Original: T OWARDS U NDERSTANDING R EGULARIZATION IN B ATCH N ORMALIZATION
Revised: T OWARDS U NDERSTANDING R EGULARIZATION IN B ATCH N ORMALIZATION

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: Batch Normalization (BN) improves both convergence and generalization in training neural networks.
Revised: Batch Normalization (BN) improves both convergence and generalization in training neural networks.

Original: This work understands these phenomena theoretically.
Revised: This work understands these phenomena theoretically.

Original: We analyze BN by using a basic block of neural networks, consisting of a kernel layer, a BN layer, and a nonlinear activation function.
Revised: We analyze BN by using a basic block of neural networks, consisting of a kernel layer, a BN layer, and a nonlinear activation function.

Original: This basic network helps us understand the impacts of BN in three aspects
Revised: This basic network helps us understand the impacts of BN in three aspects

Original: First, by viewing BN as an implicit regularizer
Revised: First, by viewing BN as an implicit regularizer

Original: BN can be decomposed into population normalization (
Revised: BN can be decomposed into population normalization (

Original: PN) and gamma decay as an explicit regularization.
Revised: PN) and gamma decay as an explicit regularization.

Original: Second learning dynamics of
Revised: Second learning dynamics of

Original: BN and the regularization show that training converged with large maximum and effective learning rate.
Revised: BN and the regularization show that training converged with large maximum and effective learning rate.

Original: Third
Revised: Third

Original: , generalization of
Revised: , generalization of

Original: BN is explored by using statistical mechanics.
Revised: BN is explored by using statistical mechanics.

Original: Experiments demonstrate that BN in convolutional neural networks share the same traits of regularization as the above analyses.
Revised: Experiments demonstrate that BN in convolutional neural networks share the same traits of regularization as the above analyses.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Batch Normalization (BN) is an indispensable component in many deep neural networks (He et al., 2016; Huang et al., 2017).
Revised: Batch Normalization (BN) is an indispensable component in many deep neural networks (He et al., 2016; Huang et al., 2017).

Original: BN has been widely used in various areas such as machine vision, speech and natural language processing.
Revised: BN has been widely used in various areas such as machine vision, speech and natural language processing.

Original: Experimental studies (Ioffe & Szegedy, 2015) suggested that BN improves convergence and generalization by enabling large learning rate and preventing overﬁtting when training deep networks.
Revised: Experimental studies (Ioffe & Szegedy, 2015) suggested that BN improves convergence and generalization by enabling large learning rate and preventing overﬁtting when training deep networks.

Original: Understanding BN theoretically is a key question.
Revised: Understanding BN theoretically is a key question.

Original: This work investigates regularization of BN as well as its optimization and generalization in a singlelayer perceptron, which is a building block of deep models, consisting of a kernel layer, a BN layer, and a nonlinear activation function such as ReLU.
Revised: This work investigates regularization of BN as well as its optimization and generalization in a singlelayer perceptron, which is a building block of deep models, consisting of a kernel layer, a BN layer, and a nonlinear activation function such as ReLU.

Original: The computation of BN is written by
Revised: The computation of BN is written by

Original: [Equation]
Revised: [Equation]

Original: This work denotes a scalar and a vector by using lowercase letter ( e.g. x ) and bold lowercase letter ( e.g. x ) respectively.
Revised: This work denotes a scalar and a vector by using lowercase letter ( e.g. x ) and bold lowercase letter ( e.g. x ) respectively.

Original: In Eqn.(1), y is the output of a neuron, g ( · ) denotes an activation function, h and ˆ h are hidden values before and after batch normalization, w and x are kernel weight vector and network input respectively.
Revised: In Eqn.(1), y is the output of a neuron, g ( · ) denotes an activation function, h and ˆ h are hidden values before and after batch normalization, w and x are kernel weight vector and network input respectively.

Original: In BN, µ B and σ B represent the mean and standard deviation of h .
Revised: In BN, µ B and σ B represent the mean and standard deviation of h .

Original: They are estimated within a batch of samples for each neuron independently.
Revised: They are estimated within a batch of samples for each neuron independently.

Original: γ is a scale parameter and β is a shift parameter.
Revised: γ is a scale parameter and β is a shift parameter.

Original: In what follows, Sec.1.1 overviews assumptions and main results, and Sec.1.
Revised: In what follows, Sec.1.1 overviews assumptions and main results, and Sec.1.

Original: presents relationships with previous work.
Revised: presents relationships with previous work.

Original: 1. O VERVIEW OF R ESULTS
Revised: 1. O VERVIEW OF R ESULTS

Original: We overview results in three aspects.
Revised: We overview results in three aspects.

Original: • First, Sec.2 decomposes BN into population normalization (PN) and gamma decay.
Revised: • First, Sec.2 decomposes BN into population normalization (PN) and gamma decay.

Original: To better understand BN, we treat a single-layer perceptron with ReLU activation function as an illustrative case.
Revised: To better understand BN, we treat a single-layer perceptron with ReLU activation function as an illustrative case.

Original: Despite the simplicity of this case, it is a building block of deep networks and has been widely adopted in theoretical analyses such as proper initialization (Krogh & Hertz, 1992; Advani & Saxe, 2017), dropout (Wager et al., 2013), weight decay and data augmentation (B¨os, 1998).
Revised: Despite the simplicity of this case, it is a building block of deep networks and has been widely adopted in theoretical analyses such as proper initialization (Krogh & Hertz, 1992; Advani & Saxe, 2017), dropout (Wager et al., 2013), weight decay and data augmentation (B¨os, 1998).

Original: The results in Sec.2 can be extended to deep neural networks as presented in Appendix C.4.
Revised: The results in Sec.2 can be extended to deep neural networks as presented in Appendix C.4.

Original: Our analyses assume that neurons at the BN layer are independent similar to (Salimans & Kingma, 2016; van Laarhoven, 2017; Teye et al., 2018) as
Revised: Our analyses assume that neurons at the BN layer are independent similar to (Salimans & Kingma, 2016; van Laarhoven, 2017; Teye et al., 2018) as

Original: the mean and the variance of
Revised: the mean and the variance of

Original: BN are estimated individually
Revised: BN are estimated individually

Original: for each neuron of each
Revised: for each neuron of each

Original: layer.
Revised: layer.

Original: The form of regularization in this study does not rely on Gaussian assumption on the network input and the weight vector, meaning our assumption is milder
Revised: The form of regularization in this study does not rely on Gaussian assumption on the network input and the weight vector, meaning our assumption is milder

Original: than
Revised: than

Original: those in
Revised: those in

Original: (Yoshida et al.,
Revised: (Yoshida et al.,

Original: ; Ba et al., 2016; Salimans & Kingma, 2016).
Revised: ; Ba et al., 2016; Salimans & Kingma, 2016).

Original: Sec.2 tells us that BN has an explicit regularization form, gamma decay, where µ B and σ B have different impacts: (1) µ B discourages reliance on a single neuron and encourages different neurons to have equal magnitude, in the sense that corrupting individual neuron does not harm generalization.
Revised: Sec.2 tells us that BN has an explicit regularization form, gamma decay, where µ B and σ B have different impacts: (1) µ B discourages reliance on a single neuron and encourages different neurons to have equal magnitude, in the sense that corrupting individual neuron does not harm generalization.

Original: This phenomenon was also found empirically in a recent work (Morcos et al., 2018), but has not been established analytically.
Revised: This phenomenon was also found empirically in a recent work (Morcos et al., 2018), but has not been established analytically.

Original: (2) σ B reduces kurtosis of the input distribution as well as correlations between neurons.
Revised: (2) σ B reduces kurtosis of the input distribution as well as correlations between neurons.

Original: The regularization strengths of these statistics are inversely proportional to the batch size M , indicating that BN with large batch would decrease generalization.
Revised: The regularization strengths of these statistics are inversely proportional to the batch size M , indicating that BN with large batch would decrease generalization.

Original: (4) Removing either one of µ B and σ B could imped convergence and generalization.
Revised: (4) Removing either one of µ B and σ B could imped convergence and generalization.

Original: • Second, by using ordinary differential equations (ODEs), Sec.3 shows that gamma decay enables the network trained with BN to converge with large maximum learning rate and effective learning rate, compared to the network trained without BN or trained with weight normalization (WN) (Salimans & Kingma, 2016) that is a counterpart of BN.
Revised: • Second, by using ordinary differential equations (ODEs), Sec.3 shows that gamma decay enables the network trained with BN to converge with large maximum learning rate and effective learning rate, compared to the network trained without BN or trained with weight normalization (WN) (Salimans & Kingma, 2016) that is a counterpart of BN.

Original: The maximum learning rate (LR) represents the largest LR value that allows training to converge to a ﬁxed point without diverging, while effective LR represents the actual LR in training.
Revised: The maximum learning rate (LR) represents the largest LR value that allows training to converge to a ﬁxed point without diverging, while effective LR represents the actual LR in training.

Original: Larger maximum and effective LRs imply faster convergence rate.
Revised: Larger maximum and effective LRs imply faster convergence rate.

Original: • Third, Sec.4 compares generalization errors of BN, WN, and vanilla SGD by using statistical mechanics.
Revised: • Third, Sec.4 compares generalization errors of BN, WN, and vanilla SGD by using statistical mechanics.

Original: The “large-scale” regime is of interest, where number of samples P and number of neurons N are both large but their ratio P/N is ﬁnite.
Revised: The “large-scale” regime is of interest, where number of samples P and number of neurons N are both large but their ratio P/N is ﬁnite.

Original: In this regime, the generalization errors are quantiﬁed both analytically and empirically.
Revised: In this regime, the generalization errors are quantiﬁed both analytically and empirically.

Original: Numerical results in Sec.5 show that BN
Revised: Numerical results in Sec.5 show that BN

Original: in
Revised: in

Original: CNNs has the same traits of regularization as disclosed
Revised: CNNs has the same traits of regularization as disclosed

Original: above.
Revised: above.

Original: 1. R ELATED W ORK
Revised: 1. R ELATED W ORK

Original: Neural Network Analysis .
Revised: Neural Network Analysis .

Original: Many studies conducted theoretical analyses of neural networks (Opper et al., 1990; Saad & Solla, 1996; Bs & Opper, 1998; Pennington & Bahri, 2017; Zhang et al., 2017b; Brutzkus & Globerson, 2017; Raghu et al., 2017; Mei et al., 2016; Tian, 2017).
Revised: Many studies conducted theoretical analyses of neural networks (Opper et al., 1990; Saad & Solla, 1996; Bs & Opper, 1998; Pennington & Bahri, 2017; Zhang et al., 2017b; Brutzkus & Globerson, 2017; Raghu et al., 2017; Mei et al., 2016; Tian, 2017).

Original: For example, for a multilayer network with linear activation function, Glorot & Bengio (2010) explored its SGD dynamics and Kawaguchi (2016) showed that every local minimum is global.
Revised: For example, for a multilayer network with linear activation function, Glorot & Bengio (2010) explored its SGD dynamics and Kawaguchi (2016) showed that every local minimum is global.

Original: Tian (2017) studied the critical points and convergence behaviors of a 2-layered network with ReLU units.
Revised: Tian (2017) studied the critical points and convergence behaviors of a 2-layered network with ReLU units.

Original: Zhang et al.
Revised: Zhang et al.

Original: (2017b) investigated a teacher-student model when the activation function is harmonic.
Revised: (2017b) investigated a teacher-student model when the activation function is harmonic.

Original: In (Saad &
Revised: In (Saad &

Original: Solla, 1996), the learning dynamics of a committee machine were discussed when the activation function is error function erf( x
Revised: Solla, 1996), the learning dynamics of a committee machine were discussed when the activation function is error function erf( x

Original: Unlike previous work, this work analyzes regularization emerged in
Revised: Unlike previous work, this work analyzes regularization emerged in

Original: BN and its impact to both learning and generalization, which are still unseen in the literature.
Revised: BN and its impact to both learning and generalization, which are still unseen in the literature.

Original: Normalization .
Revised: Normalization .

Original: Many normalization methods have been proposed recently.
Revised: Many normalization methods have been proposed recently.

Original: For example, BN (Ioffe & Szegedy, 2015) was introduced to stabilize the distribution of input data of each hidden layer.
Revised: For example, BN (Ioffe & Szegedy, 2015) was introduced to stabilize the distribution of input data of each hidden layer.

Original: Weight normalization (WN) (Salimans & Kingma, 2016) decouples the lengths of the network parameter vectors from their directions, by normalizing the parameter vectors to unit length.
Revised: Weight normalization (WN) (Salimans & Kingma, 2016) decouples the lengths of the network parameter vectors from their directions, by normalizing the parameter vectors to unit length.

Original: The dynamic of WN was studied by using a single-layer network (Yoshida et al., 2017).
Revised: The dynamic of WN was studied by using a single-layer network (Yoshida et al., 2017).

Original: Li et al.
Revised: Li et al.

Original: (2018) diagnosed the compatibility of BN and dropout (Srivastava et al., 2014) by reducing the variance shift produced by them.
Revised: (2018) diagnosed the compatibility of BN and dropout (Srivastava et al., 2014) by reducing the variance shift produced by them.

Original: Moreover, van Laarhoven (2017) showed that weight decay has no regularization effect when using together with BN or WN.
Revised: Moreover, van Laarhoven (2017) showed that weight decay has no regularization effect when using together with BN or WN.

Original: Ba et al.
Revised: Ba et al.

Original: (2016) demonstrated when BN or WN is employed, backpropagating gradients through a hidden layer is scale-invariant with respect to the network parameters.
Revised: (2016) demonstrated when BN or WN is employed, backpropagating gradients through a hidden layer is scale-invariant with respect to the network parameters.

Original: Santurkar et al.
Revised: Santurkar et al.

Original: (2018) gave another perspective of the role of BN during training instead of reducing the covariant shift.
Revised: (2018) gave another perspective of the role of BN during training instead of reducing the covariant shift.

Original: They argued that BN results in a smoother optimization landscape and the
Revised: They argued that BN results in a smoother optimization landscape and the

Original: Lipschitzness is strengthened in networks trained with BN.
Revised: Lipschitzness is strengthened in networks trained with BN.

Original: However, both analytical and empirical results of regularization in BN are still desirable.
Revised: However, both analytical and empirical results of regularization in BN are still desirable.

Original: Our study explores regularization, optimization, and generalization of BN in the scenario of online learning.
Revised: Our study explores regularization, optimization, and generalization of BN in the scenario of online learning.

Original: Regularization .
Revised: Regularization .

Original: Ioffe & Szegedy (2015) conjectured that BN implicitly regularizes training to prevent overﬁtting.
Revised: Ioffe & Szegedy (2015) conjectured that BN implicitly regularizes training to prevent overﬁtting.

Original: Zhang et al.
Revised: Zhang et al.

Original: (2017a) categorized BN as an implicit regularizer from experimental results.
Revised: (2017a) categorized BN as an implicit regularizer from experimental results.

Original: Szegedy et al.
Revised: Szegedy et al.

Original: (2015) also conjectured that in the Inception network, BN behaves similar to dropout to improve the generalization ability.
Revised: (2015) also conjectured that in the Inception network, BN behaves similar to dropout to improve the generalization ability.

Original: Gitman & Ginsburg (2017) experimentally compared
Revised: Gitman & Ginsburg (2017) experimentally compared

Original: BN and WN, and also conﬁrmed the better generalization of BN.
Revised: BN and WN, and also conﬁrmed the better generalization of BN.

Original: In the literature there are also implicit regularization schemes other than BN.
Revised: In the literature there are also implicit regularization schemes other than BN.

Original: For instance, random noise in the input layer for data augmentation has long been discovered equivalent to a weight decay method, in the sense that the inverse of the signal-to-noise ratio acts as the decay factor (Krogh & Hertz, 1992; Rifai et al., 2011).
Revised: For instance, random noise in the input layer for data augmentation has long been discovered equivalent to a weight decay method, in the sense that the inverse of the signal-to-noise ratio acts as the decay factor (Krogh & Hertz, 1992; Rifai et al., 2011).

Original: Dropout (Srivastava et al., 2014) was also proved able to regularize training by using the generalized linear model (Wager et al., 2013).
Revised: Dropout (Srivastava et al., 2014) was also proved able to regularize training by using the generalized linear model (Wager et al., 2013).

Original:  A P ROBABILISTIC I NTERPRETATION OF BN
Revised:  A P ROBABILISTIC I NTERPRETATION OF BN

Original: The notations in this work are summarized in Appendix Table 2 for reference.
Revised: The notations in this work are summarized in Appendix Table 2 for reference.

Original: Training the above single-layer perceptron with BN  typically involves minimizing a negative log likelihood function with respect to a set of network parameters θ = { w , γ, β } .
Revised: Training the above single-layer perceptron with BN in Eqn.(1) typically involves minimizing a negative log likelihood function with respect to a set of network parameters θ = { w , γ, β } .

Original: Then the loss function is deﬁned by
Revised: Then the loss function is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where p ( y j | ˆ h j ; θ ) represents the likelihood function of the network and P is number of training samples.
Revised: where p ( y j | ˆ h j ; θ ) represents the likelihood function of the network and P is number of training samples.

Original: As Gaussian distribution is often employed as prior distribution for the network parameters, we have a regularization term ζ (cid:107) θ (cid:107) 22 known as weight decay (Krizhevsky et al., 2012) that is a popular technique in deep learning, where ζ is a coefﬁcient.
Revised: As Gaussian distribution is often employed as prior distribution for the network parameters, we have a regularization term ζ (cid:107) θ (cid:107) 22 known as weight decay (Krizhevsky et al., 2012) that is a popular technique in deep learning, where ζ is a coefﬁcient.

Original: To derive regularization of BN, we treat µ B and σ B as random variables.
Revised: To derive regularization of BN, we treat µ B and σ B as random variables.

Original: Since one sample x is seen many times in the entire training course, and at each time x is presented with the other samples in a batch that is drawn randomly, µ B and σ B can be treated as injected random noise for x .
Revised: Since one sample x is seen many times in the entire training course, and at each time x is presented with the other samples in a batch that is drawn randomly, µ B and σ B can be treated as injected random noise for x .

Original: Prior of µ B , σ B .
Revised: Prior of µ B , σ B .

Original: By following (Teye et al., 2018), we ﬁnd that BN also induces Gaussian priors for µ B and σ B .
Revised: By following (Teye et al., 2018), we ﬁnd that BN also induces Gaussian priors for µ B and σ B .

Original: We have µ B ∼ N ( µ P , σ 2 P M ) and σ B ∼ N ( σ P , ρ +24 M ) , where M is batch size, µ P and σ P are population mean and standard deviation respectively, and ρ is kurtosis that measures the peakedness of the distribution of h .
Revised: We have µ B ∼ N ( µ P , σ 2 P M ) and σ B ∼ N ( σ P , ρ +24 M ) , where M is batch size, µ P and σ P are population mean and standard deviation respectively, and ρ is kurtosis that measures the peakedness of the distribution of h .

Original: These priors tell us that µ B and σ B would produce Gaussian noise in training.
Revised: These priors tell us that µ B and σ B would produce Gaussian noise in training.

Original: There is a tradeoff regarding this noise.
Revised: There is a tradeoff regarding this noise.

Original: For example, when M is small, training could diverge because the noise is large.
Revised: For example, when M is small, training could diverge because the noise is large.

Original: This is supported by experiment of BN (Wu & He, 2018) where training diverges when M = 2 in ImageNet (Russakovsky et al., 2015).
Revised: This is supported by experiment of BN (Wu & He, 2018) where training diverges when M = 2 in ImageNet (Russakovsky et al., 2015).

Original: When M is large, the noise is small because µ B and σ B get close to µ P and σ P .
Revised: When M is large, the noise is small because µ B and σ B get close to µ P and σ P .

Original: It is known that M > 30 would provide a moderate noise, as the sample statistics converges in probability to the population statistics by the weak Law of Large Numbers.
Revised: It is known that M > 30 would provide a moderate noise, as the sample statistics converges in probability to the population statistics by the weak Law of Large Numbers.

Original: This is also supported by experiment (Ioffe & Szegedy, 2015) where
Revised: This is also supported by experiment (Ioffe & Szegedy, 2015) where

Original: BN with M = 32 already works well in ImageNet.
Revised: BN with M = 32 already works well in ImageNet.

Original: 2. A R EGULARIZATION F ORM
Revised: 2. A R EGULARIZATION F ORM

Original: The loss function in Eqn.(2) can be written as an expected loss by integrating over the priors of µ B and σ B , that is, 1 P (cid:80)
Revised: The loss function in Eqn.(2) can be written as an expected loss by integrating over the priors of µ B and σ B , that is, 1 P (cid:80)

Original: Pj =
Revised: Pj =

Original: 1 E µ B ,σ B [ (cid:96) (ˆ h j )] where E [ · ] denotes expectation.
Revised: 1 E µ B ,σ B [ (cid:96) (ˆ h j )] where E [ · ] denotes expectation.

Original: We show that µ B and σ B impose regularization on the scale parameter γ by decomposing BN into population normalization (PN) and gamma decay.
Revised: We show that µ B and σ B impose regularization on the scale parameter γ by decomposing BN into population normalization (PN) and gamma decay.

Original: To see this, we employ a single-layer perceptron and ReLU activation function as an illustrative example.
Revised: To see this, we employ a single-layer perceptron and ReLU activation function as an illustrative example.

Original: A more rigorous description is provided in Appendix C.1.
Revised: A more rigorous description is provided in Appendix C.1.

Original: Regularization of µ B , σ B .
Revised: Regularization of µ B , σ B .

Original: Let (cid:96) ( ˆ h ) be the loss function deﬁned in Eqn.(2) and ReLU be the activation function.
Revised: Let (cid:96) ( ˆ h ) be the loss function deﬁned in Eqn.(2) and ReLU be the activation function.

Original: We have
Revised: We have

Original: [Equation] and ζ ( h ) =
Revised: (cid:125) gamma decay , and ζ ( h ) =

Original: ρ + 2 8 M I ( γ ) (cid:124) (cid:123)(cid:122) (cid:125) from σ B + 1 2 M 1 P
Revised: ρ + 2 8 M I ( γ ) (cid:124) (cid:123)(cid:122) (cid:125) from σ B + 1 2 M 1 P

Original: P (cid:88)
Revised: P (cid:88)

Original: j = σ (¯ h j ) (cid:124) (cid:123)(cid:122) (cid:125) from µ B , (3) where ¯ h j = γ h j − µ P σ P + β and h j = w T x j represent the computations of PN.
Revised: j = σ (¯ h j ) (cid:124) (cid:123)(cid:122) (cid:125) from µ B , (3) where ¯ h j = γ h j − µ P σ P + β and h j = w T x j represent the computations of PN.

Original: ζ ( h ) γ 2 represents gamma decay, where ζ ( h ) is an adaptive decay factor depended on the hidden value h .
Revised: ζ ( h ) γ 2 represents gamma decay, where ζ ( h ) is an adaptive decay factor depended on the hidden value h .

Original: Moreover, ρ is the kurtosis of distribution of h , I ( γ ) represents an estimation of the Fisher information of γ and
Revised: Moreover, ρ is the kurtosis of distribution of h , I ( γ ) represents an estimation of the Fisher information of γ and

Original: I ( γ ) =
Revised: I ( γ ) =

Original: 1 P (cid:80) P j =1 ( ∂(cid:96) (ˆ h j ) ∂γ ) 2 , and σ ( · ) is a sigmoid function.
Revised: 1 P (cid:80) P j =1 ( ∂(cid:96) (ˆ h j ) ∂γ ) 2 , and σ ( · ) is a sigmoid function.

Original: From Eqn.(3), we have several observations that have both theoretical and practical values.
Revised: From Eqn.(3), we have several observations that have both theoretical and practical values.

Original: • First, PN replaces the batch statistics µ B , σ B in BN by the population statistics µ P , σ P .
Revised: • First, PN replaces the batch statistics µ B , σ B in BN by the population statistics µ P , σ P .

Original: In gamma decay, computation of ζ ( h ) is data-dependent, making it differed from weight decay where the coefﬁcient is determined manually.
Revised: In gamma decay, computation of ζ ( h ) is data-dependent, making it differed from weight decay where the coefﬁcient is determined manually.

Original: In fact, Eqn.(3) recasts the randomness of BN in a deterministic manner, not only enabling us to apply methodologies such as ODEs and statistical mechanics to
Revised: In fact, Eqn.(3) recasts the randomness of BN in a deterministic manner, not only enabling us to apply methodologies such as ODEs and statistical mechanics to

Original: analyze BN, but also inspiring us to imitate BN
Revised: analyze BN, but also inspiring us to imitate BN

Original: ’s performance by WN without computing batch statistics in empirical study.
Revised: ’s performance by WN without computing batch statistics in empirical study.

Original: • Second, PN is closely connected to WN, which is independent from sample mean and variance.
Revised: • Second, PN is closely connected to WN, which is independent from sample mean and variance.

Original: WN (Salimans & Kingma, 2016) is deﬁned by υ w T x || w || 2 that normalizes the weight vector w to have unit variance, where υ is a learnable parameter.
Revised: WN (Salimans & Kingma, 2016) is deﬁned by υ w T x || w || 2 that normalizes the weight vector w to have unit variance, where υ is a learnable parameter.

Original: Let each diagonal element of the covariance matrix of x be a and all the off-diagonal elements be zeros.
Revised: Let each diagonal element of the covariance matrix of x be a and all the off-diagonal elements be zeros.

Original: ¯ h j in Eqn.(3) can be rewritten as
Revised: ¯ h j in Eqn.(3) can be rewritten as

Original: [Equation]
Revised: [Equation]

Original: where υ = γa and b = − γµ P a || w || 2 + β . Eqn.(4) removes the estimations of statistics and eases our analyses of regularization for BN.
Revised: where υ = γa and b = − γµ P a || w || 2 + β . Eqn.(4) removes the estimations of statistics and eases our analyses of regularization for BN.

Original: • Third, µ B and σ B produce different strengths in ζ ( h ) .
Revised: • Third, µ B and σ B produce different strengths in ζ ( h ) .

Original: As shown in Eqn.(3), the strength from µ B depends on the expectation of σ (¯ h j ) ∈ [0 , 1] , which represents excitation or inhibition of a neuron, meaning that a neuron with larger output may exposure to larger regularization, encouraging different neurons to have equal magnitude.
Revised: As shown in Eqn.(3), the strength from µ B depends on the expectation of σ (¯ h j ) ∈ [0 , 1] , which represents excitation or inhibition of a neuron, meaning that a neuron with larger output may exposure to larger regularization, encouraging different neurons to have equal magnitude.

Original: This is consistent with empirical result (Morcos et al., 2018) which prevented reliance on single neuron to improve generalization.
Revised: This is consistent with empirical result (Morcos et al., 2018) which prevented reliance on single neuron to improve generalization.

Original: The strength from σ B works as a complement for µ B .
Revised: The strength from σ B works as a complement for µ B .

Original: For a single neuron, I ( γ ) represents the norm of gradient, implying that BN punishes large gradient norm.
Revised: For a single neuron, I ( γ ) represents the norm of gradient, implying that BN punishes large gradient norm.

Original: For multiple neurons, I ( γ ) is the Fisher information matrix of γ , meaning that BN would penalize correlations among neurons.
Revised: For multiple neurons, I ( γ ) is the Fisher information matrix of γ , meaning that BN would penalize correlations among neurons.

Original: Both σ B and µ B are important, removing either one of them would imped performance.
Revised: Both σ B and µ B are important, removing either one of them would imped performance.

Original: Extensions to Deep Networks.
Revised: Extensions to Deep Networks.

Original: The above results can be extended to deep networks as shown in
Revised: The above results can be extended to deep networks as shown in

Original: Appendix C.4 by decomposing the expected loss at a certain hidden layer.
Revised: Appendix C.4 by decomposing the expected loss at a certain hidden layer.

Original: We also demonstrate the results empirically in Sec.5, where we observe that CNNs trained with BN share similar traits of regularization as discussed above.
Revised: We also demonstrate the results empirically in Sec.5, where we observe that CNNs trained with BN share similar traits of regularization as discussed above.

Original:  O PTIMIZATION WITH R EGULARIZATION
Revised:  O PTIMIZATION WITH R EGULARIZATION

Original: Now we show that BN converges with large maximum and effective learning rate (LR), where the former one is the largest LR when training converged, while the latter one is the actual LR during training.
Revised: Now we show that BN converges with large maximum and effective learning rate (LR), where the former one is the largest LR when training converged, while the latter one is the actual LR during training.

Original: With BN, we ﬁnd that both LRs would be larger than a network trained without BN.
Revised: With BN, we ﬁnd that both LRs would be larger than a network trained without BN.

Original: Our result explains why BN enables large learning rates used in practice (Ioffe & Szegedy, 2015).
Revised: Our result explains why BN enables large learning rates used in practice (Ioffe & Szegedy, 2015).

Original: Our analyses are conducted in three stages.
Revised: Our analyses are conducted in three stages.

Original: First, we establish dynamical equations of a teacherstudent model in the thermodynamic limit and acquire the ﬁxed point.
Revised: First, we establish dynamical equations of a teacherstudent model in the thermodynamic limit and acquire the ﬁxed point.

Original: Second, we investigate the eigenvalues of the corresponding Jacobian matrix at this ﬁxed point.
Revised: Second, we investigate the eigenvalues of the corresponding Jacobian matrix at this ﬁxed point.

Original: Finally, we calculate the maximum and the effective LR.
Revised: Finally, we calculate the maximum and the effective LR.

Original: Teacher-Student Model .
Revised: Teacher-Student Model .

Original: We ﬁrst introduce useful techniques from statistical mechanics (SM).
Revised: We ﬁrst introduce useful techniques from statistical mechanics (SM).

Original: With
Revised: With

Original: SM, a student network is dedicated to learn relationship between a Gaussian input and an output by using a weight vector w as parameters.
Revised: SM, a student network is dedicated to learn relationship between a Gaussian input and an output by using a weight vector w as parameters.

Original: It is useful to characterize behavior of the student by using a teacher network with w ∗ as a ground-truth parameter vector.
Revised: It is useful to characterize behavior of the student by using a teacher network with w ∗ as a ground-truth parameter vector.

Original: We treat single-layer perceptron as a student, which is optimized by minimizing the euclidian distance between its output and the supervision provided by a teacher without BN.
Revised: We treat single-layer perceptron as a student, which is optimized by minimizing the euclidian distance between its output and the supervision provided by a teacher without BN.

Original: The student and the teacher have identical activation function.
Revised: The student and the teacher have identical activation function.

Original: Loss Function.
Revised: Loss Function.

Original: We deﬁne a loss function of the above teacher-student model by 1 P (cid:80) Pj =1 (cid:96) ( x j ) =
Revised: We deﬁne a loss function of the above teacher-student model by 1 P (cid:80) Pj =1 (cid:96) ( x j ) =

Original: 1 P (cid:80) Pj =1 (cid:2) g
Revised: 1 P (cid:80) Pj =1 (cid:2) g

Original: ( w ∗ T x j ) − g ( √ Nγ w T x j (cid:107) w (cid:107) 2 ) (cid:3) 2 + ζγ 2 , where g ( w ∗
Revised: ( w ∗ T x j ) − g ( √ Nγ w T x j (cid:107) w (cid:107) 2 ) (cid:3) 2 + ζγ 2 , where g ( w ∗

Original: T x j ) represents supervision from the teacher, while g ( √ Nγ w T x j (cid:107) w (cid:107) 2 ) is the output of student trained to mimic the teacher.
Revised: T x j ) represents supervision from the teacher, while g ( √ Nγ w T x j (cid:107) w (cid:107) 2 ) is the output of student trained to mimic the teacher.

Original: This student is deﬁned by following Eqn.(4) with ν = √ Nγ and the bias term is absorbed into w .
Revised: This student is deﬁned by following Eqn.(4) with ν = √ Nγ and the bias term is absorbed into w .

Original: The above loss function represents BN by using WN with gamma decay, and it is sufﬁcient to study the learning rates of different approaches.
Revised: The above loss function represents BN by using WN with gamma decay, and it is sufﬁcient to study the learning rates of different approaches.

Original: Let θ = { w , γ } be a set of parameters updated by SGD,
Revised: Let θ = { w , γ } be a set of parameters updated by SGD,

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: e.
Revised: e.

Original: [Equation]
Revised: [Equation]

Original: j where η denotes learning rate.
Revised: j where η denotes learning rate.

Original: The update rules for w and γ are
Revised: The update rules for w and γ are

Original: [Equation]
Revised: [Equation]

Original: where ˜ w j denotes a normalized weight vector of the student, that is, ˜ w j = √
Revised: where ˜ w j denotes a normalized weight vector of the student, that is, ˜ w j = √

Original: Nγ j w j (cid:107) w j (cid:107) 2 , and δ j = g (cid:48)
Revised: Nγ j w j (cid:107) w j (cid:107) 2 , and δ j = g (cid:48)

Original: ( ˜ w j T x j )[ g ( w ∗ T x j ) − g ( ˜ w j T x j )] represents the gradient 1 for clarity of notations.
Revised: ( ˜ w j T x j )[ g ( w ∗ T x j ) − g ( ˜ w j T x j )] represents the gradient 1 for clarity of notations.

Original: Order Parameters .
Revised: Order Parameters .

Original: As we are interested in the “large-scale” regime where both N and P are large and their ratio P/N is ﬁnite, it is difﬁcult to examine a student with parameters in high dimensions directly.
Revised: As we are interested in the “large-scale” regime where both N and P are large and their ratio P/N is ﬁnite, it is difﬁcult to examine a student with parameters in high dimensions directly.

Original: Therefore, we transform the weight vectors to order parameters that fully characterize interactions between the student and the teacher network.
Revised: Therefore, we transform the weight vectors to order parameters that fully characterize interactions between the student and the teacher network.

Original: In this case, the parameter vector can be reparameterized by using a vector of three elements including γ , R , and L .
Revised: In this case, the parameter vector can be reparameterized by using a vector of three elements including γ , R , and L .

Original: In particular, γ measures length of the normalized weight vector ˜ w , that is, ˜ w T ˜ w =
Revised: In particular, γ measures length of the normalized weight vector ˜ w , that is, ˜ w T ˜ w =

Original: Nγ 2 w T w (cid:107) w (cid:107) 

Revised: Nγ 2 w T w (cid:107) w (cid:107)

Original: =
Revised: =

Original: Nγ 2 .
Revised: Nγ 2 .

Original: The parameter R measures angle (overlapping ratio) between the weight vectors of student and teacher.
Revised: The parameter R measures angle (overlapping ratio) between the weight vectors of student and teacher.

Original: We have
Revised: We have

Original: R =
Revised: R =

Original: [Equation]
Revised: [Equation]

Original: w ∗ , where the norm of the ground-truth vector is 1 N w ∗ T w ∗ = 1 .
Revised: w ∗ , where the norm of the ground-truth vector is 1 N w ∗ T w ∗ = 1 .

Original: Moreover,
Revised: Moreover,

Original: L represents length of the original weight vector w and L 2 = 1 N w T w .
Revised: L represents length of the original weight vector w and L 2 = 1 N w T w .

Original: Learning Dynamics.
Revised: Learning Dynamics.

Original: The update equations (5) can be transformed into a set of differential equations (ODEs) by using the above order parameters.
Revised: The update equations (5) can be transformed into a set of differential equations (ODEs) by using the above order parameters.

Original: This is achieved by treating the update step j as a continuous time variable t = jN .
Revised: This is achieved by treating the update step j as a continuous time variable t = jN .

Original: They can be turned into differential equations because the contiguous step ∆ t = 1 N approaches zero in the thermodynamic limit when N → ∞ .
Revised: They can be turned into differential equations because the contiguous step ∆ t = 1 N approaches zero in the thermodynamic limit when N → ∞ .

Original: We obtain a dynamical system of three order parameters
Revised: We obtain a dynamical system of three order parameters

Original: [Equation]
Revised: [Equation]

Original: where I 1 =
Revised: where I 1 =

Original: E x [ δ ˜ w T x ] , I 2 =
Revised: E x [ δ ˜ w T x ] , I 2 =

Original: E x [ δ 2 x T x ] , and I 3 =
Revised: E x [ δ 2 x T x ] , and I 3 =

Original: E x [ δ w ∗ T x ] are deﬁned to simplify notations.
Revised: E x [ δ w ∗ T x ] are deﬁned to simplify notations.

Original: The derivations of Eqn.(6) can be found in Appendix C.5.
Revised: The derivations of Eqn.(6) can be found in Appendix C.5.

Original: 3. F IXED P OINT OF THE D YNAMICAL S YSTEM
Revised: 3. F IXED P OINT OF THE D YNAMICAL S YSTEM

Original: [Table]
Revised: [Table]

Original: To ﬁnd the ﬁxed points of (6), we set dγ/dt = dR/dt = dL/dt = 0 .
Revised: To ﬁnd the ﬁxed points of (6), we set dγ/dt = dR/dt = dL/dt = 0 .

Original: The ﬁxed points of BN, WN, and vanilla SGD (without BN and WN) are given in Table 1.
Revised: The ﬁxed points of BN, WN, and vanilla SGD (without BN and WN) are given in Table 1.

Original: In the thermodynamic limit, the optima denoted as ( γ 0 , R 0 , L 0 ) would be ( γ 0 , R 0 , L 0 ) =
Revised: In the thermodynamic limit, the optima denoted as ( γ 0 , R 0 , L 0 ) would be ( γ 0 , R 0 , L 0 ) =

Original: Our main interest is the overlapping ratio R 0 between the student and the teacher, because it optimizes the direction of the weight vector regardless of its length.
Revised: Our main interest is the overlapping ratio R 0 between the student and the teacher, because it optimizes the direction of the weight vector regardless of its length.

Original: We see that R 0 for all three approaches attain optimum ‘1’.
Revised: We see that R 0 for all three approaches attain optimum ‘1’.

Original: Intuitively, in BN and WN, this optimal solution does not depend on the value of L 0 because their weight vectors are normalized.
Revised: Intuitively, in BN and WN, this optimal solution does not depend on the value of L 0 because their weight vectors are normalized.

Original: In other words,
Revised: In other words,

Original: WN and BN are easier to optimize than vanilla SGD, unlike SGD where both R 0 and L 0 have to be optimized to ‘1’.
Revised: WN and BN are easier to optimize than vanilla SGD, unlike SGD where both R 0 and L 0 have to be optimized to ‘1’.

Original: Furthermore, γ 0 in BN depends on the activation function.
Revised: Furthermore, γ 0 in BN depends on the activation function.

Original: For ReLU, we have γ bn 

Revised: For ReLU, we have γ bn

Original: = 1 2 ζ +1 (see Proposition 1 in Appendix C.5), meaning that norm of the normalized weight vector relies on the decay factor ζ .
Revised: = 1 2 ζ +1 (see Proposition 1 in Appendix C.5), meaning that norm of the normalized weight vector relies on the decay factor ζ .

Original: In WN, we have γ wn 

Revised: In WN, we have γ wn

Original: = 1 as WN has no regularization on γ .
Revised: = 1 as WN has no regularization on γ .

Original: 3. M AXIMUM AND E FFECTIVE L EARNING R ATES
Revised: 3. M AXIMUM AND E FFECTIVE L EARNING R ATES

Original: With the above ﬁxed points, we derive the maximum and the effective LR.
Revised: With the above ﬁxed points, we derive the maximum and the effective LR.

Original: Speciﬁcally, we analyze eigenvalues and eigenvectors of the Jacobian matrix corresponding to Eqn.(6).
Revised: Speciﬁcally, we analyze eigenvalues and eigenvectors of the Jacobian matrix corresponding to Eqn.(6).

Original: We are interested in the LR to approach R 0 .
Revised: We are interested in the LR to approach R 0 .

Original: We ﬁnd that this optimum value only depends on its corresponding eigenvalue denoted as λ R .
Revised: We ﬁnd that this optimum value only depends on its corresponding eigenvalue denoted as λ R .

Original: We have λ R = ∂I 2 ∂R ηγ 0 

Revised: We have λ R = ∂I 2 ∂R ηγ 0

Original: L 20 ( η max − η eﬀ ) , where η max and η eﬀ represent the maximum and effective LR (proposition 2 in Appendix C.5), which are given in Table 1.
Revised: L 20 ( η max − η eﬀ ) , where η max and η eﬀ represent the maximum and effective LR (proposition 2 in Appendix C.5), which are given in Table 1.

Original: We demonstrate that λ R < 

Revised: We demonstrate that λ R <

Original: if and only if η max > η eﬀ , such that the ﬁxed point R 0 is stable for all approaches (proposition 3 in Appendix C.6).
Revised: if and only if η max > η eﬀ , such that the ﬁxed point R 0 is stable for all approaches (proposition 3 in Appendix C.6).

Original: Moreover, it is also able to show that η max of BN ( η bn max ) is larger than WN and SGD, enabling R to converge with a larger learning rate.
Revised: Moreover, it is also able to show that η max of BN ( η bn max ) is larger than WN and SGD, enabling R to converge with a larger learning rate.

Original: For ReLU as an example, we ﬁnd that η bn max ≥ η { wn,sgd } max + 2 ζ
Revised: For ReLU as an example, we ﬁnd that η bn max ≥ η { wn,sgd } max + 2 ζ

Original: proposition 4 in Appendix C.7).
Revised: proposition 4 in Appendix C.7).

Original: The larger maximum LRs enables the network to be trained more stably and has the potential to be combined with other stabilization techniques (Fagan & Iyengar, 2018) during optimization.
Revised: The larger maximum LRs enables the network to be trained more stably and has the potential to be combined with other stabilization techniques (Fagan & Iyengar, 2018) during optimization.

Original: The effective LRs shown in Table 1 are consistent with previous work (van Laarhoven, 2017).
Revised: The effective LRs shown in Table 1 are consistent with previous work (van Laarhoven, 2017).

Original:  G ENERALIZATION A NALYSIS
Revised:  G ENERALIZATION A NALYSIS

Original: Here we investigate generalization of BN by using a teacher-student model that minimizes a loss function 1 P (cid:80) Pj =1 (( y ∗ ) j − y j ) 2 , where y ∗ represents the teacher’s output and y is the student’s output.
Revised: Here we investigate generalization of BN by using a teacher-student model that minimizes a loss function 1 P (cid:80) Pj =1 (( y ∗ ) j − y j ) 2 , where y ∗ represents the teacher’s output and y is the student’s output.

Original: We compare BN with WN+gamma decay and vanilla SGD.
Revised: We compare BN with WN+gamma decay and vanilla SGD.

Original: All of them share the same teacher network whose output is a noise-corrupted linear function y ∗ =
Revised: All of them share the same teacher network whose output is a noise-corrupted linear function y ∗ =

Original: w ∗ T x + s , where x is drawn from
Revised: w ∗ T x + s , where x is drawn from

Original: N (0 , 1 N ) and s is an unobserved Gaussian noise.
Revised: N (0 , 1 N ) and s is an unobserved Gaussian noise.

Original: We are interested to see how the above methods resist this noise by using student networks with both identity (linear) and ReLU activation functions.
Revised: We are interested to see how the above methods resist this noise by using student networks with both identity (linear) and ReLU activation functions.

Original: For vanilla SGD , the student is computed by y = g ( w T x ) with g ( · ) being either identity or ReLU, and w being the weight vector to optimize, where w has the same dimension as w ∗ .
Revised: For vanilla SGD , the student is computed by y = g ( w T x ) with g ( · ) being either identity or ReLU, and w being the weight vector to optimize, where w has the same dimension as w ∗ .

Original: The loss function of vanilla SGD is (cid:96) sgd = 1 P (cid:80) Pj =1 (cid:0) y ∗ − g ( w T x j ) (cid:1) 2 .
Revised: The loss function of vanilla SGD is (cid:96) sgd = 1 P (cid:80) Pj =1 (cid:0) y ∗ − g ( w T x j ) (cid:1) 2 .

Original: For BN , the student is deﬁned as y = γ w T x − µ B σ B + β .
Revised: For BN , the student is deﬁned as y = γ w T x − µ B σ B + β .

Original: As our main interest is the weight vector, we freeze the bias by setting β = 0 .
Revised: As our main interest is the weight vector, we freeze the bias by setting β = 0 .

Original: Therefore, the batch average term µ B is also unnecessary to avoid additional parameters, and the loss function is written as (cid:96) bn = 1 P (cid:80)
Revised: Therefore, the batch average term µ B is also unnecessary to avoid additional parameters, and the loss function is written as (cid:96) bn = 1 P (cid:80)

Original: Pj =1 (cid:0) ( y ∗ )
Revised: Pj =1 (cid:0) ( y ∗ )

Original: j − γ w T x j /σ B (cid:1) 2 .
Revised: j − γ w T x j /σ B (cid:1) 2 .

Original: For WN+gamma decay , the student is computed similar to Eqn.(4) by using y = √
Revised: For WN+gamma decay , the student is computed similar to Eqn.(4) by using y = √

Original: Nγ w T x (cid:107) w (cid:107) 2 .
Revised: Nγ w T x (cid:107) w (cid:107) 2 .

Original: Then the loss function is deﬁned by (cid:96)
Revised: Then the loss function is deﬁned by (cid:96)

Original: [Equation]
Revised: [Equation]

Original: γ (cid:107) 22 .
Revised: γ (cid:107) 22 .

Original: With the above deﬁnitions, the three approaches are studied under the same teacher-student framework, where their generalization errors can be strictly compared with the other factors ruled out.
Revised: With the above deﬁnitions, the three approaches are studied under the same teacher-student framework, where their generalization errors can be strictly compared with the other factors ruled out.

Original: 4. G ENERALIZATION E RRORS
Revised: 4. G ENERALIZATION E RRORS

Original: [Figure]
Revised: [Figure]

Original: We provide closed-form solutions of the generalization errors (see Appendix D.1) for vanilla SGD with both linear and ReLU student networks.
Revised: We provide closed-form solutions of the generalization errors (see Appendix D.1) for vanilla SGD with both linear and ReLU student networks.

Original: The theoretical solution of WN+gamma decay can also be solved for the linear student, but still remains difﬁcult for ReLU student whose numerical veriﬁcation is provided instead.
Revised: The theoretical solution of WN+gamma decay can also be solved for the linear student, but still remains difﬁcult for ReLU student whose numerical veriﬁcation is provided instead.

Original: Both vanilla SGD and WN+gamma decay are compared with numerical solutions of BN. vanilla SGD.
Revised: Both vanilla SGD and WN+gamma decay are compared with numerical solutions of BN. vanilla SGD.

Original: In an identity (linear) student, the solution of generalization error depends on the rank of correlation matrix Σ
Revised: In an identity (linear) student, the solution of generalization error depends on the rank of correlation matrix Σ

Original: = x T x .
Revised: = x T x .

Original: Here we deﬁne an effective load α = P/N that is the ratio between number of samples P and number of input neurons N (number of learnable parameters).
Revised: Here we deﬁne an effective load α = P/N that is the ratio between number of samples P and number of input neurons N (number of learnable parameters).

Original: The generalization error of the identity student is denoted as (cid:15) sgd id , which can be acquired by using the distribution of eigenvalues of Σ following (Advani & Saxe, 2017).
Revised: The generalization error of the identity student is denoted as (cid:15) sgd id , which can be acquired by using the distribution of eigenvalues of Σ following (Advani & Saxe, 2017).

Original: If α < 1 , (cid:15) sgd id = 1 − α + αS/ (1 − α ) .
Revised: If α < 1 , (cid:15) sgd id = 1 − α + αS/ (1 − α ) .

Original: Otherwise, (cid:15) sgd id = S/
Revised: Otherwise, (cid:15) sgd id = S/

Original: ( α − 1) where S is the variance of the injected noise to the teacher network.
Revised: ( α − 1) where S is the variance of the injected noise to the teacher network.

Original: The values of (cid:15) sgd id with respect to α are plotted in blue curve of Fig.1(a).
Revised: The values of (cid:15) sgd id with respect to α are plotted in blue curve of Fig.1(a).

Original: It ﬁrst decreases
Revised: It ﬁrst decreases

Original: but then increases as α increases from 0 to 1.
Revised: but then increases as α increases from 0 to 1.

Original: (cid:15) sgd id diverges at α = 1 .
Revised: (cid:15) sgd id diverges at α = 1 .

Original: And it would decrease again when α > 1 .
Revised: And it would decrease again when α > 1 .

Original: In a ReLU student, the nonlinear activation yields difﬁculty to derive the theoretical solution.
Revised: In a ReLU student, the nonlinear activation yields difﬁculty to derive the theoretical solution.

Original: Here we utilize the statistical mechanics and calculate that (cid:15) sgd relu = 1 − α/ 4 + αS 2(2 − α ) and α < 2 (see AppendixD.2).
Revised: Here we utilize the statistical mechanics and calculate that (cid:15) sgd relu = 1 − α/ 4 + αS 2(2 − α ) and α < 2 (see AppendixD.2).

Original: When comparing to the lower bound (trained without noisy supervision) shown as the red curve in
Revised: When comparing to the lower bound (trained without noisy supervision) shown as the red curve in

Original: Fig.1(b), we see that (cid:15) sgd relu (blue curve) diverges at α = 2 .
Revised: Fig.1(b), we see that (cid:15) sgd relu (blue curve) diverges at α = 2 .

Original: This is because the student overﬁts the noise in the teacher’s output.
Revised: This is because the student overﬁts the noise in the teacher’s output.

Original: The curve of numerical solution is also plotted in dashed line in Fig.1(b) and it captures the diverging trend well.
Revised: The curve of numerical solution is also plotted in dashed line in Fig.1(b) and it captures the diverging trend well.

Original: It should be noted that obtaining the theoretical curve empirically requires an inﬁnitely long time of training and an inﬁnitely small learning rate.
Revised: It should be noted that obtaining the theoretical curve empirically requires an inﬁnitely long time of training and an inﬁnitely small learning rate.

Original: This unreachable limit explains the discrepancies between the theoretical and the numerical solution.
Revised: This unreachable limit explains the discrepancies between the theoretical and the numerical solution.

Original: WN+gamma decay.
Revised: WN+gamma decay.

Original: In a linear student, the gamma decay term turns the correlation matrix to Σ = (cid:0) x
Revised: In a linear student, the gamma decay term turns the correlation matrix to Σ = (cid:0) x

Original: T x + ζ I (cid:1) , which is positive deﬁnite.
Revised: T x + ζ I (cid:1) , which is positive deﬁnite.

Original: Following statistical mechanics (Krogh & Hertz, 1992), the generalization error is (cid:15) wn id = δ 2 ∂ ( ζG ) ∂ζ − ζ 2 ∂G∂ζ where G = 1 − α − ζ + (cid:0) ζ + (1 + √ α ) 2 (cid:1) 12 (cid:0) ζ + (1 − √ α ) 2 (cid:1) 12 (cid:14) 2 ζ.
Revised: Following statistical mechanics (Krogh & Hertz, 1992), the generalization error is (cid:15) wn id = δ 2 ∂ ( ζG ) ∂ζ − ζ 2 ∂G∂ζ where G = 1 − α − ζ + (cid:0) ζ + (1 + √ α ) 2 (cid:1) 12 (cid:0) ζ + (1 − √ α ) 2 (cid:1) 12 (cid:14) 2 ζ.

Original: We see that (cid:15) wn id can be computed quantitatively given the values of ζ and α .
Revised: We see that (cid:15) wn id can be computed quantitatively given the values of ζ and α .

Original: Let the variance of noise injected to the teacher be 0 .
Revised: Let the variance of noise injected to the teacher be 0 .

Original: Fig.1(a) shows that no other curves could outperform the red curve when ζ = 0 .
Revised: Fig.1(a) shows that no other curves could outperform the red curve when ζ = 0 .

Original: 25 , a value equal to the noise magnitude.
Revised: 25 , a value equal to the noise magnitude.

Original: The ζ smaller than 0 .
Revised: The ζ smaller than 0 .

Original: 25 (green curve ζ = 12 M and M = 32 ) would exhibit overtraining around α = 1 , but they still perform signiﬁcantly better than vanilla SGD.
Revised: 25 (green curve ζ = 12 M and M = 32 ) would exhibit overtraining around α = 1 , but they still perform signiﬁcantly better than vanilla SGD.

Original: For the ReLU student in Fig,1(b), a direct solution of the generalization error (cid:15) wn relu remains an open problem.
Revised: For the ReLU student in Fig,1(b), a direct solution of the generalization error (cid:15) wn relu remains an open problem.

Original: Therefore, the numerical results of ‘WN+gamma decay’ (green curve) are run at each α value.
Revised: Therefore, the numerical results of ‘WN+gamma decay’ (green curve) are run at each α value.

Original: It effectively reduces over-ﬁtting compared to vanilla SGD.
Revised: It effectively reduces over-ﬁtting compared to vanilla SGD.

Original: Numerical Solutions of BN.
Revised: Numerical Solutions of BN.

Original: In the linear student, we employ SGD with M = 32 to ﬁnd solutions of w for BN.
Revised: In the linear student, we employ SGD with M = 32 to ﬁnd solutions of w for BN.

Original: The number of input neurons is 4096 and the number of training samples can be varied to change α .
Revised: The number of input neurons is 4096 and the number of training samples can be varied to change α .

Original: The results are marked as black squares in Fig.1(a).
Revised: The results are marked as black squares in Fig.1(a).

Original: After applying the analyses for linear student (Appendix C.3), BN is equivalent to ‘WN+gamma decay’ when ζ = 12 M (green curve).
Revised: After applying the analyses for linear student (Appendix C.3), BN is equivalent to ‘WN+gamma decay’ when ζ = 12 M (green curve).

Original: It is seen that BN gets in line with the curve ‘ ζ = 1 / 2 M ’ ( M = 32 ) and thus quantitatively validates our derivations.
Revised: It is seen that BN gets in line with the curve ‘ ζ = 1 / 2 M ’ ( M = 32 ) and thus quantitatively validates our derivations.

Original: In the ReLU student, the setting is mostly the same as the linear case, except that we employ a smaller batch size M = 16 .
Revised: In the ReLU student, the setting is mostly the same as the linear case, except that we employ a smaller batch size M = 16 .

Original: The results are shown as black squares in Fig.1(b).
Revised: The results are shown as black squares in Fig.1(b).

Original: For ReLU units, the equivalent ζ of gamma decay is ζ = 14 M .
Revised: For ReLU units, the equivalent ζ of gamma decay is ζ = 14 M .

Original: If one compares the generalization error of BN with ‘WN+gamma decay’ (green curve), a clear correspondence is found, which also validates the derivations for the ReLU activation function.
Revised: If one compares the generalization error of BN with ‘WN+gamma decay’ (green curve), a clear correspondence is found, which also validates the derivations for the ReLU activation function.

Original:  E XPERIMENTS IN CNN S
Revised:  E XPERIMENTS IN CNN S

Original: This section shows that BN in CNNs follows similar traits of regularization as the above analyses.
Revised: This section shows that BN in CNNs follows similar traits of regularization as the above analyses.

Original: To compare different methods, the CNN architectures are ﬁxed while only the normalization layers are changed.
Revised: To compare different methods, the CNN architectures are ﬁxed while only the normalization layers are changed.

Original: We adopt CIFAR10 (Krizhevsky, 2009) that contains 60k images of 10 categories (50k images for training and 10k images for test).
Revised: We adopt CIFAR10 (Krizhevsky, 2009) that contains 60k images of 10 categories (50k images for training and 10k images for test).

Original: All models are trained by using SGD with momentum, while the initial learning rates are scaled proportionally (Goyal et al., 2017) when different batch sizes are presented.
Revised: All models are trained by using SGD with momentum, while the initial learning rates are scaled proportionally (Goyal et al., 2017) when different batch sizes are presented.

Original: More empirical setting can be found in Appendix B.
Revised: More empirical setting can be found in Appendix B.

Original: Evaluation of PN+Gamma Decay .
Revised: Evaluation of PN+Gamma Decay .

Original: This work shows that BN can be decomposed into PN and gamma decay.
Revised: This work shows that BN can be decomposed into PN and gamma decay.

Original: We empirically compare ‘PN+gamma decay’ with BN by using ResNet18 (He et al., 2016).
Revised: We empirically compare ‘PN+gamma decay’ with BN by using ResNet18 (He et al., 2016).

Original: For ‘PN+gamma decay’, the population statistics of PN and the decay factor of gamma decay are estimated by using sufﬁcient amount of training samples.
Revised: For ‘PN+gamma decay’, the population statistics of PN and the decay factor of gamma decay are estimated by using sufﬁcient amount of training samples.

Original: For BN, BN trained with a normal batch size M = 128 is treated as baseline as shown in Fig.2(a&b).
Revised: For BN, BN trained with a normal batch size M = 128 is treated as baseline as shown in Fig.2(a&b).

Original: We see that when batch size increases,
Revised: We see that when batch size increases,

Original: BN would imped both loss and accuracy.
Revised: BN would imped both loss and accuracy.

Original: For example, when increasing M to 1024 , performance decreases because the regularization from the batch statistics reduces in large batch, resulting in overtraining (see the gap between train and validation loss in (a) when M = 1024 ).
Revised: For example, when increasing M to 1024 , performance decreases because the regularization from the batch statistics reduces in large batch, resulting in overtraining (see the gap between train and validation loss in (a) when M = 1024 ).

Original: In comparison, we train PN by using 10k training samples to estimate the population statistics.
Revised: In comparison, we train PN by using 10k training samples to estimate the population statistics.

Original: Note that this further reduces regularization.
Revised: Note that this further reduces regularization.

Original: We see that the release of regularization can be complemented by gamma decay, making PN outperformed BN.
Revised: We see that the release of regularization can be complemented by gamma decay, making PN outperformed BN.

Original: This empirical result veriﬁes our derivation of regularization for BN.
Revised: This empirical result veriﬁes our derivation of regularization for BN.

Original: Similar trend can be observed by experiment in a down-sampled version of
Revised: Similar trend can be observed by experiment in a down-sampled version of

Original: ImageNet (see Appendix B.1).
Revised: ImageNet (see Appendix B.1).

Original: We would like to point out that ‘PN+gamma decay’ is of interest in theoretical analyses, but it is computation-demanding when applied in practice because evaluating µ P , σ P and ζ ( h ) may require sufﬁciently large number of samples.
Revised: We would like to point out that ‘PN+gamma decay’ is of interest in theoretical analyses, but it is computation-demanding when applied in practice because evaluating µ P , σ P and ζ ( h ) may require sufﬁciently large number of samples.

Original: Comparisons of Regularization.
Revised: Comparisons of Regularization.

Original: We study the regulation strengths of vanilla SGD, BN, WN,
Revised: We study the regulation strengths of vanilla SGD, BN, WN,

Original: WN+mean-only BN, and WN+variance-only BN.
Revised: WN+mean-only BN, and WN+variance-only BN.

Original: At ﬁrst, the strength of regularization terms from both µ B and σ B are compared by using a simpler network with 4 convolutional and 2 fully connected layers as used in (Salimans & Kingma, 2016).
Revised: At ﬁrst, the strength of regularization terms from both µ B and σ B are compared by using a simpler network with 4 convolutional and 2 fully connected layers as used in (Salimans & Kingma, 2016).

Original: Fig.2(c&d) compares their training and validation losses.
Revised: Fig.2(c&d) compares their training and validation losses.

Original: We see that the generalization error of BN is much lower than WN and vanilla SGD.
Revised: We see that the generalization error of BN is much lower than WN and vanilla SGD.

Original: The reason has been disclosed in this work: stochastic behaviors of µ B and σ B in BN improves generalization.
Revised: The reason has been disclosed in this work: stochastic behaviors of µ B and σ B in BN improves generalization.

Original: To investigate µ B and σ B individually, we decompose their contributions by running a WN with mean-only BN as well as a WN with variance-only BN, to simulate their respective regularization.
Revised: To investigate µ B and σ B individually, we decompose their contributions by running a WN with mean-only BN as well as a WN with variance-only BN, to simulate their respective regularization.

Original: [Figure]
Revised: [Figure]

Original: As shown in Fig.2(c&d), improvements from the mean-only and the variance-only BN over WN verify our conclusion that noises from µ B and σ B have different regularization strengths.
Revised: As shown in Fig.2(c&d), improvements from the mean-only and the variance-only BN over WN verify our conclusion that noises from µ B and σ B have different regularization strengths.

Original: Both of them are essential to produce good result.
Revised: Both of them are essential to produce good result.

Original: Regularization and parameter norm.
Revised: Regularization and parameter norm.

Original: We further demonstrate impact of BN to the norm of parameters.
Revised: We further demonstrate impact of BN to the norm of parameters.

Original: We compare BN with vanilla SGD.
Revised: We compare BN with vanilla SGD.

Original: A network is ﬁrst trained by BN in order to converge to a local minima where the parameters do not change much.
Revised: A network is ﬁrst trained by BN in order to converge to a local minima where the parameters do not change much.

Original: At this local minima, the weight vector is frozen and denoted as w bn .
Revised: At this local minima, the weight vector is frozen and denoted as w bn .

Original: Then this network is ﬁnetuned by using vanilla SGD with a small learning rate 10 − 3 and its kernel parameters are initialized by w sgd
Revised: Then this network is ﬁnetuned by using vanilla SGD with a small learning rate 10 − 3 and its kernel parameters are initialized by w sgd

Original: = γ w bn σ , where σ is the moving average of σ B .
Revised: = γ w bn σ , where σ is the moving average of σ B .

Original: Fig.4 in Appendix B.2 visualizes the results.
Revised: Fig.4 in Appendix B.2 visualizes the results.

Original: As µ B and σ B are removed in vanilla SGD, it is found that the training loss decreases while the validation loss increases, implying that reduction in regularization makes the network converged to a sharper local minimum that generalizes less well.
Revised: As µ B and σ B are removed in vanilla SGD, it is found that the training loss decreases while the validation loss increases, implying that reduction in regularization makes the network converged to a sharper local minimum that generalizes less well.

Original: The magnitudes of kernel parameters w sgd at different layers are also observed to increase after freezing BN, due to the release of regularization on these parameters.
Revised: The magnitudes of kernel parameters w sgd at different layers are also observed to increase after freezing BN, due to the release of regularization on these parameters.

Original: Batch size.
Revised: Batch size.

Original: To study BN with different batch sizes, we train different networks but only add BN at one layer at a time.
Revised: To study BN with different batch sizes, we train different networks but only add BN at one layer at a time.

Original: The regularization on the γ parameter is compared in Fig.2(e) when BN is located at different layers.
Revised: The regularization on the γ parameter is compared in Fig.2(e) when BN is located at different layers.

Original: The values of γ 2 increase along with the batch size M due to the weaker regularization for the larger batches.
Revised: The values of γ 2 increase along with the batch size M due to the weaker regularization for the larger batches.

Original: The increase of γ 2 also makes all validation losses increased as shown in Fig.2(f).
Revised: The increase of γ 2 also makes all validation losses increased as shown in Fig.2(f).

Original: BN and WN trained with dropout .
Revised: BN and WN trained with dropout .

Original: As PN and gamma decay requires estimating the population statistics that increases computations, we utilize dropout as an alternative to improve regularization of
Revised: As PN and gamma decay requires estimating the population statistics that increases computations, we utilize dropout as an alternative to improve regularization of

Original: BN.
Revised: BN.

Original: We add a dropout after each BN layer.
Revised: We add a dropout after each BN layer.

Original: Fig.2(g&h) plot the classiﬁcation results using ResNet18.
Revised: Fig.2(g&h) plot the classiﬁcation results using ResNet18.

Original: The generalization of BN deteriorates signiﬁcantly when M increases from 128 to 1024.
Revised: The generalization of BN deteriorates signiﬁcantly when M increases from 128 to 1024.

Original: This is observed by the much higher validation loss (Fig.2(g)) and lower validation accuracy (Fig.2(h)) when M = 1024 .
Revised: This is observed by the much higher validation loss (Fig.2(g)) and lower validation accuracy (Fig.2(h)) when M = 1024 .

Original: If a dropout layer with ratio 0 .
Revised: If a dropout layer with ratio 0 .

Original: 1 is added after each residual block layer for
Revised: 1 is added after each residual block layer for

Original: M = 1024 in ResNet18, the validation loss is suppressed and accuracy increased by a great margin.
Revised: M = 1024 in ResNet18, the validation loss is suppressed and accuracy increased by a great margin.

Original: This superﬁcially contradicts with the original claim that BN reduces the need for dropout (Ioffe &
Revised: This superﬁcially contradicts with the original claim that BN reduces the need for dropout (Ioffe &

Original: Szegedy, 2015).
Revised: Szegedy, 2015).

Original: As discussed in Appendix B.3, we ﬁnd that there are two differences between our study and previous work (Ioffe & Szegedy, 2015).
Revised: As discussed in Appendix B.3, we ﬁnd that there are two differences between our study and previous work (Ioffe & Szegedy, 2015).

Original: Fig.2(g&h) also show that WN can also be regularized by dropout.
Revised: Fig.2(g&h) also show that WN can also be regularized by dropout.

Original: We apply dropout after each WN layer with ratio 0.2 and the dropout is applied at the same layers as that for BN.
Revised: We apply dropout after each WN layer with ratio 0.2 and the dropout is applied at the same layers as that for BN.

Original: We found that the improvement on both validation accuracy and loss is surprising.
Revised: We found that the improvement on both validation accuracy and loss is surprising.

Original: The accuracy increases from 0.
Revised: The accuracy increases from 0.

Original: to 0.93, even close to the results of BN.
Revised: to 0.93, even close to the results of BN.

Original: Nevertheless, additional regularization on WN still cannot make WN on par with the performance BN.
Revised: Nevertheless, additional regularization on WN still cannot make WN on par with the performance BN.

Original: In deep neural networks the distribution after each layer would be far from a Gaussian distribution, in which case WN is not a good substitute for PN.
Revised: In deep neural networks the distribution after each layer would be far from a Gaussian distribution, in which case WN is not a good substitute for PN.

Original:  C ONCLUSIONS
Revised:  C ONCLUSIONS

Original: This work investigated an explicit regularization form of BN, which was decomposed into PN and gamma decay where the regularization strengths from µ B and σ B were explored.
Revised: This work investigated an explicit regularization form of BN, which was decomposed into PN and gamma decay where the regularization strengths from µ B and σ B were explored.

Original: Moreover, optimization and generalization of BN with regularization were derived and compared with vanilla
Revised: Moreover, optimization and generalization of BN with regularization were derived and compared with vanilla

Original: SGD, WN and WN+gamma decay, showing that BN enables training to converge with large maximum and effective learning rate, as well as leads to better generalization.
Revised: SGD, WN and WN+gamma decay, showing that BN enables training to converge with large maximum and effective learning rate, as well as leads to better generalization.

Original: Our analytical results explain many existing empirical phenomena.
Revised: Our analytical results explain many existing empirical phenomena.

Original: Experiments in CNNs showed that BN in deep networks share the same traits of regularization.
Revised: Experiments in CNNs showed that BN in deep networks share the same traits of regularization.

Original: In future work, we are interested in analyzing optimization and generalization of BN in deep networks, which is still an open problem
Revised: In future work, we are interested in analyzing optimization and generalization of BN in deep networks, which is still an open problem

Original: A PPENDICES A N OTATIONS
Revised: A PPENDICES A N OTATIONS

Original: [Table]
Revised: [Table]

Original: B M ORE E MPIRICAL S ETTINGS AND R ESULTS
Revised: B M ORE E MPIRICAL S ETTINGS AND R ESULTS

Original: All experiments in Sec.5 are conducted in CIFAR10 by using ResNet18 and a CNN architecture similar to (Salimans & Kingma, 2016) that is summarized as ‘conv(3,32)-conv(3,32)-conv(3,64)- conv(3,64)-pool(2,2)-fc(512)-fc(10)’, where ‘conv(3,32)’ represents a convolution with kernel size 3 and 32 channels, ‘pool(2,2)’ is max-pooling with kernel size 2 and stride
Revised: All experiments in Sec.5 are conducted in CIFAR10 by using ResNet18 and a CNN architecture similar to (Salimans & Kingma, 2016) that is summarized as ‘conv(3,32)-conv(3,32)-conv(3,64)- conv(3,64)-pool(2,2)-fc(512)-fc(10)’, where ‘conv(3,32)’ represents a convolution with kernel size 3 and 32 channels, ‘pool(2,2)’ is max-pooling with kernel size 2 and stride

Original: [Table]
Revised: [Table]

Original: and ‘fc’ indicates a full connection.
Revised: and ‘fc’ indicates a full connection.

Original: We follow a conﬁguration for training by using SGD with a momentum value of 0. and continuously decaying the learning rate by a factor of 10 − 4 each step.
Revised: We follow a conﬁguration for training by using SGD with a momentum value of 0. and continuously decaying the learning rate by a factor of 10 − 4 each step.

Original: For different batch sizes, the initial learning rate is scaled proportionally with the batch size to maintain a similar learning dynamics (Goyal et al., 2017)
Revised: For different batch sizes, the initial learning rate is scaled proportionally with the batch size to maintain a similar learning dynamics (Goyal et al., 2017)

Original: [Table]
Revised: [Table]

Original: B. R ESULTS IN DOWNSAMPLED I MAGE N ET
Revised: B. R ESULTS IN DOWNSAMPLED I MAGE N ET

Original: Besides CIFAR10, we also evaluate ‘PN+gamma decay’ by employing a downsampled version of
Revised: Besides CIFAR10, we also evaluate ‘PN+gamma decay’ by employing a downsampled version of

Original: ImageNet (Loshchilov & Hutter, 2016), which contains identical 1.2 million data and 1k categories as the original ImageNet, but each image is scaled to 32 × 32.
Revised: ImageNet (Loshchilov & Hutter, 2016), which contains identical 1.2 million data and 1k categories as the original ImageNet, but each image is scaled to 32 × 32.

Original: We train ResNet18 in downsampled
Revised: We train ResNet18 in downsampled

Original: ImageNet by following the training protocol used in (He et al., 2016).
Revised: ImageNet by following the training protocol used in (He et al., 2016).

Original: In particular, ResNet18 is trained by using SGD with momentum of 0.9 and the initial learning rate is 0.1, which is then decayed by a factor of 10 after 30, 60, and 90 training epochs.
Revised: In particular, ResNet18 is trained by using SGD with momentum of 0.9 and the initial learning rate is 0.1, which is then decayed by a factor of 10 after 30, 60, and 90 training epochs.

Original: In downsampled ImageNet, we observe similar trends as those presented in CIFAR10.
Revised: In downsampled ImageNet, we observe similar trends as those presented in CIFAR10.

Original: For example, we see that BN would imped both loss and accuracy when batch size increases.
Revised: For example, we see that BN would imped both loss and accuracy when batch size increases.

Original: When increasing M to 1024 as shown in Fig.3, both the loss and validation accuracy decrease because the regularization from the random batch statistics reduces in large batch size, resulting in overtraining.
Revised: When increasing M to 1024 as shown in Fig.3, both the loss and validation accuracy decrease because the regularization from the random batch statistics reduces in large batch size, resulting in overtraining.

Original: This can be seen by the gap between the training and the validation loss.
Revised: This can be seen by the gap between the training and the validation loss.

Original: Nevertheless, we see that the reduction of regularization can be complemented when PN is trained with adaptive gamma decay, which makes
Revised: Nevertheless, we see that the reduction of regularization can be complemented when PN is trained with adaptive gamma decay, which makes

Original: PN performed comparably to BN in downsampled ImageNet.
Revised: PN performed comparably to BN in downsampled ImageNet.

Original: B. I MPACT OF BN TO THE N ORM OF P ARAMETERS
Revised: B. I MPACT OF BN TO THE N ORM OF P ARAMETERS

Original: We demonstrate the impact of BN to the norm of parameters.
Revised: We demonstrate the impact of BN to the norm of parameters.

Original: We compare BN with vanilla SGD, where a network is ﬁrst trained by BN in order to converge to a local minima when the parameters do not change much.
Revised: We compare BN with vanilla SGD, where a network is ﬁrst trained by BN in order to converge to a local minima when the parameters do not change much.

Original: At this local minima, the weight vector is frozen and denoted as w bn .
Revised: At this local minima, the weight vector is frozen and denoted as w bn .

Original: Then this
Revised: Then this

Original: [Figure]
Revised: [Figure]

Original: network is ﬁnetuned by using vanilla SGD with a small learning rate 10 − 3 with the kernel parameters initialized by w sgd = γ w bn σ , where σ is the moving average of σ B .
Revised: network is ﬁnetuned by using vanilla SGD with a small learning rate 10 − 3 with the kernel parameters initialized by w sgd = γ w bn σ , where σ is the moving average of σ B .

Original: Fig.4 below visualizes the results.
Revised: Fig.4 below visualizes the results.

Original: As µ B and σ B are removed in the vanilla SGD, it is found from the last two ﬁgures that the training loss decreases while the validation loss increases, meaning that the reduction in regularization makes the network converged to a sharper local minimum that generalizes less well.
Revised: As µ B and σ B are removed in the vanilla SGD, it is found from the last two ﬁgures that the training loss decreases while the validation loss increases, meaning that the reduction in regularization makes the network converged to a sharper local minimum that generalizes less well.

Original: The magnitudes of kernel parameters w sgd at different layers are also displayed in the ﬁrst four ﬁgures.
Revised: The magnitudes of kernel parameters w sgd at different layers are also displayed in the ﬁrst four ﬁgures.

Original: All of them increase after freezing BN, due to the release of regularization on these parameters.
Revised: All of them increase after freezing BN, due to the release of regularization on these parameters.

Original: [Figure]
Revised: [Figure]

Original: B. BN AND WN WITH DROPOUT
Revised: B. BN AND WN WITH DROPOUT

Original: BN+dropout .
Revised: BN+dropout .

Original: Despite the better generalization of BN with smaller batch sizes, large-batch training is more efﬁcient in real cases.
Revised: Despite the better generalization of BN with smaller batch sizes, large-batch training is more efﬁcient in real cases.

Original: Therefore, improving generalization of BN with large batch is more desiring.
Revised: Therefore, improving generalization of BN with large batch is more desiring.

Original: However, gamma decay requires estimating the population statistics that increases computations.
Revised: However, gamma decay requires estimating the population statistics that increases computations.

Original: We also found that treating the decay factor as a constant hardly improves generalization for large batch.
Revised: We also found that treating the decay factor as a constant hardly improves generalization for large batch.

Original: Therefore, we utilize dropout as an alternative to compensate for the insufﬁcient regularization.
Revised: Therefore, we utilize dropout as an alternative to compensate for the insufﬁcient regularization.

Original: Dropout has also been analytically viewed as a regularizer (Wager et al., 2013).
Revised: Dropout has also been analytically viewed as a regularizer (Wager et al., 2013).

Original: We add a dropout after each BN layer to impose regularization.
Revised: We add a dropout after each BN layer to impose regularization.

Original: Fig.2(g&h) in the main paper plot the classiﬁcation results using ResNet18.
Revised: Fig.2(g&h) in the main paper plot the classiﬁcation results using ResNet18.

Original: The generalization of
Revised: The generalization of

Original: BN deteriorates signiﬁcantly when M increases from 128 to 1024.
Revised: BN deteriorates signiﬁcantly when M increases from 128 to 1024.

Original: This is observed by the much higher validation loss (Fig.2(g)) and lower validation accuracy (Fig.2(h)) when M = 1024 .
Revised: This is observed by the much higher validation loss (Fig.2(g)) and lower validation accuracy (Fig.2(h)) when M = 1024 .

Original: If a dropout layer with ratio 0 .
Revised: If a dropout layer with ratio 0 .

Original: 1 is added after each residual block layer for M = 1024 in ResNet18, the validation loss is suppressed and accuracy increased by a great margin.
Revised: 1 is added after each residual block layer for M = 1024 in ResNet18, the validation loss is suppressed and accuracy increased by a great margin.

Original: This superﬁcially contradicts with the original claim that BN reduces the need for dropout (Ioffe & Szegedy, 2015).
Revised: This superﬁcially contradicts with the original claim that BN reduces the need for dropout (Ioffe & Szegedy, 2015).

Original: We ﬁnd that there are two differences between our study and (Ioffe & Szegedy, 2015).
Revised: We ﬁnd that there are two differences between our study and (Ioffe & Szegedy, 2015).

Original: First, in pervious study the batch size was ﬁxed at a quite small value ( e.g. 32), at which the regularization was already quite strong.
Revised: First, in pervious study the batch size was ﬁxed at a quite small value ( e.g. 32), at which the regularization was already quite strong.

Original: Therefore, an additional dropout could not further cause better regularization, but on the contrary increases the instability in training and yields a lower accuracy.
Revised: Therefore, an additional dropout could not further cause better regularization, but on the contrary increases the instability in training and yields a lower accuracy.

Original: However, our study explores relatively large batch that degrades the regularization of BN, and thus dropout with a small ratio can complement.
Revised: However, our study explores relatively large batch that degrades the regularization of BN, and thus dropout with a small ratio can complement.

Original: Second, usual trials put dropout before BN and cause BN to have different variances during training and test.
Revised: Second, usual trials put dropout before BN and cause BN to have different variances during training and test.

Original: In contrast, dropout follows BN in this study and the distance between two dropout layers is large (a residual block separation), thus the problem can be alleviated.
Revised: In contrast, dropout follows BN in this study and the distance between two dropout layers is large (a residual block separation), thus the problem can be alleviated.

Original: The improvement by applying dropout after BN has also been observed by a recent work (Li et al., 2018).
Revised: The improvement by applying dropout after BN has also been observed by a recent work (Li et al., 2018).

Original: WN+dropout .
Revised: WN+dropout .

Original: Since BN can be treated as WN trained with regularization as shown in this study, combining WN with regularization should be able to match the performance of BN.
Revised: Since BN can be treated as WN trained with regularization as shown in this study, combining WN with regularization should be able to match the performance of BN.

Original: As WN outperforms BN in running speed (without calculating statistics) and it suits better in RNNs than
Revised: As WN outperforms BN in running speed (without calculating statistics) and it suits better in RNNs than

Original: BN, an improvement of its generalization is also of great importance.
Revised: BN, an improvement of its generalization is also of great importance.

Original: Fig.2(g&h) also show that
Revised: Fig.2(g&h) also show that

Original: WN can also be regularized by dropout.
Revised: WN can also be regularized by dropout.

Original: We apply dropout after each WN layer with ratio 0.2 and the dropout is applied at the same layers as that for BN.
Revised: We apply dropout after each WN layer with ratio 0.2 and the dropout is applied at the same layers as that for BN.

Original: We found that the improvement on both validation accuracy and loss is surprising.
Revised: We found that the improvement on both validation accuracy and loss is surprising.

Original: The accuracy increases from 0.90 to 0.93, even close to the results of BN.
Revised: The accuracy increases from 0.90 to 0.93, even close to the results of BN.

Original: Nevertheless, additional regularization on WN still cannot make WN on par with the performance BN.
Revised: Nevertheless, additional regularization on WN still cannot make WN on par with the performance BN.

Original: In deep neural networks the distribution after each layer would be far from a
Revised: In deep neural networks the distribution after each layer would be far from a

Original: Gaussian distribution, in which case WN is not a good substitute for PN.
Revised: Gaussian distribution, in which case WN is not a good substitute for PN.

Original: A potential substibute of
Revised: A potential substibute of

Original: BN would require us for designing better estimations of the distribution to improve the training speed and performance of deep networks.
Revised: BN would require us for designing better estimations of the distribution to improve the training speed and performance of deep networks.

Original: C P ROOF OF R ESULTS C. P ROOF OF E QN .(3)
Revised: C P ROOF OF R ESULTS C. P ROOF OF E QN .(3)

Original: Theorem 1 (Regularization of µ B , σ B ) .
Revised: Theorem 1 (Regularization of µ B , σ B ) .

Original: Let a single-layer perceptron with BN and ReLU activation function be deﬁned by y = max(0 , ˆ h ) , ˆ h = γ h − µ B σ B + β and h = w T x , where x and y are the network input and output respectively, h and ˆ h are the hidden values before and after batch normalization, and w is the weight vector.
Revised: Let a single-layer perceptron with BN and ReLU activation function be deﬁned by y = max(0 , ˆ h ) , ˆ h = γ h − µ B σ B + β and h = w T x , where x and y are the network input and output respectively, h and ˆ h are the hidden values before and after batch normalization, and w is the weight vector.

Original: Let (cid:96) (ˆ h ) be the loss function.
Revised: Let (cid:96) (ˆ h ) be the loss function.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where ¯ h j = γ w T x j − µ P σ P + β represents population normalization (PN), ζ ( h ) γ 2 represents gamma decay and ζ ( h ) is a data-dependent decay factor.
Revised: where ¯ h j = γ w T x j − µ P σ P + β represents population normalization (PN), ζ ( h ) γ 2 represents gamma decay and ζ ( h ) is a data-dependent decay factor.

Original: ρ is the kurtosis of the distribution of h , I ( γ ) is an estimation of the Fisher information of γ and I ( γ )
Revised: ρ is the kurtosis of the distribution of h , I ( γ ) is an estimation of the Fisher information of γ and I ( γ )

Original: = 1 P (cid:80)
Revised: = 1 P (cid:80)

Original: Pj =1 ( ∂(cid:96) (ˆ h j ) ∂γ ) 2 , and σ ( · ) is a sigmoid function.
Revised: Pj =1 ( ∂(cid:96) (ˆ h j ) ∂γ ) 2 , and σ ( · ) is a sigmoid function.

Original: Proof.
Revised: Proof.

Original: We have ˆ h j = γ w T x j − µ B σ B +
Revised: We have ˆ h j = γ w T x j − µ B σ B +

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: + β .
Revised: + β .

Original: We prove theorem 1 by performing a Taylor expansion on a function
Revised: We prove theorem 1 by performing a Taylor expansion on a function

Original: A (ˆ h j ) at ¯ h j , where A (ˆ h j ) is a function of ˆ h j deﬁned according to a particular activation function.
Revised: A (ˆ h j ) at ¯ h j , where A (ˆ h j ) is a function of ˆ h j deﬁned according to a particular activation function.

Original: The negative log likelihood function of the above single-layer perceptron can be generally deﬁned as − log p ( y j | ˆ h j )
Revised: The negative log likelihood function of the above single-layer perceptron can be generally deﬁned as − log p ( y j | ˆ h j )

Original: =
Revised: =

Original: A (ˆ h j ) − y j ˆ h j , which is similar to the loss function of the generalized linear models with different activation functions.
Revised: A (ˆ h j ) − y j ˆ h j , which is similar to the loss function of the generalized linear models with different activation functions.

Original: Therefore, we have
Revised: Therefore, we have

Original: [Equation]
Revised: [Equation]

Original: where A (cid:48) ( · ) and A (cid:48)(cid:48) ( · ) denote the ﬁrst and second derivatives of function A ( · ) .
Revised: where A (cid:48) ( · ) and A (cid:48)(cid:48) ( · ) denote the ﬁrst and second derivatives of function A ( · ) .

Original: The ﬁrst and second order terms in the expansion are represented by R f and R q respectively.
Revised: The ﬁrst and second order terms in the expansion are represented by R f and R q respectively.

Original: To derive the analytical forms of R f and R q
Revised: To derive the analytical forms of R f and R q

Original: [Equation]
Revised: [Equation]

Original: we take a second-order Taylor expansion of of 1 σ B and 1 σ 2 B around σ P , it sufﬁces to have
Revised: we take a second-order Taylor expansion of of 1 σ B and 1 σ 2 B around σ P , it sufﬁces to have

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: By applying the distributions of µ B and σ B introduced in section
Revised: By applying the distributions of µ B and σ B introduced in section

Original: [Equation]
Revised: [Equation]

Original: we have µ B ∼ N ( µ P , σ 2 P M ) and σ B ∼ N ( σ P , ρ +24 M ) .
Revised: we have µ B ∼ N ( µ P , σ 2 P M ) and σ B ∼ N ( σ P , ρ +24 M ) .

Original: Hence
Revised: Hence

Original: [Equation]
Revised: [Equation]

Original: R f can be derived as in the paper
Revised: R f can be derived as in the paper

Original: [Equation]
Revised: [Equation]

Original: R f can be derived as
Revised: R f can be derived as

Original: [Equation]
Revised: [Equation]

Original: This R f term can be understood as below.
Revised: This R f term can be understood as below.

Original: Let h = w T x − µ P σ P and the distribution of the population data be p xy .
Revised: Let h = w T x − µ P σ P and the distribution of the population data be p xy .

Original: We establish the following relationship
Revised: We establish the following relationship

Original: [Equation]
Revised: [Equation]

Original: Since the sample mean converges in probability to the population mean by the Weak Law of
Revised: Since the sample mean converges in probability to the population mean by the Weak Law of

Original: Large Numbers, for all (cid:15) > 0 and a constant number K ( ∃ K > 0 and ∀ P > K ), we have
Revised: Large Numbers, for all (cid:15) > 0 and a constant number K ( ∃ K > 0 and ∀ P > K ), we have

Original: [Equation]
Revised: [Equation]

Original: = 0 .
Revised: = 0 .

Original: This equation implies that R f is sufﬁciently small with a probability of 1 given moderately large number of data points P (the above inequality holds when P > 30 )
Revised: This equation implies that R f is sufﬁciently small with a probability of 1 given moderately large number of data points P (the above inequality holds when P > 30 )

Original: [Equation]
Revised: [Equation]

Original: On the other hand
Revised: On the other hand

Original: [Equation]
Revised: [Equation]

Original: R q can be derived as
Revised: R q can be derived as

Original: [Equation]
Revised: [Equation]

Original: Note that ∂ 2 l (¯ h j ) ∂γ
Revised: Note that ∂ 2 l (¯ h j ) ∂γ

Original: [Equation]
Revised: [Equation]

Original: ) 2 , we have I ( γ )
Revised: ) 2 , we have I ( γ )

Original: =
Revised: =

Original: 1 P (cid:80) Pj
Revised: 1 P (cid:80) Pj

Original: [Equation]
Revised: [Equation]

Original: (¯ h j )(
Revised: (¯ h j )(

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: P ) 2 been an estimator of the Fisher information with respect to the scale parameter γ .
Revised: P ) 2 been an estimator of the Fisher information with respect to the scale parameter γ .

Original: Then, by neglecting
Revised: Then, by neglecting

Original: O (1 /M 2 ) high-order term in R q , we get
Revised: O (1 /M 2 ) high-order term in R q , we get

Original: [Equation]
Revised: [Equation]

Original: where µ d 2 A indicates the mean of the second derivative of A ( h ) .
Revised: where µ d 2 A indicates the mean of the second derivative of A ( h ) .

Original: The results of both ReLU activation function and identity function are provided as below.
Revised: The results of both ReLU activation function and identity function are provided as below.

Original: C. R E LU A CTIVATION F UNCTION
Revised: C. R E LU A CTIVATION F UNCTION

Original: For the ReLU non-linear activation function, that is g ( h ) =
Revised: For the ReLU non-linear activation function, that is g ( h ) =

Original: max( h, 0) , we use its continuous approximation softplus function g ( h )
Revised: max( h, 0) , we use its continuous approximation softplus function g ( h )

Original: =
Revised: =

Original: log(1+exp( h )) to derive the partition function A ( h ) .
Revised: log(1+exp( h )) to derive the partition function A ( h ) .

Original: In this case, we have µ d 2 A = 1 P (cid:80) Pj =1 σ (¯ h j ) .
Revised: In this case, we have µ d 2 A = 1 P (cid:80) Pj =1 σ (¯ h j ) .

Original: Therefore, we have ζ ( h ) =
Revised: Therefore, we have ζ ( h ) =

Original: ρ +28 M I
Revised: ρ +28 M I

Original: ( γ )
Revised: ( γ )

Original: +
Revised: +

Original: 12 M 1 P (cid:80) Pj =1 σ (¯ h j ) as shown in Eqn.(3).
Revised: 12 M 1 P (cid:80) Pj =1 σ (¯ h j ) as shown in Eqn.(3).

Original: C. L INEAR S TUDENT N ETWORK WITH I DENTITY A CTIVATION F UNCTION
Revised: C. L INEAR S TUDENT N ETWORK WITH I DENTITY A CTIVATION F UNCTION

Original: For a loss function with identity (linear) units, 1 P (cid:80) Pj =1 (cid:0) w ∗ T x j − γ ( w T x j − µ B ) /σ B (cid:1) 2 , we have
Revised: For a loss function with identity (linear) units, 1 P (cid:80) Pj =1 (cid:0) w ∗ T x j − γ ( w T x j − µ B ) /σ B (cid:1) 2 , we have

Original: I ( γ ) = 2 λ and ρ = 0 for Gaussian input distribution.
Revised: I ( γ ) = 2 λ and ρ = 0 for Gaussian input distribution.

Original: The exact expression of Eqn.(3) is also possible for such linear regression problem.
Revised: The exact expression of Eqn.(3) is also possible for such linear regression problem.

Original: Under the condition of Gaussian input x ∼ N (0 , 1 /N ) , h = w T x is also a random variable satisfying a normal distribution N (0 , 1) .
Revised: Under the condition of Gaussian input x ∼ N (0 , 1 /N ) , h = w T x is also a random variable satisfying a normal distribution N (0 , 1) .

Original: It can be derived that
Revised: It can be derived that

Original: E (cid:0) σ − 1 B (cid:1) = √ M √ 2 σ P Γ ( M − 2 2 ) Γ ( M − 1 2 ) and E (cid:0) σ − 2 B (cid:1) =
Revised: E (cid:0) σ − 1 B (cid:1) = √ M √ 2 σ P Γ ( M − 2 2 ) Γ ( M − 1 2 ) and E (cid:0) σ − 2 B (cid:1) =

Original: Mσ 2 P Γ ( M − 1 2 − 1 ) Γ ( M − 1 2 ) .
Revised: Mσ 2 P Γ ( M − 1 2 − 1 ) Γ ( M − 1 2 ) .

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: Furthermore, the expression of ζ can be simpliﬁed as ζ = 34 M .
Revised: Furthermore, the expression of ζ can be simpliﬁed as ζ = 34 M .

Original: If the bias term is neglected in a simple linear regression, contributions from µ B to the regularization term is neglected and thus ζ = 14 M .
Revised: If the bias term is neglected in a simple linear regression, contributions from µ B to the regularization term is neglected and thus ζ = 14 M .

Original: Note that if one uses mean square error without being divided by 2 during linear regression, the values for ζ should be multiplied by 2 as well, where ζ = 1 2 M .
Revised: Note that if one uses mean square error without being divided by 2 during linear regression, the values for ζ should be multiplied by 2 as well, where ζ = 1 2 M .

Original: C. BN R EGULARIZATION IN A D EEP N ETWORK
Revised: C. BN R EGULARIZATION IN A D EEP N ETWORK

Original: The previous derivation is based on the single-layer perceptron.
Revised: The previous derivation is based on the single-layer perceptron.

Original: In deep neural networks, the forward computation inside one basic building block of a deep network is written by
Revised: In deep neural networks, the forward computation inside one basic building block of a deep network is written by

Original: [Equation]
Revised: [Equation]

Original: where the superscript l ∈ [1 , L ] is the index of a building block in a deep neural network, and i ∈ [1 , N l ] indexes each neuron inside a layer.
Revised: where the superscript l ∈ [1 , L ] is the index of a building block in a deep neural network, and i ∈ [1 , N l ] indexes each neuron inside a layer.

Original: z 

Revised: z

Original: and z L are synonyms of input x and output y , respectively.
Revised: and z L are synonyms of input x and output y , respectively.

Original: In order to analyze the regularization of BN from a speciﬁc layer, one needs to isolate its input and focus on the noise introduced by the BN layer in this block.
Revised: In order to analyze the regularization of BN from a speciﬁc layer, one needs to isolate its input and focus on the noise introduced by the BN layer in this block.

Original: Therefore, the loss function (cid:96) (ˆ h l ) can also be expanded at (cid:96) (¯ h l ) .
Revised: Therefore, the loss function (cid:96) (ˆ h l ) can also be expanded at (cid:96) (¯ h l ) .

Original: In BN, the batch variance is calculated with regard to each neuron under the assumption of mutual independence of neurons inside a layer.
Revised: In BN, the batch variance is calculated with regard to each neuron under the assumption of mutual independence of neurons inside a layer.

Original: By following this assumption and the above derivation in Appendix C.1, the loss function with BN in deep networks can also be similarly decomposed.
Revised: By following this assumption and the above derivation in Appendix C.1, the loss function with BN in deep networks can also be similarly decomposed.

Original: Regularization of µ l B , σ l B in a deep network.
Revised: Regularization of µ l B , σ l B in a deep network.

Original: Let ζ l be the strength (coefﬁcient) of the regularization at the l -th layer.
Revised: Let ζ l be the strength (coefﬁcient) of the regularization at the l -th layer.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: and ζ li = 1 P
Revised: and ζ li = 1 P

Original: [Equation]
Revised: [Equation]

Original: M
Revised: M

Original: [Equation]
Revised: [Equation]

Original: where i is the index of a neuron in the layer, (¯ h li )
Revised: where i is the index of a neuron in the layer, (¯ h li )

Original: j = γ li ( w li ) T ( z l − 1 ) j − ( µ P ) li ( σ P ) li + β li represents population normalization (PN), H (cid:96) (¯ h l ) is the Hessian matrix at ¯ h l regarding to the loss (cid:96) and diag( · ) represents the diagonal vector of a matrix.
Revised: j = γ li ( w li ) T ( z l − 1 ) j − ( µ P ) li ( σ P ) li + β li represents population normalization (PN), H (cid:96) (¯ h l ) is the Hessian matrix at ¯ h l regarding to the loss (cid:96) and diag( · ) represents the diagonal vector of a matrix.

Original: It is seen that the above equation is compatible with the results from the single-layer perceptron.
Revised: It is seen that the above equation is compatible with the results from the single-layer perceptron.

Original: The main difference of the regularization term in a deep model is that the Hessian matrix is not guaranteed to be positive semi-deﬁnite during training.
Revised: The main difference of the regularization term in a deep model is that the Hessian matrix is not guaranteed to be positive semi-deﬁnite during training.

Original: However, this form of regularization is also seen from other regularization such as noise injection (Rifai et al., 2011) and dropout (Wager et al., 2013), and has long been recognized as a Tikhonov regularization term (Bishop, 1995).
Revised: However, this form of regularization is also seen from other regularization such as noise injection (Rifai et al., 2011) and dropout (Wager et al., 2013), and has long been recognized as a Tikhonov regularization term (Bishop, 1995).

Original: In fact, it has been reported that in common neural networks, where convex activation functions such as ReLU and convex loss functions such as common cross entropy are adopted, the Hessian matrix
Revised: In fact, it has been reported that in common neural networks, where convex activation functions such as ReLU and convex loss functions such as common cross entropy are adopted, the Hessian matrix

Original: H (cid:96) (¯ h l ) can be seen as ‘locally’ positive semideﬁnite (Santurkar et al., 2018).
Revised: H (cid:96) (¯ h l ) can be seen as ‘locally’ positive semideﬁnite (Santurkar et al., 2018).

Original: Especially, as training converges to its mimimum training loss, the Hessian matrix of the loss can be viewed as positive semi-deﬁnite and thus the regularization term on γ l is positive.
Revised: Especially, as training converges to its mimimum training loss, the Hessian matrix of the loss can be viewed as positive semi-deﬁnite and thus the regularization term on γ l is positive.

Original: C. D YNAMICAL E QUATIONS
Revised: C. D YNAMICAL E QUATIONS

Original: Here we discuss the dynamical equations of BN.
Revised: Here we discuss the dynamical equations of BN.

Original: Let the length of teacher’s weight vector be 1, that is, 1 N w ∗ T w ∗ = 1 .
Revised: Let the length of teacher’s weight vector be 1, that is, 1 N w ∗ T w ∗ = 1 .

Original: We introduce a normalized weight vector of the student as (cid:101) w = √
Revised: We introduce a normalized weight vector of the student as (cid:101) w = √

Original: Nγ w (cid:107) w (cid:107) .
Revised: Nγ w (cid:107) w (cid:107) .

Original: Then the overlapping ratio between teacher and student, the length of student’s vector, and the length of student’s normalized weight vector are 1 N (cid:101) w T w ∗ =
Revised: Then the overlapping ratio between teacher and student, the length of student’s vector, and the length of student’s normalized weight vector are 1 N (cid:101) w T w ∗ =

Original: QR = γR , 1 N (cid:101) w T (cid:101)
Revised: QR = γR , 1 N (cid:101) w T (cid:101)

Original: w = Q 2 = γ 2 , and
Revised: w = Q 2 = γ 2 , and

Original: N w T w =
Revised: N w T w =

Original: L 2 respectively, where Q = γ .
Revised: L 2 respectively, where Q = γ .

Original: And we have 1 N w T w = LR .
Revised: And we have 1 N w T w = LR .

Original: We transform update equations (5) by using order parameters.
Revised: We transform update equations (5) by using order parameters.

Original: The update rule for variable Q 2 can be obtained by (cid:0) Q 2 (cid:1)
Revised: The update rule for variable Q 2 can be obtained by (cid:0) Q 2 (cid:1)

Original: j +1 − (cid:0) Q 2 (cid:1)
Revised: j +1 − (cid:0) Q 2 (cid:1)

Original: j = 1 N (cid:2) 2 ηδ j ˜ w j T x j − 2 ηζ (cid:0) Q 2 (cid:1) j (cid:3) following update rule of γ .
Revised: j = 1 N (cid:2) 2 ηδ j ˜ w j T x j − 2 ηζ (cid:0) Q 2 (cid:1) j (cid:3) following update rule of γ .

Original: Similarly, the update rules for variables RL and L 2 are calculated as follow:
Revised: Similarly, the update rules for variables RL and L 2 are calculated as follow:

Original: [Equation]
Revised: [Equation]

Original: Let t = jN is a normalized sample index that can be treated as a continuous time variable.
Revised: Let t = jN is a normalized sample index that can be treated as a continuous time variable.

Original: We have ∆ t = 1 N that approaches zero in the thermodynamic limit when N → ∞ .
Revised: We have ∆ t = 1 N that approaches zero in the thermodynamic limit when N → ∞ .

Original: In this way, the learning dynamic of Q 2 , RL and L 2 can be formulated as the following differential equations:
Revised: In this way, the learning dynamic of Q 2 , RL and L 2 can be formulated as the following differential equations:

Original: [Equation]
Revised: [Equation]

Original: where I 1 = (cid:104) δ ˜ w T x (cid:105) x , I 2 = (cid:104) δ 2 x T x (cid:105) x , and I 3 = (cid:104) δ w ∗ T x (cid:105) x , which are the terms presented in dQ 2 dt , dRLdt , and dL 2 dt and (cid:104)·(cid:105) x denotes expectation over the distribution of x .
Revised: where I 1 = (cid:104) δ ˜ w T x (cid:105) x , I 2 = (cid:104) δ 2 x T x (cid:105) x , and I 3 = (cid:104) δ w ∗ T x (cid:105) x , which are the terms presented in dQ 2 dt , dRLdt , and dL 2 dt and (cid:104)·(cid:105) x denotes expectation over the distribution of x .

Original: They are used to simplify notations.
Revised: They are used to simplify notations.

Original: Note that we neglect the last term of dL 2 /dt in Eqn.(8) since η 2 N ( L 2 ) δ 2 ( ˜ w T x )can be approximately equal to zero when N approaches inﬁnity.
Revised: Note that we neglect the last term of dL 2 /dt in Eqn.(8) since η 2 N ( L 2 ) δ 2 ( ˜ w T x )can be approximately equal to zero when N approaches inﬁnity.

Original: On the other hand, we have dQ 2 = 2 QdQ, dRL = RdL + LdR and dL 2 = 2 LdL .
Revised: On the other hand, we have dQ 2 = 2 QdQ, dRL = RdL + LdR and dL 2 = 2 LdL .

Original: Hence, Eqn.(9) can be reduced to
Revised: Hence, Eqn.(9) can be reduced to

Original: [Equation]
Revised: [Equation]

Original: Proposition 1.
Revised: Proposition 1.

Original: Let ( Q 0 , R 0 , L 0 ) denote a ﬁxed point with parameters Q , R and L of Eqn.(10).
Revised: Let ( Q 0 , R 0 , L 0 ) denote a ﬁxed point with parameters Q , R and L of Eqn.(10).

Original: Assume the learning rate η is sufﬁciently small when training converges and x ∼ N (0 , 1 N I ) .
Revised: Assume the learning rate η is sufﬁciently small when training converges and x ∼ N (0 , 1 N I ) .

Original: If activation function g is ReLU , then we have Q 0 = 1 2 ζ +1 , R 

Revised: If activation function g is ReLU , then we have Q 0 = 1 2 ζ +1 , R

Original: = 

Revised: =

Original: and L 0 could be arbitrary.
Revised: and L 0 could be arbitrary.

Original: Proof.
Revised: Proof.

Original: First, L has no inﬂuence on the output of student model since w is normalized, which implies that if ( Q 0 , R 0 , L 0 ) is a ﬁxed point of Eqn.(10), L 0 could be arbitrary.
Revised: First, L has no inﬂuence on the output of student model since w is normalized, which implies that if ( Q 0 , R 0 , L 0 ) is a ﬁxed point of Eqn.(10), L 0 could be arbitrary.

Original: Besides, we have η (cid:29) ηbecause the learning rate η is sufﬁciently small.
Revised: Besides, we have η (cid:29) ηbecause the learning rate η is sufﬁciently small.

Original: Therefore, the terms in Eqn.(10) proportional to ηcan be neglected.
Revised: Therefore, the terms in Eqn.(10) proportional to ηcan be neglected.

Original: If ( Q 0 , R 0 , L 0 ) is a ﬁxed point, it sufﬁces to have
Revised: If ( Q 0 , R 0 , L 0 ) is a ﬁxed point, it sufﬁces to have

Original: [Equation]
Revised: [Equation]

Original: To calculate I 

Revised: To calculate I

Original: and I 3 , we deﬁne s and t as ˜ w T x and w T x .
Revised: and I 3 , we deﬁne s and t as ˜ w T x and w T x .

Original: Since x ∼ N (0 , 1 N I ) , we can acquire
Revised: Since x ∼ N (0 , 1 N I ) , we can acquire

Original: [Equation]
Revised: [Equation]

Original: so probability measure of [ s, t ] T can be written as
Revised: so probability measure of [ s, t ] T can be written as

Original: [Equation]
Revised: [Equation]

Original: Then,
Revised: Then,

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: By substituting Eqn.(13) and (14) into Eqn.(11) and (12), we get Q 0 =
Revised: By substituting Eqn.(13) and (14) into Eqn.(11) and (12), we get Q 0 =

Original: [Equation]
Revised: [Equation]

Original: and R 0 = 1 .
Revised: and R 0 = 1 .

Original: Proposition 2.
Revised: Proposition 2.

Original: Given conditions in proposition1, let λ bnQ , λ bnR be the eigenvalues of the Jacobian matrix at ﬁxed point ( Q 0 , R 0 , L 0 ) corresponding to the order parameters Q and R respectively in
Revised: Given conditions in proposition1, let λ bnQ , λ bnR be the eigenvalues of the Jacobian matrix at ﬁxed point ( Q 0 , R 0 , L 0 ) corresponding to the order parameters Q and R respectively in

Original: BN.
Revised: BN.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where η bn max and η bn eﬀ are the maximum and effective learning rates respectively in BN.
Revised: where η bn max and η bn eﬀ are the maximum and effective learning rates respectively in BN.

Original: Proof.
Revised: Proof.

Original: At ﬁxed point ( Q 0 , R 0 , L 0 ) =
Revised: At ﬁxed point ( Q 0 , R 0 , L 0 ) =

Original: ( 1 2 ζ +1 , 1 , L 0 ) obtained in proposition1, the Jacobian of dynamic equations of BN can be derived as
Revised: ( 1 2 ζ +1 , 1 , L 0 ) obtained in proposition1, the Jacobian of dynamic equations of BN can be derived as

Original: [Equation]
Revised: [Equation]

Original: , and the eigenvalues of J bn can be obtained by inspection
Revised: , and the eigenvalues of J bn can be obtained by inspection

Original: [Equation]
Revised: [Equation]

Original: Since γ 0 = Q 0 , we have η bn max = ( ∂ ( γ 

Revised: Since γ 0 = Q 0 , we have η bn max = ( ∂ ( γ

Original: I 3 − I 1 ) γ 0 ∂R − ζγ 0 ) / ∂I 2 2 ∂R and η bn eﬀ = ηγ 0 L 20 .
Revised: I 3 − I 1 ) γ 0 ∂R − ζγ 0 ) / ∂I 2 2 ∂R and η bn eﬀ = ηγ 0 L 20 .

Original: C. STABLE FIXED POINTS OF BN
Revised: C. STABLE FIXED POINTS OF BN

Original: Proposition 3.
Revised: Proposition 3.

Original: Given conditions in proposition1, when activation function is ReLU, then (i) λ bnQ < 0 , and (ii) λ bnR < 0 iff η bn max > η bn eﬀ .
Revised: Given conditions in proposition1, when activation function is ReLU, then (i) λ bnQ < 0 , and (ii) λ bnR < 0 iff η bn max > η bn eﬀ .

Original: Proof.
Revised: Proof.

Original: When activation function is ReLU, we derive I 1 = Q (
Revised: When activation function is ReLU, we derive I 1 = Q (

Original: [Equation]
Revised: [Equation]

Original: 2 R
Revised: 2 R

Original: [Equation]
Revised: [Equation]

Original: 4 π − Q 2 2 , which gives
Revised: 4 π − Q 2 2 , which gives

Original: [Equation]
Revised: [Equation]

Original: Therefore at the ﬁxed point of BN ( Q 0 , R 0 , L 0 )
Revised: Therefore at the ﬁxed point of BN ( Q 0 , R 0 , L 0 )

Original: = ( 1 2 ζ +1 , 1 , L 0 ) , we have
Revised: = ( 1 2 ζ +1 , 1 , L 0 ) , we have

Original: [Equation]
Revised: [Equation]

Original: Note that x T x approximately equals 1.
Revised: Note that x T x approximately equals 1.

Original: We get
Revised: We get

Original: I 2 =
Revised: I 2 =

Original: [Equation]
Revised: [Equation]

Original: At the ﬁxed point we have ∂I 2 ∂R = − Q 0 < 0 .
Revised: At the ﬁxed point we have ∂I 2 ∂R = − Q 0 < 0 .

Original: Therefore, we conclude that λ bnR < 

Revised: Therefore, we conclude that λ bnR <

Original: iff η bn max > η bn eﬀ .
Revised: iff η bn max > η bn eﬀ .

Original: C. M AXIMUM L EARNING R ATE OF BN
Revised: C. M AXIMUM L EARNING R ATE OF BN

Original: Proposition 4.
Revised: Proposition 4.

Original: When the activation function is ReLU, then η bn max ≥ η { wn,sgd } max + 2 ζ , where η bn max and η { wn,sgd } max indicate the maximum learning rates of BN, WN, and vanilla SGD respectively.
Revised: When the activation function is ReLU, then η bn max ≥ η { wn,sgd } max + 2 ζ , where η bn max and η { wn,sgd } max indicate the maximum learning rates of BN, WN, and vanilla SGD respectively.

Original: Proof.
Revised: Proof.

Original: From the above results, we have I 1 = Q ( πR +2 √ 1 − R 2 +

Revised: From the above results, we have I 1 = Q ( πR +2 √ 1 − R 2 +

Original: R arcsin( R )) 4 π − Q 2 2 , which gives ∂I 1 /∂R ≥ 0 at the ﬁxed point of BN.
Revised: R arcsin( R )) 4 π − Q 2 2 , which gives ∂I 1 /∂R ≥ 0 at the ﬁxed point of BN.

Original: Then it can be derived that ∂I 2 ∂R < 0 .
Revised: Then it can be derived that ∂I 2 ∂R < 0 .

Original: Furthermore, at the ﬁxed point of BN, Q 0 = γ 

Revised: Furthermore, at the ﬁxed point of BN, Q 0 = γ

Original: = 1 

Revised: = 1

Original: ζ +1 < 1 , then we have
Revised: ζ +1 < 1 , then we have

Original: [Equation]
Revised: [Equation]

Original: where the inequality sign holds because (1 − 1 γ 0 ) ∂I
Revised: where the inequality sign holds because (1 − 1 γ 0 ) ∂I

Original: 1 ∂R / ∂I 2 2 ∂R is positive.
Revised: 1 ∂R / ∂I 2 2 ∂R is positive.

Original: Note that ∂
Revised: Note that ∂

Original: ( I 3 − I 1 ) ∂R / ∂I 2 2 ∂R is also deﬁned as maximum learning rates of WN, and vanilla SGD in Yoshida et al.
Revised: ( I 3 − I 1 ) ∂R / ∂I 2 2 ∂R is also deﬁned as maximum learning rates of WN, and vanilla SGD in Yoshida et al.

Original: Hence, we conclude that η bn max ≥ η { wn,sgd } max + 2 ζ .
Revised: Hence, we conclude that η bn max ≥ η { wn,sgd } max + 2 ζ .

Original: D P ROOFS REGARDING GENERALIZATION AND STATISTICAL MECHANICS (SM)
Revised: D P ROOFS REGARDING GENERALIZATION AND STATISTICAL MECHANICS (SM)

Original: In this section, we build an analytical model for the generalization ability of a single-layer network.
Revised: In this section, we build an analytical model for the generalization ability of a single-layer network.

Original: The framework is based on the Teacher-Student model, where the teacher network output y ∗ =
Revised: The framework is based on the Teacher-Student model, where the teacher network output y ∗ =

Original: g ∗ (cid:0) w ∗ T · x + s (cid:1) is learned by a student network.
Revised: g ∗ (cid:0) w ∗ T · x + s (cid:1) is learned by a student network.

Original: The weight parameter of the teacher network satisﬁes 1 N ( w ∗ ) T · w ∗ =1 and the bias term s is a random variable s ∼ N (0 , S ) ﬁxed for each training example x to represent static errors in training data from observations.
Revised: The weight parameter of the teacher network satisﬁes 1 N ( w ∗ ) T · w ∗ =1 and the bias term s is a random variable s ∼ N (0 , S ) ﬁxed for each training example x to represent static errors in training data from observations.

Original: In the generalization analysis, the input is assumed to be drawn from x ∼ N (cid:0) 0 , 1 N I (cid:1) .
Revised: In the generalization analysis, the input is assumed to be drawn from x ∼ N (cid:0) 0 , 1 N I (cid:1) .

Original: The output of the student can also be written as a similar form y = g
Revised: The output of the student can also be written as a similar form y = g

Original: ( (cid:101) w · x ) , where the activation function g ( · ) can be either linear or ReLU in the analysis and (cid:101) w is a general weight parameter which can be used in either WN or common linear perceptrons.
Revised: ( (cid:101) w · x ) , where the activation function g ( · ) can be either linear or ReLU in the analysis and (cid:101) w is a general weight parameter which can be used in either WN or common linear perceptrons.

Original: Here we take WN for example, since it has been derived in this study that BN can be decomposed into WN with a regularization term on γ.
Revised: Here we take WN for example, since it has been derived in this study that BN can be decomposed into WN with a regularization term on γ.

Original: In WN (cid:101) w = γ w (cid:107) w (cid:107) 

Revised: In WN (cid:101) w = γ w (cid:107) w (cid:107)

Original: and we deﬁned the same order parameter as the previous section that γ 2 = 1 N (cid:101) w T · (cid:101) w and γR = 1 N (cid:101) w T · w ∗ .
Revised: and we deﬁned the same order parameter as the previous section that γ 2 = 1 N (cid:101) w T · (cid:101) w and γR = 1 N (cid:101) w T · w ∗ .

Original: D. G ENERALIZATION ERROR
Revised: D. G ENERALIZATION ERROR

Original: Since the learning task is a regression problem with teacher output biased by a Gaussian noise, it comes natural that we can use the the average mean square error loss (cid:15) t
Revised: Since the learning task is a regression problem with teacher output biased by a Gaussian noise, it comes natural that we can use the the average mean square error loss (cid:15) t

Original: = 1 P (cid:80) j (cid:0) y ∗ j − y j (cid:1) 2 for the regression.
Revised: = 1 P (cid:80) j (cid:0) y ∗ j − y j (cid:1) 2 for the regression.

Original: The generalization error deﬁned as the estimation over the distribution of input x and is written as
Revised: The generalization error deﬁned as the estimation over the distribution of input x and is written as

Original: [Equation]
Revised: [Equation]

Original: where (cid:104)·(cid:105) x denotes an average over the distribution over x .
Revised: where (cid:104)·(cid:105) x denotes an average over the distribution over x .

Original: The generalization error is a function of its weight parameter and can be converted to a function only with regard to the aformentioned order parameters, detailed derivation can be seen in (B¨os, 1998; Krogh & Hertz, 1992).
Revised: The generalization error is a function of its weight parameter and can be converted to a function only with regard to the aformentioned order parameters, detailed derivation can be seen in (B¨os, 1998; Krogh & Hertz, 1992).

Original: [Equation]
Revised: [Equation]

Original: where h 1 and h 2 are variables drawn from standard Gaussian distribution and Dh 1 := N (0 , 1) dh 1 .
Revised: where h 1 and h 2 are variables drawn from standard Gaussian distribution and Dh 1 := N (0 , 1) dh 1 .

Original: When both the teacher network and student network have a linear activation function, the above integration can be easily solved and
Revised: When both the teacher network and student network have a linear activation function, the above integration can be easily solved and

Original: [Equation]
Revised: [Equation]

Original: As for the case where the teacher network is linear and the student network has a ReLU activation, it can still be solved ﬁrst by decomposing the loss function
Revised: As for the case where the teacher network is linear and the student network has a ReLU activation, it can still be solved ﬁrst by decomposing the loss function

Original: [Equation]
Revised: [Equation]

Original: It should be noted that the last two terms should only be integrated over the half space γRh 1 + γ √ 1 − R 2 h 2 > 0 , and therefore if we deﬁne the angle of this line with the h the integration is transformed to polar coordinate (cid:15) gen
Revised: It should be noted that the last two terms should only be integrated over the half space γRh 1 + γ √ 1 − R 2 h 2 > 0 , and therefore if we deﬁne the angle of this line with the h the integration is transformed to polar coordinate (cid:15) gen

Original: [Equation]
Revised: [Equation]

Original: D. E QUILIBRIUM ORDER PARAMETERS
Revised: D. E QUILIBRIUM ORDER PARAMETERS

Original: Following studies on statistical mechanics, the learning process of a neural network resembles a
Revised: Following studies on statistical mechanics, the learning process of a neural network resembles a

Original: Langevin process (Mandt et al., 2017) and at the equilibrium the network parameters θ follow a Gibbs distribution.
Revised: Langevin process (Mandt et al., 2017) and at the equilibrium the network parameters θ follow a Gibbs distribution.

Original: That is, the weight vector that yields lower training error produces higher probability.
Revised: That is, the weight vector that yields lower training error produces higher probability.

Original: We have p ( θ ) =
Revised: We have p ( θ ) =

Original: Z − 1 exp {− β(cid:15) t ( θ ; x ) } , where β
Revised: Z − 1 exp {− β(cid:15) t ( θ ; x ) } , where β

Original: = 1 /T and T is temperature, representing the variance of noise during training and implicitly controlling the learning process.
Revised: = 1 /T and T is temperature, representing the variance of noise during training and implicitly controlling the learning process.

Original: (cid:15) t ( θ ; x ) is an energy term of the training loss function, Z = (cid:82) d P ( θ ) exp {− (cid:15) t ( θ ; x ) /T } is the partition function, and
Revised: (cid:15) t ( θ ; x ) is an energy term of the training loss function, Z = (cid:82) d P ( θ ) exp {− (cid:15) t ( θ ; x ) /T } is the partition function, and

Original: P ( θ ) is a prior distribution.
Revised: P ( θ ) is a prior distribution.

Original: Instead of directly minimizing the energy term above, statistical mechanics ﬁnds the minima of free energy, f , which is a function over T , considering the ﬂuctuations of θ at ﬁnite temperatures.
Revised: Instead of directly minimizing the energy term above, statistical mechanics ﬁnds the minima of free energy, f , which is a function over T , considering the ﬂuctuations of θ at ﬁnite temperatures.

Original: We have − βf =
Revised: We have − βf =

Original: (cid:104)
Revised: (cid:104)

Original: ln Z (cid:105) x .
Revised: ln Z (cid:105) x .

Original: By substituting the parameters that minimize f back into the generalization errors calculated above, we are able to calculate the averaged generalization error, at a certain temperature.
Revised: By substituting the parameters that minimize f back into the generalization errors calculated above, we are able to calculate the averaged generalization error, at a certain temperature.

Original: The solution of SM requires the differentiation of f with respect to the order parameters.
Revised: The solution of SM requires the differentiation of f with respect to the order parameters.

Original: In general, the expression of free energy under the replica theory is expressed as(Seung
Revised: In general, the expression of free energy under the replica theory is expressed as(Seung

Original: et al., 1992)
Revised: et al., 1992)

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: In the above expression, h 1 , h 2 , h 3 are three independent variables following the standard Gaussian distribution and α = P/N represents the ratio of the number of training samples P to number of unknown parameters N in the network, R = 1 N w (cid:107) w (cid:107) 2 · w ∗ , q is the prior value of γ ..
Revised: In the above expression, h 1 , h 2 , h 3 are three independent variables following the standard Gaussian distribution and α = P/N represents the ratio of the number of training samples P to number of unknown parameters N in the network, R = 1 N w (cid:107) w (cid:107) 2 · w ∗ , q is the prior value of γ ..

Original: The above equation can be utilized for a general SM solution of a network.
Revised: The above equation can be utilized for a general SM solution of a network.

Original: However, the solution is notoriously difﬁcult to solve and only a few linear settings for the student network have close-form solutions(B¨os, 1998).
Revised: However, the solution is notoriously difﬁcult to solve and only a few linear settings for the student network have close-form solutions(B¨os, 1998).

Original: Here we extend the previous analysis of linear activations to a non-linear one, though still under the condition that β → ∞ , which means that the student network undergoes a exhaustive learning that minimizes the training error.
Revised: Here we extend the previous analysis of linear activations to a non-linear one, though still under the condition that β → ∞ , which means that the student network undergoes a exhaustive learning that minimizes the training error.

Original: In the current setting, the student network is a nonlinear ReLU network while the teacher is a noise-corrupted linear one.
Revised: In the current setting, the student network is a nonlinear ReLU network while the teacher is a noise-corrupted linear one.

Original: Proposition 5.
Revised: Proposition 5.

Original: Given a single-layer linear teacher y ∗ =
Revised: Given a single-layer linear teacher y ∗ =

Original: w ∗ x + s and a student ReLU network y
Revised: w ∗ x + s and a student ReLU network y

Original: =
Revised: =

Original: g ( γ w (cid:107) w (cid:107) 2 x ) linear student network with g being a ReLU activation function, x ∼ N (0 , I N ) the free energy f satisﬁes as β → ∞
Revised: g ( γ w (cid:107) w (cid:107) 2 x ) linear student network with g being a ReLU activation function, x ∼ N (0 , I N ) the free energy f satisﬁes as β → ∞

Original: [Equation]
Revised: [Equation]

Original: where S is the variance of the Gaussian noise s injected to the output of the teacher.
Revised: where S is the variance of the Gaussian noise s injected to the output of the teacher.

Original: Proof.
Revised: Proof.

Original: The most difﬁcult process in Eqn.19 is to solve the inner integration over h 3 .
Revised: The most difﬁcult process in Eqn.19 is to solve the inner integration over h 3 .

Original: Here as β → ∞ , it is noted that the function exp ( − βx ) only notches up only at x = 0 and is 0 elsewhere.
Revised: Here as β → ∞ , it is noted that the function exp ( − βx ) only notches up only at x = 0 and is 0 elsewhere.

Original: Therefore, the integration (cid:82) Dh
Revised: Therefore, the integration (cid:82) Dh

Original: [Equation]
Revised: [Equation]

Original: (cid:16) −
Revised: (cid:16) −

Original: [Equation]
Revised: [Equation]

Original: ∗
Revised: ∗

Original: [Equation]
Revised: [Equation]

Original: 2(cid:17) depends on the value of g ∗ .
Revised: 2(cid:17) depends on the value of g ∗ .

Original: If g ∗ < 0 , no solution exists for g − g ∗ = 0 as g is a ReLU activation, and thus the integration is equivalent to the maximum value of the integral under the limit of β → ∞ .
Revised: If g ∗ < 0 , no solution exists for g − g ∗ = 0 as g is a ReLU activation, and thus the integration is equivalent to the maximum value of the integral under the limit of β → ∞ .

Original: As g ∗ > 0 , the integration over the “notch” is equivalent to the one at full range.
Revised: As g ∗ > 0 , the integration over the “notch” is equivalent to the one at full range.

Original: That is,
Revised: That is,

Original: [Equation]
Revised: [Equation]

Original: The above equation can be readity integrated out and we obtain
Revised: The above equation can be readity integrated out and we obtain

Original: [Equation]
Revised: [Equation]

Original: Substituting it back to Eqn.19, we have its third term equivalent
Revised: Substituting it back to Eqn.19, we have its third term equivalent

Original: [Equation]
Revised: [Equation]

Original: To solve the above integration, we ﬁrst realize that s is a random variable to corrupt the output of the teacher output and the above integration should be averaged out over s .
Revised: To solve the above integration, we ﬁrst realize that s is a random variable to corrupt the output of the teacher output and the above integration should be averaged out over s .

Original: Given that s ∼ N (0 , S ) , it is easy to realize that
Revised: Given that s ∼ N (0 , S ) , it is easy to realize that

Original: [Equation]
Revised: [Equation]

Original: Through simple Gaussian integraions
Revised: Through simple Gaussian integraions

Original: [Equation]
Revised: [Equation]

Original: we get
Revised: we get

Original: [Equation]
Revised: [Equation]

Original: Substituting Term3 back yields the results of the free energy.
Revised: Substituting Term3 back yields the results of the free energy.

Original: Therefore, by locating the values that minimizes f in the above proposition
Revised: Therefore, by locating the values that minimizes f in the above proposition

Original: [Equation]
Revised: [Equation]

Original: we have equilibrium order parameters
Revised: we have equilibrium order parameters

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: 23) where a is deﬁned as a = 1+ β ( q 2 − γ 2 )
Revised: 23) where a is deﬁned as a = 1+ β ( q 2 − γ 2 )

Original: β ( q 2 − γ 2 ) .
Revised: β ( q 2 − γ 2 ) .

Original: Substituting the order parameters back to the generalization error, we have
Revised: Substituting the order parameters back to the generalization error, we have

Original: [Equation]
Revised: [Equation]

Original: When α < 2 and β → ∞ , a = 1 , the generalization error is
Revised: When α < 2 and β → ∞ , a = 1 , the generalization error is

Original: [Equation]
Revised: [Equation]