To get a more intuitive sense of how feature norm can affect the model conﬁdence, we plot the value of the model conﬁdence for two scenarios: θ w 1 , x < θ w 2 , x and θ w 1 , x > θ w 2 , x .
Revised: To get a more intuitive sense of how feature norm can affect the model conﬁdence, we plot the value of the model conﬁdence for two scenarios: θ w 1 , x < θ w 2 , x and θ w 1 , x > θ w 2 , x .

Original: Under the case that the sample x belongs to class 1, once we have θ w 1 , x < θ w 2 , x , then we only need to increase the feature norm and can easily get nearly perfect conﬁdence on this sample.
Revised: Under the case that the sample x belongs to class 1, once we have θ w 1 , x < θ w 2 , x , then we only need to increase the feature norm and can easily get nearly perfect conﬁdence on this sample.

Original: In contrast, AVH will stay unchanged during the entire process and therefore is a more robust indicator for visual hardness than model conﬁdence.
Revised: In contrast, AVH will stay unchanged during the entire process and therefore is a more robust indicator for visual hardness than model conﬁdence.

Original: [Figure]
Revised: [Figure]

Original: When θ w 1 , x < θ w 2 , x , we use θ 1 = π/ 4 − 0 .
Revised: When θ w 1 , x < θ w 2 , x , we use θ 1 = π/ 4 − 0 .

Original: 05 and θ 2 = π/ 4 + 0 .
Revised: 05 and θ 2 = π/ 4 + 0 .

Original: When θ w 1 , x > θ w 2 , x , we use θ 1 = π/ 4 + 0 .
Revised: When θ w 1 , x > θ w 2 , x , we use θ 1 = π/ 4 + 0 .

Original: 05 and θ 2 = π/ 4 − 0 .
Revised: 05 and θ 2 = π/ 4 − 0 .

Original: Note that, unlike model conﬁdence, the smaller AVH is, the more conﬁdent the network is ( i.e. ,
Revised: Note that, unlike model conﬁdence, the smaller AVH is, the more conﬁdent the network is ( i.e. ,