Original: U NDERSTANDING W HY N EURAL N ETWORKS G ENERALIZE W ELL T HROUGH GSNR OF P ARAMETERS
Revised: U NDERSTANDING W HY N EURAL N ETWORKS G ENERALIZE W ELL T HROUGH GSNR OF P ARAMETERS

Original: A BSTRACT
Revised: A BSTRACT

Original: As deep neural networks (DNNs) achieve tremendous success across many application domains, researchers tried to explore in many aspects on why they generalize well.
Revised: As deep neural networks (DNNs) achieve tremendous success across many application domains, researchers tried to explore in many aspects on why they generalize well.

Original: In this paper, we provide a novel perspective on these issues using the gradient signal to noise ratio (GSNR) of parameters during training process of DNNs.
Revised: In this paper, we provide a novel perspective on these issues using the gradient signal to noise ratio (GSNR) of parameters during training process of DNNs.

Original: The GSNR of a parameter is deﬁned as the ratio between its gradient’s squared mean and variance, over the data distribution.
Revised: The GSNR of a parameter is deﬁned as the ratio between its gradient’s squared mean and variance, over the data distribution.

Original: Based on several approximations, we establish a quantitative relationship between model parameters’
Revised: Based on several approximations, we establish a quantitative relationship between model parameters’

Original: GSNR and the generalization gap.
Revised: GSNR and the generalization gap.

Original: This relationship indicates that larger GSNR during training process leads to better generalization performance.
Revised: This relationship indicates that larger GSNR during training process leads to better generalization performance.

Original: Moreover, we show that, different from that of shallow models (e.g. logistic regression, support vector machines), the gradient descent optimization dynamics of DNNs naturally produces large GSNR during training, which is probably the key to DNNs’ remarkable generalization ability.
Revised: Moreover, we show that, different from that of shallow models (e.g. logistic regression, support vector machines), the gradient descent optimization dynamics of DNNs naturally produces large GSNR during training, which is probably the key to DNNs’ remarkable generalization ability.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Deep neural networks typically contain far more trainable parameters than training samples, which seems to easily cause a poor generalization performance.
Revised: Deep neural networks typically contain far more trainable parameters than training samples, which seems to easily cause a poor generalization performance.

Original: However, in fact they usually exhibit remarkably small generalization gaps.
Revised: However, in fact they usually exhibit remarkably small generalization gaps.

Original: Traditional generalization theories such as VC dimension (Vapnik & Chervonenkis, 1991) or Rademacher complexity (Bartlett & Mendelson, 2002) cannot explain its mechanism.
Revised: Traditional generalization theories such as VC dimension (Vapnik & Chervonenkis, 1991) or Rademacher complexity (Bartlett & Mendelson, 2002) cannot explain its mechanism.

Original: Extensive research focuses on the generalization ability of DNNs (Neyshabur et al., 2017; Arora et al., 2018;
Revised: Extensive research focuses on the generalization ability of DNNs (Neyshabur et al., 2017; Arora et al., 2018;

Original: Keskar et al., 2016; Dinh et al., 2017; Hoffer et al., 2017; Novak et al., 2018; Dziugaite & Roy, 2017; Jakubovitz et al., 2019; Kawaguchi et al., 2017; Advani &
Revised: Keskar et al., 2016; Dinh et al., 2017; Hoffer et al., 2017; Novak et al., 2018; Dziugaite & Roy, 2017; Jakubovitz et al., 2019; Kawaguchi et al., 2017; Advani &

Original: Saxe, 2017).
Revised: Saxe, 2017).

Original: Unlike that of shallow models such as logistic regression or support vector machines, the global minimum of high-dimensional and non-convex DNNs cannot be found analytically, but can only be approximated by gradient descent and its variants (Zeiler, 2012; Kingma & Ba, 2014; Graves, 2013).
Revised: Unlike that of shallow models such as logistic regression or support vector machines, the global minimum of high-dimensional and non-convex DNNs cannot be found analytically, but can only be approximated by gradient descent and its variants (Zeiler, 2012; Kingma & Ba, 2014; Graves, 2013).

Original: Previous work (Zhang et al., 2016; Hardt et al., 2015; Dziugaite & Roy, 2017) suggests that the generalization ability of DNNs is closely related to gradient descent optimization.
Revised: Previous work (Zhang et al., 2016; Hardt et al., 2015; Dziugaite & Roy, 2017) suggests that the generalization ability of DNNs is closely related to gradient descent optimization.

Original: For example,
Revised: For example,

Original: Hardt et al.
Revised: Hardt et al.

Original: (2015) claims that any model trained with stochastic gradient descent (SGD) for reasonable epochs would exhibit small generalization error.
Revised: (2015) claims that any model trained with stochastic gradient descent (SGD) for reasonable epochs would exhibit small generalization error.

Original: Their analysis is based on the smoothness of loss function.
Revised: Their analysis is based on the smoothness of loss function.

Original: In this work, we attempt to understand the generalization behavior of DNNs through
Revised: In this work, we attempt to understand the generalization behavior of DNNs through

Original: GSNR and reveal how GSNR affects the training dynamics of gradient descent.
Revised: GSNR and reveal how GSNR affects the training dynamics of gradient descent.

Revised: Stanislav Fort (2019) studied a new gradient alignment measure called stiffness in order to understand generalization better and stiffness is related to our work.

Original: The GSNR of a parameter is deﬁned as the ratio between its gradient’s squared mean and variance over the data distribution.
Revised: The GSNR of a parameter is deﬁned as the ratio between its gradient’s squared mean and variance over the data distribution.

Original: Previous work tried to use GSNR to conduct theoretical analysis on deep learning.
Revised: Previous work tried to use GSNR to conduct theoretical analysis on deep learning.

Original: For example, Rainforth et al.
Revised: For example, Rainforth et al.

Original: (2018) used GSNR to analyze variational bounds in unsupervised DNNs such as variational auto-encoder (VAE).
Revised: (2018) used GSNR to analyze variational bounds in unsupervised DNNs such as variational auto-encoder (VAE).

Original: Here we focus on analyzing the relation between GSNR and the generalization gap.
Revised: Here we focus on analyzing the relation between GSNR and the generalization gap.

Original: Intuitively, GSNR measures the similarity of a parameter’s gradients among different training samples.
Revised: Intuitively, GSNR measures the similarity of a parameter’s gradients among different training samples.

Original: Large GSNR implies that most training samples agree on the optimization direction of this parameter, thus the parameter is more likely to be associated with a meaningful “pattern” and we assume its update could lead to a better generalization.
Revised: Large GSNR implies that most training samples agree on the optimization direction of this parameter, thus the parameter is more likely to be associated with a meaningful “pattern” and we assume its update could lead to a better generalization.

Original: In this work, we prove that the GSNR is strongly related to the generalization performance, and larger GSNR means a better generalization.
Revised: In this work, we prove that the GSNR is strongly related to the generalization performance, and larger GSNR means a better generalization.

Original: To reveal the mechanism of DNNs’ good generalization ability, we show that the gradient descent optimization dynamics of DNN naturally leads to large GSNR of model parameters and therefore good generalization.
Revised: To reveal the mechanism of DNNs’ good generalization ability, we show that the gradient descent optimization dynamics of DNN naturally leads to large GSNR of model parameters and therefore good generalization.

Original: Furthermore, we give a complete analysis and a detailed interpretation to this phenomenon.
Revised: Furthermore, we give a complete analysis and a detailed interpretation to this phenomenon.

Original: We believe this is probably the key to DNNs remarkable generalization ability.
Revised: We believe this is probably the key to DNNs remarkable generalization ability.

Original: In the remainder of this paper we ﬁrst analyze the relation between GSNR and generalization (Section 2).
Revised: In the remainder of this paper we ﬁrst analyze the relation between GSNR and generalization (Section 2).

Original: We then show how the training dynamics lead to large GSNR of model parameters experimentally and analytically in Section 3.
Revised: We then show how the training dynamics lead to large GSNR of model parameters experimentally and analytically in Section 3.

Original:  L ARGER GSNR L EADS TO B ETTER G ENERALIZATION
Revised:  L ARGER GSNR L EADS TO B ETTER G ENERALIZATION

Original: In this section, we establish a quantitative relation between the GSNR of model parameters and generalization gap, showing that larger GSNR during training leads to better generalization.
Revised: In this section, we establish a quantitative relation between the GSNR of model parameters and generalization gap, showing that larger GSNR during training leads to better generalization.

Original: 2. G RADIENTS S IGNAL TO N OISE R ATIO
Revised: 2. G RADIENTS S IGNAL TO N OISE R ATIO

Original: Consider a data distribution Z =
Revised: Consider a data distribution Z =

Original: X
Revised: X

Original: × Y , from which each sample ( x, y ) is drawn; a model ˆ y = f
Revised: × Y , from which each sample ( x, y ) is drawn; a model ˆ y = f

Original: ( x, θ ) parameterized by θ ; and a loss function L .
Revised: ( x, θ ) parameterized by θ ; and a loss function L .

Original: The parameters’ gradient w.r.t.
Revised: The parameters’ gradient w.r.t.

Original: L and sample ( x i , y i ) is denoted by 
Revised: L and sample ( x i , y i ) is denoted by g ( x i , y i , θ ) or g i ( θ ) :=

Original: [Equation]
Revised: [Equation]

Original: whose j -th element is g i ( θ j ) .
Revised: whose j -th element is g i ( θ j ) .

Original: Note that throughout this paper we always use i to index data examples and j to index model parameters.
Revised: Note that throughout this paper we always use i to index data examples and j to index model parameters.

Original: Given the data distribution Z , we have the (sample-wise) mean and variance of g i ( θ ) .
Revised: Given the data distribution Z , we have the (sample-wise) mean and variance of g i ( θ ) .

Original: We denote them as ˜g ( θ ) =
Revised: We denote them as ˜g ( θ ) =

Original: E ( x,y ) ∼Z ( g ( x, y, θ )) and ρ 2 ( θ ) =
Revised: E ( x,y ) ∼Z ( g ( x, y, θ )) and ρ 2 ( θ ) =

Original: Var ( x,y ) ∼Z ( g ( x, y, θ )) , respectively.
Revised: Var ( x,y ) ∼Z ( g ( x, y, θ )) , respectively.

Original: The gradient signal to noise ratio (GSNR) of one model parameter θ j is deﬁned as:
Revised: The gradient signal to noise ratio (GSNR) of one model parameter θ j is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: At a particular point of the parameter space, GSNR measures the consistency of a parameter’s gradients across different data samples.
Revised: At a particular point of the parameter space, GSNR measures the consistency of a parameter’s gradients across different data samples.

Original: Figure 1 intuitively shows that if GSNR is large, the parameter gradient space tends to be distributed in the similar direction and if GSNR is small, the gradient vectors are then scatteredly distributed.
Revised: Figure 1 intuitively shows that if GSNR is large, the parameter gradient space tends to be distributed in the similar direction and if GSNR is small, the gradient vectors are then scatteredly distributed.

Original: 2. O NE -S TEP G ENERALIZATION R ATIO
Revised: 2. O NE -S TEP G ENERALIZATION R ATIO

Original: In this section we introduce a new concept to help measure the generalization performance during gradient descent optimization, which we call one-step generalization ratio (OSGR).
Revised: In this section we introduce a new concept to help measure the generalization performance during gradient descent optimization, which we call one-step generalization ratio (OSGR).

Original: Consider training set D
Revised: Consider training set D

Original: = { ( x 1 , y 1 ) , ..., ( x n , y n ) } ∼ Z n with n samples drawn from Z , and a test set
Revised: = { ( x 1 , y 1 ) , ..., ( x n , y n ) } ∼ Z n with n samples drawn from Z , and a test set

Original: D ′
Revised: D ′

Original: = { ( x ′ 1 , y ′ 1 ) , ..., ( x ′ n ′ , y ′ n ′ ) } ∼ Z n ′ .
Revised: = { ( x ′ 1 , y ′ 1 ) , ..., ( x ′ n ′ , y ′ n ′ ) } ∼ Z n ′ .

Original: In practice we use the loss on D ′ to measure generalization.
Revised: In practice we use the loss on D ′ to measure generalization.

Original: For simplicity, we assume the sizes of training and test datasets are equal, i.e. n = n ′ .
Revised: For simplicity, we assume the sizes of training and test datasets are equal, i.e. n = n ′ .

Original: We denote the empirical training and test loss as:
Revised: We denote the empirical training and test loss as:

Original: [Equation]
Revised: [Equation]

Original: respectively.
Revised: respectively.

Original: Then the empirical generalization gap is given by L [ D ′ ] − L
Revised: Then the empirical generalization gap is given by L [ D ′ ] − L

Original: [ D ] .
Revised: [ D ] .

Original: In gradient descent optimization, both the training and test loss would decrease step by step.
Revised: In gradient descent optimization, both the training and test loss would decrease step by step.

Original: We use ∆ L [ D ] and ∆ L [ D ′ ] to denote the one-step training and test loss decrease during training, respectively.
Revised: We use ∆ L [ D ] and ∆ L [ D ′ ] to denote the one-step training and test loss decrease during training, respectively.

Original: Let’s consider the ratio between the expectations of ∆ L [ D ′ ] and ∆ L [ D ] of one single training step, which we denote as R ( Z , n ) .
Revised: Let’s consider the ratio between the expectations of ∆ L [ D ′ ] and ∆ L [ D ] of one single training step, which we denote as R ( Z , n ) .

Original: [Equation]
Revised: [Equation]

Original: Note that this ratio also depends on current model parameters θ and learning rate λ .
Revised: Note that this ratio also depends on current model parameters θ and learning rate λ .

Original: We are not including them in the above notation as we will not explicitly model these dependencies, but rather try to quantitatively characterize R for very small λ and for θ at the early stage of training (satisfying
Revised: We are not including them in the above notation as we will not explicitly model these dependencies, but rather try to quantitatively characterize R for very small λ and for θ at the early stage of training (satisfying

Original: Assumption 2.3.1).
Revised: Assumption 2.3.1).

Original: Also note that the expectation of ∆ L [ D ′ ] is over D and D ′ .
Revised: Also note that the expectation of ∆ L [ D ′ ] is over D and D ′ .

Original: This is because the optimization step is performed on D .
Revised: This is because the optimization step is performed on D .

Original: We refer to R ( Z , n ) as OSGR of gradient descent optimization.
Revised: We refer to R ( Z , n ) as OSGR of gradient descent optimization.

Original: Statistically the training loss decreases faster than the test loss and 0 <
Revised: Statistically the training loss decreases faster than the test loss and 0 <

Original: OSGR ( t ) < 

Revised: OSGR ( t ) <

Original: ( Middle panel of Figure 2), which usually results in a non-zero generalization gap at the end of training.
Revised: ( Middle panel of Figure 2), which usually results in a non-zero generalization gap at the end of training.

Original: If OSGR ( t ) is large ( ≈ 1 ) in the whole training process ( Right panel of Figure 2), generalization gap would be small when training completes, implying good generalization ability of the model.
Revised: If OSGR ( t ) is large ( ≈ 1 ) in the whole training process ( Right panel of Figure 2), generalization gap would be small when training completes, implying good generalization ability of the model.

Original: If OSGR ( t ) is small ( = 0 ), the test loss will not decrease while the training loss normally drops ( Left panel of Figure 2), corresponding to a large generalization gap.
Revised: If OSGR ( t ) is small ( = 0 ), the test loss will not decrease while the training loss normally drops ( Left panel of Figure 2), corresponding to a large generalization gap.

Original: 2. R ELATION BETWEEN GSNR AND OSGR
Revised: 2. R ELATION BETWEEN GSNR AND OSGR

Original: In this section, we derive a relation between the OSGR during training and the GSNR of model parameters.
Revised: In this section, we derive a relation between the OSGR during training and the GSNR of model parameters.

Original: This relation indicates that, for the ﬁrst time as far as we know, the sample-wise gradient distribution of parameters is related to the generalization performance of gradient descent optimization.
Revised: This relation indicates that, for the ﬁrst time as far as we know, the sample-wise gradient distribution of parameters is related to the generalization performance of gradient descent optimization.

Original: In gradient descent optimization, we take the average gradient over training set D , which we denote as g D ( θ ) .
Revised: In gradient descent optimization, we take the average gradient over training set D , which we denote as g D ( θ ) .

Original: Note that we have used g i ( θ ) to denote gradient evaluated on one data sample and ˜ g
Revised: Note that we have used g i ( θ ) to denote gradient evaluated on one data sample and ˜ g

Original: ( θ ) to denote its expectation over the entire data distribution.
Revised: ( θ ) to denote its expectation over the entire data distribution.

Original: Similarly we deﬁne g D ′
Revised: Similarly we deﬁne g D ′

Original: ( θ ) to be the average gradient over test set D ′ .
Revised: ( θ ) to be the average gradient over test set D ′ .

Original: [Equation]
Revised: [Equation]

Original: Both the training and test dataset are randomly generated from the same distribution Z n , so we can treat g D ( θ ) and g D
Revised: Both the training and test dataset are randomly generated from the same distribution Z n , so we can treat g D ( θ ) and g D

Original: ′ ( θ ) as random variables.
Revised: ′ ( θ ) as random variables.

Original: At the beginning of the optimization process, θ is randomly initialized thus independent of D , so g D ( θ ) and g D ′ ( θ ) would obey the same distribution.
Revised: At the beginning of the optimization process, θ is randomly initialized thus independent of D , so g D ( θ ) and g D ′ ( θ ) would obey the same distribution.

Original: After a period of training, the model parameters begin to ﬁt the training dataset and become a function of D , i.e. θ = θ ( D ) , therefore distributions of g D ( θ ( D )) and g D ′ ( θ ( D )) become different.
Revised: After a period of training, the model parameters begin to ﬁt the training dataset and become a function of D , i.e. θ = θ ( D ) , therefore distributions of g D ( θ ( D )) and g D ′ ( θ ( D )) become different.

Original: However we choose not to model this dependency and make the following assumption for our analysis:
Revised: However we choose not to model this dependency and make the following assumption for our analysis:

Original: Assumption 2.3.1 (Non-overﬁtting limit approximation)
Revised: Assumption 2.3.1 (Non-overﬁtting limit approximation)

Original: The average gradient over the training dataset and test dataset g D ( θ ) and g D
Revised: The average gradient over the training dataset and test dataset g D ( θ ) and g D

Original: ′ ( θ ) obey the same distribution.
Revised: ′ ( θ ) obey the same distribution.

Original: Obviously the mean of g D ( θ ) and g D ′ ( θ ) is just the mean gradient over the data distribution ˜ g
Revised: Obviously the mean of g D ( θ ) and g D ′ ( θ ) is just the mean gradient over the data distribution ˜ g

Original: ( θ ) .
Revised: ( θ ) .

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

Original: We denote their variance as σ 2 ( θ ) , i.e.
Revised: We denote their variance as σ 2 ( θ ) , i.e.

Original: [Equation]
Revised: [Equation]

Original: It is straightforward to show that:
Revised: It is straightforward to show that:

Original: [Equation]
Revised: [Equation]

Original: where σ 2 ( θ ) is the variance of the average gradient over the dataset of size n , and ρ 2 ( θ ) is the variance of the gradient of a single data sample.
Revised: where σ 2 ( θ ) is the variance of the average gradient over the dataset of size n , and ρ 2 ( θ ) is the variance of the gradient of a single data sample.

Original: In one gradient descent step, the model parameter is updated by ∆ θ = θ t +1 − θ t = − λ g D ( θ ) where λ is the learning rate.
Revised: In one gradient descent step, the model parameter is updated by ∆ θ = θ t +1 − θ t = − λ g D ( θ ) where λ is the learning rate.

Original: If λ is small enough, the one-step training and test loss decrease can be approximated by
Revised: If λ is small enough, the one-step training and test loss decrease can be approximated by

Original: [Equation]
Revised: [Equation]

Original: Usually there are some differences between the directions of g D ( θ ) and g D ′ ( θ ) , so statistically ∆ L [ D ] tends to be larger than ∆ L [ D ′ ] and the generalization gap would increase during training.
Revised: Usually there are some differences between the directions of g D ( θ ) and g D ′ ( θ ) , so statistically ∆ L [ D ] tends to be larger than ∆ L [ D ′ ] and the generalization gap would increase during training.

Original: When λ → 0 , in one single training step the empirical generalization gap increases by ∆ L [ D ] − ∆ L
Revised: When λ → 0 , in one single training step the empirical generalization gap increases by ∆ L [ D ] − ∆ L

Original: [ D ′ ] , for simplicity we denote this quantity as ▽ :
Revised: [ D ′ ] , for simplicity we denote this quantity as ▽ :

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Here we replaced the random variables by g D ( θ ) =
Revised: Here we replaced the random variables by g D ( θ ) =

Original: ˜ g
Revised: ˜ g

Original: ( θ ) + ϵ and g D ′ ( θ ) =
Revised: ( θ ) + ϵ and g D ′ ( θ ) =

Original: ˜ g
Revised: ˜ g

Original: ( θ ) + ϵ ′ , where ϵ and ϵ ′ are random variables with zero mean and variance σ 2 ( θ ) .
Revised: ( θ ) + ϵ ′ , where ϵ and ϵ ′ are random variables with zero mean and variance σ 2 ( θ ) .

Original: Since E ( ϵ ′ ) =
Revised: Since E ( ϵ ′ ) =

Original: E ( ϵ ) = 0 , ϵ and ϵ ′ are independent, the expectation of ▽ is
Revised: E ( ϵ ) = 0 , ϵ and ϵ ′ are independent, the expectation of ▽ is

Original: [Equation]
Revised: [Equation]

Original: where σ 2 ( θ j ) is the variance the of average gradient of the parameter θ j .
Revised: where σ 2 ( θ j ) is the variance the of average gradient of the parameter θ j .

Original: For simplicity, when it involves a single model parameter θ j , we will use only a subscript j instead of the full notation.
Revised: For simplicity, when it involves a single model parameter θ j , we will use only a subscript j instead of the full notation.

Original: For example, we use σ 2 j , r j , and g D,j to denote σ 2 ( θ j ) , r ( θ j ) , and g D ( θ j ) respectively.
Revised: For example, we use σ 2 j , r j , and g D,j to denote σ 2 ( θ j ) , r ( θ j ) , and g D ( θ j ) respectively.

Original: Consider the expectation of ∆ L [ D ] and ∆ L [ D ′ ] when λ →
Revised: Consider the expectation of ∆ L [ D ] and ∆ L [ D ′ ] when λ →

Original: [Equation]
Revised: [Equation]

Original: Substituting (18) and (15) into (4) we have:
Revised: Substituting (18) and (15) into (4) we have:

Original: [Equation]
Revised: [Equation]

Original: Although we derived eq.
Revised: Although we derived eq.

Original: (19) from simpliﬁed assumptions, we can empirically verify it by estimating two sides of the equation on real data.
Revised: (19) from simpliﬁed assumptions, we can empirically verify it by estimating two sides of the equation on real data.

Original: We will elaborate on this estimation method in section 2.4.
Revised: We will elaborate on this estimation method in section 2.4.

Original: We can rewrite eq.
Revised: We can rewrite eq.

Original: (19) as:
Revised: (19) as:

Original: [Equation]
Revised: [Equation]

Original: (21) where E D ∼Z n ( g 2 D
Revised: (21) where E D ∼Z n ( g 2 D

Original: [Equation]
Revised: [Equation]

Original: j
Revised: j

Original: [Equation]
Revised: [Equation]

Original: = V ar D ∼Z n
Revised: = V ar D ∼Z n

Original: [Equation]
Revised: [Equation]

Original: g
Revised: g

Original: [Equation]
Revised: [Equation]

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: ∼Z n ( g D,j ) =
Revised: ∼Z n ( g D,j ) =

Original: 1 n ρ 2 j + ˜ g 2 j .
Revised: 1 n ρ 2 j + ˜ g 2 j .

Original: We deﬁne ∆ L j [ D ] to be the training loss decrease caused by updating θ j .
Revised: We deﬁne ∆ L j [ D ] to be the training loss decrease caused by updating θ j .

Original: We can show that when λ is very small ∆ L j [ D ] =
Revised: We can show that when λ is very small ∆ L j [ D ] =

Original: λ g 2 D,j + O ( λ 2 ) .
Revised: λ g 2 D,j + O ( λ 2 ) .

Original: Therefore when λ → 0 , we have
Revised: Therefore when λ → 0 , we have

Original: R ( Z , n ) = 1 − 1 n ∑ j
Revised: R ( Z , n ) = 1 − 1 n ∑ j

Original: W j r j + 1 n , where W j := E D ∼Z n (∆ L j [ D ])
Revised: W j r j + 1 n , where W j := E D ∼Z n (∆ L j [ D ])

Original: E D ∼Z n (∆ L [ D ]) with ∑ j
Revised: E D ∼Z n (∆ L [ D ]) with ∑ j

Original: W j =(22)
Revised: W j =(22)

Original: Eq.
Revised: Eq.

Original: (22) shows that the GSNR r j plays a crucial role in the model’s generalization ability—the onestep generalization ratio in gradient descent equals one minus the weighted average of 1 r j + 1 n over all model parameters divided by n .
Revised: (22) shows that the GSNR r j plays a crucial role in the model’s generalization ability—the onestep generalization ratio in gradient descent equals one minus the weighted average of 1 r j + 1 n over all model parameters divided by n .

Original: The weight is proportional to the expectation of the training loss decrease resulted from updating that parameter.
Revised: The weight is proportional to the expectation of the training loss decrease resulted from updating that parameter.

Original: This implies that larger GSNR of model parameters during training leads to smaller generalization gap growth thus better generalization performance of the trained model.
Revised: This implies that larger GSNR of model parameters during training leads to smaller generalization gap growth thus better generalization performance of the trained model.

Original: Also note when n → ∞ , we have R ( Z , n ) → 1 , meaning that training on more data helps generalization.
Revised: Also note when n → ∞ , we have R ( Z , n ) → 1 , meaning that training on more data helps generalization.

Original: 2. E XPERIMENTAL VERIFICATION OF THE RELATION BETWEEN GSNR AND OSGR
Revised: 2. E XPERIMENTAL VERIFICATION OF THE RELATION BETWEEN GSNR AND OSGR

Original: The relation between GSNR and OSGR, i.e. eq.
Revised: The relation between GSNR and OSGR, i.e. eq.

Original: (19) or (22) can be empirically veriﬁed using any dataset if: (1) The dataset includes enough samples to construct many training sets and a large enough test set so that we can reliably estimate ρ 2 j , E D ∼Z n ( g 2 D,j ) and OSGR.
Revised: (19) or (22) can be empirically veriﬁed using any dataset if: (1) The dataset includes enough samples to construct many training sets and a large enough test set so that we can reliably estimate ρ 2 j , E D ∼Z n ( g 2 D,j ) and OSGR.

Original: The learning rate is small enough.
Revised: The learning rate is small enough.

Original: In the early training stage of gradient descent.
Revised: In the early training stage of gradient descent.

Original: To empirically verify eq.
Revised: To empirically verify eq.

Original: (19), we show how to estimate its left and right hand sides, i.e. OSGR by deﬁnition and OSGR as a function of GSNR.
Revised: (19), we show how to estimate its left and right hand sides, i.e. OSGR by deﬁnition and OSGR as a function of GSNR.

Original: Suppose we have M training sets each with size n , and a test set of size n ′ .
Revised: Suppose we have M training sets each with size n , and a test set of size n ′ .

Original: We initialize a model and train it separately on the M training sets and test it with the same test set.
Revised: We initialize a model and train it separately on the M training sets and test it with the same test set.

Original: For the t -th training iteration, we denote the training loss and test loss of the model trained on the m -th training dataset as L ( m ) t and L ′ ( m ) t , respectively.
Revised: For the t -th training iteration, we denote the training loss and test loss of the model trained on the m -th training dataset as L ( m ) t and L ′ ( m ) t , respectively.

Original: Then the left hand side, i.e. OSGR by
Revised: Then the left hand side, i.e. OSGR by

Original: , of the t -th iteration can be estimated by
Revised: , of the t -th iteration can be estimated by

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: For the model trained on the m -th training set, we can compute the t -th step average gradient and sample-wise gradient variance of θ j on the corresponding training set, denoted as g m,j,t and ρ 2 m,j,t , respectively.
Revised: For the model trained on the m -th training set, we can compute the t -th step average gradient and sample-wise gradient variance of θ j on the corresponding training set, denoted as g m,j,t and ρ 2 m,j,t , respectively.

Original: Therefore the right hand side of eq.
Revised: Therefore the right hand side of eq.

Original: (19) can be estimated by
Revised: (19) can be estimated by

Original: [Equation]
Revised: [Equation]

Original: We performed the above estimations on MNIST with a simple CNN structure consists of
Revised: We performed the above estimations on MNIST with a simple CNN structure consists of

Original: Conv-Relu-MaxPooling blocks and 2 fully-connected layers.
Revised: Conv-Relu-MaxPooling blocks and 2 fully-connected layers.

Original: First, to estimate eq.
Revised: First, to estimate eq.

Original: (24) with
Revised: (24) with

Original: M = 10 , we randomly sample 10 training sets with size n and a test set with size 10,000.
Revised: M = 10 , we randomly sample 10 training sets with size n and a test set with size 10,000.

Original: To cover different conditions, we (1) choose n ∈ { 1000 , 2000 , 4000 , 6000 , 8000 , 10000 , 15000 } , respectively; (2) inject noise by randomly changing the labels with probability p random ∈ { 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 5 } ; (3) change the model structure by varying number of channels in the layers, ch ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .
Revised: To cover different conditions, we (1) choose n ∈ { 1000 , 2000 , 4000 , 6000 , 8000 , 10000 , 15000 } , respectively; (2) inject noise by randomly changing the labels with probability p random ∈ { 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 5 } ; (3) change the model structure by varying number of channels in the layers, ch ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .

Original: See Appendix A for more details of the setup.
Revised: See Appendix A for more details of the setup.

Original: We use the gradient descent training (not SGD), with a small learning rate of 0 . 001 .
Revised: We use the gradient descent training (not SGD), with a small learning rate of 0 . 001 .

Original: The left and right hand sides of 19 at different epochs are shown in Figure 3, where each point represents one speciﬁc choice of the above settings.
Revised: The left and right hand sides of 19 at different epochs are shown in Figure 3, where each point represents one speciﬁc choice of the above settings.

Original: At the beginning of training, the data points are closely distributed along the dashed line corresponding to LHS=
Revised: At the beginning of training, the data points are closely distributed along the dashed line corresponding to LHS=

Original: RHS.
Revised: RHS.

Original: This shows that eq.
Revised: This shows that eq.

Original: (19) ﬁts quite well under a variety of different settings.
Revised: (19) ﬁts quite well under a variety of different settings.

Original: As training proceeds, the points become more scattered as the non-overﬁtting limit approximation no longer holds, but correlation between the LHS and RHS remains high even when the training converges (at epoch 2,500).
Revised: As training proceeds, the points become more scattered as the non-overﬁtting limit approximation no longer holds, but correlation between the LHS and RHS remains high even when the training converges (at epoch 2,500).

Original: We also conducted the same experiment on CIFAR10 A.2 and a toy dataset
Revised: We also conducted the same experiment on CIFAR10 A.2 and a toy dataset

Original: A.3 observed the same behavior.
Revised: A.3 observed the same behavior.

Original: See Appendix for these experiments.
Revised: See Appendix for these experiments.

Original: The empirical evidence together with our previous derivation of eq.
Revised: The empirical evidence together with our previous derivation of eq.

Original: (19) clearly show the relation between GSNR and OSGR and its implication in the model’s generalization ability.
Revised: (19) clearly show the relation between GSNR and OSGR and its implication in the model’s generalization ability.

Original:  T RAINING DYNAMICS OF DNN S NATURALLY LEADS TO LARGE GSNR
Revised:  T RAINING DYNAMICS OF DNN S NATURALLY LEADS TO LARGE GSNR

Original: In this section, we analyze and explain one interesting phenomenon: the parameters’ GSNR of
Revised: In this section, we analyze and explain one interesting phenomenon: the parameters’ GSNR of

Original: DNNs rises in the early stages of training, whereas the GSNR of shallow models such as logistic regression or support vector machines declines during the entire training process.
Revised: DNNs rises in the early stages of training, whereas the GSNR of shallow models such as logistic regression or support vector machines declines during the entire training process.

Original: This difference gives rise to GSNR’s large practical values during training, which in turn is associated with good generalization.
Revised: This difference gives rise to GSNR’s large practical values during training, which in turn is associated with good generalization.

Original: We analyze the dynamics behind this phenomenon both experimentally and theoretically.
Revised: We analyze the dynamics behind this phenomenon both experimentally and theoretically.

Original: 3. GSNR BEHAVIOR OF DNN S TRAINING
Revised: 3. GSNR BEHAVIOR OF DNN S TRAINING

Original: For shallow models, the GSNR of parameters decreases in the whole training process because gradients become small as learning converges.
Revised: For shallow models, the GSNR of parameters decreases in the whole training process because gradients become small as learning converges.

Original: But for DNNs it is not the case.
Revised: But for DNNs it is not the case.

Original: We trained DNNs on the CIFAR datasets and computed the GSNR averaged over all model parameters.
Revised: We trained DNNs on the CIFAR datasets and computed the GSNR averaged over all model parameters.

Original: Because
Revised: Because

Original: E D ∼Z n ( g 2 D,j ) =
Revised: E D ∼Z n ( g 2 D,j ) =

Original: 1 n ρ 2 j + ˜ g
Revised: 1 n ρ 2 j + ˜ g

Original: 2 j
Revised: 2 j

Original: and we assume n is large, E D ∼Z n ( g 2 D,j )
Revised: and we assume n is large, E D ∼Z n ( g 2 D,j )

Original: ≈ ˜
Revised: ≈ ˜

Original: g 2 j .
Revised: g 2 j .

Original: In the case of only one large training datasets, we estimate GSNR of t -th iteration by
Revised: In the case of only one large training datasets, we estimate GSNR of t -th iteration by

Original: [Equation]
Revised: [Equation]

Original: As shown in Figure 4, the GSNR starts out low with randomly initialized parameters.
Revised: As shown in Figure 4, the GSNR starts out low with randomly initialized parameters.

Original: As learning progresses, the GSNR increases in the early training stage and stays at a high level in the whole learning process.
Revised: As learning progresses, the GSNR increases in the early training stage and stays at a high level in the whole learning process.

Original: For each model parameter, we also computed the proportion of the samples with the same gradient sign, denoted as p same _ sign .
Revised: For each model parameter, we also computed the proportion of the samples with the same gradient sign, denoted as p same _ sign .

Original: In Figure 4c, we plot the mean of time series of this proportion for all the parameters.
Revised: In Figure 4c, we plot the mean of time series of this proportion for all the parameters.

Original: This value increases from about 50% (half positive half negetive due to random initialization) to about 56% ﬁnally, which indicates that for most parameters, the gradient signs on different samples become more consistent.
Revised: This value increases from about 50% (half positive half negetive due to random initialization) to about 56% ﬁnally, which indicates that for most parameters, the gradient signs on different samples become more consistent.

Original: This is because meaningful features begin to emerge in the learning process and the gradients of the weights on these features tend to have the same sign among different samples.
Revised: This is because meaningful features begin to emerge in the learning process and the gradients of the weights on these features tend to have the same sign among different samples.

Original: Previous research (Zhang et al., 2016) showed that DNNs achieved zero training loss by memorizing training samples even if the labels were randomized.
Revised: Previous research (Zhang et al., 2016) showed that DNNs achieved zero training loss by memorizing training samples even if the labels were randomized.

Original: We also plot the average GSNR for model trained using data with randomized labels in Figure 4 and ﬁnd that the GSNR stays at a low level throughout the training process.
Revised: We also plot the average GSNR for model trained using data with randomized labels in Figure 4 and ﬁnd that the GSNR stays at a low level throughout the training process.

Original: Although the training loss of both the original and randomized labels go to zero (not shown), the GSNR curves clearly distinguish between these two cases and reveal the lack of meaningful patterns in the latter one.
Revised: Although the training loss of both the original and randomized labels go to zero (not shown), the GSNR curves clearly distinguish between these two cases and reveal the lack of meaningful patterns in the latter one.

Original: We believe this is the reason why DNNs trained on real and random data lead to completely different generalization behaviors.
Revised: We believe this is the reason why DNNs trained on real and random data lead to completely different generalization behaviors.

Original: 3. T RAINING D YNAMICS BEHIND THE GSNR BEHAVIOR
Revised: 3. T RAINING D YNAMICS BEHIND THE GSNR BEHAVIOR

Original: In this section we show that the feature learning ability of DNNs is the key reason why the GSNR curve behavior of DNNs is different from that of shallow models during the gradient descent training.
Revised: In this section we show that the feature learning ability of DNNs is the key reason why the GSNR curve behavior of DNNs is different from that of shallow models during the gradient descent training.

Original: To demonstrate this, a simple two-layer perceptron regression model is constructed.
Revised: To demonstrate this, a simple two-layer perceptron regression model is constructed.

Original: A synthetic dataset is generated as following.
Revised: A synthetic dataset is generated as following.

Original: Each data point is constructed i.i.d.
Revised: Each data point is constructed i.i.d.

Original: using y = x 0 x 1 + ϵ , where x 0 and x 1 are drawn from uniform distribution [ − 1 , 1] and ϵ is drawn from uniform distribution [ − 0 .
Revised: using y = x 0 x 1 + ϵ , where x 0 and x 1 are drawn from uniform distribution [ − 1 , 1] and ϵ is drawn from uniform distribution [ − 0 .

Original: The training set and test set sizes are 200 and 10,000, respectively.
Revised: The training set and test set sizes are 200 and 10,000, respectively.

Original: We use a very simple two-layer MLP structure with 2 inputs, 20 hidden neurons and 1 output.
Revised: We use a very simple two-layer MLP structure with 2 inputs, 20 hidden neurons and 1 output.

Original: We randomly initialized the model parameters and trained the model on the synthetic training dataset.
Revised: We randomly initialized the model parameters and trained the model on the synthetic training dataset.

Original: As a control setup we also tried to freeze model weights in the ﬁrst layer to prevent it from learning features.
Revised: As a control setup we also tried to freeze model weights in the ﬁrst layer to prevent it from learning features.

Original: Note that a two layer MLP with the ﬁrst layer frozen is equivalent to a linear regression model.
Revised: Note that a two layer MLP with the ﬁrst layer frozen is equivalent to a linear regression model.

Original: That is, regression weights are learned on the second layer using ﬁxed features extracted by the ﬁrst layer.
Revised: That is, regression weights are learned on the second layer using ﬁxed features extracted by the ﬁrst layer.

Original: We plot the average GSNR of the second layer parameters for both the frozen and non-frozen cases.
Revised: We plot the average GSNR of the second layer parameters for both the frozen and non-frozen cases.

Original: Figure 5 shows that in the non-frozen case, the average GSNR over parameters of the second layer shows a signiﬁcant upward process, whereas in the frozen case the average GSNR decreases in the beginning and remains at a low level during the whole training process.
Revised: Figure 5 shows that in the non-frozen case, the average GSNR over parameters of the second layer shows a signiﬁcant upward process, whereas in the frozen case the average GSNR decreases in the beginning and remains at a low level during the whole training process.

Original: In the non-frozen case, GSNR curve of individual parameters of the second layer are shown in Figure 5.
Revised: In the non-frozen case, GSNR curve of individual parameters of the second layer are shown in Figure 5.

Original: The GSNR for some parameters show a signiﬁcant upward process.
Revised: The GSNR for some parameters show a signiﬁcant upward process.

Original: To measure the quality of these features, we computed the Pearson correlation between them and the target output y , both at the beginning of training and at the maximum point of their GSNR curves.
Revised: To measure the quality of these features, we computed the Pearson correlation between them and the target output y , both at the beginning of training and at the maximum point of their GSNR curves.

Original: We can see that the learning process learns “good” features (high correlation value, i.e. with stronger correlation with y ) from random initialized ones, as shown in Table 1.
Revised: We can see that the learning process learns “good” features (high correlation value, i.e. with stronger correlation with y ) from random initialized ones, as shown in Table 1.

Original: This shows that the GSNR increasing process is related to feature learning.
Revised: This shows that the GSNR increasing process is related to feature learning.

Original: 3. A NALYSIS OF TRAINING DYNAMICS BEHIND DNN S ’ GSNR BEHAVIOR
Revised: 3. A NALYSIS OF TRAINING DYNAMICS BEHIND DNN S ’ GSNR BEHAVIOR

Original: In this section, we will investigate the training dynamics behind the GSNR curve behavior.
Revised: In this section, we will investigate the training dynamics behind the GSNR curve behavior.

Original: In the case of fully connected network structure, we can analytically show that the numerator of GSNR, i.e. the squared gradient mean of model parameters, tends to increase in the early training stage through feature learning.
Revised: In the case of fully connected network structure, we can analytically show that the numerator of GSNR, i.e. the squared gradient mean of model parameters, tends to increase in the early training stage through feature learning.

Original: Consider a fully connected network, whose parameters are θ = { W (1) , b (1) , ..., W ( l max ) , b ( l max ) } , where W (1) , b (1) are the weight matrix and bias of the ﬁrst layer, and so on.
Revised: Consider a fully connected network, whose parameters are θ = { W (1) , b (1) , ..., W ( l max ) , b ( l max ) } , where W (1) , b (1) are the weight matrix and bias of the ﬁrst layer, and so on.

Original: We denote the activations of the l -th layer as a ( l ) =
Revised: We denote the activations of the l -th layer as a ( l ) =

Original: { a ( l ) s ( θ ( l − ) )
Revised: { a ( l ) s ( θ ( l − ) )

Original: } , where s is the index for nodes/channels of this layer, and θ ( l − ) is the collection of model parameters in the layers before l , i.e. θ ( l − )
Revised: } , where s is the index for nodes/channels of this layer, and θ ( l − ) is the collection of model parameters in the layers before l , i.e. θ ( l − )

Original: = { W (1) , b (1) , ..., W ( l − 1) , b ( l − 1) } .
Revised: = { W (1) , b (1) , ..., W ( l − 1) , b ( l − 1) } .

Original: In the forward pass on data sample i , { a ls ( θ ( l − ) ) } is multiplied by the weight matrix W ( l ) :
Revised: In the forward pass on data sample i , { a ls ( θ ( l − ) ) } is multiplied by the weight matrix W ( l ) :

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where o ( l ) = { o ( l ) i,c } is the output of the matrix multiplication, for the i -th data sample, on the l -th layer
Revised: where o ( l ) = { o ( l ) i,c } is the output of the matrix multiplication, for the i -th data sample, on the l -th layer

Original: , c = { 1 , 2 , ..., C } is the index of nodes/channels in the ( l + 1) -th layer.
Revised: , c = { 1 , 2 , ..., C } is the index of nodes/channels in the ( l + 1) -th layer.

Original: We use g ( l ) D to denote the average gradient of weights of the l -th layer W ( l ) , i.e. g ( l ) D = 1 n ∑ ni =1 ∂L i ∂ W ( l ) , where L i is the loss of the i -th sample.
Revised: We use g ( l ) D to denote the average gradient of weights of the l -th layer W ( l ) , i.e. g ( l ) D = 1 n ∑ ni =1 ∂L i ∂ W ( l ) , where L i is the loss of the i -th sample.

Original: Here we show that the feature learning ability of DNNs plays a crucial role in the GSNR increasing process.
Revised: Here we show that the feature learning ability of DNNs plays a crucial role in the GSNR increasing process.

Original: More precisely, we show that the learning of features a ( l ) ( θ ( l − ) ) , i.e. the learning of parameters θ ( l − ) tends to increase the absolute value of g ( l ) D .
Revised: More precisely, we show that the learning of features a ( l ) ( θ ( l − ) ) , i.e. the learning of parameters θ ( l − ) tends to increase the absolute value of g ( l ) D .

Original: Consider the one-step change of gradient mean ∆ g ( l ) D = g ( l ) D,t +1 − g ( l ) D,t with the learning rate λ → 0 .
Revised: Consider the one-step change of gradient mean ∆ g ( l ) D = g ( l ) D,t +1 − g ( l ) D,t with the learning rate λ → 0 .

Original: In one training step, θ is updated by ∆ θ = θ t +1 − θ t = − λ g D ( θ ) .
Revised: In one training step, θ is updated by ∆ θ = θ t +1 − θ t = − λ g D ( θ ) .

Original: Using linear approximation with λ → 0 , we have
Revised: Using linear approximation with λ → 0 , we have

Original: [Equation]
Revised: [Equation]

Original: where θ ( l − ) and θ ( l +) denote model parameters before and after the l -the layer (including the l -th), respectively.
Revised: where θ ( l − ) and θ ( l +) denote model parameters before and after the l -the layer (including the l -th), respectively.

Original: We focus on the ﬁrst term of eq.
Revised: We focus on the ﬁrst term of eq.

Original: (27), i.e. the one-step change of g ( l ) D caused by learning θ ( l − ) .
Revised: (27), i.e. the one-step change of g ( l ) D caused by learning θ ( l − ) .

Original: Substituting g ( l ) D = 1 n ∑ ni =1 ∂L i ∂ W ( l ) and ∆ θ j = ( − λ 1 n ∑ ni =1 ∂L i ∂θ j ) into eq.
Revised: Substituting g ( l ) D = 1 n ∑ ni =1 ∂L i ∂ W ( l ) and ∆ θ j = ( − λ 1 n ∑ ni =1 ∂L i ∂θ j ) into eq.

Original: (27), we have
Revised: (27), we have

Original: [Equation]
Revised: [Equation]

Original: g ( l ) D,s,
Revised: g ( l ) D,s,

Original: [Equation]
Revised: [Equation]

Original: The detailed derivation of eq.
Revised: The detailed derivation of eq.

Original: (28) can be found in Appendix B. We can see the ﬁrst term (which is a summation over parameters in θ ( l − ) ) in eq.
Revised: (28) can be found in Appendix B. We can see the ﬁrst term (which is a summation over parameters in θ ( l − ) ) in eq.

Original: (28) has opposite sign with W ( l ) s,c .
Revised: (28) has opposite sign with W ( l ) s,c .

Original: This term will make ∆ g ( l ) D,s,c negatively correlated with W ( l ) s,c .
Revised: This term will make ∆ g ( l ) D,s,c negatively correlated with W ( l ) s,c .

Original: We plot the correlation between ∆ g ( l ) D,s,c with W ( l ) s,c for a model trained on MNIST for 200 epochs in Figure 6a.
Revised: We plot the correlation between ∆ g ( l ) D,s,c with W ( l ) s,c for a model trained on MNIST for 200 epochs in Figure 6a.

Original: In the early training stage, they are indeed negatively correlated.
Revised: In the early training stage, they are indeed negatively correlated.

Original: For top-10% weights with larger absolute values, the negative correlation is even more signiﬁcant.
Revised: For top-10% weights with larger absolute values, the negative correlation is even more signiﬁcant.

Original: Here we show that this negative correlation between ∆ g ( l ) D,s,c and W ( l ) s,c tends to increase the absolute value of g ( l ) D through an interesting mechanism.
Revised: Here we show that this negative correlation between ∆ g ( l ) D,s,c and W ( l ) s,c tends to increase the absolute value of g ( l ) D through an interesting mechanism.

Original: Consider the weights W ( l ) s,c with { W ( l ) s,c > 0 , g ( l ) D,s,c < 0 } .
Revised: Consider the weights W ( l ) s,c with { W ( l ) s,c > 0 , g ( l ) D,s,c < 0 } .

Original: Learning θ l − would decrease g ( l ) D,s,c and thus increase its absolute value because the ﬁrst term in eq.
Revised: Learning θ l − would decrease g ( l ) D,s,c and thus increase its absolute value because the ﬁrst term in eq.

Original: (28) is negative.
Revised: (28) is negative.

Original: On the other hand, learning W ( l ) s,c would increase W ( l ) s,c and its absolute value because ∆ W ( l ) s,c = − λ g ( l ) D,s,c is positive.
Revised: On the other hand, learning W ( l ) s,c would increase W ( l ) s,c and its absolute value because ∆ W ( l ) s,c = − λ g ( l ) D,s,c is positive.

Original: This will form a positive feedback process, in which the numerator of GSNR, ( g ( l ) D,s,c ) 2 , would increase and so is the GSNR.
Revised: This will form a positive feedback process, in which the numerator of GSNR, ( g ( l ) D,s,c ) 2 , would increase and so is the GSNR.

Original: Similar analysis can be done for the case with { W ( l ) s,c < 0 , g ( l ) D,s,c > 0 } .
Revised: Similar analysis can be done for the case with { W ( l ) s,c < 0 , g ( l ) D,s,c > 0 } .

Original: On the other hand, when { W ( l ) s,c g ( l ) D,s,c > 0 } , we show that the weights tend to change into the earlier case, i.e. { W ( l ) s,c g ( l ) D,s,c < 0 } during training.
Revised: On the other hand, when { W ( l ) s,c g ( l ) D,s,c > 0 } , we show that the weights tend to change into the earlier case, i.e. { W ( l ) s,c g ( l ) D,s,c < 0 } during training.

Original: Consider the case of { W ( l ) s,c > 0 , g ( l ) D,s,c > 0 } , the ﬁrst term in eq.
Revised: Consider the case of { W ( l ) s,c > 0 , g ( l ) D,s,c > 0 } , the ﬁrst term in eq.

Original: (28) is negative, learning θ ( l − ) tends to decrease g ( l ) D,s,c or even change its sign.
Revised: (28) is negative, learning θ ( l − ) tends to decrease g ( l ) D,s,c or even change its sign.

Original: Another posibility is that learning W ( l ) s,
Revised: Another posibility is that learning W ( l ) s,

Original: c changes the sign of W ( l ) s,c because ∆ W ( l ) s,c = − λ g ( l ) D,s,c is negative.
Revised: c changes the sign of W ( l ) s,c because ∆ W ( l ) s,c = − λ g ( l ) D,s,c is negative.

Original: In both cases the weights change into the earlier case with { W ( l ) s,c g ( l ) D,s,c < 0 } .
Revised: In both cases the weights change into the earlier case with { W ( l ) s,c g ( l ) D,s,c < 0 } .

Original: Similar analysis can be done for the case of { W ( l ) s,c < 0 , g ( l ) D,s,c < 0 } .
Revised: Similar analysis can be done for the case of { W ( l ) s,c < 0 , g ( l ) D,s,c < 0 } .

Original: Therefore { W ( l ) s,c g
Revised: Therefore { W ( l ) s,c g

Original: ( l ) D,s,c < 0 } is a more stable state in the training process.
Revised: ( l ) D,s,c < 0 } is a more stable state in the training process.

Original: For a simple model trained on MNIST, We plot the proportion of weights satisfying { W ( l ) s,c g ( l ) D,s,c < 0 } in Figure 6b and ﬁnd that there are indeed more weights with { W ( l ) s,c g ( l ) D,s,c < 0 } than the opposite.
Revised: For a simple model trained on MNIST, We plot the proportion of weights satisfying { W ( l ) s,c g ( l ) D,s,c < 0 } in Figure 6b and ﬁnd that there are indeed more weights with { W ( l ) s,c g ( l ) D,s,c < 0 } than the opposite.

Original: Because weights with small absolute value easily change sign during training, we also plot this proportion for the top-10% weights with larger absolute values.
Revised: Because weights with small absolute value easily change sign during training, we also plot this proportion for the top-10% weights with larger absolute values.

Original: We can see that for the weights with large absolute values, nearly 80% of them have opposite signs with their gradient mean, conﬁrming our earlier analysis.
Revised: We can see that for the weights with large absolute values, nearly 80% of them have opposite signs with their gradient mean, conﬁrming our earlier analysis.

Original: For these weights, the numerator of GSNR, ( g ( l ) D,s,c ) 2 , tends to increase through the positive feedback process as discussed above.
Revised: For these weights, the numerator of GSNR, ( g ( l ) D,s,c ) 2 , tends to increase through the positive feedback process as discussed above.

Original: [Table]
Revised: [Table]

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

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

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

Original:  S UMMARY
Revised:  S UMMARY

Original: In this paper we performed a series of analysis on the role of model parameters’ GSNR in deep neural networks’ generalization ability.
Revised: In this paper we performed a series of analysis on the role of model parameters’ GSNR in deep neural networks’ generalization ability.

Original: We showed that large GSNR is a key to small generalization gap, and gradient descent training naturally incurs and exploits large GSNR as the model discovers useful features in learning
Revised: We showed that large GSNR is a key to small generalization gap, and gradient descent training naturally incurs and exploits large GSNR as the model discovers useful features in learning

Original: A A PPENDIX A A. M ODEL S TRUCTURE IN S ECTION 2. 
Revised: A A PPENDIX A A. M ODEL S TRUCTURE IN S ECTION 2.

Original: As shown in Table 2, all models in the experiment consist of 2 Conv-Relu-MaxPooling blocks and 2 fully-connected layers, but they are different in the number of channels.
Revised: As shown in Table 2, all models in the experiment consist of 2 Conv-Relu-MaxPooling blocks and 2 fully-connected layers, but they are different in the number of channels.

Original: We choose the number of channels p from { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .
Revised: We choose the number of channels p from { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .

Original: [Table]
Revised: [Table]

Original: A. E XPERIMENT ON CIFAR 
Revised: A. E XPERIMENT ON CIFAR

Original: Different from the experiment on MNIST, we use a deeper network on CIFAR10.
Revised: Different from the experiment on MNIST, we use a deeper network on CIFAR10.

Original: We also include the Batch Normalization (BN) layer
Revised: We also include the Batch Normalization (BN) layer

Original: [Table]
Revised: [Table]

Original: because we ﬁnd that it’s difﬁcult for the network to converge in the absence of it.
Revised: because we ﬁnd that it’s difﬁcult for the network to converge in the absence of it.

Original: The network consists of 4 Conv-BN-Relu-Conv-BN-Relu-MaxPooling blocks and 3 fully-connected layers.
Revised: The network consists of 4 Conv-BN-Relu-Conv-BN-Relu-MaxPooling blocks and 3 fully-connected layers.

Original: More details are shown in Table
Revised: More details are shown in Table

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

Original: [Equation]
Revised: [Equation]

Original: The experiment is conducted under a similar setting as that of MNIST in section 2.4.
Revised: The experiment is conducted under a similar setting as that of MNIST in section 2.4.

Original: We choose n ∈ { 2000 , 4000 , 6000 , 8000 , 10000 } , p random ∈ { 0 . 0 , 0 . 2 , 0 . 4 } , ch ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 } .
Revised: We choose n ∈ { 2000 , 4000 , 6000 , 8000 , 10000 } , p random ∈ { 0 . 0 , 0 . 2 , 0 . 4 } , ch ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 } .

Original: We use the gradient descent training (Not SGD), with a small learning rate of 0 . 001 .
Revised: We use the gradient descent training (Not SGD), with a small learning rate of 0 . 001 .

Original: The left and right hand sides of 19 at different epochs are shown in Figure 7, where each point represents one speciﬁc combination of the above settings.
Revised: The left and right hand sides of 19 at different epochs are shown in Figure 7, where each point represents one speciﬁc combination of the above settings.

Original: Note that at the evaluation step of every epoch, we use
Revised: Note that at the evaluation step of every epoch, we use

Original: At the beginning of training, compared to that of MNIST, the data points no longer perfectly resides on the diagonal dashed line.
Revised: At the beginning of training, compared to that of MNIST, the data points no longer perfectly resides on the diagonal dashed line.

Original: We suppose that’s beacuse of the presence of BN layer, whose internal parameters, i.e. running mean and running variance, are not regular learnable parameters in the optimization process, but change their values in a different way.
Revised: We suppose that’s beacuse of the presence of BN layer, whose internal parameters, i.e. running mean and running variance, are not regular learnable parameters in the optimization process, but change their values in a different way.

Original: Their change affects the OSGR, yet we could not include them in the estimation of OSGR.
Revised: Their change affects the OSGR, yet we could not include them in the estimation of OSGR.

Original: However, the strong positive correlation between the left and right hand sides of eq.
Revised: However, the strong positive correlation between the left and right hand sides of eq.

Original: (19) can always be observed until the training begins to converge.
Revised: (19) can always be observed until the training begins to converge.

Original: A. E XPERIMENT ON T OY D ATASET
Revised: A. E XPERIMENT ON T OY D ATASET

Original: In this section we show a simple two-layer regression model consists of a FC-Relu structure with only 2 inputs, 1 hidden layer with N neurons and 1 output.
Revised: In this section we show a simple two-layer regression model consists of a FC-Relu structure with only 2 inputs, 1 hidden layer with N neurons and 1 output.

Original: A similar synthetic dataset with the training data used in the experiment of Section 3.2 is generated as follows.
Revised: A similar synthetic dataset with the training data used in the experiment of Section 3.2 is generated as follows.

Original: Each data point is constructed i.i.d.
Revised: Each data point is constructed i.i.d.

Original: using y = x 0 x 1 + ϵ , where x 0 and x 1 are drawn from uniform distribution of [ − 1 , 1] and ϵ is drawn from uniform distribution of [ − η noise , η noise ] .
Revised: using y = x 0 x 1 + ϵ , where x 0 and x 1 are drawn from uniform distribution of [ − 1 , 1] and ϵ is drawn from uniform distribution of [ − η noise , η noise ] .

Original: To estimate eq.
Revised: To estimate eq.

Original: (24), we randomly generate 100 training sets with n samples each, i.e. M =100, and a test set with 20,000 samples.
Revised: (24), we randomly generate 100 training sets with n samples each, i.e. M =100, and a test set with 20,000 samples.

Original: To cover different conditions, we (1) choose n ∈ { 50 , 100 , 300 , 600 , 1000 , 2000 , 6000 } ; (2) inject noise with η noise ∈ { 0 . 2 , 2 , 4 , 6 , 8 } ; (3) perturb model structures by choosing N ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .
Revised: To cover different conditions, we (1) choose n ∈ { 50 , 100 , 300 , 600 , 1000 , 2000 , 6000 } ; (2) inject noise with η noise ∈ { 0 . 2 , 2 , 4 , 6 , 8 } ; (3) perturb model structures by choosing N ∈ { 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 } .

Original: We use gradient descent with learning rate of 0.001.
Revised: We use gradient descent with learning rate of 0.001.

Original: Figure 8 shows a similar behavior as Fig. 3.
Revised: Figure 8 shows a similar behavior as Fig. 3.

Original: During the early training stages, the LHS and RHS of eq.
Revised: During the early training stages, the LHS and RHS of eq.

Original: (19) are very close.
Revised: (19) are very close.

Original: Their highly correlated relation remains until training converges, whereas the RHS of eq.
Revised: Their highly correlated relation remains until training converges, whereas the RHS of eq.

Original: (19) decreases signiﬁcantly.
Revised: (19) decreases signiﬁcantly.

Original: B A PPENDIX B Derivation of eq. (28)
Revised: B A PPENDIX B Derivation of eq. (28)

Original: [Equation]
Revised: [Equation]

Original: + other terms (33)
Revised: + other terms (33)

Original: Above we used ∂o ( l ) i,c ′ ∂a (
Revised: Above we used ∂o ( l ) i,c ′ ∂a (

Original: [Equation]
Revised: [Equation]

Original: ) i,s
Revised: ) i,s

Original: [Equation]
Revised: [Equation]

Original: = [Equation] ( l ) s ′ ,c ′ and ∂o ( l ) i,c
Revised: = W ( l ) s ′ ,c ′ and ∂o ( l ) i,c

Original: ∂ W ( l ) s,c = a ( l ) i,s that can both be derived from eq.
Revised: ∂ W ( l ) s,c = a ( l ) i,s that can both be derived from eq.

Original: Consider the ﬁrst term of eq.
Revised: Consider the ﬁrst term of eq.

Original: When s ′ =
Revised: When s ′ =

Original: s, c ′ =
Revised: s, c ′ =

Original: c , we have
Revised: c , we have

Original: [Equation]
Revised: [Equation]

Original: Note that the term related to ∂ 2 L i ∂o ( l ) i,c
Revised: Note that the term related to ∂ 2 L i ∂o ( l ) i,c

Original: ∂θ j a ( l ) i,s and the terms when s ′ ̸
Revised: ∂θ j a ( l ) i,s and the terms when s ′ ̸

Original: = s or c ′ ̸
Revised: = s or c ′ ̸

Original: = c in eq.
Revised: = c in eq.

Original: (33) are merged into other terms of eq.
Revised: (33) are merged into other terms of eq.

Original: C A PPENDIX C
Revised: C A PPENDIX C

Original: Notations
Revised: Notations

Original: [Table]
Revised: [Table]

Original: Training set consists of n samples drawn from Z
Revised: Training set consists of n samples drawn from Z

Original: [Table]
Revised: [Table]

Original: Test set consists of n ′ samples drawn from Z
Revised: Test set consists of n ′ samples drawn from Z

Original: [Table]
Revised: [Table]

Original: Model parameters, whose components are denoted as θ j
Revised: Model parameters, whose components are denoted as θ j

Original: [Table]
Revised: [Table]

Original: Parameters’ gradient w.r.t.
Revised: Parameters’ gradient w.r.t.

Original: a single data sample s or ( x i , y i )
Revised: a single data sample s or ( x i , y i )

Original: [Table]
Revised: [Table]

Original: of
Revised: of

Original: [Table]
Revised: [Table]

Original: over a
Revised: over a

Original: [Table]
Revised: [Table]

Original: - tion,
Revised: - tion,

Original: i
Revised: i

Original: [Table]
Revised: [Table]

Original: e
Revised: e

Original: [Table]
Revised: [Table]

Original: Note that,
Revised: Note that,

Original: [Table]
Revised: [Table]

Original: we assume
Revised: we assume

Original: [Table]
Revised: [Table]

Original: n
Revised: n

Original: [Table]
Revised: [Table]

Original: parameters’ gradient of a single sample, i.e. ,
Revised: parameters’ gradient of a single sample, i.e. ,

Original: [Table]
Revised: [Table]

Original: training
Revised: training

Original: [Table]
Revised: [Table]

Original: of
Revised: of

Original: [Table]
Revised: [Table]

Original: Gradient signal to noise ratio (GSNR) of model parameter θ
Revised: Gradient signal to noise ratio (GSNR) of model parameter θ

Original: [Table]
Revised: [Table]

Original: L
Revised: L

Original: [Table]
Revised: [Table]

Original: eter
Revised: eter

Original: [Table]
Revised: [Table]

Original: ratio (OSGR) for the training and test sets of size n sampled from data distribution Z , i.e. , E
Revised: ratio (OSGR) for the training and test sets of size n sampled from data distribution Z , i.e. , E

Original: [Table]
Revised: [Table]

Original: One-step generalization gap increment, i.e. , ∆ L [ D ] - ∆ L [ D ′ ]
Revised: One-step generalization gap increment, i.e. , ∆ L [ D ] - ∆ L [ D ′ ]

Original: [Table]
Revised: [Table]

Original: Collection of model parameters over all the layers before the l -th layer
Revised: Collection of model parameters over all the layers before the l -th layer

Original: [Table]
Revised: [Table]

Original: Average gradient of W ( 
Revised: Average gradient of W ( l

Original: [Table]
Revised: [Table]

Original:  the training dataset
Revised: over the training dataset

Original: [Table]
Revised: [Table]

Original: Collection of model parameters over all the layers after the l -th layer, including the l -th layer
Revised: Collection of model parameters over all the layers after the l -th layer, including the l -th layer

Original: [Table]
Revised: [Table]

Original: Activations of the l -th layer, where s = {
Revised: Activations of the l -th layer, where s = {

Revised: , ..., S }

Original: , S } is the index of nodes/channels
Revised: is the index of nodes /channels

Original: in the l -th layer
Revised: in the l -th layer

Original: [Table]
Revised: [Table]

Original: Outputs of
Revised: Outputs of

Original: matrix multiplication of the l -th layer, where c = { 1 , 2 , ..., C } is index of nodes/channels in the ( l + 1) -th layer
Revised: matrix multiplication of the l -th layer, where c = { 1 , 2 , ..., C } is index of nodes/channels in the ( l + 1) -th layer

Original: [Table]
Revised: [Table]

Original: l
Revised: l

Original: [Table]
Revised: [Table]

Original: evaluated on data sample i
Revised: evaluated on data sample i