Original: D ECISION BOUNDARY VARIABILITY AND GENERALIZATION IN NEURAL NETWORKS
Revised: D ECISION BOUNDARY VARIABILITY AND GENERALIZATION IN NEURAL NETWORKS

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: This paper discovers that the neural network with lower decision boundary (DB) variability has better generalizability.
Revised: This paper discovers that the neural network with lower decision boundary (DB) variability has better generalizability.

Original: Two new notions, algorithm DB variability and ( (cid:15), η ) -data DB variability , are proposed to measure the decision boundary variability from the algorithm and data perspectives.
Revised: Two new notions, algorithm DB variability and ( (cid:15), η ) -data DB variability , are proposed to measure the decision boundary variability from the algorithm and data perspectives.

Original: Extensive experiments show signiﬁcant negative correlations between the decision boundary variability and the generalizability.
Revised: Extensive experiments show signiﬁcant negative correlations between the decision boundary variability and the generalizability.

Original: From the theoretical view, two lower bounds based on algorithm DB variability are proposed and do not explicity depend on the sample size.
Revised: From the theoretical view, two lower bounds based on algorithm DB variability are proposed and do not explicity depend on the sample size.

Original: By Chebyshev’s inequality, we also prove two upper bounds of
Revised: By Chebyshev’s inequality, we also prove two upper bounds of

Original: O (cid:16)(cid:112) AV ( f Q , D ) /m
Revised: O (cid:16)(cid:112) AV ( f Q , D ) /m

Original: ( k − 1) (cid:17) and O (cid:16) 1 √ m + (cid:15) + η log 1 η (cid:17) based on algorithm and data DB variability, respectively.
Revised: ( k − 1) (cid:17) and O (cid:16) 1 √ m + (cid:15) + η log 1 η (cid:17) based on algorithm and data DB variability, respectively.

Original: The algorithm DB variability upper bound is easier to calculate in practice, while the data DB variability upper bound relies on less assumptions.
Revised: The algorithm DB variability upper bound is easier to calculate in practice, while the data DB variability upper bound relies on less assumptions.

Original: Moreover, the bounds do not explicitly depend on the networkweight and size, which is usually prohibitively large in deep learning, and thus have advantages in empirically approximation.
Revised: Moreover, the bounds do not explicitly depend on the network size, which is usually prohibitively large in deep learning.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Neural networks (NNs) have achieved signiﬁcant success in vast applications (Krizhevsky et al., 2012; Vaswani et al., 2017).
Revised: Neural networks (NNs) have achieved signiﬁcant success in vast applications (Krizhevsky et al., 2012; Vaswani et al., 2017).

Original: However, the advance of NNs is arduous to be characterized by canonical statistical hypothesis complexity, such as VC-dimension (Vapnik et al., 1994) or Rademacher complexity (Bartlett & Mendelson, 2002).
Revised: However, the advance of NNs is arduous to be characterized by canonical statistical hypothesis complexity, such as VC-dimension (Vapnik et al., 1994) or Rademacher complexity (Bartlett & Mendelson, 2002).

Original: Considering the intimate connection between low variance and signiﬁcant performance in learning theory, we investigate decision boundaries (DBs) from the perspective of variability in this paper.
Revised: Considering the intimate connection between low variance and signiﬁcant performance in learning theory, we investigate decision boundaries (DBs) from the perspective of variability in this paper.

Original: For neural networks, the decision boundary variability is largely caused by (1) the randomness from the training algorithm, and (2) the training data.
Revised: For neural networks, the decision boundary variability is largely caused by (1) the randomness from the training algorithm, and (2) the training data.

Original: In this paper, they are measured by two new terms, algorithm DB variability and ( (cid:15), η ) -data DB variability , and networks with lower DB variability are proved to possess better generalization performance.
Revised: In this paper, they are measured by two new terms, algorithm DB variability and ( (cid:15), η ) -data DB variability , and networks with lower DB variability are proved to possess better generalization performance.

Original: Algorithm DB variability measures the variability of DBs in different training repeats, and extensive experiments are conducted onthe two half moon (Pedregosa et al., 2011), CIFAR-10, and CIFAR-100 (Krizhevsky & Hinton, 2009) datasets to explore the factors impact algorithm DB variability.
Revised: Algorithm DB variability measures the variability of DBs in different training repeats, and extensive experiments are conducted on CIFAR-10, and CIFAR-100 (Krizhevsky & Hinton, 2009) datasets to explore the factors inﬂuence algorithm DB variability.

Original: We visualize the trend of the algorithm DB variability with respect to ( w.r.t. ) different training strategies, training epochs, sample size, and label noise.
Revised: We visualize the trend of the algorithm DB variability with respect to ( w.r.t. ) different training strategies, training time, sample sizes, and label noise.

Original: The empirical results present the negative correlation between algorithm DB variability and the test accuracy in all scenarios, which suggests that the algorithm decision boundary variability largely indicates the generalization of neural networks.
Revised: The empirical results present the negative correlation between algorithm DB variability and the test accuracy in all scenarios, which suggests that the algorithm DB variability largely indicates the generalization of neural networks.

Original: From the theoretical view, two lower bounds and an upper bound of the generalization error are proved based on the algorithm DB variability.
Revised: From the theoretical view, two lower bounds and an upper bound of the generalization error are proved based on the algorithm DB variability.

Original: The upper bound has a rate of O (cid:16)(cid:112) AV ( f Q , D ) /m
Revised: The upper bound has a rate of O (cid:16)(cid:112) AV ( f Q , D ) /m

Original: ( k − 1) (cid:17) , where AV ( f Q , D ) is the algorithm DB variability for f Q on data generation distribution D , m is the sample size, and k is the number of classes.
Revised: ( k − 1) (cid:17) , where AV ( f Q , D ) is the algorithm DB variability for f Q on data generating distribution D , m is the sample size, and k is the number of classes.

Original: To present how the training data affects the decision boundary variability, the ( (cid:15), η ) -data DB variability is introduced by employing the η -subset to reconstruct decision boundaries with error (cid:15) (see Deﬁnition 5.1).
Revised: To present how the training data inﬂuences the decision boundary variability, the ( (cid:15), η ) -data DB variability is introduced by employing the η -subset to reconstruct decision boundaries with error (cid:15) (see Deﬁnition 5.1).

Original: If a decision boundary can be reconstructed by training on a smaller η -subset with a smaller reconstruction error (cid:15) , the decision boundary possesses smaller data DB variability.
Revised: If a decision boundary can be reconstructed by training on a smaller η -subset with a smaller reconstruction error (cid:15) , the decision boundary possesses smaller data DB variability.

Original: Moreover, an η - (cid:15) curve can be drawn by varying the value of η and numerically evaluating the label mismatch rate.
Revised: Moreover, an η - (cid:15) curve can be drawn by varying the value of η and numerically evaluating the label mismatch rate.

Original: The area under the η - (cid:15) curve could be a more elaborate indicator for the generalization of NNs because the curve contains richer information than the algorithm DB variability.
Revised: The area under the η - (cid:15) curve could be a more elaborate indicator for the generalization of NNs because the curve contains richer information than the algorithm DB variability.

Original: An
Revised: An

Original: O (cid:16) 1 √ m + (cid:15) + η log 1 η (cid:17) generalization bound based on the ( (cid:15), η ) -data DB variability is established theoretically to enhance the relationship between the generalization of NNs and DB variability.
Revised: O (cid:16) 1 √ m + (cid:15) + η log 1 η (cid:17) generalization bound based on the ( (cid:15), η ) -data DB variability is established theoretically to enhance the relationship between the generalization of NNs and DB variability.

Original: Compared to the algorithm DB variability upper bound, the data DB variability bound does not depend on additional Assumption 2, while the algorithm DB variability is easier to calculate than data
Revised: Compared to the algorithm DB variability upper bound, the data DB variability bound does not depend on additional Assumption 2, while the algorithm DB variability is easier to calculate than data

Original: DB variability in practice.
Revised: DB variability in practice.

Original: Moreover, in contrast to many existing generalization bounds based on hypothesis complexity (Bartlett et al., 2019; Golowich et al., 2018; Bartlett et al., 2017) that require access to the network weight, our generalization bounds based on DB variability only demand the network predictions and thus have advantages in empirically approximating the generalization bound in (1) black-model settings, where model parameters are unavailable; and (2) over-parameterized settings, where calculating the weight norm is of prohibitively high computing burden.
Revised: Moreover, in contrast to many existing generalization bounds based on hypothesis complexity (Bartlett et al., 2019; Golowich et al., 2018; Bartlett et al., 2017) that require access to the network weight, our generalization bounds based on DB variability only demand the network predictions and thus have advantages in empirically approximating the generalization bound in (1) black-model settings, where model parameters are unavailable; and (2) over-parameterized settings, where calculating the weight norm is of prohibitively high computing burden.

Original:  R ELATED WORKS
Revised:  R ELATED WORKS

Original: Deep learning theory.
Revised: Deep learning theory.

Original: Due to the bias-variance trade-off, model complexity faces a dilemma in conventional wisdom (Mohri et al., 2018).
Revised: Due to the bias-variance trade-off, model complexity faces a dilemma in conventional wisdom (Mohri et al., 2018).

Original: Recently, Zhang et al.
Revised: Recently, Zhang et al.

Original: (2021) reveal the surprising ability of NNs in ﬁtting noise, but the NNs still have impressive generalization performance in practice.
Revised: (2021) reveal the surprising ability of neural networks in ﬁtting noise, but the networks still have impressive generalization performance in practice.

Original: This gap between the well-known bias-variance trade-off and the universal approximation ability for NNs draws attention to numerous researchers (Belkin et al., 2019; Nakkiran et al., 2019).
Revised: This gap between the well-known bias-variance trade-off and the universal approximation ability for NNs draws attention to numerous researchers (Belkin et al., 2019; Nakkiran et al., 2019).

Original: Many works attribute the success of NNs to the effectiveness of the stochastic gradient descent (SGD) algorithm (Bottou, 2010; Hardt et al., 2016; Jin et al., 2017).
Revised: Many works attribute the success of NNs to the effectiveness of the stochastic gradient descent (SGD) algorithm (Bottou, 2010; Hardt et al., 2016; Jin et al., 2017).

Original: Some empirical studies also explain the decent performance of networks by uncovering their learning properties (Nakkiran et al., 2020; Jiang et al., 2021;
Revised: Some empirical studies also explain the decent performance of networks by uncovering their learning properties (Nakkiran et al., 2020; Jiang et al., 2021;

Original: He et al., 2021).
Revised: He et al., 2021).

Original: For instance, neural networks tend to ﬁt the low-frequency information ﬁrst
Revised: For instance, neural networks tend to ﬁt the low-frequency information ﬁrst

Original: (Rahaman et al., 2019; Xu et al., 2019) and gradually learn a more complex function (Kalimeris et al., 2019) during the training procedure.
Revised: (Rahaman et al., 2019; Xu et al., 2019) and gradually learn a more complex function (Kalimeris et al., 2019) during the training procedure.

Original: Decision boundaries and generalization.
Revised: Decision boundaries and generalization.

Original: Recent studies attempt to understand neural networks from the aspect of decision boundaries (He et al., 

Revised: Recent studies attempt to understand neural networks from the aspect of decision boundaries (He et al.,

Original: ; Karimi et al., 2019).
Revised: ; Karimi et al., 2019).

Original: Guan & Loew (2020) empirically show the negative correlation between the complexity of decision boundary and generalization performance.
Revised: Guan & Loew (2020) empirically show the negative correlation between the complexity of decision boundary and generalization performance.

Original: Mickisch et al.
Revised: Mickisch et al.

Original: (2020) reveals the phenomenon that the distance from data to decision boundaries continuously decrease during the training procedure.
Revised: (2020) reveal the phenomenon that the distance from data to decision boundaries continuously decreases during the training procedure.

Original: More recently, researchers ﬁnd that NNs only rely on the most discriminative or simplest features to construct the
Revised: More recently, researchers ﬁnd that NNs only rely on the most discriminative or simplest features to construct the

Original: DBs (Ortiz-Jimenez et al., 2020; Shah et al., 2020).
Revised: DBs (Ortiz-Jimenez et al., 2020; Shah et al., 2020).

Original: Instead, our approach is different from these former methods by considering decision boundary variability, which is empirically and theoretically shown to closely correlate with the generalization in neural networks.
Revised: Instead, our approach is different from these former methods by considering decision boundary variability, which is empirically and theoretically shown to closely correlate with the generalization in neural networks.

Original: Adversarial training and generalization.
Revised: Adversarial training and generalization.

Original: It has been shown that the adversarial examples, which are created by adding non-perceivable perturbation on the input data, can completely mislead the
Revised: It has been shown that the adversarial examples, which are created by adding non-perceivable perturbation on the input data, can completely mislead the

Original: NNs (Szegedy et al., 2013; Goodfellow et al., 2014).
Revised: NNs (Szegedy et al., 2013; Goodfellow et al., 2014).

Original: To tackle this problem, adversarial training is proposed to improve the robustness of the NNs through continuous training on adversarial examples (Madry et al., 2017).
Revised: To tackle this problem, adversarial training is proposed to improve the robustness of the NNs through continuous training on adversarial examples (Madry et al., 2017).

Original: Nevertheless, Su et al.
Revised: Nevertheless, Su et al.

Original: (2018) and Zhang et al.
Revised: (2018) and Zhang et al.

Original: (2019) show a trade-off between the robustness and the generalization performance of NNs.
Revised: (2019) show a trade-off between the robustness and the generalization performance of NNs.

Original:  P RELIMINARIES3. N OTATION
Revised:  P RELIMINARIES

Original:  We denote the training set by S =
Revised: Notation: We denote the training set by S =

Original: { ( x i , y i ) , i = 1 , . . .
Revised: { ( x i , y i ) , i = 1 , . . .

Original: , m } , where x i ∈ R n , n is the dimension of input data, y i ∈ { 1 , . .
Revised: , m } , where x i ∈ R n , n is the dimension of input data, y i ∈ { 1 , . .

Original: . , k } , k is the number of classes, and m = |S| is the training sample size.
Revised: . , k } , k is the number of classes, and m = |S| is the training sample size.

Original: We assume that ( x i , y i ) are independent and identically distributed (i.i.d.)
Revised: We assume that ( x i , y i ) are independent and identically distributed (i.i.d.)

Original: random variables drawn from the data generating distribution D .
Revised: random variables drawn from the data generating distribution D .

Original: Denote the classiﬁer as f θ ( x ) : R n → R k , which is a neural network parameterized by θ .
Revised: Denote the classiﬁer as f θ ( x ) : R n → R k , which is a neural network parameterized by θ .

Original: The output of f θ ( x ) is a k -dimensional vector and is assumed to be a discrete probability density function.
Revised: The output of f θ ( x ) is a k -dimensional vector and is assumed to be a discrete probability density function.

Original: Let f ( i ) θ ( x ) be the i -th component of f θ ( x ) , hence (cid:80) ki
Revised: Let f ( i ) θ ( x ) be the i -th component of f θ ( x ) , hence (cid:80) k

Original: =1 f ( i ) θ ( x ) = 1 .
Revised: i = 1 f ( i ) θ ( x ) = 1 .

Original: We deﬁne T ( f θ , x ) =
Revised: We deﬁne T ( f θ , x ) =

Original: { i ∈ { 1 , · · · , k }| f
Revised: { i ∈ { 1 , · · · , k }| f

Original: ( i ) θ ( x ) = max j f ( j ) θ ( x ) } to denote the set of predicted labels by f θ on x .
Revised: ( i ) θ ( x ) = max j f ( j ) θ ( x ) } to denote the set of predicted labels by f θ on x .

Original: Due to the randomness of the learning algorithm A , let Q ( θ ) =
Revised: Due to the randomness of the learning algorithm A , let Q ( θ ) =

Original: A ( S ) denote the posteriori distribution returned by the learning algorithm
Revised: A ( S ) denote the posteriori distribution returned by the learning algorithm

Original: A leveraged on the training set S .
Revised: A leveraged on the training set S .

Original: Hence, we focus on the Gibbs classiﬁer (a.k.a.
Revised: Hence, we focus on the Gibbs classiﬁer (a.k.a.

Original: random classiﬁer)
Revised: random classiﬁer)

Original: f Q = {
Revised: f Q = {

Original: f θ | θ ∼ Q } . {+
+}
Revised: f θ | θ ∼ Q } . {+
+}

Original: 0 − 1 loss is employed in this paper, and the expected risks in terms of θ and Q are deﬁned as:
Revised: − 1 loss is employed in this paper, and the expected risks in terms of θ and Q are deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: respectively.
Revised: respectively.

Original: Here I ( · ) is the indicator function.
Revised: Here I ( · ) is the indicator function.

Original: Since the data generating distribution D is unknown, evaluating the expected risk R D is not practical.
Revised: Since the data generating distribution D is unknown, evaluating the expected risk R D is not practical.

Original: Therefore, it is a practical way to estimate the expected risk by the empirical risk R S , which is deﬁned as:
Revised: Therefore, it is a practical way to estimate the expected risk by the empirical risk R S , which is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where ( x i , y i ) ∈ S and m = |S| .
Revised: where ( x i , y i ) ∈ S and m = |S| .

Original: 3. D ECISION BOUNDARY
Revised: 3. D ECISION BOUNDARY

Original: If the output k -dimensional vector f θ ( x ) on the input example x has a tie, i.e. , the maximum value of the vector is not unique, then x is considered to locate on the decision boundary of f θ .
Revised: If the output k -dimensional vector f θ ( x ) on the input example x has a tie, i.e. , the maximum value of the vector is not unique, then x is considered to locate on the decision boundary of f θ .

Original: With this idea, the decision boundary can be formally deﬁned as below:
Revised: With this idea, the decision boundary can be formally deﬁned as below:

Original: Deﬁnition 3.1 (decision boundary) .
Revised: Deﬁnition 3.1 (decision boundary) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ .
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ .

Original: Then the decision boundary of f θ is deﬁned by
Revised: Then the decision boundary of f θ is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: After deﬁning the decision boundary, we have the following remark:
Revised: After deﬁning the decision boundary, we have the following remark:

Original: Remark1.
Revised: Remark

Original:  If an input example ( x , y ) is not located on the decision boundary of f θ , T ( f θ , x ) is a singleton, and we have
Revised: (1) If an input example ( x , y ) is not located on the decision boundary of f θ , T ( f θ , x ) is a singleton, and we have

Original: [Equation]
Revised: [Equation]

Original: (2) If the input x is a boundary point, in practice, we randomly draw a label from the set T ( f θ , x ) as the prediction of f θ on x .
Revised: (2) If the input x is a boundary point, in practice, we randomly draw a label from the set T ( f θ , x ) as the prediction of f θ on x .

Original: 3. A DVERSARIAL TRAINING
Revised: 3. A DVERSARIAL TRAINING

Original: Adversarial training (AT) is a popular strategy to enhance the adversarial robustness of neural networks against adversarial examples, which is generated through projected gradient descent (PGD) (Madry et al., 2018) in our empirical studies.
Revised: Adversarial training (AT) is a popular strategy to enhance the adversarial robustness of neural networks against adversarial examples, which is generated through projected gradient descent (PGD) (Madry et al., 2018) in our empirical studies.

Original: More speciﬁcally, adversarial training can be formulated as solving the minimax-loss problem as follows:
Revised: More speciﬁcally, adversarial training can be formulated as solving the minimax-loss problem as follows:

Original: [Equation]
Revised: [Equation]

Original: where γ is the radius to limit the distance between adversarial examples and original examples.
Revised: where γ is the radius to limit the distance between adversarial examples and original examples.

Original: Adversarial training can be considered to enlarge the minimum distances from training examples to decision boundaries to at least γ .
Revised: Adversarial training can be considered to enlarge the minimum distances from training examples to decision boundaries to at least γ .

Original:  A LGORITHM DECISION BOUNDARY VARIABILITY
Revised:  A LGORITHM DECISION BOUNDARY VARIABILITY

Original: Due to the randomness of learning algorithms, there is no doubt that the parameters have a substantial variation when the network is repeatedly trained.
Revised: Due to the randomness of learning algorithms, there is no doubt that the parameters have a substantial variation when the network is repeatedly trained.

Original: However, do the decision boundaries in these repeated-training networks have a large discrepancy?
Revised: However, do the decision boundaries in these repeated-training networks have a large discrepancy?

Original: With this question, we deﬁne the algorithm decision boundary variability (AV) to measure the variability of DBs caused by the randomness of algorithms in different training repeats. 
Revised: With this question, we deﬁne the algorithm decision boundary variability (AV) to measure the variability of DBs caused by the randomness of algorithms in different training repeats. Deﬁnition 4.1 (algorithm decision boundary variability) .

Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .

Revised: Then, the algorithm decision boundary variability for f Q on D is

Original: [Figure]
Revised: [Equation]

Original: A l g o r i t h m v a r i a
Revised: = T ( f θ (cid:48) , x ))] , (8) where T ( f θ , x ) = { i ∈ { 1 , · · · , k }| f ( i ) θ ( x ) = max j f ( j ) θ ( x ) } .

Original: [Figure]
Revised: [Figure]

Original: ili ty
Revised: r i t

Original: S N A
Revised: [Figure] m [Figure]

Original: CIFAR
Revised: i

Original: [Figure]
Revised: [Figure]

Original: VGGResNWRN
Revised: VGGResNWRN

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure]
Revised: r i

Original: [Figure]
Revised: [Figure]

Revised: m v a r i a

Original: [Figure]
Revised: [Figure]

Revised: li t y

Original: [Figure]
Revised: [Figure]

Original: A l g o r i t h m v a r i a b li t y

Original: S N A
Revised: N A

Original: CIFAR
Revised: CIFAR

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

Original: Deﬁnition 4.1 (algorithm decision boundary variability) . Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ . Then, the algorithm decision boundary variability for f Q on D is [Equation]
Revised: In the light of Deﬁnition 4.

Original: ) where T ( f θ , x ) = { i ∈ { 1 , · · · , k }| f ( i ) θ ( x ) = max j f ( j ) θ ( x ) } . In the light of Deﬁnition 4.1, smaller algorithm DB variability AV ( f Q , D ) means that the network produces more stable DBs or converges to more similar DBs during different training repeats.
Revised: smaller algorithm DB variability AV ( f Q , D ) means that the network produces more stable DBs or converges to more similar DBs during different training repeats.

Original: The algorithm DB variability is also a good notion for the “entropy” of decision boundaries because it measures the decision uncertainty over the whole data generating distribution; we provide a detailed discussion in Appendix B.1 due to space limitation.
Revised: The algorithm DB variability is also a good notion for the “entropy” of decision boundaries because it measures the decision uncertainty over the whole data generating distribution; we provide a detailed discussion in Appendix B.1 due to space limitation.

Original: 4. A LGORITHM DB VARIABILITY AND GENERALIZATION
Revised: 4. A LGORITHM DB VARIABILITY AND GENERALIZATION

Original: To explore the relationship between the algorithm DB variability and generalization in neural networks, we conduct experiments with different network architectures and training strategies on
Revised: To explore the relationship between the algorithm DB variability and generalization in neural networks, we conduct experiments with different network architectures and training strategies on

Original: CIFAR-10 and CIFAR-100.
Revised: CIFAR-10 and CIFAR-100.

Original: In detail, VGG-16 (Simonyan & Zisserman, 2014), ResNet-18 (He et al., 2016), and WideResNet-28 (Zagoruyko & Komodakis, 2016b) are optimized by standard, non-data-augmentation and adversarial training, respectively, until the training procedure converges.
Revised: In detail, VGG-16 (Simonyan & Zisserman, 2014), ResNet-18 (He et al., 2016), and WideResNet-28 (Zagoruyko & Komodakis, 2016b) are optimized by standard, non-data-augmentation and adversarial training, respectively, until the training procedure converges.

Original: We clarify that basic data augmentation (crop and ﬂip) (Zagoruyko & Komodakis, 2016a) is adopted in both standard and adversarial training, and only the basic data augmentation is considered in our experiments and analysis.
Revised: We clarify that basic data augmentation (crop and ﬂip) (Zagoruyko & Komodakis, 2016a) is adopted in both standard and adversarial training, and only the basic data augmentation is considered in our experiments and analysis.

Original: Each training setting (dataset, architecture, training strategy) is repeated for 10 trials with different random seeds to estimate the parameter distribution Q ( θ ) .
Revised: Each training setting (dataset, architecture, training strategy) is repeated for 10 trials with different random seeds to estimate the parameter distribution Q ( θ ) .

Original: As for simulating the data generating distribution, we trained two conditional BigGAN (Zhao et al., 2020), an advanced generative network architecture, to produce 100 , 000 fake images for CIFAR-10 and
Revised: As for simulating the data generating distribution, we trained two conditional BigGAN (Zhao et al., 2020), an advanced generative network architecture, to produce 100 , 000 fake images for CIFAR-10 and

Original: CIFAR-100, respectively.
Revised: CIFAR-100, respectively.

Original: Examples of fake images have been shown in Figure 1(a) and 1(b).
Revised: Examples of fake images have been shown in Figure 1(a) and 1(b).

Original: Through these generative fake images, we can estimate the algorithm DB variability w.r.t.
Revised: Through these generative fake images, we can estimate the algorithm DB variability w.r.t.

Original: these well-trained models.
Revised: these well-trained models.

Original: For each training setting, we plot both the average test accuracy and algorithm
Revised: For each training setting, we plot both the average test accuracy and algorithm

Original: DB variability; see Figure 1(c).
Revised: DB variability; see Figure 1(c).

Original: From the plots, we have several observations: (1) adversarial training dramatically decreases the test accuracy and promotes the algorithm DB variability compared to standard training.
Revised: From the plots, we have several observations: (1) adversarial training dramatically decreases the test accuracy and promotes the algorithm DB variability compared to standard training.

Original: (2) data augmentation decreases the algorithm decision boundary variability by comparing the standard training and non-data-augmentation training.
Revised: (2) data augmentation decreases the algorithm decision boundary variability by comparing the standard training and non-data-augmentation training.

Original: The reason is that the augmented images by cropping or ﬂipping are still located on the data generating distribution, so data augmentation is considered to expand the size of the training set.
Revised: The reason is that the augmented images by cropping or ﬂipping are still located on the data generating distribution, so data augmentation is considered to expand the size of the training set.

Original: Hence, the expanded training set can characterize longer or larger decision boundaries on the data generating distribution; (3)
Revised: Hence, the expanded training set can characterize longer or larger decision boundaries on the data generating distribution; (3)

Original: WideResNet has better test accuracy and lower algorithm DB variability than ResNet and VGG; and (4) there is a negative correlation between the test accuracy and the algorithm DB variability.
Revised: WideResNet has better test accuracy and lower algorithm DB variability than ResNet and VGG; and (4) there is a negative correlation between the test accuracy and the algorithm DB variability.

Original: Therefore, based on these ﬁndings, we propose the following conjecture:
Revised: Therefore, based on these ﬁndings, we propose the following conjecture:

Original: [Figure]

Original: Hypothesis 1.
Revised: Hypothesis 1.

Original: Neural networks with smaller algorithm decision boundary variability on the data generating distributions possess better generalization performance.
Revised: Neural networks with smaller algorithm decision boundary variability on the data generating distributions possess better generalization performance.

Original: We then conduct experiments on the algorithm DB variability with respect to training epochs, sample size, and label noise to concrete the hypothesis. 
Revised: We then conduct experiments on the algorithm DB variability w.r.t. training time, sample size, and label noise to concrete this hypothesis. [Figure] [Figure]

Original: 4. A LGORITHM DB VARIABILITY AND TRAINING PROCESS
Revised: 4. A LGORITHM DB VARIABILITY AND TRAINING TIME

Original: To explore the relationship between algorithm DB variability and training epochs, we train
Revised: To investigate the relationship between algorithm DB variability and training time, we train

Original: ResNet-18 with different initial learning rates of 0 . 1 and 0 .
Revised: ResNet-18 with different initial learning rates of 0 . 1 and 0 .

Original: 01 on CIFAR-10 and CIFAR-10.
Revised: 01 on CIFAR-10 and CIFAR-100.

Original: Then, the algorithm DB variability and test error are calculated at each epoch; see Figure 2(a).
Revised: Then, the algorithm DB variability and test error are calculated at each epoch; see Figure 2(a).

Original: From the plots, we can observe that (1) algorithm DB variability and test error share a very similar curve with respect to the training time; and (2) algorithm DB variability decreases during the training process.
Revised: From the plots, we can observe that (1) algorithm DB variability and test error share a very similar curve w.r.t. the training time; and (2) algorithm DB variability decreases during the training process.

Original: The decline of algorithm DB variability shows that the interpolation on examples reduces the variability of decision boundaries on data generating distribution.
Revised: The decline of algorithm DB variability shows that the interpolation on examples reduces the variability of decision boundaries on data generating distribution.

Original: As shown in Figure 2(b), we also plot the scatter plots of (algorithm DB variability, test error), which present a signiﬁcant positive correlation between test error and the algorithm DB variability, and thus supports Hypothesis 1.
Revised: As shown in Figure 2(b), we collect the points of (algorithm DB variability, test error) from different epochs, and the scatter plots present a signiﬁcant positive correlation between test error and the algorithm DB variability, and thus supports Hypothesis 1.

Original: 4. A LGORITHM DB VARIABILITY AND SAMPLE SIZE
Revised: 4. A LGORITHM DB VARIABILITY AND SAMPLE SIZE

Original: We next investigate how sample size impacts the algorithm DB variability.
Revised: We next investigate how sample size impacts the algorithm DB variability.

Original: We trained 100 ResNet18 on ﬁve training sample sets of different sizes randomly drawn fromof CIFAR-10 and CIFAR-100, respectively, while all irrelevant variables are strictly controlled.
Revised: 100 ResNet-18 are trained on ﬁve training sample sets of different sizes randomly drawn from CIFAR-10 and CIFAR100, while all irrelevant variables are strictly controlled.

Original: We calculate the algorithm DB variability and test error  in all cases; See Figure 3(a).
Revised: Then, the algorithm DB variability and test error are calculated in all cases; see Figure 3(a).

Original: From the plots, we have several observations: (1) test error and algorithm DB variability share a very similar curve with respect to the training sample size; (2) larger sample size, which intuitively helps obtain a more smooth estimation of the decision boundary, also contributes to smaller decision boundary variability; and (3) there is a signiﬁcant positive correlation between test error andthe algorithm DB variability by varying training sample size, which fully support Hypothesis 
Revised: From the plots, we have the following observations: (1) test error and algorithm DB variability share a very similar curve w.r.t. the training sample size; (2) larger sample size, which intuitively helps obtain a more smooth estimation of the decision boundary, also contributes to smaller algorithm DB variability; and (3) there is a signiﬁcant positive correlation between test error and algorithm DB variability, which fully supports Hypothesis 1.

Original: . 4. A LGORITHM DB VARIABILITY AND LABEL NOISE
Revised: 4. A LGORITHM DB VARIABILITY AND LABEL NOISE

Original: Belkin et al.
Revised: Belkin et al.

Original: ; Nakkiran et al.
Revised: ; Nakkiran et al.

Original: (2019) show the surprising epoch-wise double descent of test error, especially with the existence of label noise.
Revised: (2019) show the surprising epoch-wise double descent of test error, especially with the existence of label noise.

Original: In this section, we explore the trend of algorithm
Revised: In this section, we explore the trend of algorithm

Original: [Figure]

Original: averaged on 10 trials.

Original: DB variability when the label noise is injected.
Revised: DB variability when the label noise exists.

Original: We train 20 ResNet-10 for 500 epoch with a constant learning rate of 0 . 0001 on CIFAR-10 and CIFAR-100 with 20% label noise, respectively.
Revised: We train 20 ResNet-10 for 500 epoch with a constant learning rate of 0 . 0001 on CIFAR-10 and CIFAR-100 with 20% label noise.

Original: We clarify that the noise labels remain unchanged during different training repeats, which is necessary for estimate the algorithm DB variability.
Revised: We clarify that the noise labels remain unchanged during different training repeats, which is necessary to estimate the algorithm DB variability.

Original: Then, the average test error and algorithm DB variability are calculate at each training epoch, as shown in Figure 3(b).
Revised: Then, the average test error and algorithm DB variability are calculated at each training epoch, as shown in Figure 3(b).

Revised: From the plots, we observe that: (1) test error and

Original: From the plots, we observe that: (1) test error and  algorithm DB variability still share a very similar curve with respect to the training time with the existence of label noise; and (2) the algorithm DB variability also undergoes an epoch-wise double descent during the training process, especially in the left panel of Figure 3(b), which implies that factors inﬂuence the generalization of networks can also have an impact on the algorithm DB variability.
Revised: [Figure] and then averaged on 10 trials. algorithm DB variability still share a very similar curve w.r.t. the training time with the existence of label noise; and (2) the algorithm DB variability also undergoes an epoch-wise double descent during the training process, especially in the left panel of Figure 3(b), which implies that factors inﬂuence the generalization of networks can also have an inﬂuence on the algorithm DB variability.

Original: Therefore, algorithm DB variability is a satisﬁed indicator for the generalization ability ofneural networks.
Revised: Hence, algorithm DB variability is an excellent indicator for the generalization ability of networks.

Original: 4. T HEORETICAL E VIDENCE
Revised: 4. T HEORETICAL E VIDENCE

Original: In this section, we explore and develop the theoretical foundations for the algorithm decision boundary variability on data generating distributions.
Revised: In this section, we explore and develop the theoretical foundations for the algorithm decision boundary variability on data generating distributions.

Original: Assumption 1.
Revised: Assumption 1.

Original: The decision boundary of the classiﬁer network f θ on data generation distribution
Revised: The decision boundary of the classiﬁer network f θ on data generation distribution

Original: D is a set with measure zero.
Revised: D is a set with measure zero.

Original: Corollary 1.
Revised: Corollary 1.

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ .
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ .

Original: If Assumptionholds for all θ ∼ Q , then, for all i ∈ { 1 , · · · , k } , we haveE [Equation] ∼D
Revised: If Assumptionholds for all θ ∼ Q , then, for all i ∈ { 1 , · · · , k } , we have

Original: [Equation]

Original: [Equation]

Original: ( f

Original: [Equation]

Original: x

Original: [Equation]
Revised: [Equation]

Original: i )

Original: [Equation]

Original: (9) and
Revised: 9) and

Original: E( x ,y ) ∼D [Equation] / ∈
Revised: E

Original: [Equation]

Original: ( f

Original: [Equation]

Original: [Equation]

Original: ] =

Original: [Equation]

Original: ( x ,y )

Original: ∼D [ I ( T ( f θ

Original: [Equation]
Revised: [Equation]

Original: x

Original: [Equation]

Original: (cid:54) = i )] .

Original: 4.5. A LGORITHM DB VARIABILITY - BASED LOWER BOUNDS
Revised: 4.5. A LGORITHM DB VARIABILITY - BASED LOWER BOUNDS

Original: Theorem 1 (lower bound on expected risk) .
Revised: Theorem 1 (lower bound on expected risk) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .

Original: Then, if Assumption 1 holds for all θ ∼ Q , we have
Revised: Then, if Assumption 1 holds for all θ ∼ Q , we have

Original: [Equation]
Revised: [Equation]

Original:  where AV ( f Q , D ) =
Revised: 11) where AV ( f Q , D ) =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) (cid:54) =
Revised: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) (cid:54) =

Original: T ( f θ (cid:48) , x ))] is the algorithm DB variability for f Q on data generating distribution D .
Revised: T ( f θ (cid:48) , x ))] is the algorithm DB variability for f Q on data generating distribution D .

Original: The proof is given in Appendix C.1.
Revised: The proof is given in Appendix C.1.

Original: Theorem 1 provides a lower bound on the expected risk R D ( Q ) based on the algorithm DB variability AV ( f Q , D ) , and the lower bound presents that the Gibbs classiﬁer f Q possesses a signiﬁcant expected risk when its algorithm DB variability is large.
Revised: Theorem 1 provides a lower bound on the expected risk R D ( Q ) based on the algorithm DB variability AV ( f Q , D ) , and the lower bound presents that the Gibbs classiﬁer f Q possesses a signiﬁcant expected risk when its algorithm DB variability is large.

Original: Moreover, when we consider the binary classiﬁcation, i.e. , k = 2 , there is a tighter lower bound:
Revised: Moreover, when we consider the binary classiﬁcation, i.e. , k = 2 , there is a tighter lower bound:

Original: Theorem 2 (lower bound for binary case) .
Revised: Theorem 2 (lower bound for binary case) .

Original: Let f θ ( x ) : R n → R 2 be a binary classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .
Revised: Let f θ ( x ) : R n → R 2 be a binary classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .

Original: Suppose the expected risk R D ( Q ) ≤ 1 and Assumption 1 hold for all θ ∼ Q , then we have
Revised: Suppose the expected risk R D ( Q ) ≤and Assumption 1 hold for all θ ∼ Q , then we have

Original: [Equation]
Revised: [Equation]

Original: 4.5. A N ALGORITHM DB VARIABILITY - BASED UPPER BOUND
Revised: 4.5. A N ALGORITHM DB VARIABILITY - BASED UPPER BOUND

Original: Before we present the upper bound based on the algorithm DB variability, let us introduce the assumption used in this section.
Revised: Before we present the upper bound based on the algorithm DB variability, let us introduce the assumption used in this section.

Original: Assumption 2.
Revised: Assumption 2.

Original: [Equation]
Revised: [Equation]

Original: where i ∈ { 1 , 2 , · · · , k } and k is the number of potential categories.
Revised: where i ∈ { 1 , 2 , · · · , k } and k is the number of potential categories.

Original: With Assumption 2, we can derive the upper bound on the variance of risk:
Revised: With Assumption 2, we can derive the upper bound on the variance of risk:

Original: Lemma 1 (upper bound on risk variance) .
Revised: Lemma 1 (upper bound on risk variance) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ .

Original: Suppose Assumption 1 and 2 hold, then we have
Revised: Suppose Assumption 1 and 2 hold, then we have

Original: [Equation]
Revised: [Equation]

Original: where AV ( f Q , D ) =
Revised: where AV ( f Q , D ) =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ [Equation] x ) 
Revised: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) (cid:54) =

Original: [Equation]

Original: = T ( f θ [Equation] , x ))] is the algorithm DB variability of f Q on data generating distribution D .
Revised: T ( f θ (cid:48) , x ))] is the algorithm DB variability of f Q on data generating distribution D .

Original: The proof is given in Appendix C.3.
Revised: The proof is given in Appendix C.3.

Original: Hence, the next theorem is a direct consequence of the onesided Chebyshev’s inequality : Pr (cid:2) E
Revised: Hence, the next theorem is a direct consequence of the onesided Chebyshev’s inequality : Pr (cid:2) E

Original: [ Z 1 ] − 1 m (cid:80) mi
Revised: [ Z 1 ] − 1 m (cid:80) mi

Original: =

Revised: =

Original: Z i > a (cid:3)
Revised: Z i > a (cid:3)

Original: ≤ Var[ Z 1 ] 2 ma 2 for any a > 0 .
Revised: ≤ Var[ Z 1 ] 2 ma 2 for any a > 0 .

Original: Theorem 3 (PAC upper bound on expected risk) .
Revised: Theorem 3 (PAC upper bound on expected risk) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ w.r.t.
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ and let Q ( θ ) be the distribution over θ w.r.t.

Original: the training set S .
Revised: the training set S .

Original: Suppose Assumption 1 and 2 hold.
Revised: Suppose Assumption 1 and 2 hold.

Original: Then, with the probability of at least 1 − δ over a sample of size m , we have
Revised: Then, with the probability of at least 1 − δ over a sample of size m , we have

Original: [Equation]
Revised: [Equation]

Original: The above theorem presents that a smaller algorithm DB variability AV ( f Q , D ) contributes to a tighter upper bound on the true risk, which theoretically veriﬁes Hypothesis 1 that neural networks with smaller algorithm decision boundary variability possess better generalization performance.
Revised: The above theorem presents that a smaller algorithm DB variability AV ( f Q , D ) contributes to a tighter upper bound on the true risk, which theoretically veriﬁes Hypothesis 1 that neural networks with smaller algorithm decision boundary variability possess better generalization performance.

Original:  D ATA DECISION BOUNDARY VARIABILITY
Revised:  D ATA DECISION BOUNDARY VARIABILITY

Original: In the previous sections, we introduced the algorithm DB variability, which measures the decision boundary variability caused by the randomness of learning algorithms.
Revised: In the previous sections, we introduced the algorithm DB variability, which measures the decision boundary variability caused by the randomness of learning algorithms.

Original: However, the algorithm DB variability hardly shows the decision boundary variability caused by changes in training data.
Revised: However, the algorithm DB variability hardly shows the decision boundary variability caused by changes in training data.

Original: To remedy the problem and complete our theorem about decision boundary variability, we deﬁne the data DB variability as below:
Revised: To remedy the problem and complete our theorem about decision boundary variability, we deﬁne the data DB variability as below:

Original: Deﬁnition 5.1 (data decision boundary variability) .
Revised: Deﬁnition 5.1 (data decision boundary variability) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ , where θ ∼ A ( S ) is returned by leveraging the stochastic learning
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ , where θ ∼ A ( S ) is returned by leveraging the stochastic learning

Original: algorithm A on the training set S , which is sampled from the data generating distribution D .
Revised: algorithm A on the training set S , which is sampled from the data generating distribution D .

Original: We term S η ⊂ S as a η -subset of S if |S η | |S| = η .
Revised: We term S η ⊂ S as a η -subset of S if |S η | |S| = η .

Original: Then, if we ﬁxed η and
Revised: Then, if we ﬁxed η and

Original: [Equation]
Revised: [Equation]

Original: (16) the decision boundary of f A ( S ) is said to possess a ( (cid:15), η ) -data decision boundary variability .
Revised: the decision boundary of f A ( S ) is said to possess a ( (cid:15), η ) -data decision boundary variability .

Original: The data decision boundary variability contains two parameters of (cid:15) and η , respectively.
Revised: The data decision boundary variability contains two parameters of (cid:15) and η , respectively.

Original: That Gibbs classiﬁer f A ( S ) has a ( (cid:15), η ) -data DB variability means that only the proportion of η of S , i.e. , S η , which can be considered as “support vector set”, is enough to reconstruct a similar decision boundary with the reconstruction error (cid:15) .
Revised: That Gibbs classiﬁer f A ( S ) has a ( (cid:15), η ) -data DB variability means that only the proportion of η of S , i.e. , S η , which can be considered as “support vector set”, is enough to reconstruct a similar decision boundary with the reconstruction error (cid:15) .

Original: The data DB variability can also be connected with the complexity of decision boundaries if we assume that simpler decision boundaries rely on fewer “support vectors”; we provide a detailed discussion in Appendix B.2 due to space limitation.
Revised: The data DB variability can also be connected with the complexity of decision boundaries if we assume that simpler decision boundaries rely on fewer “support vectors”; we provide a detailed discussion in Appendix B.2 due to space limitation.

Original: [Figure]
Revised: [Figure]

Original:  η - (cid:15) CURVES ABOUT DATA DB VARIABILITY
Revised: 5. η - (cid:15) CURVES ABOUT DATA DB VARIABILITY

Original: According to Deﬁnition 5. the data DB variability degrades to the algorithm DB variability
Revised: According to Deﬁnition 5.1, the data DB variability degrades to the algorithm DB variability

Original: AV ( f Q , D ) when S η = S .
Revised: AV ( f Q , D ) when S η = S .

Original: In other words, the algorithm DB variability is a special case of the data DB variability with η = 1 .
Revised: In other words, the algorithm DB variability is a special case of the data DB variability with η = 1 .

Original: Therefore, the data DB variability could present more detailed information on reﬂecting how the decision boundary variability depends on the training set, especially when we observe the variation of the reconstruction error (cid:15) w.r.t.
Revised: Therefore, the data DB variability could present more detailed information on reﬂecting how the decision boundary variability depends on the training set, especially when we observe the variation of the reconstruction error (cid:15) w.r.t.

Original: different η .
Revised: different η .

Original: To explore the relationship between the reconstruction error (cid:15) and the proportion of subset η , we train 1000 networks of ResNet-18 on CIFAR-10 and CIFAR-100 of different sample sizes m , respectively.
Revised: To explore the relationship between the reconstruction error (cid:15) and the proportion of subset η , we train 1000 networks of ResNet-18 on CIFAR-10 and CIFAR-100 of different sample sizes m , respectively.

Original: Albeit ﬁnding the most suitable η -subset is intractable, we adopt a coreset selection approach named selection via proxy (Coleman et al., 2020), which can rank the importance of training examples, to estimate the η -subset for a given training set S and proportion η (full list of η involved in our experiments is presented in Appendix A.3.6).
Revised: Albeit ﬁnding the most suitable η -subset is intractable, we adopt a coreset selection approach named selection via proxy (Coleman et al., 2020), which can rank the importance of training examples, to estimate the η -subset for a given training set S and proportion η (full list of η involved in our experiments is presented in Appendix A.3.6).

Original: Then, through repeatedly training the network on
Revised: Then, through repeatedly training the network on

Original: S η , we can estimate the reconstruction error (cid:15) .
Revised: S η , we can estimate the reconstruction error (cid:15) .

Original: The η - (cid:15) curves of CIFAR-10 and CIFAR-100 are presented in Figure 4(a) and 4(b), respectively.
Revised: The η - (cid:15) curves of CIFAR-10 and CIFAR-100 are presented in Figure 4(a) and 4(b), respectively.

Original: From the plots, we have an observation that there is a more rapid decline in (cid:15) along with small η and also a smaller algorithm DB variability when the training sample size m is larger.
Revised: From the plots, we have an observation that there is a more rapid decline in (cid:15) along with small η and also a smaller algorithm DB variability when the training sample size m is larger.

Original: Furthermore, we plot the schematic diagram of η - (cid:15) curves w.r.t.
Revised: Furthermore, we plot the schematic diagram of η - (cid:15) curves w.r.t.

Original: different sample size m , as shown in Figure 4(c).
Revised: different sample size m , as shown in Figure 4(c).

Original: When η = 0 , f A ( S n ) cannot be better than random guess, and hence (cid:15) =
Revised: When η = 0 , f A ( S n ) cannot be better than random guess, and hence (cid:15) =

Original: k − 1 k , where k is the number of potential categories.
Revised: k − 1 k , where k is the number of potential categories.

Original: We have noticed that (cid:15) has a sharper drop along with η when the sample size m is larger.
Revised: We have noticed that (cid:15) has a sharper drop along with η when the sample size m is larger.

Original: Therefore, we rationally propose the following assumption, which is also shown by the right angle with m ∞ in Figure 4(c).
Revised: Therefore, we rationally propose the following assumption, which is also shown by the right angle with m ∞ in Figure 4(c).

Original: Assumption 3.
Revised: Assumption 3.

Original: If m → ∞ , then we have (cid:15) → 0 when η → 0 .
Revised: If m → ∞ , then we have (cid:15) → 0 when η → 0 .

Original: These plots indicate that the area under the η - (cid:15) curve could be a more meticulous predictor for the generalization ability of neural networks compared to the algorithm DB variability, which is only a point on the η - (cid:15) curve when η = 1 .
Revised: These plots indicate that the area under the η - (cid:15) curve could be a more meticulous predictor for the generalization ability of neural networks compared to the algorithm DB variability, which is only a point on the η - (cid:15) curve when η = 1 .

Original: Hence, the area under the η - (cid:15) curve can also be considered as an extension of the algorithm DB variability: if the Gibbs classiﬁer f A ( S ) possesses a smaller area under the η - (cid:15) curve, it produces more stable decision boundaries with varying training subsets.
Revised: Hence, the area under the η - (cid:15) curve can also be considered as an extension of the algorithm DB variability: if the Gibbs classiﬁer f A ( S ) possesses a smaller area under the η - (cid:15) curve, it produces more stable decision boundaries with varying training subsets.

Original: In the following section, we will theoretically establish the connection between the parameters of η and (cid:15) and the generalization ability of neural networks.
Revised: In the following section, we will theoretically establish the connection between the parameters of η and (cid:15) and the generalization ability of neural networks.

Original: 5. T HEORETICAL EVIDENCE
Revised: 5. T HEORETICAL EVIDENCE

Original: In this section, we develop the theoretical foundations for the data decision boundary variability, and present that neural networks with better data DB variability possess better generalization .
Revised: In this section, we develop the theoretical foundations for the data decision boundary variability, and present that neural networks with better data DB variability possess better generalization .

Original: Lemma 2.
Revised: Lemma 2.

Original: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability, then we have
Revised: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability, then we have

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Lemma 2 shows that the difference between the expected risk of A ( S ) and A ( S η ) can be bounded by their difference in decision boundaries.
Revised: Lemma 2 shows that the difference between the expected risk of A ( S ) and A ( S η ) can be bounded by their difference in decision boundaries.

Original: Then, by applying the concentration inequality, we canbound the differencebetween their empirical risk on the same training set Swith the probability 1 − δ :
Revised: Then, by applying the concentration inequality, we can difference S δ :

Original: Lemma 3.
Revised: Lemma 3.

Original: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability, then, with probability of at least 1 − δ over a sample of size m , we have
Revised: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability, then, with probability of at least 1 − δ over a sample of size m , we have

Original: [Equation]
Revised: [Equation]

Original: where m = |S| is the training sample size.
Revised: where m = |S| is the training sample size.

Original: If we further assume the training error R S ( A ( S ))
Revised: If we further assume the training error R S ( A ( S ))

Original: = 0 , then, with the probability of at least 1 − δ over a sample of size m , we have
Revised: = 0 , then, with the probability of at least 1 − δ over a sample of size m , we have

Original: [Equation]
Revised: [Equation]

Original: Theorem 4 (data DB variability-based upper bound on expected risk) .
Revised: Theorem 4 (data DB variability-based upper bound on expected risk) .

Original: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability on the data generating distribution D , and assume η ≤ 0 .and
Revised: If the decision boundaries of f A ( S ) possess a ( (cid:15), η ) -data DB variability on the data generating distribution D , and assume η ≤ 0 .and

Original: the empirical risks R S
Revised: the empirical risks R S

Original: ( A ( S ))
Revised: ( A ( S ))

Original: =
Revised: =

Original: R S η
Revised: R S η

Original: ( A ( S η ))
Revised: ( A ( S η ))

Original: = 0 , then, with the probability of at least 1 − δ over a sample of size m , we have
Revised: = 0 , then, with the probability of at least 1 − δ over a sample of size m , we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Moreover, for sufﬁcient large m , we have
Revised: Moreover, for sufﬁcient large m , we have

Original: [Equation]
Revised: [Equation]

Original: The proof is omitted here and is given in Appendix C.7.
Revised: The proof is omitted here and is given in Appendix C.7.

Original: According to Assumption 3, when m → ∞ , then η → 0 and (cid:15) → 0 , and according to Eq. 23, R D ( A ( S ))
Revised: According to Assumption 3, when m → ∞ , then η → 0 and (cid:15) → 0 , and according to Eq. 23, R D ( A ( S ))

Original: → 0 .
Revised: → 0 .

Original: Therefore, the generalization bound is asymptotically converged.
Revised: Therefore, the generalization bound is asymptotically converged.

Original: Theorem 4 presents that smaller data DB variability, i.e. , smaller (cid:15) and η , contributes to a tighter upper bound on the expected risk, and also theoretically veriﬁes the relationship between the data DB variability and the generalization ability of neural networks.
Revised: Theorem 4 presents that smaller data DB variability, i.e. , smaller (cid:15) and η , contributes to a tighter upper bound on the expected risk, and also theoretically veriﬁes the relationship between the data DB variability and the generalization ability of neural networks.

Original:  C ONCLUSION AND DISCUSSION
Revised:  C ONCLUSION AND DISCUSSION

Original: In this paper, we empirically and theoretically explored the relationship between decision boundary variability and generalization in neural networks, through the algorithm DB variability and data DB variability, respectively.
Revised: In this paper, we empirically and theoretically explored the relationship between decision boundary variability and generalization in neural networks, through the algorithm DB variability and data DB variability, respectively.

Original: A signiﬁcant negative correlation between the decision boundary  and generalization performance is observed in our experiments.
Revised: A signiﬁcant negative correlation between the decision boundary variability and generalization performance is observed in our experiments.

Original: As for the theoretical results, two lower bounds and two upper bounds were proposed based on algorithm DB variability and data DB variability to enhance our ﬁndings.
Revised: As for the theoretical results, two lower bounds and two upper bounds were proposed based on algorithm DB variability and data DB variability to enhance our ﬁndings.

Original: Compared to the algorithm DB variability upperbound, the data DB variability bounddoes not depend on additional assumption, while the algorithm DB variability is easier to calculate than data DB variability in practice.
Revised: The algorithm DB variability upper bound is easier to calculate in practice, while the data DB variability upper bound relies on less assumptions.

Original: Moreover, in contrast to many existing generalization bounds based on hypothesis complexity that require access to the network weight, our generalization bounds based on decision boundary variability only demand access to the network predictions and thus have advantages in empirically approximating the generalization bound in (1) black-model settings, where model parameters are unavailable; and (2) over-parameterized settings, where calculating the weight norm is of prohibitively high computing burden.
Revised: Moreover, in contrast to many existing generalization bounds based on hypothesis complexity that require access to the network weight, our generalization bounds based on decision boundary variability only demand access to the network predictions and thus have advantages in empirically approximating the generalization bound in (1) black-model settings, where model parameters are unavailable; and (2) over-parameterized settings, where calculating the weight norm is of prohibitively high computing burden.

Original: Furthermore, the analysis in Section 5.1 mentioned that the algorithm DB variability is a special case of the data DB variability.
Revised: Furthermore, the analysis in Section 5.1 mentioned that the algorithm DB variability is a special case of the data DB variability.

Original: Therefore, unifying the theoretical works about algorithm DB variability and data DB variability will be a promising direction in future works. 
Revised: Therefore, unifying the theoretical works about algorithm DB variability and data DB variability will be a promising direction in future works. A CKNOWLEDGEMENT

Revised: The authors sincerely appreciate the anonymous reviewers for their valuable suggestions.

Original: A A DDITIONAL EXPERIMENTS DETAILS
Revised: A A DDITIONAL EXPERIMENTS DETAILS

Original: This section provides all the additional details of our experiments.
Revised: This section provides all the additional details of our experiments.

Original: A. D ATASETS
Revised: A. D ATASETS

Original: Our experiments are conducted on two public datasets (CIFAR-10 and CIFAR-100 (Krizhevsky &
Revised: Our experiments are conducted on two public datasets (CIFAR-10 and CIFAR-100 (Krizhevsky &

Original: Hinton, 2009)) and two manufactured datasets (fake CIFAR-10, and fake CIFAR-100).
Revised: Hinton, 2009)) and two manufactured datasets (fake CIFAR-10, and fake CIFAR-100).

Original: The detail of these datasets are shown as follows
Revised: The detail of these datasets are shown as follows

Original: • CIFAR-10 consists of 50 , 000 training images and 10 , 000 test images from 10 different classes, and CIFAR-100 consists of 50 , 000 training images and 10 , 000 test images from 100 different classes.
Revised: • CIFAR-10 consists of 50 , 000 training images and 10 , 000 test images from 10 different classes, and CIFAR-100 consists of 50 , 000 training images and 10 , 000 test images from 100 different classes.

Original: One can download CIFAR-10 and CIFAR-100 from https:// www.cs.toronto.edu/˜kriz/cifar.html .
Revised: One can download CIFAR-10 and CIFAR-100 from https:// www.cs.toronto.edu/˜kriz/cifar.html .

Original: • Fake CIFAR-10 consists of 100 , 000 test images from 10 different classes.
Revised: • Fake CIFAR-10 consists of 100 , 000 test images from 10 different classes.

Original: The fake CIFAR-10 is generated by conditional BigGAN (Zhao et al., 2020).
Revised: The fake CIFAR-10 is generated by conditional BigGAN (Zhao et al., 2020).

Original: The pre-trained BigGAN is provided by Kang & Park (2020) and can be download from https://drive.
Revised: The pre-trained BigGAN is provided by Kang & Park (2020) and can be download from https://drive.

Original: google.com/drive/folders/1xVN7dQPWMLi8gDZEb5FThkjbFtIdzb6b .
Revised: google.com/drive/folders/1xVN7dQPWMLi8gDZEb5FThkjbFtIdzb6b .

Original: • Fake CIFAR-100 consists of 100 , 000 test images from 100 different classes.
Revised: • Fake CIFAR-100 consists of 100 , 000 test images from 100 different classes.

Original: The generation of fake CIFAR-100 is similar to it of fake CIFAR-10, while the training set for the BigGAN is replaced from CIFAR-10 to CIFAR-100.
Revised: The generation of fake CIFAR-100 is similar to it of fake CIFAR-10, while the training set for the BigGAN is replaced from CIFAR-10 to CIFAR-100.

Original: A. M ODEL ARCHITECTURES
Revised: A. M ODEL ARCHITECTURES

Original: We use different neural network architectures in our experiments, including VGG-16, ResNet-18, and WideResNet-28.
Revised: We use different neural network architectures in our experiments, including VGG-16, ResNet-18, and WideResNet-28.

Original: The architectures are presented in Table 1.
Revised: The architectures are presented in Table 1.

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

Original: A. IMPLEMENTATION DETAILS
Revised: A. IMPLEMENTATION DETAILS

Original: This section provides all the additional implementation details for our experiments.
Revised: This section provides all the additional implementation details for our experiments.

Original: A.3. M ODEL TRAINING
Revised: A.3. M ODEL TRAINING

Original: We employ SGD to optimize all the models and the momentum factor is
Revised: We employ SGD to optimize all the models and the momentum factor is

Original: [Table]
Revised: [Table]

Original: The weight decay factor is set to 5 e-
Revised: The weight decay factor is set to 5 e-

Original: [Table]
Revised: [Table]

Original: and the learning rate is decayed by 0 .
Revised: and the learning rate is decayed by 0 .

Original: 2 every 50 epochs.
Revised: 2 every 50 epochs.

Original: A.3. A DDITIONAL DETAILS IN S ECTION 4. 
Revised: A.3. A DDITIONAL DETAILS IN S ECTION 4.

Original: We train VGG-16, ResNet-18, Wide-ResNet-28 on CIFAR-10 and CIFAR-100.
Revised: We train VGG-16, ResNet-18, Wide-ResNet-28 on CIFAR-10 and CIFAR-100.

Original: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 .
Revised: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 .

Original: There are three training strategies included in this experiment: standard training, non-data-augmentation training, and adversarial training.
Revised: There are three training strategies included in this experiment: standard training, non-data-augmentation training, and adversarial training.

Original: In the adversarial training, the radius of the adversarial perturbation is set as 10 / 255 and l ∞ distance is selected.
Revised: In the adversarial training, the radius of the adversarial perturbation is set as 10 / 255 and l ∞ distance is selected.

Original: The basic data augmentation (cropping and ﬂipping) in the standard training and adversarial training is achieved by the following Pytorch code:
Revised: The basic data augmentation (cropping and ﬂipping) in the standard training and adversarial training is achieved by the following Pytorch code:

Original: 1 transforms.
Revised: 1 transforms.

Original: RandomCrop(32, padding=4) 2 transforms.
Revised: RandomCrop(32, padding=4) 2 transforms.

Original: RandomHorizontalFlip()
Revised: RandomHorizontalFlip()

Original: The experiment is repeated for 10 trials for each (dataset, architecture, training strategy) setting.
Revised: The experiment is repeated for 10 trials for each (dataset, architecture, training strategy) setting.

Original: A.3. A DDITIONAL DETAILS IN S ECTION 4. 
Revised: A.3. A DDITIONAL DETAILS IN S ECTION 4.

Original: We repeatedly train 10 ResNet-18 on CIFAR-10 and CIFAR-100, respectively, with different random seeds.
Revised: We repeatedly train 10 ResNet-18 on CIFAR-10 and CIFAR-100, respectively, with different random seeds.

Original: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 and 0 .
Revised: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 and 0 .

Original: 01 , respectively.
Revised: 01 , respectively.

Original: Basic data augmentation is included during the training process.
Revised: Basic data augmentation is included during the training process.

Original: A.3. A DDITIONAL DETAILS IN S ECTION 4. 
Revised: A.3. A DDITIONAL DETAILS IN S ECTION 4.

Original: We randomly sample examples from the training set of CIFAR-10 and CIFAR-100 to form ﬁve datasets with different sizes of [2000 , 5000 , 10000 , 20000 , 50000] , respectively.
Revised: We randomly sample examples from the training set of CIFAR-10 and CIFAR-100 to form ﬁve datasets with different sizes of [2000 , 5000 , 10000 , 20000 , 50000] , respectively.

Original: 10 ResNet-18 are trained for each dataset.
Revised: 10 ResNet-18 are trained for each dataset.

Original: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 .
Revised: In the training procedure, the model is trained for 200 epochs, in which the batch size is set to 128 , and the learning rate is initialized as 0 . 1 .

Original: Basic data augmentation is included during the training process.
Revised: Basic data augmentation is included during the training process.

Original: A.3. A DDITIONAL DETAILS IN S ECTION 4. 
Revised: A.3. A DDITIONAL DETAILS IN S ECTION 4.

Original: We randomly change the labels of 20% examples in the training set of CIFAR-10 and CIFAR-100.
Revised: We randomly change the labels of 20% examples in the training set of CIFAR-10 and CIFAR-100.

Original: Then, 10 ResNet-18 are optimize by SGD for 500 epochs on the noise CIFAR-10 and CIFAR-100, respectively.
Revised: Then, 10 ResNet-18 are optimize by SGD for 500 epochs on the noise CIFAR-10 and CIFAR-100, respectively.

Original: the momentum factor is 0 . 9 , and the learning rate is 0 .
Revised: the momentum factor is 0 . 9 , and the learning rate is 0 .

Original: 001 and does not decay during the training process.
Revised: 001 and does not decay during the training process.

Original: A.3. A DDITIONAL DETAILS IN S ECTION 5. 
Revised: A.3. A DDITIONAL DETAILS IN S ECTION 5.

Original: We randomly sample examples from the training set of CIFAR-10 and CIFAR-100 to form ﬁve datasets with different sizes of [2000 , 5000 , 10000 , 20000 , 50000] , respectively.
Revised: We randomly sample examples from the training set of CIFAR-10 and CIFAR-100 to form ﬁve datasets with different sizes of [2000 , 5000 , 10000 , 20000 , 50000] , respectively.

Original: For each dataset, we obtain 10 η -subsets with different η of [0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 .
Revised: For each dataset, we obtain 10 η -subsets with different η of [0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 .

Original: via a coreset selection approach named selection via proxy (Coleman et al., 2020).
Revised: via a coreset selection approach named selection via proxy (Coleman et al., 2020).

Original: The related code can be downloaded from https://github.com/stanford-futuredata/ selection-via-proxy .
Revised: The related code can be downloaded from https://github.com/stanford-futuredata/ selection-via-proxy .

Original: The ResNet-18 is repeatedly trained for 10 trials to estimate the complexity of decision boundaries for each η -subset.
Revised: The ResNet-18 is repeatedly trained for 10 trials to estimate the complexity of decision boundaries for each η -subset.

Original: B A DDITIONAL DISCUSSION ON DECISION BOUNDARY VARIABILITY B. A LGORITHM DB VARIABILITY AND THE ENTROPY OF DECISION BOUNDARIES
Revised: B A DDITIONAL DISCUSSION ON DECISION BOUNDARY VARIABILITY B. A LGORITHM DB VARIABILITY AND THE ENTROPY OF DECISION BOUNDARIES

Original: If Assumption 1 holds for all θ ∼ Q , 1 − AV ( f Q , D ) can be rewritten as
Revised: If Assumption 1 holds for all θ ∼ Q , 1 − AV ( f Q , D ) can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: The term (cid:80) ki
Revised: The term (cid:80) ki

Original: =1 E 2 θ ∼ Q
Revised: =1 E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] can be considered to measure the degree of prediction uncertainty for the given voxel x in the input space R n .
Revised: [ I ( T ( f θ , x ) = i )] can be considered to measure the degree of prediction uncertainty for the given voxel x in the input space R n .

Original: If we leverage − log( · ) on the term (cid:80) ki =1 E 2 θ ∼ Q
Revised: If we leverage − log( · ) on the term (cid:80) ki =1 E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] , then − log (cid:80) ki
Revised: [ I ( T ( f θ , x ) = i )] , then − log (cid:80) ki

Original: =1 E 2 θ ∼ Q
Revised: =1 E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] denotes the collision entropy of prediction made by the Gibbs classiﬁer f Q on x .
Revised: [ I ( T ( f θ , x ) = i )] denotes the collision entropy of prediction made by the Gibbs classiﬁer f Q on x .

Original: We can also replace the collision entropy with canonical Shannon entropy − (cid:80) ki
Revised: We can also replace the collision entropy with canonical Shannon entropy − (cid:80) ki

Original: =1 E θ ∼ Q [ I ( T ( f θ , x ) = i )] log E θ ∼ Q
Revised: =1 E θ ∼ Q [ I ( T ( f θ , x ) = i )] log E θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] in the future research.
Revised: [ I ( T ( f θ , x ) = i )] in the future research.

Original: Hence, the algorithm DB variability is closely related to the “entropy of decision boundary”, and the uncanny generalization in neural networks might be further uncovered by investigating this low entropy of decision boundary.
Revised: Hence, the algorithm DB variability is closely related to the “entropy of decision boundary”, and the uncanny generalization in neural networks might be further uncovered by investigating this low entropy of decision boundary.

Original: B. D ATA DB VARIABILITY AND THE COMPLEXITY OF DB S
Revised: B. D ATA DB VARIABILITY AND THE COMPLEXITY OF DB S

Original: According to Guan & Loew (2020), a complex decision boundary is considered to own large curvatures and conjectured to indicate inferior generalization.
Revised: According to Guan & Loew (2020), a complex decision boundary is considered to own large curvatures and conjectured to indicate inferior generalization.

Original: Nevertheless, from the perspective of causality, we argue that the large curvature or non-linearity of DBs is the result other than the cause for the classiﬁcation task, and the primary reason for shaping a complex DB during the training procedure should be the signiﬁcant non-linearity of the training data.
Revised: Nevertheless, from the perspective of causality, we argue that the large curvature or non-linearity of DBs is the result other than the cause for the classiﬁcation task, and the primary reason for shaping a complex DB during the training procedure should be the signiﬁcant non-linearity of the training data.

Original: If only the geometric properties of decision boundaries are analysed without investigating the data, the results might be incomplete and even misleading.
Revised: If only the geometric properties of decision boundaries are analysed without investigating the data, the results might be incomplete and even misleading.

Original: Another obstacle for describing the complexity of DBs via its geometric properties is the huge dimensional input space, which makes the geometric properties of DBs hard to quantify and estimate.
Revised: Another obstacle for describing the complexity of DBs via its geometric properties is the huge dimensional input space, which makes the geometric properties of DBs hard to quantify and estimate.

Original: Therefore, deﬁning the complexity of DBs based on its curvature is not rational and impractical.
Revised: Therefore, deﬁning the complexity of DBs based on its curvature is not rational and impractical.

Original: Here we consider the complexity of DBs from the perspective of the training set.
Revised: Here we consider the complexity of DBs from the perspective of the training set.

Original: During the training procedure, a small part of training examples, considered as “support vectors”, play a more critical role in supervising the formation of decision boundaries and compelling the DB to be gradually more complicated.
Revised: During the training procedure, a small part of training examples, considered as “support vectors”, play a more critical role in supervising the formation of decision boundaries and compelling the DB to be gradually more complicated.

Original: If the construction of decision boundaries relies on fewer “support vectors”, the decision boundary should be simpler.
Revised: If the construction of decision boundaries relies on fewer “support vectors”, the decision boundary should be simpler.

Original: In other words, if these “support vectors” are excluded from the training sample, the DB will be notably dissimilar when the network is retrained on the modiﬁed training set.
Revised: In other words, if these “support vectors” are excluded from the training sample, the DB will be notably dissimilar when the network is retrained on the modiﬁed training set.

Original: Hence, the complexity of DBs can be also deﬁned with the notion of the data DB variability:
Revised: Hence, the complexity of DBs can be also deﬁned with the notion of the data DB variability:

Original: Deﬁnition B.1 (complexity of decision boundaries) .
Revised: Deﬁnition B.1 (complexity of decision boundaries) .

Original: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ , where θ ∼ A ( S ) is returned by leveraging the stochastic learning
Revised: Let f θ ( x ) : R n → R k be a classiﬁer network parameterized by θ , where θ ∼ A ( S ) is returned by leveraging the stochastic learning

Original: algorithm A on the training set S , which is sampled from the data generating distribution D .
Revised: algorithm A on the training set S , which is sampled from the data generating distribution D .

Original: We term S η ⊂ S as a η -subset of S if |S η | |S| = η .
Revised: We term S η ⊂ S as a η -subset of S if |S η | |S| = η .

Original: Then, if we ﬁxed η and
Revised: Then, if we ﬁxed η and

Original: [Equation]
Revised: [Equation]

Original: (25) the decision boundary of f A ( S ) is said to possess a ( (cid:15), η ) -complexity .
Revised: (25) the decision boundary of f A ( S ) is said to possess a ( (cid:15), η ) -complexity .

Original: Implications about the complexity of DB.
Revised: Implications about the complexity of DB.

Original: With Deﬁnition B.1, we could more intuitively understand the relationship between the complexity of DBs and the generalization ability in deep learning: (1) difﬁcult tasks generally have more complex decision boundaries, since their datum are more nonlinear and contain more “support vectors”; (2) in adversarial training, each data point is converted to a “data ball” with the radius of the adversarial perturbation and has more impact on forming
Revised: With Deﬁnition B.1, we could more intuitively understand the relationship between the complexity of DBs and the generalization ability in deep learning: (1) difﬁcult tasks generally have more complex decision boundaries, since their datum are more nonlinear and contain more “support vectors”; (2) in adversarial training, each data point is converted to a “data ball” with the radius of the adversarial perturbation and has more impact on forming

Original: the
Revised: the

Original: DBs.
Revised: DBs.

Original: Hence, adversarial training contributes to a more complex decision boundary by enlarging the “support vector set”; (3) for data augmentation, generated images are also considered to obey the data generation distribution D .
Revised: Hence, adversarial training contributes to a more complex decision boundary by enlarging the “support vector set”; (3) for data augmentation, generated images are also considered to obey the data generation distribution D .

Original: Hence, data augmentation decreases the complexity of decision boundaries by greatly expanding the training set S , while |S η | has only a slight growth.
Revised: Hence, data augmentation decreases the complexity of decision boundaries by greatly expanding the training set S , while |S η | has only a slight growth.

Original: Besides, according to our theoretical results in Section 5.2, it can be veriﬁed that neural networks with simpler decision boundaries possess better generalization performance .
Revised: Besides, according to our theoretical results in Section 5.2, it can be veriﬁed that neural networks with simpler decision boundaries possess better generalization performance .

Original: C P ROOFS
Revised: C P ROOFS

Original: To avoid technicalities, the measurability/integrability issues are ignored throughout this paper.
Revised: To avoid technicalities, the measurability/integrability issues are ignored throughout this paper.

Original: Moreover, Fubini’s theorem is assumed to be applicable for any integration w.r.t.
Revised: Moreover, Fubini’s theorem is assumed to be applicable for any integration w.r.t.

Original: multiple variables.
Revised: multiple variables.

Original: In other words, the order of integrations is exchangeable.
Revised: In other words, the order of integrations is exchangeable.

Original: We also use E 2 [ · ] to denote [ E [ · ]]in our proof for simplicity.
Revised: We also use E 2 [ · ] to denote [ E [ · ]]in our proof for simplicity.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Proof.
Revised: Proof.

Original: If Assumption 1 holds for all θ ∼ Q , we have
Revised: If Assumption 1 holds for all θ ∼ Q , we have

Original: [Equation]
Revised: [Equation]

Original: Hence,
Revised: Hence,

Original: AV
Revised: AV

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: The proof is completed.
Revised: The proof is completed.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Proof.
Revised: Proof.

Original: When the classiﬁcation is binary, i.e. , k = 2 , with Assumption 1, we have
Revised: When the classiﬁcation is binary, i.e. , k = 2 , with Assumption 1, we have

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: Plugging in Var ( x ,y )
Revised: Plugging in Var ( x ,y )

Original: ∼D [ E θ ∼ Q [ I ( y ∈ T ( f θ , x ))]]
Revised: ∼D [ E θ ∼ Q [ I ( y ∈ T ( f θ , x ))]]

Original: =
Revised: =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E 2 θ ∼ Q
Revised: ∼D E 2 θ ∼ Q

Original: [ I ( y ∈ T ( f θ , x ))] − (
Revised: [ I ( y ∈ T ( f θ , x ))] − (

Original: [Equation]
Revised: [Equation]

Original: R D ( Q ))
Revised: R D ( Q ))

Original: 2 yields
Revised: 2 yields

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: Plugging in AV ( f Q , D ) = 1 − E ( x ,y )
Revised: Plugging in AV ( f Q , D ) = 1 − E ( x ,y )

Original: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) =
Revised: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) =

Original: T ( f θ (cid:48) , x ))] and solving the inequality yield the desired inequality and ﬁnishes the proof of Theorem 2.
Revised: T ( f θ (cid:48) , x ))] and solving the inequality yield the desired inequality and ﬁnishes the proof of Theorem 2.

Original: C. P ROOF OF L EMMA
Revised: C. P ROOF OF L EMMA

Original: Proof.
Revised: Proof.

Original: Without loss of generality, we assume that all examples are belong to the ﬁrst class, i.e. , y = 1 .
Revised: Without loss of generality, we assume that all examples are belong to the ﬁrst class, i.e. , y = 1 .

Original: Hence, for 1 − AV ( f Q , D ) =
Revised: Hence, for 1 − AV ( f Q , D ) =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) =
Revised: ∼D E θ , θ (cid:48) ∼ Q [ I ( T ( f θ , x ) =

Original: T ( f θ (cid:48) , x ))] , we have
Revised: T ( f θ (cid:48) , x ))] , we have

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: The penultimate inequality holds because 2 p i p j ≤ p 2 i + p 2 j .
Revised: The penultimate inequality holds because 2 p i p j ≤ p 2 i + p 2 j .

Original: Moreover, the variance of
Revised: Moreover, the variance of

Original: E θ ∼ Q [ I ( y / ∈ T ( f θ , x ))] is
Revised: E θ ∼ Q [ I ( y / ∈ T ( f θ , x ))] is

Original: [Equation]
Revised: [Equation]

Original: plugging the equality into Eq.
Revised: plugging the equality into Eq.

Original: 44 yields
Revised: 44 yields

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: y )
Revised: y )

Original: [Equation]
Revised: [Equation]

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: Q [ I ( T ( f θ , x ) (cid:54) =
Revised: Q [ I ( T ( f θ , x ) (cid:54) =

Original: y )]]
Revised: y )]]

Original: + ( k − 1) E
Revised: + ( k − 1) E

Original: ( x ,y )
Revised: ( x ,y )

Original: ∼D E 2 θ ∼ Q
Revised: ∼D E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) (cid:54) =
Revised: [ I ( T ( f θ , x ) (cid:54) =

Original: y )] − ( k − 1) E 2( x ,y )
Revised: y )] − ( k − 1) E 2( x ,y )

Original: ∼D E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) =
Revised: ∼D E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) =

Original: y )]
Revised: y )]

Original: + ( k − 1) k
Revised: + ( k − 1) k

Original: (cid:88) i =
Revised: (cid:88) i =

Original: E 2( x ,y )
Revised: E 2( x ,y )

Original: ∼D E θ ∼ Q
Revised: ∼D E θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] +k (cid:88) i =
Revised: [ I ( T ( f θ , x ) = i )] +k (cid:88) i =

Original: E ( x
Revised: E ( x

Original: [Equation]
Revised: [Equation]

Original: y )
Revised: y )

Original: ∼D
Revised: ∼D

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( f
Revised: ( f

Original: [Equation]
Revised: [Equation]

Original: k (cid:88) i =
Revised: k (cid:88) i =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E θ ∼ Q
Revised: ∼D E θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] (47)
Revised: [ I ( T ( f θ , x ) = i )] (47)

Original: Then,
Revised: Then,

Original: because E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) = y )]
Revised: because E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) = y )]

Original: = (cid:80) ki
Revised: = (cid:80) ki

Original: =2 E θ ∼ Q [ I ( T ( f θ , x ) = i )] , we have k (cid:88) i =
Revised: =2 E θ ∼ Q [ I ( T ( f θ , x ) = i )] , we have k (cid:88) i =

Original: E 2( x ,y )
Revised: E 2( x ,y )

Original: ∼D E θ ∼ Q
Revised: ∼D E θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] ≤ E 2( x ,y )
Revised: [ I ( T ( f θ , x ) = i )] ≤ E 2( x ,y )

Original: ∼D E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) =
Revised: ∼D E θ ∼ Q [ I ( T ( f θ , x ) (cid:54) =

Original: y )] (48) and k (cid:88) i =
Revised: y )] (48) and k (cid:88) i =

Original: E ( x ,y )
Revised: E ( x ,y )

Original: ∼D E 2 θ ∼ Q
Revised: ∼D E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) = i )] ≤ E ( x ,y )
Revised: [ I ( T ( f θ , x ) = i )] ≤ E ( x ,y )

Original: ∼D E 2 θ ∼ Q
Revised: ∼D E 2 θ ∼ Q

Original: [ I ( T ( f θ , x ) (cid:54) =
Revised: [ I ( T ( f θ , x ) (cid:54) =

Original: y )] .
Revised: y )] .

Original: Therefore, Eq. 47 can be scaled to
Revised: Therefore, Eq. 47 can be scaled to

Original: [Equation]
Revised: [Equation]

Original: Then, with Assumption 2 that E ( x ,y )
Revised: Then, with Assumption 2 that E ( x ,y )

Original: ∼D E 2 θ ∼ Q
Revised: ∼D E 2 θ ∼ Q

Original: [ I ( y / ∈ T ( f θ , x ))] ≤ 2 k +1 R D ( Q ) , we get the following inequality w.r.t.
Revised: [ I ( y / ∈ T ( f θ , x ))] ≤ 2 k +1 R D ( Q ) , we get the following inequality w.r.t.

Original: the variance:
Revised: the variance:

Original: [Equation]
Revised: [Equation]

Original: The proof is completed.
Revised: The proof is completed.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: Proof.
Revised: Proof.

Original: For the random variable Z i , recall the one-sided Chebyshev’s inequality that
Revised: For the random variable Z i , recall the one-sided Chebyshev’s inequality that

Original: [Equation]
Revised: [Equation]

Original: Let Z i =
Revised: Let Z i =

Original: E θ ∼ Q
Revised: E θ ∼ Q

Original: [ I ( y i / ∈ T ( f θ , x i ))] , so we have
Revised: [ I ( y i / ∈ T ( f θ , x i ))] , so we have

Original: [Equation]
Revised: [Equation]

Original: Plugging Lemma 1 into the above inequality yields
Revised: Plugging Lemma 1 into the above inequality yields

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Plugging in δ = AV ( f Q , D ) 2 m ( k − 1) a 2 yields the desired inequality and completes the proof.
Revised: Plugging in δ = AV ( f Q , D ) 2 m ( k − 1) a 2 yields the desired inequality and completes the proof.

Original: C. P ROOF OF L EMMA
Revised: C. P ROOF OF L EMMA

Original: Proof.
Revised: Proof.

Original: From the deﬁnition of the complexity of DB, there exists a η -subset S η s.t.
Revised: From the deﬁnition of the complexity of DB, there exists a η -subset S η s.t.

Original: [Equation]
Revised: [Equation]

Original: Recall the LHS =
Revised: Recall the LHS =

Original: |R D ( A ( S ))
Revised: |R D ( A ( S ))

Original: − R D ( A ( S η ))
Revised: − R D ( A ( S η ))

Original: | , then
Revised: | , then

Original: LHS =
Revised: LHS =

Original: (cid:12)(cid:12) E ( x ,y )
Revised: (cid:12)(cid:12) E ( x ,y )

Original: ∼D E θ ∼A ( S )
Revised: ∼D E θ ∼A ( S )

Original: [ I ( y / ∈ T ( f θ , x ))] − E ( x ,y )
Revised: [ I ( y / ∈ T ( f θ , x ))] − E ( x ,y )

Original: ∼D E θ ∼A ( S η )
Revised: ∼D E θ ∼A ( S η )

Original: [ I ( y / ∈ T ( f θ , x ))] (cid:12)(cid:12) (56) = (cid:12)(cid:12) E ( x ,y )
Revised: [ I ( y / ∈ T ( f θ , x ))] (cid:12)(cid:12) (56) = (cid:12)(cid:12) E ( x ,y )

Original: ∼D E θ ∼A ( S ) , θ (cid:48)
Revised: ∼D E θ ∼A ( S ) , θ (cid:48)

Original: ∼A ( S η ) [ I ( y / ∈ T ( f θ , x )) − I
Revised: ∼A ( S η ) [ I ( y / ∈ T ( f θ , x )) − I

Original: ( y / ∈ T ( f θ (cid:48) , x ))] (cid:12)(cid:12) (57) ≤ E ( x ,y )
Revised: ( y / ∈ T ( f θ (cid:48) , x ))] (cid:12)(cid:12) (57) ≤ E ( x ,y )

Original: ∼D E θ ∼A ( S ) , θ (cid:48)
Revised: ∼D E θ ∼A ( S ) , θ (cid:48)

Original: ∼A ( S η ) [ | I ( y / ∈ T ( f θ , x )) − I
Revised: ∼A ( S η ) [ | I ( y / ∈ T ( f θ , x )) − I

Original: ( y / ∈ T ( f θ (cid:48) , x ))
Revised: ( y / ∈ T ( f θ (cid:48) , x ))

Original: | ] (58) ≤ E ( x ,y )
Revised: | ] (58) ≤ E ( x ,y )

Original: ∼D E θ ∼A ( S ) , θ (cid:48)
Revised: ∼D E θ ∼A ( S ) , θ (cid:48)

Original: ∼A ( S η ) [ I ( T ( f θ , x ) (cid:54) =
Revised: ∼A ( S η ) [ I ( T ( f θ , x ) (cid:54) =

Original: T ( f θ (cid:48) , x ))]
Revised: T ( f θ (cid:48) , x ))]

Original: = (cid:15) (59)
Revised: = (cid:15) (59)

Original: The proof is completed.
Revised: The proof is completed.

Original: C. P ROOF OF L EMMA
Revised: C. P ROOF OF L EMMA

Original: Proof.
Revised: Proof.

Original: Let
Revised: Let

Original: [Equation]
Revised: [Equation]

Original: By applying Hoeffding’s
Revised: By applying Hoeffding’s

Original: Inequality, we have
Revised: Inequality, we have

Original: [Equation]
Revised: [Equation]

Original: Plug in δ = exp( − 2 mt 2 ) into Eq.
Revised: Plug in δ = exp( − 2 mt 2 ) into Eq.

Original: 62, thus, with the probability of at least 1 − δ , we have
Revised: 62, thus, with the probability of at least 1 − δ , we have

Original: [Equation]
Revised: [Equation]

Original: Following the proof of Lemma 2, we can derive the inequality of |R S ( A ( S η ))
Revised: Following the proof of Lemma 2, we can derive the inequality of |R S ( A ( S η ))

Original: − R S
Revised: − R S

Original: ( A ( S ))
Revised: ( A ( S ))

Original: | ≤ ˆ µ .
Revised: | ≤ ˆ µ .

Original: Plugging this into Eq.
Revised: Plugging this into Eq.

Original: 63 yields the desired inequality and concludes the proof of Lemma 3.
Revised: 63 yields the desired inequality and concludes the proof of Lemma 3.

Original: C. P ROOF OF T HEOREM
Revised: C. P ROOF OF T HEOREM

Original: We ﬁrst introduce Lemma 4 and Lemma 5 as below.
Revised: We ﬁrst introduce Lemma 4 and Lemma 5 as below.

Original: Lemma 4 (Lemma 30.1 in (Shalev-Shwartz & Ben-David, 2014)) .
Revised: Lemma 4 (Lemma 30.1 in (Shalev-Shwartz & Ben-David, 2014)) .

Original: Assume T and V are two datasets independently sampled from the data generating distribution D , then, with the probability of at least 1 − δ , we have
Revised: Assume T and V are two datasets independently sampled from the data generating distribution D , then, with the probability of at least 1 − δ , we have

Original: [Equation]
Revised: [Equation]

Original: Lemma 5 (Theorem 30.2 in (Shalev-Shwartz & Ben-David, 2014)) .
Revised: Lemma 5 (Theorem 30.2 in (Shalev-Shwartz & Ben-David, 2014)) .

Original: Let S η be a η -subnet of the dataset S , which is sampled from the data generation distribution D and the sample size |S| = m .
Revised: Let S η be a η -subnet of the dataset S , which is sampled from the data generation distribution D and the sample size |S| = m .

Original: Let S\S η = S − S η and assume η ≤ 0 .
Revised: Let S\S η = S − S η and assume η ≤ 0 .

Original: Then, with the probability of at least 1 − δ over a sample of size m , we have
Revised: Then, with the probability of at least 1 − δ over a sample of size m , we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Proof of Lemma 5.
Revised: Proof of Lemma 5.

Original: Pr
Revised: Pr

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Plug in δ (cid:48)
Revised: Plug in δ (cid:48)

Original: = (cid:16) eη (cid:17)
Revised: = (cid:16) eη (cid:17)

Original: ηm δ , and use the assumption η ≤ 12 , which implies |S\S η | ≥ m 2 , then, with the probability of at least 1 − δ (cid:48) , we have that
Revised: ηm δ , and use the assumption η ≤ 12 , which implies |S\S η | ≥ m 2 , then, with the probability of at least 1 − δ (cid:48) , we have that

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof.
Revised: which concludes the proof.

Original: With the above lemmas, we can derive the generalization bound based on the complexity of decision boundary.
Revised: With the above lemmas, we can derive the generalization bound based on the complexity of decision boundary.

Original: Proof of Theorem 4.
Revised: Proof of Theorem 4.

Original: With the assumption of zero-training error that R S ( A ( S ))
Revised: With the assumption of zero-training error that R S ( A ( S ))

Original: =
Revised: =

Original: R S η
Revised: R S η

Original: ( A ( S η ))
Revised: ( A ( S η ))

Original: = 0 , we can derive the follow inequality:
Revised: = 0 , we can derive the follow inequality:

Original: [Equation]
Revised: [Equation]

Original: Then, by plugging the above equation into to Lemma 3, with the probability of at least 1 − δ , we have
Revised: Then, by plugging the above equation into to Lemma 3, with the probability of at least 1 − δ , we have

Original: [Equation]
Revised: [Equation]

Original: Through combining this with Lemma 5, with the probability of at least 1 − 2 δ , we have
Revised: Through combining this with Lemma 5, with the probability of at least 1 − 2 δ , we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Plugging the equality of R D ( A ( S )) ≤ R D
Revised: Plugging the equality of R D ( A ( S )) ≤ R D

Original: ( A ( S η ))
Revised: ( A ( S η ))

Original: + (cid:15) in Lemma 2 into Eq.
Revised: + (cid:15) in Lemma 2 into Eq.

Original: 73, with the probability of at least 1 − 2 δ , we have
Revised: 73, with the probability of at least 1 − 2 δ , we have

Original: [Equation]
Revised: [Equation]

Original: Plugging in δ (cid:48)
Revised: Plugging in δ (cid:48)

Original: = 2 δ yields the desired inequality of Eq.
Revised: = 2 δ yields the desired inequality of Eq.

Original: When m is sufﬁcient large, √ 4Ω∆ can be dropped due to √ 4Ω∆ ≤ Ω + ∆ .
Revised: When m is sufﬁcient large, √ 4Ω∆ can be dropped due to √ 4Ω∆ ≤ Ω + ∆ .

Original: Considering η ≤ 0 .
Revised: Considering η ≤ 0 .

Original: 5 , Ω ≤ 2 (cid:16)(cid:113) 12 m log 1 δ + (cid:15) (cid:17) =
Revised: 5 , Ω ≤ 2 (cid:16)(cid:113) 12 m log 1 δ + (cid:15) (cid:17) =

Original: O ( 1 √ m + (cid:15) ) .
Revised: O ( 1 √ m + (cid:15) ) .

Original: The term 1 m log 1 δ in ∆ can be dropped because it has a faster convergence speed compared to (cid:113) 12 m log 1 δ in Ω .
Revised: The term 1 m log 1 δ in ∆ can be dropped because it has a faster convergence speed compared to (cid:113) 12 m log 1 δ in Ω .

Original: Because log 1 δ is considered as a constant, we have
Revised: Because log 1 δ is considered as a constant, we have

Original: [Equation]
Revised: [Equation]

Original: The proof of Theorem 4 is ﬁnished.
Revised: The proof of Theorem 4 is ﬁnished.

Revised: D A DDITIONAL EXPERIMENTAL RESULTS

Revised: This appendix collect experimental results omitted from the main text due to the space limitation.

Revised: D. A DDITIONAL RESULTS FOR LEARNING RATE

Revised: Accroding to Li et al.

Revised: that learning rate is a major factor of affecting generalization of networks, we investigate the relationship between algorithm DB variability and learning rate by training 20 ResNet-18 with different constant learning rates (no decay) of 0 .

Original: {+001 and 0 . 
+}
Revised: {+001 and 0 . 
+}

Revised: on CIFARuntil the training procedure converges.

Revised: Then, the average test error and algorithm DB variability are calculated at each epoch, as shown in Figure 5.

Revised: From the plots, we observe that: (1) the larger learning rate 0 .

Revised: 001 contributes to better generalization performance; and (2) algorithm DB variability with the larger learning rate 0 .

Revised: 001 is smaller during the training process, compared to the learning rate 0 . 0001 .

Revised: Therefore, there is still a negative correlation between algorithm DB variability and generalization by varying the learning rate, and thus Hypothesis 1 is well supported.

Revised: [Figure]