Original: R ETHINKING S OFT L ABELS FOR K NOWLEDGE D ISTILLATION : A B IAS -V ARIANCE T RADEOFF P ERSPECTIVE
Revised: R ETHINKING S OFT L ABELS FOR K NOWLEDGE D ISTILLATION : A B IAS -V ARIANCE T RADEOFF P ERSPECTIVE

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: Knowledge distillation is an effective approach to leverage a well-trained network or an ensemble of them, named as the teacher, to guide the training of a student network.
Revised: Knowledge distillation is an effective approach to leverage a well-trained network or an ensemble of them, named as the teacher, to guide the training of a student network.

Original: The outputs from the teacher network are used as soft labels for supervising the training of a new network.
Revised: The outputs from the teacher network are used as soft labels for supervising the training of a new network.

Original: Recent studies (M¨uller et al., 2019; Yuan et al., 2020) revealed an intriguing property of the soft labels that making labels soft serves as a good regularization to the student network.
Revised: Recent studies (M¨uller et al., 2019; Yuan et al., 2020) revealed an intriguing property of the soft labels that making labels soft serves as a good regularization to the student network.

Original: From the perspective of statistical learning, regularization aims to reduce the variance, however how bias and variance change is not clear for training with soft labels.
Revised: From the perspective of statistical learning, regularization aims to reduce the variance, however how bias and variance change is not clear for training with soft labels.

Original: In this paper, we investigate the bias-variance tradeoff brought by distillation with soft labels.
Revised: In this paper, we investigate the bias-variance tradeoff brought by distillation with soft labels.

Original: Speciﬁcally, we observe that during training the bias-variance tradeoff varies sample-wisely.
Revised: Speciﬁcally, we observe that during training the bias-variance tradeoff varies sample-wisely.

Original: Further, under the same distillation temperature setting, we observe that the distillation performance is negatively associated with the number of some speciﬁc samples, which are named as regularization samples since these samples lead to bias increasing and variance decreasing.
Revised: Further, under the same distillation temperature setting, we observe that the distillation performance is negatively associated with the number of some speciﬁc samples, which are named as regularization samples since these samples lead to bias increasing and variance decreasing.

Original: Nevertheless, we empirically ﬁnd that completely ﬁltering out regularization samples also deteriorates distillation performance.
Revised: Nevertheless, we empirically ﬁnd that completely ﬁltering out regularization samples also deteriorates distillation performance.

Original: Our discoveries inspired us to propose the novel weighted soft labels to help the network adaptively handle the sample-wise biasvariance tradeoff.
Revised: Our discoveries inspired us to propose the novel weighted soft labels to help the network adaptively handle the sample-wise biasvariance tradeoff.

Original: Experiments on standard evaluation benchmarks validate the effectiveness of our method.
Revised: Experiments on standard evaluation benchmarks validate the effectiveness of our method.

Original: Our code is available in the supplementary.
Revised: Our code is available in the supplementary.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: For deep neural networks (Goodfellow et al., 2016), knowledge distillation (KD) (Ba & Caruana, 2014; Hinton et al., 2015) refers to the technique that uses well-trained networks to guide the training of another network.
Revised: For deep neural networks (Goodfellow et al., 2016), knowledge distillation (KD) (Ba & Caruana, 2014; Hinton et al., 2015) refers to the technique that uses well-trained networks to guide the training of another network.

Original: Typically, the well-trained network is named as the teacher network while the network to be trained is named as the student network.
Revised: Typically, the well-trained network is named as the teacher network while the network to be trained is named as the student network.

Original: For distillation, the predictions from the teacher network are leveraged and referred to as the soft labels (Balan et al., 2015; M ¨ uller et al., 2019).
Revised: For distillation, the predictions from the teacher network are leveraged and referred to as the soft labels (Balan et al., 2015; M ¨ uller et al., 2019).

Original: Soft labels generated by the teacher network have been proven effective in large-scale empirical studies (Liang et al., 2019; Tian et al., 2020; Zagoruyko & Komodakis, 2017; Romero et al., 2015) as well as recent theoretical studies (Phuong & Lampert, 2019).
Revised: Soft labels generated by the teacher network have been proven effective in large-scale empirical studies (Liang et al., 2019; Tian et al., 2020; Zagoruyko & Komodakis, 2017; Romero et al., 2015) as well as recent theoretical studies (Phuong & Lampert, 2019).

Original: However, the reason why soft labels are beneﬁcial to the student network is still not well explained.
Revised: However, the reason why soft labels are beneﬁcial to the student network is still not well explained.

Original: Giving a clear theoretical explanation is challenging:
Revised: Giving a clear theoretical explanation is challenging:

Original: The optimization details of a deep network with the common one-hot labels are still not well-studied (Nagarajan & Kolter, 2019), not to mention training with the soft labels.
Revised: The optimization details of a deep network with the common one-hot labels are still not well-studied (Nagarajan & Kolter, 2019), not to mention training with the soft labels.

Original: Nevertheless, two recent studies (M¨uller et al., 2019; Yuan et al., 2020) shed light on the intuitions about how the soft labels work.
Revised: Nevertheless, two recent studies (M¨uller et al., 2019; Yuan et al., 2020) shed light on the intuitions about how the soft labels work.

Original: Speciﬁcally, label smoothing, which is a special case of soft labels based training, is shown to regularize the activations of the penultimate layer to the network (M¨uller et al., 2019).
Revised: Speciﬁcally, label smoothing, which is a special case of soft labels based training, is shown to regularize the activations of the penultimate layer to the network (M¨uller et al., 2019).

Original: The regularization property of soft labels is further explored in (Yuan et al., 2020).
Revised: The regularization property of soft labels is further explored in (Yuan et al., 2020).

Original: They hypothesize that in KD, one main reason why the soft labels work is the regularization introduced by soft labels.
Revised: They hypothesize that in KD, one main reason why the soft labels work is the regularization introduced by soft labels.

Original: Based on the assumption, the authors design a teacher-free distillation method by turning the predictions of the student network into soft labels.
Revised: Based on the assumption, the authors design a teacher-free distillation method by turning the predictions of the student network into soft labels.

Original: Considering that soft labels are targets for distillation, the evidence of the regularization brought by soft labels drives us to rethink soft labels for KD:
Revised: Considering that soft labels are targets for distillation, the evidence of the regularization brought by soft labels drives us to rethink soft labels for KD:

Original: Soft labels are both supervisory signals and regularizers.
Revised: Soft labels are both supervisory signals and regularizers.

Original: Meanwhile, it is known that there is a tradeoff between ﬁtting the data and imposing regularizations, i.e., the bias-variance dilemma (Kohavi & Wolpert, 1996; Bishop, 2006), but it is unclear how bias and variance change for distillation with soft labels.
Revised: Meanwhile, it is known that there is a tradeoff between ﬁtting the data and imposing regularizations, i.e., the bias-variance dilemma (Kohavi & Wolpert, 1996; Bishop, 2006), but it is unclear how bias and variance change for distillation with soft labels.

Original: Since the bias-variance tradeoff is an important issue in statistical learning, we investigate whether the bias-variance tradeoff exists for soft labels and how the tradeoff affects distillation performance.
Revised: Since the bias-variance tradeoff is an important issue in statistical learning, we investigate whether the bias-variance tradeoff exists for soft labels and how the tradeoff affects distillation performance.

Original: We ﬁrst compare the bias and variance decomposition of direct training with that of distillation with soft labels, noticing that distillation results in a larger bias error and a smaller variance.
Revised: We ﬁrst compare the bias and variance decomposition of direct training with that of distillation with soft labels, noticing that distillation results in a larger bias error and a smaller variance.

Original: Then, we rewrite distillation loss into the form of a regularization loss adding the direct training loss.
Revised: Then, we rewrite distillation loss into the form of a regularization loss adding the direct training loss.

Original: Through inspecting the gradients of the two terms during training, we notice that for soft labels, the biasvariance tradeoff varies sample-wisely.
Revised: Through inspecting the gradients of the two terms during training, we notice that for soft labels, the biasvariance tradeoff varies sample-wisely.

Original: Moreover, by looking into a conclusion from (M¨uller et al., 2019), we observe that under the same temperature setting, the distillation performance is negatively associated with the number of some certain samples.
Revised: Moreover, by looking into a conclusion from (M¨uller et al., 2019), we observe that under the same temperature setting, the distillation performance is negatively associated with the number of some certain samples.

Original: These samples lead to bias increase and variance decrease and we name them as regularization samples.
Revised: These samples lead to bias increase and variance decrease and we name them as regularization samples.

Original: To investigate how regularization samples affect distillation, we ﬁrst examine if we can design ad hoc ﬁlters for soft labels to avoid training with regularization samples.
Revised: To investigate how regularization samples affect distillation, we ﬁrst examine if we can design ad hoc ﬁlters for soft labels to avoid training with regularization samples.

Original: But completely ﬁltering out regularization samples also deteriorates distillation performance, leading us to speculate that regularization samples are not well handled by standard KD.
Revised: But completely ﬁltering out regularization samples also deteriorates distillation performance, leading us to speculate that regularization samples are not well handled by standard KD.

Original: In the light of these ﬁndings, we propose weighted soft labels for distillation to handle the sample-wise bias-variance tradeoff, by adaptively assigning a lower weight to regularization samples and a larger weight to the others.
Revised: In the light of these ﬁndings, we propose weighted soft labels for distillation to handle the sample-wise bias-variance tradeoff, by adaptively assigning a lower weight to regularization samples and a larger weight to the others.

Original: To sum up, our contributions are:
Revised: To sum up, our contributions are:

Original: • For knowledge distillation, we analyze how the soft labels work from a perspective of biasvariance tradeoff.
Revised: • For knowledge distillation, we analyze how the soft labels work from a perspective of biasvariance tradeoff.

Original: • We discover that the bias-variance tradeoff varies sample-wisely.
Revised: • We discover that the bias-variance tradeoff varies sample-wisely.

Original: Also, we discover that if we ﬁx the distillation temperature, the number of regularization samples is negatively associated with the distillation performance.
Revised: Also, we discover that if we ﬁx the distillation temperature, the number of regularization samples is negatively associated with the distillation performance.

Original: • We design straightforward schemes to alleviate negative impacts from regularization samples and then propose the novel weighted soft labels for distillation.
Revised: • We design straightforward schemes to alleviate negative impacts from regularization samples and then propose the novel weighted soft labels for distillation.

Original: Experiments on large scale datasets validate the effectiveness of the proposed weighted soft labels.
Revised: Experiments on large scale datasets validate the effectiveness of the proposed weighted soft labels.

Original:  R ELATED WORKS
Revised:  R ELATED WORKS

Original: Knowledge distillation.
Revised: Knowledge distillation.

Original: Hinton et al.
Revised: Hinton et al.

Original: (2015) proposed to distill outputs from large and cumbersome models into smaller and faster models, which is named as knowledge distillation.
Revised: (2015) proposed to distill outputs from large and cumbersome models into smaller and faster models, which is named as knowledge distillation.

Original: The outputs for large networks are averaged and formulated as soft labels.
Revised: The outputs for large networks are averaged and formulated as soft labels.

Original: Also, other kinds of soft labels have been widely used for training deep neural networks (Szegedy et al., 2016; Pereyra et al., 2017).
Revised: Also, other kinds of soft labels have been widely used for training deep neural networks (Szegedy et al., 2016; Pereyra et al., 2017).

Original: Treating soft labels as regularizers were pointed out in (Hinton et al., 2015) since a lot of helpful information can be carried in soft labels.
Revised: Treating soft labels as regularizers were pointed out in (Hinton et al., 2015) since a lot of helpful information can be carried in soft labels.

Original: More recently, M ¨ uller et al.
Revised: More recently, M ¨ uller et al.

Original: (2019) showed the adverse effect of label smoothing upon distillation.
Revised: (2019) showed the adverse effect of label smoothing upon distillation.

Original: It is a thought-provoking discovery for the reason that both label smoothing and distillation are exploiting the regularization property behind soft labels.
Revised: It is a thought-provoking discovery for the reason that both label smoothing and distillation are exploiting the regularization property behind soft labels.

Original: Yuan et al.
Revised: Yuan et al.

Original: (2020) further investigated the regularization property of soft labels and then proposed a teacher free distillation scheme.
Revised: (2020) further investigated the regularization property of soft labels and then proposed a teacher free distillation scheme.

Original: Distillation loss.
Revised: Distillation loss.

Original: One of our main contributions is that we improve the distillation loss.
Revised: One of our main contributions is that we improve the distillation loss.

Original: For adaptively adjusting the distillation loss, Tang et al.
Revised: For adaptively adjusting the distillation loss, Tang et al.

Original: (2019) pays attention to hard-to-learn and hard-tomimic samples, and the latter is weighted based on the prediction gap between teacher and student.
Revised: (2019) pays attention to hard-to-learn and hard-tomimic samples, and the latter is weighted based on the prediction gap between teacher and student.

Original: However, it does not consider that the teacher may give an incorrect guide to the student, under which the prediction gap is still large and such a method may lead to the performance being hurt.
Revised: However, it does not consider that the teacher may give an incorrect guide to the student, under which the prediction gap is still large and such a method may lead to the performance being hurt.

Original: Saputra et al.
Revised: Saputra et al.

Original: (2019) transfers teacher’s guidance only on the samples where the performance of the teacher surpasses the student, while Wen et al.
Revised: (2019) transfers teacher’s guidance only on the samples where the performance of the teacher surpasses the student, while Wen et al.

Original: (2019) deals with the incorrect guidance by probability shifting strategy.
Revised: (2019) deals with the incorrect guidance by probability shifting strategy.

Original: Our approach is different from the above methods, in terms of motivations as well as the proposed solutions.
Revised: Our approach is different from the above methods, in terms of motivations as well as the proposed solutions.

Original: Bias-variance tradeoff.
Revised: Bias-variance tradeoff.

Original: Bias-variance tradeoff is a well-studied topic in machine learning (Kohavi & Wolpert, 1996; Domingos, 2000;  Bishop, 2006) and for neural networks (Geman et al., 1992; Neal et al., 2018; Belkin et al., 2019; Yang et al., 2020).
Revised: Bias-variance tradeoff is a well-studied topic in machine learning (Kohavi & Wolpert, 1996; Domingos, 2000; Valentini & Dietterich, 2004; Bishop, 2006) and for neural networks (Geman et al., 1992; Neal et al., 2018; Belkin et al., 2019; Yang et al., 2020).

Original: Existing methods are mainly concerned with the variance brought by the choice of network models.
Revised: Existing methods are mainly concerned with the variance brought by the choice of network models.

Original: Our perspective is different from the previous methods since we focus on the behavior of samples during training.
Revised: Our perspective is different from the previous methods since we focus on the behavior of samples during training.

Revised: In our work, based on the results from Heskes (1998), we present the decomposition of distillation loss, which is deﬁned by Kullback-Leibler divergence.

Original: Besides, our main contribution is not to study how to theoretically analyze the tradeoff, but how to adaptively tune the sample-wise tradeoff during training.
Revised: Besides, our main contribution is not to study how to theoretically analyze the tradeoff, but how to adaptively tune the sample-wise tradeoff during training.

Original:  B IAS - VARIANCE TRADEOFF FOR SOFT LABELS
Revised:  B IAS - VARIANCE TRADEOFF FOR SOFT LABELS

Original: [Figure]
Revised: [Figure]

Revised: Label

Revised: [Figure]

Original:  Soft labels play the role of supervisory signals and regularizations at the same time, which inspires us to rethink soft labels from the perspective of the bias-variance tradeoff.
Revised: A Bias Variance Soft labels play the role of supervisory signals and regularizations at the same time, which inspires us to rethink soft labels from the perspective of the bias-variance tradeoff.

Original: We begin our analysis with some mathematical descriptions.
Revised: We begin our analysis with some mathematical descriptions.

Original: For a sample x labeled as i -th class, let the ground-truth label be a one-hot vector y where y i = 1 and other entries are 0.
Revised: For a sample x labeled as i -th class, let the ground-truth label be a one-hot vector y where y i =and other entries are 0.

Original: Then for x and softmax output temperature τ , the soft label predicted by the teacher network is denoted as T ( x, τ ) and the output from the student is denoted as S ( x, τ ) .
Revised: Then for x and softmax output temperature τ , the soft label predicted by the teacher network is denoted as ˆ y tτ and the output from the student is denoted as ˆ y sτ .

Original: The soft label T ( x, τ ) is then used for training the student by the distillation loss,i.e.
Revised: The soft label ˆ y tτ is then used for training the student by the distillation loss,

Original: L kd = −
Revised: i

Original: [Equation]
Revised: [Equation]

Original: 2 (cid:80) k ˆ y tk,τ log ˆ y sk,τ , where ˆ y sk,τ , ˆ y tk,τ means the k -th element of the student’s output S ( x, τ ) and the teacher’s output T ( x, τ ) , respectively.
Revised: where ˆ y sk,τ , ˆ y tk,τ means the k -th element of the student’s output ˆ y sτ and the teacher’s output ˆ y tτ , respectively.

Original: With the above notations, the cross-entropy loss for training with one-hot labels is L ce = − y k log ˆ y sk, 1 .
Revised: With the above notations, the cross-entropy loss for training with one-hot labels is L ce = − y k log ˆ y sk, 1 .

Original: We now present the bias-variance decomposition for L ce and L kd , based on the deﬁnition and notations from Heskes (1998).
Revised: We now present the bias-variance decomposition for L ce and L kd , based on the deﬁnition and notations from Heskes (1998).

Original: First, we denote the  output distribution on  sample x of a student network trainedwith distillation as ˆ y kd and without distillation as ˆ y ce  . 
Revised: First, we denote the train dataset as D and the output distribution on a sample x of the network trained without distillation as ˆ y ce = f ce ( x ; D ) . For the network trained with distillation, the model also depends on the teacher network, so we deﬁne the output on x as ˆ y kd = f kd ( x ; D , T ) , where T is the selected teacher network.

Original: Then, let the averaged output of ˆ y kd and ˆ y ce be ¯ y kd and ¯ y ce , that is,
Revised: Then, let the averaged output of ˆ y kd and ˆ y ce be ¯ y kd and ¯ y ce , that is,

Original: [Equation]
Revised: [Equation]

Original: where Z ce , Z kd are two normalization constant.
Revised: where Z ce , Z kd are two normalization constant.

Original: Then  we have 
Revised: Then according to Heskes (1998), we have the following decomposition for the expected error on the sample x and y = t ( x ) is the ground truth label:

Original: [Equation]
Revised: [Equation]

Original: where D KL is the Kullback-Leibler divergence.
Revised: where D KL is the Kullback-Leibler divergence.

Original: For direct training, the variance term is caused by different training sets for the task. For training with distillation, the variance E [ D KL (¯ y kd , ˆ y kd )] is caused by both different training sets and different teacher networks.
Revised: Next, we analyze the bias-variance decomposition of L kd . As mentioned above, when training with soft labels, extra randomness is introduced for the selection of a teacher network.

Original: In Fig.
Revised: In Fig.

Original: 1, we illustrate the  bias and variance  of difference teacher networks.
Revised: 1, we illustrate the corresponding bias and variance for the selection process of a set of soft labels, which are generated by a teacher network.

Revised: In this case, a high variance model indicates the model (grey point) is closer to the one-hot trained model (black point), while a low variance model indicates that the model is closer to other possible models trained with soft labels (red points).

Original: Although for KD there are more sources causing variance, the  variance brought by L kd is not necessarily higher than L ce .
Revised: Although for KD there are more sources introducing randomness, the overall variance brought by L kd is not necessarily higher than L ce .

Original: In fact, existing empirical results strongly suggest that the overall variance is smaller with KD.
Revised: In fact, existing empirical results strongly suggest that the overall variance is smaller with KD.

Original: For example, students trained with soft labels are better calibrated than one-hot baselines (M¨uller et al., 2019) and KD makes the predictions of students more consistent when facing adversarial noise (Papernot et al., 2016).
Revised: For example, students trained with soft labels are better calibrated than one-hot baselines (M¨uller et al., 2019) and KD makes the predictions of students more consistent when facing adversarial noise (Papernot et al., 2016).

Original: Here, we present these empirical evidence as an assumption:
Revised: Here, we present these empirical evidence as an assumption:

Original: Assumption 1 The variance brought by KD is smaller than direct training, that is, E
Revised: Assumption 1 The variance brought by KD is smaller than direct training, that is, E

Original: [Equation]
Revised: D , T [ D KL (¯ y kd , ˆ y kd )] (cid:54) E D [ D KL (¯ y ce , ˆ y ce )] .

Original: Similar to Eq.
Revised: Similar to Eq.

Original: (2), we write the decomposition for L kd as
Revised: (2), we write the decomposition for L kd as

Original: [Equation]
Revised: [Equation]

Original: An observation here is that ¯ y ce converges to one-hot labels while ¯ y kd converges to soft labels, so ¯ y ce is closer to the one-hot ground-truth distribution y than ¯ y kd , i.e., E y (cid:104) 
Revised: An observation here is that ¯ y ce converges to one-hot labels while ¯ y kd converges to soft labels, so ¯ y ce is closer to the one-hot ground-truth distribution y than ¯ y kd , i.e., E x (cid:104) y log (cid:16)

Original: [Equation] (cid:17)(cid:105) (cid:62) 0 .
Revised: ¯ y ce ¯ y kd (cid:17)(cid:105) (cid:62) 0 .

Original: If  we rewrite L kd as L kd =
Revised: If [Table] we rewrite L kd as L kd =

Original: L kd − L ce + L ce , then L kd − L ce causes that the bias increases by E y (cid:104)  log (cid:16) ¯ y ce ¯ y kd (cid:17)(cid:105) and the variance decreases by E  [ D KL (¯ y ce , ˆ y ce )]− E
Revised: L kd − L ce + L ce , then L kd − L ce causes that the bias increases by E x (cid:104) y log (cid:16) ¯ y ce ¯ y kd (cid:17)(cid:105) and the variance decreases by E D [ D KL (¯ y ce , ˆ y ce )]

Original:  [ D KL (¯ y kd , ˆ y kd )] .
Revised: − E D , T [ D KL (¯ y kd , ˆ y kd )] .

Original: From the above analysis, we separate L kd into two terms, and L kd − L ce leads to variance reduction, and L ce leads to bias reduction.
Revised: From the above analysis, we separate L kd into two terms, and L kd − L ce leads to variance reduction, and L ce leads to bias reduction.

Original: In the following sections, we ﬁrst analyze how L kd − L ce links to the bias-variance tradeoff during training.
Revised: In the following sections, we ﬁrst analyze how L kd − L ce links to the bias-variance tradeoff during training.

Original: Then we analyze the changes in the relative importance between bias reduction and variance reduction during training with soft labels.
Revised: Then we analyze the changes in the relative importance between bias reduction and variance reduction during training with soft labels.

Original: 3. T HE BIAS - VARIANCE TRADEOFF DURING TRAINING
Revised: 3. T HE BIAS - VARIANCE TRADEOFF DURING TRAINING

Original: It is known that bias reduction and variance reduction are often in conﬂict and we cannot minimize bias and variance together.
Revised: It is known that bias reduction and variance reduction are often in conﬂict and we cannot minimize bias and variance together.

Original: However, if we consider the change of bias and variance during the[Table] training process, the importance of tuning the tradeoff also changes during training.
Revised: However, if we consider the change of bias and variance during the training process, the importance of tuning the tradeoff also changes during training.

Original: Speciﬁcally, shortly after the training of the network starts, the bias error dominates the total error and the variance is less important.
Revised: Speciﬁcally, shortly after the training of the network starts, the bias error dominates the total error and the variance is less important.

Original: As training goes on, gradients of reducing the bias error (induced by L ce ) and reducing the variance (induced by L kd − L ce ) can be of the same scale for some samples, then we need to balance the tradeoff because reducing one term is likely to increase another one.
Revised: As training goes on, gradients of reducing the bias error (induced by L ce ) and reducing the variance (induced by L kd − L ce ) can be of the same scale for some samples, then we need to balance the tradeoff because reducing one term is likely to increase another one.

Original: Therefore for soft labels, we need to handle the bias-variance tradeoff in a sample-wise manner and take the training process into consideration.
Revised: Therefore for soft labels, we need to handle the bias-variance tradeoff in a sample-wise manner and take the training process into consideration.

Original: To study the bias-variance tradeoff during training, we consider the gradients of bias and variance reduction.
Revised: To study the bias-variance tradeoff during training, we consider the gradients of bias and variance reduction.

Original: Let z be the logits output of the student on input x and z
Revised: Let z be the logits output of the student on input x and z

Original: i is i -th element of it, then we are interested in ∂ ( L kd − L ce )
Revised: i is i -th element of it, then we are interested in ∂ ( L kd − L ce )

Original: ∂z i .
Revised: ∂z i .

Original: For simplifying analysis, we are concerned with the gradients on the ground-truth related logit, that is, the sample x is labeled as i -th class.
Revised: For simplifying analysis, we are concerned with the gradients on the ground-truth related logit, that is, the sample x is labeled as i -th class.

Original: Mathematically, for the gradients of variance reduction, we have
Revised: Mathematically, for the gradients of variance reduction, we have

Original: [Equation]
Revised: [Equation]

Original: where  y ti,τ denotes the i -th element of the teacher’s prediction, i.e., T ( x, τ ) .
Revised: where ˆ y ti,τ denotes the i -th element of the teacher’s prediction, i.e., ˆ y tτ .

Original: The term L kd − L ce is easy to understand when τ = 1 since the gradient now becomes y i − ˆ y ti, 1 .
Revised: The term L kd − L ce is easy to understand when τ = 1 since the gradient now becomes y i − ˆ y ti, 1 .

Original: Meanwhile, for the bias reduction, we have ∂L ce ∂z i
Revised: Meanwhile, for the bias reduction, we have ∂L ce ∂z i

Original: = ˆ y si, 1 − y i , so ∂L ce ∂z i and ∂ ( L kd − L ce ) ∂z i always have different signs, leading to a tradeoff.
Revised: = ˆ y si, 1 − y i , so ∂L ce ∂z i and ∂ ( L kd − L ce ) ∂z i always have different signs, leading to a tradeoff.

Original: If ∂L ce ∂z i is much higher than ∂ ( L kd − L ce )
Revised: If ∂L ce ∂z i is much higher than ∂ ( L kd − L ce )

Original: ∂z i , the bias reduction dominates the overall optimization direction.
Revised: ∂z i , the bias reduction dominates the overall optimization direction.

Original: Instead, if ∂ ( L kd − L ce ) ∂z i becomes higher, the sample is used for variance reduction.
Revised: Instead, if ∂ ( L kd − L ce ) ∂z i becomes higher, the sample is used for variance reduction.

Original: Interestingly, we discover that under a ﬁxed distillation temperature, the ﬁnal performance is worse when more training samples are used for variance reduction, which will be introduced in the next section. 
Revised: Interestingly, we discover that under a ﬁxed distillation temperature, the ﬁnal performance is worse when more training samples are used for variance reduction, which will be introduced in the next section. [Figure]

Original: 3. R EGULARIZATION SAMPLES
Revised: 3. R EGULARIZATION SAMPLES

Original: Our analysis starts with a conclusion from M¨uller et al.
Revised: Our analysis starts with a conclusion from M ¨ uller et al.

Original: (2019): if a teacher network is trained with label smoothing, knowledge distillation into a student network is much less effective.
Revised: (2019): if a teacher network is trained with label smoothing, knowledge distillation into a student network is much less effective.

Original: Inspired by the phenomenon, we gather the impact of bias and variance during training with different distillation settings.
Revised: Inspired by the phenomenon, we gather the impact of bias and variance during training with different distillation settings.

Original: Let a = ∂L ce ∂z i and b = ∂ ( L kd − L ce )
Revised: Let a = ∂L ce ∂z i and b = ∂ ( L kd − L ce )

Original: ∂z i , then as introduced before, we use a and b to represent the impact of bias and variance, respectively.
Revised: ∂z i , then as introduced before, we use a and b to represent the impact of bias and variance, respectively.

Original: If we have b > a for a sample, we name the sample as a regularization sample since the variance dominates the optimization direction.
Revised: If we have b > a for a sample, we name the sample as a regularization sample since the variance dominates the optimization direction.

Original: From the collected data, we ﬁnd that the number of regularization samples is closely related to distillation performance.
Revised: From the collected data, we ﬁnd that the number of regularization samples is closely related to distillation performance.

Original: In Tab. 1, we present the count of regularization samples for a student network trained by knowledge distillation.
Revised: In Tab. 1, we present the count of regularization samples for a student network trained by knowledge distillation.

Original: For distillation with a temperature higher than 1, which is the common setting, we observe that if the teacher network is trained with label smoothing, more samples will be involved in[Figure] [Figure] variance reduction.
Revised: For distillation with a temperature higher than 1, which is the common setting, we observe that if the teacher network is trained with label smoothing, more samples will be involved in variance reduction.

Original: Also, distillation from a teacher trained with label smoothing performs worse, which is consistent with M ¨ uller et al.
Revised: Also, distillation from a teacher trained with label smoothing performs worse, which is consistent with M ¨ uller et al.

Original: Therefore, we conclude that for distillation with soft labels, the regularization samples during training affect the ﬁnal distillation performance.
Revised: Therefore, we conclude that for distillation with soft labels, the regularization samples during training affect the ﬁnal distillation performance.

Original: Moreover, we plot the number of regularization samples with respect to different training epochs in
Revised: Moreover, we plot the number of regularization samples with respect to different training epochs in

Original: Fig.
Revised: Fig.

Original: As demonstrated in the plots, the number of such samples increases much faster when using the teacher trained with label smoothing for distillation.
Revised: As demonstrated in the plots, the number of such samples increases much faster when using the teacher trained with label smoothing for distillation.

Original: For regularization samples, the gap of their number between with and without label smoothing becomes larger for more training epochs.
Revised: For regularization samples, the gap of their number between with and without label smoothing becomes larger for more training epochs.

Original: These observations verify our motivation that the bias-variance tradeoff varies sample-wisely and evolves during the training process.
Revised: These observations verify our motivation that the bias-variance tradeoff varies sample-wisely and evolves during the training process.

Original: From the above results [Figure] we conclude that bias-variance tradeoff for soft labels varies sample-wisely, therefore the strategy for tuning the tradeoff should also be sample-wise.
Revised: From the above results, we conclude that bias-variance tradeoff for soft labels varies sample-wisely, therefore the strategy for tuning the tradeoff should also be sample-wise.

Original: In the next section, we set up ad hoc ﬁlters for soft labels and further investigate how regularization samples affect distillation.
Revised: In the next section, we set up ad hoc ﬁlters for soft labels and further investigate how regularization samples affect distillation.

Original: 3. H OW REGULARIZATION SAMPLES AFFECT DISTILLATION
Revised: 3. H OW REGULARIZATION SAMPLES AFFECT DISTILLATION

Original: The results presented in the last section suggest that we should avoid training with regularization samples.
Revised: The results presented in the last section suggest that we should avoid training with regularization samples.

Original: Hence, we design two straightforward solutions and then ﬁnd that totally ﬁltering out regularization samples deteriorates the distillation performance.
Revised: Hence, we design two straightforward solutions and then ﬁnd that totally ﬁltering out regularization samples deteriorates the distillation performance.

Original: The ﬁrst experiment we conduct is to manually resolve the conﬂicting gradient on the label related logit, as deﬁned in section 3.2.
Revised: The ﬁrst experiment we conduct is to manually resolve the conﬂicting gradient on the label related logit, as deﬁned in section 3.2.

Original: Speciﬁcally, we apply a mask to the distillation loss L kd such that ∂L kd ∂z i = 0 where i is the label.
Revised: Speciﬁcally, we apply a mask to the distillation loss L kd such that ∂L kd ∂z i = 0 where i is the label.

Original: Consequently, the loss for this sample now becomes L ∗ kd = 
Revised: Consequently, the loss for this sample now becomes L ∗ kd = [Equation] k

Original: [Equation] 
Revised: [Equation] =

Original:  ˆ y sk,τ .
Revised: i ˆ y tk,τ log ˆ y sk,τ .

Original: The motivation behind the masked distillation loss is that we only transfer the knowledge of resemblances among the labels.
Revised: The motivation behind the masked distillation loss is that we only transfer the knowledge of resemblances among the labels.

Original: Another experiment is to ﬁgure out what role in distillation those regularization samples will play.
Revised: Another experiment is to ﬁgure out what role in distillation those regularization samples will play.

Original: To investigate this, we carry out knowledge distillation on two subsets of samples: 1) L kd is not valid on regularization samples, and 2) L kd is valid only on regularization samples. 
Revised: To investigate this, we carry out knowledge distillation on two subsets of samples: 1) L kd is not valid on regularization samples, and 2) L kd is valid only on regularization samples. [Figure] [Table]

Original:  The results of the two experiments are presented in Tab. 2.
Revised: [Table] The results of the two experiments are presented in Tab. 2.

Original: We can observe that all of the three approaches are not as good as the baseline knowledge distillation performance, but are better than the direct training baseline.
Revised: We can observe that all of the three approaches are not as good as the baseline knowledge distillation performance, but are better than the direct training baseline.

Original: First, since masking L kd loss on the label related logit results in worse performance compared to standard KD, we cannot resolve the tradeoff by applying a mask on the ground truth related logit.
Revised: First, since masking L kd loss on the label related logit results in worse performance compared to standard KD, we cannot resolve the tradeoff by applying a mask on the ground truth related logit.

Original: Then, from the second experiment, we can see that ﬁltering out regulariza- [Table] [Table] tion samples deteriorates the distillation performance.
Revised: Then, from the second experiment, we can see that ﬁltering out regularization samples deteriorates the distillation performance.

Original: Moreover, the result of the third experiment is higher than the direct training baseline, indicating that regularization samples are still valuable for distillation.
Revised: Moreover, the result of the third experiment is higher than the direct training baseline, indicating that regularization samples are still valuable for distillation.

Original: The above results motivate us to think that regularization samples are not fully exploited by standard KD and we can tune the tradeoff to fulﬁll the potential of regularization samples.
Revised: The above results motivate us to think that regularization samples are not fully exploited by standard KD and we can tune the tradeoff to fulﬁll the potential of regularization samples.

Original:  W EIGHTED SOFT LABELS
Revised:  W EIGHTED SOFT LABELS

Original: From the last section

Original: [Table] we realize that the bias-variance tradeoff varies sample-wisely during training and under ﬁxed distillation settings [Table] the number of regularization samples is negatively associated with the ﬁnal distillation performance.
Revised: From the last section, we realize that the bias-variance tradeoff varies sample-wisely during training and under ﬁxed distillation settings, the number of regularization samples is negatively associated with the ﬁnal distillation performance.

Original: Yet [Table] discarding regularization samples deteriorates distillation performance and distilling knowledge from these samples is better than the direct training baseline.
Revised: Yet, discarding regularization samples deteriorates distillation performance and distilling knowledge from these samples is better than the direct training baseline.

Original: The above evidence inspires us to lower the weight of regularization samples
Revised: The above evidence inspires us to lower the weight of regularization samples.

Original: [Table]

Original: Recall that regularization samples are deﬁned by the relative value of a and b , we propose to assign importance weight to a sample according to a and b .
Revised: Recall that regularization samples are deﬁned by the relative value of a and b , we propose to assign importance weight to a sample according to a and b .

Original: However, since L kd is computed with the hyperparameter temperature, a and b are correlated with the temperature and thus bring difﬁculty to tuning the hyperparameter.
Revised: However, since L kd is computed with the hyperparameter temperature, a and b are correlated with the temperature and thus bring difﬁculty to tuning the hyperparameter.

Original: To make the weighting scheme independent of the temperature hyperparameter, we compare a and b with temperature τ = 1 .
Revised: To make the weighting scheme independent of the temperature hyperparameter, we compare a and b with temperature τ = 1 .

Original: Note that when τ = 1 , a = ˆ y si, 1 − y i and b =
Revised: Note that when τ = 1 , a = ˆ y si, 1 − y i and b =

Original: y i − ˆ y ti, 1 , so we compare ˆ y si, 1 and ˆ y ti, 1 instead.
Revised: y i − ˆ y ti, 1 , so we compare ˆ y si, 1 and ˆ y ti, 1 instead.

Original: Finally, in the light of previous works that assign sample-wise weights (Lin et al., 2017; Tang et al., 2019), we propose weighted soft labels for knowledge distillation, which is formally deﬁned as
Revised: Finally, in the light of previous works that assign sample-wise weights (Lin et al., 2017; Tang et al., 2019), we propose weighted soft labels for knowledge distillation, which is formally deﬁned as

Original: [Equation]
Revised: [Equation]

Original: where i is the ground truth class of the sample.
Revised: where i is the ground truth class of the sample.

Original: The above equation means that a weighting factor is assigned to each sample’s L kd according to the predictions of the teacher and the student.
Revised: The above equation means that a weighting factor is assigned to each sample’s L kd according to the predictions of the teacher and the student.

Original: In this way, if compared to the teacher, a student network is relatively better trained on a sample, we have ˆ y si, 1 > ˆ y ti, 1 , then a smaller weight is assigned to this sample.
Revised: In this way, if compared to the teacher, a student network is relatively better trained on a sample, we have ˆ y si, 1 > ˆ y ti, 1 , then a smaller weight is assigned to this sample.

Original: In Fig.
Revised: In Fig.

Original: 3, the whole computational graph of knowledge distillation with the proposed weighted soft labels is demonstrated.
Revised: 3, the whole computational graph of knowledge distillation with the proposed weighted soft labels is demonstrated.

Original: Finally,  we add L wsl and L ce together to supervise the network, i.e., L total =
Revised: Finally, [Table] we add L wsl and L ce together to supervise the network, i.e., L total =

Original: L ce + αL wsl , where α is a balancing hyperparameter.
Revised: L ce + αL wsl , where α is a balancing hyperparameter.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: To evaluate our weighted soft labels comprehensively, we ﬁrst conduct experiments with various teacher-student pair settings on CIFAR-100 (Krizhevsky et al., 2009).
Revised: To evaluate our weighted soft labels comprehensively, we ﬁrst conduct experiments with various teacher-student pair settings on CIFAR-100 (Krizhevsky et al., 2009).

Original: Next, we compare our method with current state-of-the-art distillation methods on ImageNet (Deng et al., 2009).
Revised: Next, we compare our method with current state-of-the-art distillation methods on ImageNet (Deng et al., 2009).

Original: To validate the effectiveness of our method in terms of handling the bias-variance tradeoff, we conduct ablation experiments by applying weighted soft labels on different subsets.
Revised: To validate the effectiveness of our method in terms of handling the bias-variance tradeoff, we conduct ablation experiments by applying weighted soft labels on different subsets.

Original: 5. D ATASET AND HYPERPARAMETER SETTINGS
Revised: 5. D ATASET AND HYPERPARAMETER SETTINGS

Original: The datasets used in our experiments are CIFAR-100 (Krizhevsky et al., 2009) and ImageNet (Deng et al., 2009).
Revised: The datasets used in our experiments are CIFAR-100 (Krizhevsky et al., 2009) and ImageNet (Deng et al., 2009).

Original: CIFAR-100 contains 50K training and 10K test images of size 32 × 32 .ImageNet [Table] [Table]
Revised: CIFAR-100 contains 50K training and 10K test images of size 32 × 32 .

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table] [Table] [Table] contains 1.2 million training and 50K validation images.
Revised: ImageNet contains 1.2 million training and 50K validation images.

Original: Except the loss function, training settings like learning rate or training epochs are the same with Tian et al.
Revised: Except the loss function, training settings like learning rate or training epochs are the same with Tian et al.

Original: (2020) for CIFAR-100 and Heo et al.
Revised: (2020) for CIFAR-100 and Heo et al.

Original: for ImageNet.
Revised: for ImageNet.

Original: For distillation, we set the temperature τ = 4 for CIFAR and τ = 2 for
Revised: For distillation, we set the temperature τ = 4 for CIFAR and τ = 2 for

Original: ImageNet.
Revised: ImageNet.

Original: For loss function, we set α = 2 .
Revised: For loss function, we set α = 2 .

Original: 25 for distillation on CIFAR and α = 2 .
Revised: 25 for distillation on CIFAR and α = 2 .

Original: 5 for ImageNet via grid search.
Revised: 5 for ImageNet via grid search.

Original: The teacher network is well-trained previously and ﬁxed during training.
Revised: The teacher network is well-trained previously and ﬁxed during training.

Original: For comparison, the following recent state-of-the-art methods are chosen:
Revised: For comparison, the following recent state-of-the-art methods are chosen:

Original: FitNet (Romero et al., 2015), AT (Zagoruyko & Komodakis, 2017), SP (Tung & Mori, 2019), CC (Peng et al., 2019), VID (Ahn et al., 2019), RKD (Park et al., 2019), PKT (Passalis & Tefas, 2018), AB (Heo et al., 2019),
Revised: FitNet (Romero et al., 2015), AT (Zagoruyko & Komodakis, 2017), SP (Tung & Mori, 2019), CC (Peng et al., 2019), VID (Ahn et al., 2019), RKD (Park et al., 2019), PKT (Passalis & Tefas, 2018), AB (Heo et al., 2019),

Original: FT (Kim et al., 2018), FSP (Yim et al., 2017), NST (Huang & Wang, 2017), Overhaul (Heo et al., 2019) and CRD (Tian et al., 2020).
Revised: FT (Kim et al., 2018), FSP (Yim et al., 2017), NST (Huang & Wang, 2017), Overhaul (Heo et al., 2019) and CRD (Tian et al., 2020).

Original: 5. M ODEL COMPRESSIONResults on CIFAR
Revised: 5. M ODEL COMPRESSION

Original:  In Tab. 3, we present the Top-1 classiﬁcation accuracy of our method and comparison methods.
Revised: Results on CIFAR-100 In Tab. 3, we present the Top-1 classiﬁcation accuracy of our method and comparison methods.

Original: The results of comparison methods are quoted from Tian et al.
Revised: The results of comparison methods are quoted from Tian et al.

Original: Teacher-student pairs of the same and different architecture styles are considered.
Revised: Teacher-student pairs of the same and different architecture styles are considered.

Original: For pairs of same architecture style, we use wide residual networks (Zagoruyko & Komodakis, 2017) and residual networks (He et al., 2016).
Revised: For pairs of same architecture style, we use wide residual networks (Zagoruyko & Komodakis, 2017) and residual networks (He et al., 2016).

Original: For pairs of different architecture style, residual networks and ShufﬂeNet  (Zhang et al., 2018) pairs are chosen for experiments.
Revised: For pairs of different architecture style, residual networks and ShufﬂeNet [Table] [Table] (Zhang et al., 2018) pairs are chosen for experiments.

Original: As shown in the table, for distillation with both same and different architecture style, our method reached new state-of-the-art results.
Revised: As shown in the table, for distillation with both same and different architecture style, our method reached new state-of-the-art results.

Original: Speciﬁcally, our method outperforms standard KD by a large margin, which veriﬁes the effectiveness of our method.
Revised: Speciﬁcally, our method outperforms standard KD by a large margin, which veriﬁes the effectiveness of our method.

Original: Results on ImageNet In Tab. 4, we compare our method with current SOTA methods on ImageNet.
Revised: Results on ImageNet In Tab. 4, we compare our method with current SOTA methods on ImageNet.

Original: Note that for the ResNet34 → ResNet-18 distillation setting, the result of CRD is trained 10 more extra epochs while ours is the same as other methods.
Revised: Note that for the ResNet34 → ResNet-18 distillation setting, the result of CRD is trained 10 more extra epochs while ours is the same as other methods.

Original: For ResNet-50 → MobileNet-v distillation setting, NST and FSP are not chosen for comparison as the two methods require too large GPU memories, so we include the accuracy of FT and AB reported in Heo et al.
Revised: For ResNet-50 → MobileNet-v distillation setting, NST and FSP are not chosen for comparison as the two methods require too large GPU memories, so we include the accuracy of FT and AB reported in Heo et al.

Original: for comparison.
Revised: for comparison.

Original: Our results outperform all the existing methods, verifying the practical value of our method.
Revised: Our results outperform all the existing methods, verifying the practical value of our method.

Original: 5. A BLATION STUDIES
Revised: 5. A BLATION STUDIES

Original: [Table] [Table]
Revised: with soft labels and our weighted soft labels. RS means regularization samples.

Original: [Table]
Revised: [Table]

Original: Weighted soft labels on different subsets.
Revised: Weighted soft labels on different subsets.

Original: Recall that we propose weighted soft labels for tuning samplewise bias-variance tradeoff, it is still unclear whether the improvements come from a well-handled samplewise bias-variance tradeoff.
Revised: Recall that we propose weighted soft labels for tuning samplewise bias-variance tradeoff, it is still unclear whether the improvements come from a well-handled samplewise bias-variance tradeoff.

Original: To investigate this issue, we compare the performance gain of weighted soft labels on different training subsets.
Revised: To investigate this issue, we compare the performance gain of weighted soft labels on different training subsets.

Original: Similar to the settings used in Tab. 2, we apply weighted soft labels on two different subsets: only the regularization samples and excluding regularization samples.
Revised: Similar to the settings used in Tab. 2, we apply weighted soft labels on two different subsets: only the regularization samples and excluding regularization samples.

Original: In Tab. 5, we show the results on subsets of only regularization samples and excluding regularization samples.
Revised: In Tab. 5, we show the results on subsets of only regularization samples and excluding regularization samples.

Original: From the signiﬁcant improvements, we can see that our method can not only improve performance on the RS subset, the improvements on excluding RS subset is also signiﬁcant.
Revised: From the signiﬁcant improvements, we can see that our method can not only improve performance on the RS subset, the improvements on excluding RS subset is also signiﬁcant.

Original: We conclude that weighted soft labels can tune sample-wise bias-variance tradeoff globally and lead to an improved distillation performance. 
Revised: We conclude that weighted soft labels can tune sample-wise bias-variance tradeoff globally and lead to an improved distillation performance. [Table]

Original: [Table]
Revised: [Table]

Original: Distillation with label smoothing trained teacher.
Revised: Distillation with label smoothing trained teacher.

Original: Our exploration of bias-variance tradeoff starts with the conclusion made in M¨uller et al.
Revised: Our exploration of bias-variance tradeoff starts with the conclusion made in M¨uller et al.

Original: (2019): a teacher network trained with the label smoothing trick is less effective for distillation.
Revised: (2019): a teacher network trained with the label smoothing trick is less effective for distillation.

Original: It is worthwhile to study whether the conclusion remains true for distillation with our weighted soft labels.
Revised: It is worthwhile to study whether the conclusion remains true for distillation with our weighted soft labels.

Original: As discussed before, we hold the opinion that too many regularization samples make the distillation less effective.
Revised: As discussed before, we hold the opinion that too many regularization samples make the distillation less effective.

Original: Since our weighted soft label is proposed to mitigate the negative effects of the regularization samples, with the same settings from Tab. 1, we conduct comparison experiments in Tab. 6 to see if the negative effects still exist.
Revised: Since our weighted soft label is proposed to mitigate the negative effects of the regularization samples, with the same settings from Tab. 1, we conduct comparison experiments in Tab. 6 to see if the negative effects still exist.

Original: It is evident that weighted soft labels signiﬁcantly improve the distillation performance, especially for distillation from the teacher trained with label smoothing.
Revised: It is evident that weighted soft labels signiﬁcantly improve the distillation performance, especially for distillation from the teacher trained with label smoothing.

Original: Besides, using the teacher trained with label smoothing still performs worse than that without label smoothing, which again veriﬁes the conclusion drawn by M¨uller et al.
Revised: Besides, using the teacher trained with label smoothing still performs worse than that without label smoothing, which again veriﬁes the conclusion drawn by M¨uller et al.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Recent studies (M¨uller et al., 2019; Yuan et al., 2020) point out that one important reason behind the effectiveness of distillation is the regularization effect brought by being soft.
Revised: Recent studies (M¨uller et al., 2019; Yuan et al., 2020) point out that one important reason behind the effectiveness of distillation is the regularization effect brought by being soft.

Original: In this paper, we rethink the soft labels for distillation from a bias-variance tradeoff perspective.
Revised: In this paper, we rethink the soft labels for distillation from a bias-variance tradeoff perspective.

Original: The tradeoff varies sample-wisely and we propose weighted soft labels to handle the tradeoff, of which the effectiveness is veriﬁed with experiments on standard evaluation benchmarks.
Revised: The tradeoff varies sample-wisely and we propose weighted soft labels to handle the tradeoff, of which the effectiveness is veriﬁed with experiments on standard evaluation benchmarks

Original: A A PPENDIX A. V ISUALIZATION OF THE RESEMBLANCES INTRODUCED BY SOFT LABEL REGULARIZER
Revised: A A PPENDIX A. V ISUALIZATION OF THE RESEMBLANCES INTRODUCED BY SOFT LABEL REGULARIZER

Original: In Sec 3. we propose to study ∂ ( L kd − L ce )
Revised: In Sec 3.1, we propose to study ∂ ( L kd − L ce )

Original: ∂z i during the training process.
Revised: ∂z i during the training process.

Original: When τ = 1 , we show that ∂ ( L kd − L ce ) ∂z
Revised: When τ = 1 , we show that ∂ ( L kd − L ce ) ∂z

Original: i equals y i −
Revised: i equals y i −

Original: ˆ y ti, 1 .
Revised: ˆ y ti, 1 .

Original: As y ti,τ is the output from the teacher network and computed by a linear mapping of the activations in the teacher’s penultimate layer, the regularization indicates that the student should follow the learned the resemblances between classes (Hinton et al., 2015;
Revised: As y ti,τ is the output from the teacher network and computed by a linear mapping of the activations in the teacher’s penultimate layer, the regularization indicates that the student should follow the learned the resemblances between classes (Hinton et al., 2015;

Original: M ¨ uller et al., 2019).
Revised: M¨uller et al., 2019).

Original: Still, two questions are unclear: 1) what the resemblances are and 2) whether the regularization still indicates resemblances if τ is set to 4, a widely adopted hyperparameter (Tian et al., 2020).
Revised: Still, two questions are unclear: 1) what the resemblances are and 2) whether the regularization still indicates resemblances if τ is set to 4, a widely adopted hyperparameter (Tian et al., 2020).

Original: Towards answering the questions, we visualize the value of gradient vector ∂ ( L kd − L ce )
Revised: Towards answering the questions, we visualize the value of gradient vector ∂ ( L kd − L ce )

Original: ∂
Revised: ∂

Original: z concerning each class.
Revised: z concerning each class.

Original: Speciﬁcally, on ImageNet (Deng et al., 2009) training set and τ = 4 , we calculate the average value of ∂ ( L kd − L ce ) ∂
Revised: Speciﬁcally, on ImageNet (Deng et al., 2009) training set and τ = 4 , we calculate the average value of ∂ ( L kd − L ce ) ∂

Original: z for each class.
Revised: z for each class.

Original: Let M be the matrix of values with the ij -th entry
Revised: Let M be the matrix of values with the ij -th entry

Original: M ij means averaged ∂ ( L kd − L ce )
Revised: M ij means averaged ∂ ( L kd − L ce )

Original: ∂z i for class j .
Revised: ∂z i for class j .

Original: Since (cid:80) i M ij = 0 , diagonal elements are ignored for visualization.
Revised: Since (cid:80) i M ij = 0 , diagonal elements are ignored for visualization.

Original: The results are visualized in Fig.
Revised: The results are visualized in Fig.

Original: We ﬁnd that plotting the common correlation matrix heatmap is ambiguous, because the matrix to be visualized is of large size ( 1000 × 1000 ) with a large variance.
Revised: We ﬁnd that plotting the common correlation matrix heatmap is ambiguous, because the matrix to be visualized is of large size ( 1000 × 1000 ) with a large variance.

Original: By treating each entry M ij as a vertex and then constructing a mesh for the matrix, we apply subdivision (Loop, 1987) to the mesh for smoothing the extreme points and  ﬁnally rendering the mesh by ray-tracing package PlotOptiX. We can observe the several facts from the ﬁgures: 1) Comparing the sub-ﬁgure (a) and (b), we can see that for distillation resemblances implied by regularizers are similar across different teacher-student pairs.
Revised: By treating each entry M ij as a vertex and then constructing a mesh for the matrix, we apply subdivision (Loop, 1987) to the mesh for smoothing the extreme points and [Figure] ﬁnally rendering the mesh by ray-tracing package PlotOptiX. We can observe the several facts from the ﬁgures: 1) Comparing the sub-ﬁgure (a) and (b), we can see that for distillation resemblances implied by regularizers are similar across different teacher-student pairs.

Original: 2) Comparing (ab) with (cd), we can see that the resemblances are consistent with the semantic similarity of image class names.
Revised: 2) Comparing (ab) with (cd), we can see that the resemblances are consistent with the semantic similarity of image class names.

Original: In a word, for τ = 4 , the variance reduction brought by soft labels still implies resemblances among labels, which are consistent with the semantic distance of class names.
Revised: In a word, for τ = 4 , the variance reduction brought by soft labels still implies resemblances among labels, which are consistent with the semantic distance of class names.

Original: In the next section, we will analyze how bias-variance tradeoff changes when training with soft labels.
Revised: In the next section, we will analyze how bias-variance tradeoff changes when training with soft labels.

Revised: A. I NTERMEDIATE STATES BETWEEN EXCLUDING AND ONLY ON REGULARIZATION SAMPLES

Revised: To further investigate the phenomenon about regularization samples, we conduct experiments to show the intermediate states between excluding and only on regularization samples.

Revised: Two settings are considered here: First, we gradually exclude regularization samples during training, from excluding all regularization samples to excluding 25% regularization samples; Second, we keep all regularization samples and then gradually add non-regularization samples.

Original: [Figure]
Revised: Since we judge a sample is regularization or not according to the training loss, we cannot pre-deﬁne a sample set such that a certain percentage samples are kept or dropped. Therefore, we propose to conduct these experiments by assigning a probability to whether backward the loss computed with regularization samples. For example, if during training, a sample is marked as regularization sample according to the value of a and b , we backward the loss of this sample by a probability p = 0 . In this way, we can get the performance of excluding 75% regularization samples.

Revised: In Tab. 7, we ﬁrst present result with KD in (a) and then present result with weighted soft labels applied in (b).

Revised: We can observe that weighted soft labels are indeed balancing the sample-wise, not on dataset scale, bias and variance.

Revised: [Table]

Revised: [Table]

Revised: A. O THER VARIANTS OF WEIGHTING .

Revised: In the work

Revised: [Table]

Revised: the weighting scheme is deﬁned as (cid:16) 1 − exp (cid:16) − L s ce

Revised: L t ce (cid:17)(cid:17) , which is inspired by Lin et al.

Revised: ; Tang et al.

Revised: The basic idea is to convert L s ce L t ce into a value in [0 , 1]

Revised: [Table]

Revised: so that the weights of regularization samples are lower than those non-regularization samples.

Revised: A straightforward baseline is that we can use the Sigmoid function to convert L s ce L t ce into a value in [0 , 1] .

Revised: Note that L s ce

Revised: L t ce is always bigger than

Revised: [Table]

Revised: so the weight needs scaling and can be deﬁned as

Revised: [Equation]

Revised: +exp( − Ls ce Lt ce ) − 1 .

Revised: In Tab. 8, we present the comparison between adopted weighting form and the Sigmoid baseline.

Revised: We can see that as long as we can adaptively tune the sample-wise bias-variance tradeoff, the performance is better than KD, i.e., without weighted soft labels.

Revised: Therefore, although the proposed weighting form is not mathematically optimal, the not-too-big or not-too-small weights for these regularization examples are not hard to tune.

Revised: These results verify our main contribution that there is sample-wise biasvariance tradeoff and we need to assign weights to the regularization examples.

Revised: A. A BLATION ON α .

Revised: In Tab.

Revised: We ﬁrst tune the value of α on CIFAR100, with four values { 1 , 2 , 3 , 4 } tested.

Revised: Then we test with three values in [2 , 3] in (b).

Revised: Finally, we tune α on ImageNet in (c).

Revised: As a conclusion, the results are not very sensitive to α and the cost of searching α in our work is not expensive.

Revised: [Table]