Title: Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: Activation function

condition number

NTK

wide neural network
Abstract: Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and intriguing property of nonlinear activations. By comparing enabling and disabling the nonlinear activations in the neural network, we demonstrate their specific effects on wide neural networks: (a) *better feature separation*, i.e.,  a larger angle separation for similar data in the feature space of model gradient, and (b) *better NTK conditioning*, i.e.,  a smaller condition number of neural tangent kernel (NTK). Furthermore, we show that the network depth (i.e., with more nonlinear activation operations) further amplifies these effects; in addition, in the infinite-width-then-depth limit, all data are equally separated with a fixed angle in the model gradient feature space, regardless of how similar they are originally in the input space. 
   Note that, without the nonlinear activation, i.e., in a linear neural network, the data separation remains the same as for the original inputs
   and NTK condition number is equivalent to the Gram matrix, regardless of the network depth. Due to the close connection between NTK condition number and convergence theories, our results imply that nonlinear activation
  helps to improve the worst-case convergence rates of gradient based methods.

Better NTK Conditioning: A Free Lunch from (ReLU) Nonlinear Activation in Wide Neural Networks

Abstract

Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and intriguing property of nonlinear activations. By comparing enabling and disabling the nonlinear activations in the neural network, we demonstrate their specific effects on wide neural networks: (a) better feature separation , i.e., a larger angle separation for similar data in the feature space of model gradient, and (b) better NTK conditioning , i.e., a smaller condition number of neural tangent kernel (NTK). Furthermore, we show that the network depth (i.e., with more nonlinear activation operations) further amplifies these effects; in addition, in the infinite-width-then-depth limit, all data are equally separated with a fixed angle in the model gradient feature space, regardless of how similar they are originally in the input space. Note that, without the nonlinear activation, i.e., in a linear neural network, the data separation remains the same as for the original inputs and NTK condition number is equivalent to the Gram matrix, regardless of the network depth. Due to the close connection between NTK condition number and convergence theories, our results imply that nonlinear activation helps to improve the worst-case convergence rates of gradient based methods.

1 Introduction

It is well known that nonlinear activation functions increase the expressivity of neural networks, which is the primary reason of their widespread implementation. A nonlinearly activated neural network can approximate any continuous function to arbitrary precision, as long as there are enough neurons in the hidden layers [13, 7, 12], while without it - as in a linear neural network , the network reduces to linear models of the input. In addition, deeper neural networks, which have more nonlinearly activated layers, have exponentially greater expressivity than shallower ones [32, 28, 29, 22, 34], indicating that the network depth promotes the power of nonlinear activation functions.

Like Hui

In this paper, we reveal a novel and interesting effect of nonlinear activations that has not been previously noticed, despite their widespread application: the nonlinearity leads to larger data separation in the feature space of model gradient, and helps to decrease the condition number of neural tangent kernel (NTK). We also show that the depth of the network further amplifies these effects, namely, a deeper neural network has a better feature separation and a smaller NTK condition number, than a shallower one. While distinct and independent from the property of increasing expressivity, this property of nonlinear activations resembles the former in the manner that the effects vanish in the absence of nonlinear activations: removing the nonlinear activations in a neural network, the data separation and NTK condition number reduce to values observed in a linear model. Hence, the effect purely attributes to the presence of nonlinear activations.

Specifically, we first show the better separation phenomenon , i.e., improved separation for similar data in the model gradient feature space. We prove that, for a wide ReLU network f , any pair of data input vectors x and z that have similar directions (i.e., small but non-zero angle θ in between x and z ) become more directionally separated in the model gradient space (i.e., model gradient angle ϕ between ∇ f ( x ) and ∇ f ( z ) is larger than θ in ) with high probability of random initialization. We also find that deeper ReLU networks result in even better feature separation, i.e., larger ϕ . Ultimately, in the infinite-width-then-depth limit, all data are equally separated with an angle ∼ 75 . 5 ◦ in the model gradient feature space, regardless of the input angle θ in , as long as θ in is non-zero. Numerical simulation also show that the better separation phenomenon generalizes to other commonly used nonlinear activation functions, including GeLU, tanh, etc.

We further show the better NTK conditioning property of nonlinear activation, i.e., smaller NTK condition number. We prove that, as a consequence of the better feature separation, the NTK condition number of a wide ReLU network is strictly smaller than that without the nonlinearity, when the training dataset is not degenerate (i.e., no pair of training inputs are parallel). Moreover, with a larger depth, the NTK condition number becomes smaller. The intuition is that, if there exists a pair of similar inputs x and z in the training set (i.e., the angle between x and z is small), which is usually the case for large datasets, then NTK of linear neural networks must have close-to-zero smallest eigenvalues, resulting in extremely large NTK condition numbers. The activation makes these similar data more separated, hence it helps to increase the smallest eigenvalues of NTK, which in turn leads to a smaller NTK condition number. We further show that, in the infinite-width-then-depth limit, the NTK condition number of ReLU network converges to a fixed number n +4 3 , which is independent of the data distribution and much smaller than typical NTK condition numbers.

Connection with optimization theory. While there could be multiple implications of the above property in various aspects, here we present its connection with existing optimization theories. Recent optimization theories showed that the NTK condition number κ , or the smallest eigenvalue of NTK, controls the theoretical convergence rate of gradient descent algorithms on wide neural networks [9, 8, 20]. Combined with these theories, our findings imply that: (a), the activation function has the effect of improving the worst-case convergence rate of gradient descent, and (b), deeper wide ReLU networks have faster convergence rate than shallower ones. Previous works often focus on accelerating convergence via a better function of κ while assuming κ is given and fixed. Our findings provide a different perspective of achieving acceleration by tuning κ itself. Experimentally, we indeed find that deeper networks converge faster than shallower ones.

Contributions. We summarize our contributions below. We find that:

Nonlinear activation functions induce better separation between similar data in the feature space of model gradient. A larger network depth amplifies this better separation phenomenon.

Nonlinear activations have the effect of decreasing the NTK condition number. A larger depth of the network further enhances this better NTK conditioning property.

This better NTK conditioning property leads to faster convergence rate of gradient descent. We empirically verify this on various real world datasets.

The paper is organized as follow: Section 2 describes the setting and defines the key quantities and concepts, and analyzes linear neural networks as the baseline for comparison; Section 3 and 4 discuss our main results on the better separation and better conditioning of nonlinear activation, respectively; Section 5 discusses the implication on theoretical convergence rates; Section 6 concludes the paper. Proofs of theorems and main corollaries can be found in the appendix.

1.1 Related work

NTK and its spectrum have been extensively studied [19, 5, 21, 10, 11, 36, 25, 4, 6], since the discovery of constant NTK for infinitely wide neural networks [17]. [33] shows that the NTK spectrum of an infinitely wide ReLU network asymptotically exhibits a power law. Its distribution is further shown to be similar to that of Laplace kernel [11, 6], and can be computed [10]. Nguyen, Mondelli, and Montufar [25] analyzed the upper and lower bounds for the smallest NTK eigenvalue in O () and Ω() , respectively. With the assumption of spherically uniformly distributed data where the spectrum of (elementary-wise) power of the Gram matrix becomes simplified, [23], utilizing Hermite polynomials and power series expansion of NTK, provides the order of the smallest eigenvalue of the NTK of two-layer ReLU network in the infinite width limit. Under the same data setting, [3] computed the NTK eigenvalues for the two-layer ReLU network. Relying on the values of off-diagonal entries of the NTK matrix in the infinite depth limit, another work [36] analyzed the asymptotic dependence of the NTK condition number on the network depth L for ReLU networks, which shows a decreasing trend as L increases, consistent with our result.

In contrast to prior works, we are able to distill the effect of ReLU activation function via a sharp comparison between scenarios with and without ReLU, at any finite depth without data distribution assumption. Note that, without an assumption on data distribution, NTK spectral analysis becomes much harder and many data-distribution-dependent results may not hold any more. Moreover, at finite depth, off-diagonal entries of the NTK matrix has not converged and are typically quite different from its infinite depth limit, which makes analysis even harder.

We are aware of a prior work [2] which has results of similar flavor. It shows that the depth of a linear neural network may help to accelerate optimization via an implicit pre-conditioning of gradient descent. We note that this prior work is in an orthogonal direction, as its analysis is based on the linear neural network, which is activation-free, while our work focus on the better-conditioning effect of activation functions.

2 Setup and Preliminaries

Notations for general purpose. We denote the set { 1 , 2 , · · · , n } by [ n ] . We use bold lowercase letters, e.g., v , to denote vectors, and capital letters, e.g., A , to denote matrices. Given a vector, ∥ · ∥ denotes its Euclidean norm. Inner product between two vectors is denoted by ⟨· , ·⟩ . Given a matrix A , we denote its i -th row by A i : , its j -th column by A : j , and its entry at i -th row and j -th column by A ij . We also denote the expectation (over a distribution) of a variable by E [ · ] , and the probability of an event by P [ · ] . For a model f ( w ; x ) which has parameters w and takes x as input, we use ∇ f to denote its first derivative w.r.t. the parameters w , i.e., ∇ f := ∂f/∂ w .

(Fully-connected) neural network. Let x ∈ R d be the input, m l be the width (i.e., number of neurons) of the l -th layer, W ( l ) ∈ R m l × m l -1 , l ∈ [ L +1] , be the matrix of the parameters at layer l , and σ ( z ) be the activation function, which is applied element-wise. A (fully-connected) neural network f , with L hidden layers, is defined as:

where c σ = ( E z ∼N (0 , 1) [ σ ( z ) 2 ]) -1 . For the special case of ReLU activation function: σ ( z ) = max { 0 , z } , c σ = 2 . We also denote ˜ α ( l ) ( x ) ≜ √ c σ √ m l W ( l ) α ( l -1) ( x ) . Following the NTK initialization scheme [17], these parameters are randomly initialized i.i.d. according to the normal distribution N (0 , 1) . The scaling factor √ c σ / √ m l is introduced to normalize the hidden neurons [8]. We denote the collection of all parameters by w .

Remark 2.1. In this paper, we consider the bias-free setting where no bias term is included when computing the hidden neurons in Eq.(1). In fact, the bias term can potentially lead to different results, as have been noticed in [16].

Without loss of generality, we set the layer widths as

and call m as the network width. In the rest of the paper, we typically consider wide neural networks, i.e., networks with large widths m and fixed depths L .

Linear neural network. For a comparison purpose, we also consider a linear neural network ¯ f , which is the same as the neural network f defined above, except that the activation function is the identity function σ ( z ) = z and that the scaling factor of Eq.(1) is 1 / √ m .

Model gradient feature and neural tangent kernel (NTK). Given a model f (e.g., a neural network) with parameters w , we call the the derivative of model f with respect to all its parameters as the model gradient feature vector ∇ f ( w ; x ) for the input x . The NTK K is defined as

where x 1 and x 2 are two arbitrary network inputs. For a given dataset D = { ( x i , y i ) } n i =1 , there is a gradient feature matrix F such that each row F i · ( w ) = ∇ f ( w ; x i ) for all i ∈ [ n ] . The n × n NTK matrix K ( w ) is defined such that its entry K ij ( w ) , i, j ∈ [ n ] , is K ( w ; x i , x j ) . It is easy to see that the NTK matrix

Note that the NTK for a linear model reduces to the Gram matrix G ∈ R d × d , where each row of the matrix X is an input feature x i , i.e., X i · = x T i .

As pointed out by [21, 19, 17], a neural network with large width m is approximately a linear model on the model gradient features ∇ f ( w 0 ; x ) :

Hence, the training dynamics of a wide neural network is largely controlled by the model gradient features of the training samples. We will see that the model gradient angle , i.e., the angle between the model gradient features of an arbitrary pair of inputs, is a key quantity that measures the mutual relations between training samples and is closely related to the NTK condition number and convergence rate.

Definition 2.2 (Model gradient angle) . Given two arbitrary inputs x , z ∈ R d , define the model gradient angle as the angle between the model gradient vectors ∇ f ( x ) and ∇ f ( z ) :

Condition number. The condition number κ of a positive definite matrix A is defined as the ratio between its maximum eigenvalue and minimum eigenvalue:

In the rest of the paper, we specifically denote the NTK matrix, NTK condition number and model gradient angle for the neural network as K , κ and ϕ , respectively, and denote their linear neural network counterparts as ¯ K , ¯ κ and ¯ ϕ , respectively. We also denote the condition number of Gram matrix G by κ 0 .

2.1 Without nonlinear activation: the baseline for comparison

To distill the effect of the nonlinear activation function, we need a activation-free case as the baseline for comparison. This baseline is the linear neural network ¯ f , with the same width and depth as f .

Theorem 2.3. Consider a linear neural network ¯ f . In the limit of infinite network width m →∞ and at network initialization w 0 , the following relations hold:

for any input x ∈ R d , ∥∇ f ( w 0 ; x ) ∥ = ( L +1) ∥ x ∥ .

for any inputs x , z ∈ R d , ¯ ϕ ( x , z ) = θ in ( x , z ) .

This theorem states that, without a nonlinear activation function, the model gradient map ∇ f : x ↦→ ∇ f ( x ) does not change the geometrical relationship between any data samples. For any input pairs, the model gradient angle ¯ ϕ remains the same as the input angle θ in . Therefore, it is not surprising that the NTK of a linear network is the same as the Gram matrix (up to a constant factor), as formally stated in the following corollary (which can also be consistently obtained using Theorem 1 of [17]).

Corollary 2.4 (NTK condition number without activation) . Consider a linear neural network ¯ f . In the limit of infinite network width m →∞ and at network initialization, the NTK matrix ¯ K = ( L +1) 2 G . Moreover, ¯ κ = κ 0 .

This corollary tells that, for a linear neural network, regardless of its depth L , the NTK condition number ¯ κ is always equal to the condition number κ 0 of the Gram matrix G . Therefore, any non-zero deviations, δϕ ≜ ϕ -θ in from the input angle θ in , and δκ ≜ κ -κ 0 from the Gram condition number κ 0 , observed for a nonlinearly activated network f , should be attributed to the corresponding nonlinear activation.

3 Better separation in model gradient space

In this section, we show that the nonlinear activation function helps data separation in the model gradient space. Our theoretical analysis will focus on the special case of ReLU, and the results will be numerically verified on other nonlinear activations as well. Specifically, for two arbitrary inputs x and z with small θ in ( x , z ) , we show that the model gradient angle ϕ ( x , z ) is strictly larger than θ in ( x , z ) , implying a better angle separation of the two data points in the model gradient space. Moreover, we show that the model gradient angle ϕ ( x , z ) monotonically increases with the number of layers L , indicating that deeper network (more ReLU nonlinearity) has better angle separation.

First, we introduce an auxiliary quantity, l -embedding angle θ ( l ) ( x , z ) , which measures the angle between two hidden vectors α ( l ) ( x ) and α ( l ) ( z ) at infinite width, and an auxiliary function g : [0 , π ) → [0 , π ) with g ( z ) = arccos ( π -z π cos z + 1 π sin z ) . We also denote the l -fold composition of g ( · ) as g ◦ l . Please see Appendix A for the plot of the function and detailed discussion about its properties. As a highlight, g has the following property: g is approximately (but less than) the identity function g ( z ) ≈ z for small z , i.e., z ≪ 1 .

The following lemma gives the relation between the model gradient angle ϕ of any two inputs and their original input angle θ in , via the embedding angles θ ( l ) and the function g .

Lemma 3.1. Consider the ReLU network defined in Eq.(1) with L hidden layers and infinite network width. Given two arbitrary inputs x and z , the angle ϕ ( x , z ) between the model gradients ∇ f ( x ) and ∇ f ( z ) satisfies

Better feature separation. Comparing with Theorem 2.3 for linear neural networks, we see that the nonlinear ReLU activation only affects the relative direction, but not the the magnitude, of the model gradient. Lemma 3.1 gives the relation between ϕ and the input angle θ in . Figure 1 plots ϕ as a function of θ in for different network depth L .

The key observation is that: for relatively small input angles (say θ in ≤ 30 ◦ , which is actually not quite small), the model gradient angle ϕ is always greater than the input angle θ in . This suggests that, after the mapping ∇ f : x ↦→∇ f ( x ) from the input space to model gradient space, data inputs becomes more (directionally) separated, if they are similar in the input space (i.e., with small θ in ). Comparing to the linear neural network case, where ¯ ϕ ( x , z ) = θ in ( x , z ) as in Theorem 2.3, we see that the ReLU nonlinearity results in a better angle separation ϕ ( x , z ) > ¯ ϕ ( x , z ) for similar data.

Another observation is that: deeper ReLU networks lead to larger model gradient angles, when θ in < 30 ◦ . This indicates that deeper ReLU networks, which has more layers of ReLU nonlinear activation, makes the model gradient more separated between inputs. Note that, in the linear network case, the depth does not affect the gradient angle ¯ ϕ .

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Very similar inputs (i.e., when θ in ( x , z ) ≪ 1 ), especially those with different labels, are often hard to distinguish and is one of the key factors that makes training difficult, because the decision boundary has to be fine-tuned to separate these very closely located inputs in order to make correct prediction. Hence, the regime of very small input angle ( θ in ≪ 1 ) is of particular interest for model training. The following theorem confirms the better separation in this regime.

Theorem 3.2 (Better separation for similar data) . Consider two arbitrary inputs x , z ∈ R d , with small input angle 0 < θ in ( x , z ) ≪ 1 , and the ReLU network defined in Eq.(1). The model angle ϕ ( x , z ) is strictly greater than the input angle θ in ( x , z ) :

with high probability of the random network initialization, if the network width m = Ω(1 /θ 2 in ) .

The following corollary quantifies the better separation in this regime.

Corollary 3.3. With the same setting as in Theorem 3.2 and with infinite width m →∞ but finite depth L = Ω(1 /θ in ) , cos ϕ ( x , z ) = ( 1 -L 2 π θ in + o ( θ in ) ) cos θ in .

Remark 3.4 (Separation in distance) . Indeed, the better angle separation discussed above implies a better separation in Euclidean distance as well. This can be easily seen by recalling from Lemma 3.1 that the model gradient mapping ∇ f preserves the norm (up to a universal factor L +1 ).

We also point out that, Figure 1 indicates that for large input angles (say θ in > 30 ◦ ) the model gradient angle ϕ is always large (greater than 30 ◦ ). Hence, non-similar data never become similar in the model gradient feature space.

Better separation in infinite width and depth limit. Now, we consider the infinite width and depth case. We took the infinite width limit a prior, this technically leads to the infinite-width-then-depth limit. The following theorem shows that, no matter how similar two inputs originally are, as long as they are not parallel, their model gradient features eventually get wide separated in large depth.

Theorem 3.5. Consider the ReLU neural network defined in Eq.(1) and two non-parallel inputs x and z , x ∦ z . In the infinite-width-then-depth limit, the model gradient angle ϕ ( x , z ) converges to a fix value arccos 1 4 , regardless the input angle θ in ( x , z ) .

Remark 3.6. The limit point value arccos 1 4 is about 75 . 5 ◦ , which means the inputs are quite wellseparated in the model gradient feature space, as network depth increase to infinity. Recall that, without the nonlinear activation, ϕ ( x , z ) = θ in , which can be arbitrarily small.

It is interesting to observe that this limit point value is independent of the input angle, which means that the data points are mutually equally-separated in the limit. We will discuss its implications on NTK in the following sections.

Beyond ReLU activation. Here, we numerically verify that the better separation phenomenon obtained for ReLU above also holds for other nonlinear activation functions. Figure 2 shows the relations between the model gradient angle ϕ and the input angle θ in for the following nonlinear activations: ReLU 2 (i.e., σ ( z ) = max { 0 , z 2 } ), GeLU and tanh. One can easily see that the better separation holds: for relatively small input angles θ in (e.g., θ in ≤ 30 ◦ ), ϕ is always greater than θ in ; and for deeper networks, ϕ is even greater. Interestingly, we observe that the gradient angle ϕ converges to 90 ◦ for these activation functions indicating gradient features become orthogonal in the limit of L →∞ , different from the 75 . 5 ◦ that we obtain for ReLU networks.

We also show that the better separation generalizes beyond the NTK setting/regime. Please see Appendix B for more discussion.

4 Better NTK conditioning

In this section, we show both theoretically and experimentally that, the nonlinear activation induces a decrease in the NTK condition number κ . Moreover, a neural network with larger depth L , which means more nonlinear activations in operation, the NTK condition number κ is generically smaller.

Connection between condition number and model gradient angle. The smallest eigenvalue and condition number of NTK are closely related to the smallest model gradient angle min i,j ∈ [ n ] ϕ ( x i , x j ) , through the gradient feature matrix F . Think about the case if ϕ ( x i , x j ) = 0 (i.e., ∇ f ( x i ) is parallel to ∇ f ( x j ) ) for some i, j ∈ [ n ] , then F , hence NTK K , is not full rank and the smallest eigenvalue λ min ( K ) is zero, leading to an infinite condition number κ . Similarly, if min i,j ∈ [ n ] ϕ ( x i , x j ) is small, the smallest eigenvalue λ min ( K ) is also small, and condition number κ is large, as stated in the following proposition (see proof in Appendix C).

Proposition 4.1. Consider a n × n positive definite matrix A = BB T , where matrix B ∈ R n × d , with d > n , is of full row rank. Suppose that there exist i, j ∈ [ n ] such that the angle ϕ between vectors B i · and B j · is small, i.e., ϕ ≪ 1 , and that there exist constant C > c > 0 such that c ≤ ∥ B k · ∥ ≤ C for all k ∈ [ n ] . Then, the smallest eigenvalue λ min ( A ) = O ( ϕ 2 ) , and the condition number κ = Ω(1 /ϕ 2 ) .

Therefore, a good data angle separation in the model gradient features, i.e., min i,j ∈ [ n ] ϕ ( x i , x j ) not too small, is a necessary condition such that the condition number κ is not too large. As is shown in the last section, the ReLU nonlinearity makes the samples more separated when mapped from the input data space to the model gradient feature space. Hence, it is expected that the NTK condition number will decrease in the presence of the ReLU nonlinearity.

Smaller NTK condition number. Theoretically, we consider the infinite width limit. We require that the dataset is not degenerated, i.e., x i ∦ x j for all i, j . This is a mild and commonly used setting in the literature, see for example [9]. We require that the weights of the first layer W (1) be trainable

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and fix the other layers in the following theorem. This is also a common setting in literature to simplify the analysis [9].

Theorem 4.2. Consider the ReLU network in Eq.(41) in the limit m → ∞ and at initialization. Let the weights of the first layer W (1) be trainable and fix the other layers. We compare the two scenarios: (a) the network with ReLU activation and (b) the network with all the ReLU activation removed. The smallest eigenvalue λ min ( K ) of its NTK in scenario (a) is larger than that of scenario (b): λ min ( K a ) > λ min ( K b ) , and the NTK condition number κ in scenario (a) is less than that in scenario (b): κ a < κ b . Moreover, for two ReLU neural networks f 1 of depth L 1 and f 2 of depth L 2 with L 1 > L 2 , we have κ f 1 < κ f 2 .

This theorem confirms the expectation that the NTK condition number κ should be decreased, as a consequence of the existence of the ReLU nonlinearity. This theorem also shows that the depth of the ReLU network enhances this better NTK conditioning.

The high-level intuition behind the proof of this theorem is that: the derivative of the ReLU function, σ ′ ( z ) = I { z ≥ 0 } , resembles a binary gate which has open and close states. When ReLU are implemented, the model gradient map ∇ f : x ↦→∇ f ( x ) increases the directional diversity of the vectors ∇ f ( x ) , due to the high dimension of the model gradient space and the different activation patterns of the hidden layer for different samples x . Hence, it is expected that the feature matrix F , as well as the NTK matrix K , is better conditioned.

In fact, fixing the weights of the top layer is not necessary and can be removed. We relax this requirement in Appendix F. In our experiments in Section 4.1 where all layers are trainable, we observe the phenomena of better separation and better NTK conditioning .

NTK condition number in infinite depth. As a consequence of the pairwise equal-separation result (Theorem 3.5), the NTK matrix got simplified in the infinite depth limit. The following theorem shows that the NTK condition number converges to a fixed value n +4 3 , which is independent of the data distribution.

̸

Theorem 4.3. Consider the ReLU neural network defined in Eq.(1) and a dataset D = { ( x i , y i ) } n i =1 . Suppose that all data inputs are normalized ∥ x i ∥ = 1 for all i , and x i ∦ x j for all i = j . In the infinite-width-then-depth limit, the NTK condition number κ converges to n +4 3 .

4.1 Experimental evidence

Here, we experimentally show that better separation and better conditioning happen in practice.

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Dataset. We use the following datasets: synthetic dataset, MNIST [18], FashionMNIST (f-MNIST) [35], SVHN [24] and Librispeech [27]. The synthetic data consists of 2000 samples which are randomly drawn from a 5-dimensional Gaussian distribution with zero-mean and unit variance. The MNIST, f-MNIST and SVHN datasets are image datasets where each input is an image. The Librispeech is a speech dataset including 100 hours of clean speeches. In the experiments, we use a subset of Librispeech with 50 , 000 samples, and each input is a 768-dimensional vector representing a frame of speech audio and we follow [15] for the feature extraction.

Models. For each of the datasets, we use a ReLU activated fully-connected neural network architecture to process. The ReLU network has L hidden layers, and has 512 neurons in each of its hidden layers. The ReLU network uses the NTK parameterization and initialization strategy (see [17]). For each dataset, we vary the network depth L from 0 to 10 . Note that L = 0 corresponding to the linear model case. In addition, for comparison, we use a linear neural network, which has the same architecture with the ReLU network except the absence of activation function.

Results. For each dataset and given network depth L , we evaluate both the smallest pairwise model gradient angle min i,j ∈ [ n ] ϕ ( x i , x j ) and the NTK condition number κ , at the network initialization. We take 5 independent runs over 5 random initialization seeds, and report the average. In each run, we used a A-100 GPU to compute the NTK, which took 4 ∼ 10 hours. The results are shown in Figure 3. We compare the two scenarios of with and without the ReLU activation function. As one can easily see from the plots, a ReLU network (depth L = 1 , 2 , · · · , 10 ) always have a better separation of data features (i.e., larger smallest pairwise model gradient angle), and a better NTK conditioning (i.e., smaller NTK condition number), than its corresponding linear network (compare the solid line and dash line of the same color). Furthermore, the monotonically decreasing NTK condition number shows that a deeper ReLU network have a better conditioning of NTK.

5 Optimization acceleration

Recently studies have shown strong connections between the NTK condition number and the theoretical convergence rate of gradient descent algorithms on wide neural networks [9, 8, 31, 1, 37, 26, 20]. In [9, 8, 20], the worst-case convergence rate has been shown to be

Although κ is evaluated on the entire optimization path, all these theories used the fact that NTK is almost constant for wide neural networks and an evaluation at initialization w 0 is enough.

As a smaller NTK condition number (or larger smallest eigenvalue of NTK) implies a faster worstcase convergence rate, our findings suggest that: (a), the ReLU activation function helps improve the worst-case convergence rate of gradient descent, and (b), deeper wide ReLU networks have faster convergence rate than shallower ones.

We experimentally verify this implication. Specifically, we train the ReLU networks, with depth L ranging from 1 to 10 , for the datasets MNIST, f-MNIST, and Librispeech. For all training tasks, we use cross-entropy loss as the objective function and use mini-batch stochastic gradient descent (SGD)

of batch size 500 to optimize. For each task, we find its optimal learning rate by grid search. On MNIST and f-MNIST, we train 500 epochs, and on Librispeech, we training 2000 epochs.

The curves of training loss against epochs are shown in Figure 4. We observe that, for all these datasets, a deeper ReLU network always converges faster than a shallower one. This is consistent with the theoretical prediction that the deeper ReLU network, which has a smaller NTK condition number, has a faster theoretical convergence rate.

Trade-off between optimization and generalization. Although a faster convergence in terms of number of iterations for deep networks, as Theorem 3.5 suggests, in the extreme case of infinite depth L →∞ , any non-parallel input pairs become equally separated in gradient features regardless of their original similarity. Even though not mutually orthogonal, this could also result in a trivial generalization: close to random guess for unseen data. The same consequence can also be obtained from [14], where they dropped the initial random guess value and obtained a zero prediction for unseen data.

As for finite depth, it is theoretically hard to predict at what depth this trade-off starts to happen. Under the same experimental setting as in Figure 4, Table 1 shows that the generalization performance starts to decrease at depth L = 8 , suggesting a optimization-generalization trade-off for large depth.

6 Conclusion and discussions

In this work, we showed the effects of nonlinear activation on better separation of similar data in feature space and on the NTK conditioning. We also showed that more sequential activation operations, i.e., larger network depth, amplifies these effects. As the NTK conditioning is closely related to theoretical convergence rate of gradient descent, our findings also suggest a positive role of activation functions in optimization theories. A limitation of the paper is that the theoretical analysis is only conducted on ReLU activation, although results have been empirically verified for other nonlinear activations. For other activations, the analysis requires analytical expressions for integrations involved, which requires a distinct type of analysis and we consider it as a future work.

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A Properties of function g

Recall that the function g : [0 , π ) → [0 , π ) is defined as (see Lemma E.5)

Figure 5 shows the plot of this function. From the plot, we can easily find the following properties.

120

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line:y=z

gfunction

0

OE

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180

Proposition A.1 (Properties of g ) . The function g defined in Eq.(10) has the following properties:

g is a monotonically increasing function;

g ( z ) ≤ z , for all z ∈ [0 , π ) ; and g ( z ) = z if and only if z = 0 ;

for any z ∈ [0 , π ) , the sequence { g l ( z ) } ∞ l =1 is monotonically decreasing, and has the limit lim l →∞ g l ( z ) = 0 .

Proof. Part 1 . First, we consider the auxiliary function ˜ g ( z ) = π -z π cos z + 1 π sin z . We see that

Hence, ˜ g ( z ) is monotonically decreasing on [0 , π ) . Combining with the monotonically decreasing nature of the arccos function, we get that g is monotonically increasing.

Part 2 . It suffices to prove that cos z ≤ ˜ g ( z ) and that the equality holds only at z = 0 . For z = 0 , it is easy to check that cos z = ˜ g ( z ) , as both z and sin z are zero. For z ∈ (0 , π/ 2) , noting that tan z -z > 0 , we have

For z = π/ 2 , we have cos π/ 2 = 0 < 1 /π = ˜ g ( π/ 2) . For z ∈ ( π/ 2 , π ) , we have the same relation as in Eq.(11). The only differences are that, in this case, cos z < 0 and tan z -z < 0 . Therefore, we still get ˜ g ( z ) > cos z for z ∈ ( π/ 2 , π ) .

Part 3. From part 2 , we see that g ( z ) < z for all z ∈ (0 , π ) . Hence, for any l , g l +1 ( z ) < g l ( z ) . Moreover, since z = 0 is the only fixed point such that g ( z ) = z , in the limit l →∞ , g l ( z ) → 0 .

It is worth to note that the last property of g function immediately implies the collapse of embedding vectors from different inputs in the infinite depth limit L → ∞ . This embedding collapse has been observed in prior works [28, 30] (although by different type of analysis) and has been widely discussed in the literature of Edge of Chaos.

Theorem A.2. Consider a ReLU neural network. Given any two inputs x , z ∈ R d , the sequence of angles { θ ( l ) ( x , z ) } L l =1 between their l -embedding vectors α ( l ) ( x ) and α ( l ) ( z ) , is monotonically decreasing. Moreover, in the limit of infinite depth,

and there exists a vector α such that, for any input x , the last layer L -embedding

B Beyond the NTK regime

Here, we show that the better separation phenomenon still holds outside of the NTK regime. We consider different initialization scales w ∼ N (0 , σ 2 w ) . Note that σ 2 w < 1 corresponds to small initialization. Figure 6 plots the model gradient angle ϕ as a function of the input θ in , for different scaling regimes: σ 2 w = 0 . 01 , 0 . 1 , 1 and 10 . It shows that the better separation phenomenon still holds for similar inputs at various network depths.

C Proof of Proposition 4.1

Proof. Consider the matrix B and the n vectors b k ≜ B k · , k ∈ [ n ] . The smallest singular value square of matrix B is defined as

̸

̸

̸

Since the angle ϕ between b i = B i · and b j = B j · is small, let v ′ be the vector such that v ′ i = ∥ b j ∥ , v ′ j = -∥ b i ∥ and v ′ k = 0 for all k = i, j . Then

Since A = BB T , the smallest eigenvalue λ min ( A ) of A is the same as σ 2 min ( B ) .

On the other hand, the largest eigenvalue λ max ( A ) of matrix A is lower bounded by tr ( A ) /n . Note that the diagonal entries A kk = ∥ b k ∥ . Hence, c ≤ λ max ( A ) ≤ C . Therefore, the condition number κ = λ max ( A ) /λ min ( A ) = Ω(1 /ϕ 2 ) .

D Proofs of Theorems without (ReLU) activation

D.1 Proof of Theorem 2.3

Proof. First of all, we provide a useful lemma.

Lemma D.1. Consider a matrix A ∈ R m × d , with each entry of A is i.i.d. drawn from N (0 , 1) . In the limit of m →∞ ,

We first consider the embedding vectors ¯ α ( l ) and the embedding angles ¯ θ ( l ) . By definition of linear neural network, we have, for all l ∈ [ L ] and input x ∈ R d ,

Note that at the network initialization entries of W ( l ) are i.i.d. and follows N (0 , 1) . Hence, the inner product

where in step (a) we recursively applied Lemma D.1 l times. Putting z = x , we get ∥ ¯ α ( l ) ( x ) ∥ = ∥ x ∥ , for all l ∈ [ L ] . By the definition of embedding angles, it is easy to check that ¯ θ ( l ) ( x , z ) = θ in ( x , z ) , for all l ∈ [ L ] .

Now, we consider the model gradient ∇ ¯ f and the model gradient angle ¯ ϕ . As we consider the model gradient only at network initialization, we don't explicitly write out the dependence on w 0 , and we write ∇ ¯ f ( w 0 , x ) simply as ∇ ¯ f ( x ) . The model gradient ∇ ¯ f can be decomposed as

Hence, the inner product and for all l ∈ [ l +1] ,

Here in step (b), we again applied Lemma D.1. Therefore,

Putting z = x , we get ∥∇ f ( x ) ∥ = ( L +1) ∥ x ∥ . By the definition of model gradient angle, it is easy to check that ¯ ϕ ( x , z ) = θ in ( x , z ) .

E Proofs of Theorems for ReLU network

E.1 Preliminary results

Before the proofs, we introduce some useful notations and lemmas. The proofs of these lemmas are deferred to Appendix G.

Given a vector v ∈ R p , we define the following diagonal indicator matrix:

Lemma E.1. Consider two vectors v 1 , v 2 ∈ R p and a p -dimensional random vector w ∼ N (0 , I p × p ) . Denote θ as the angle between v 1 and v 2 , i.e., cos θ = ⟨ v 1 , v 2 ⟩ ∥ v 1 ∥∥ v 2 ∥ . Then, the probability

Lemma E.2. Consider two arbitrary vectors v 1 , v 2 ∈ R p and a random matrix W ∈ R q × p with entries W ij i.i.d. drawn from N (0 , 1) . Denote θ as the angle between v 1 and v 2 , and define u 1 = √ 2 √ q σ ( W v 1 ) and u 2 = √ 2 √ q σ ( W v 2 ) . Then, in the limit of q →∞ ,

Lemma E.3. Consider two arbitrary vectors v 1 , v 2 ∈ R p and two random matrices U ∈ R s × q and W ∈ R q × p , where all entries U ij , i ∈ [ s ] and j ∈ [ q ] , and W kl , k ∈ [ q ] and l ∈ [ p ] , are i.i.d. drawn from N (0 , 1) . Denote θ as the angle between v 1 and v 2 , and define matrices A 1 = √ 2 √ q U I { W v 1 ≥ 0 } and A 2 = √ 2 √ q U I { W v 2 ≥ 0 } . Then, in the limit of q →∞ , the matrix

Lemma E.4. Consider matrix B = AA T with A ∈ R n × p and a random matrix W ∈ R q × p where all entries of W are i.i.d. drawn from N (0 , 1) . Define the tensor A ′ ∈ R n × p × q , such that A ′ ikl := √ 2 A ik I { W l : A i : ≥ 0 } . Let B ′ ∈ R n × n be the matrix such that each entry B ′ ij = ∑ k,l A ′ ikl A ′ jkl . Then, in the limit of q → ∞ , the smallest and largest eigenvalues satisfy: λ min ( B ′ ) > λ min ( B ) , and λ max ( B ′ ) < λ max ( B ) .

E.2 Proof of Lemma 3.1

Proof. The model gradient ∇ f ( x ) is composed of the components ∇ l f ( x ) ≜ ∂f ∂W l , for l ∈ [ L +1] . Each such component has the following expression: for l ∈ [ L +1]

with where

Note that in Eq.(22), ∇ l f ( x ) is an outer product of a column vector α ( l -1) ( x ) ∈ R m l -1 × 1 ( m l -1 = d if l = 1 , and m l -1 = m otherwise) and a row vector δ ( l ) ( x ) ∈ R 1 × m l ( m l = 1 if l = L +1 , and m l = m otherwise).

First, we consider an infinitely wide neural network f ∞ of depth L . We have the following lemma.

Lemma E.5. Consider a ReLU network f ∞ defined in Eq.(1) with infinite width. For all l ∈ [ L ] , the following relations hold:

for any input x ∈ R d , ∥ α ( l ) ( x ) ∥ = ∥ x ∥ ;

for any two inputs x , z ∈ R d , θ ( l ) ( x , z ) = g ( θ ( l -1) ( x , z ) ) . Let g l ( · ) be the l -fold composition of g ( · ) , then

We consider the inner product ⟨∇ l f ∞ ( z ) , ∇ l f ∞ ( x ) ⟩ , for l ∈ [ L +1] . 1 By Eq.(22), we have

For ⟨ α ( l -1) ( z ) , α ( l -1) ( x ) ⟩ , applying Lemma E.5, we have

For ⟨ δ ( l ) ( z ) , δ ( l ) ( x ) ⟩ , by definition Eq.(23), we have

Recalling that ˜ α ( l ) = W ( l ) ˜ α ( l -1) and applying Lemma E.3 on the the term A above, we obtain

Recursively applying the above formula for l ′ = l, l +1 , · · · , L , and noticing that δ ( L +1) = 1 , we have

Combining Eq.(25), (26) and (27), we have

For the inner product between the full model gradients, we have

Putting x = z in the above equation, we have θ ( l ) ( x , z ) = 0 for all l ∈ [ L ] , and obtain

Hence, we have, for an infinitely wide neural network,

Now, we consider the finitely wide neural network f . As have been shown by [17, 9, 21],

with high probability of random initialization of the network f . Letting z = x above, we also have

Using the above two equations, we have

with high probability of random initialization of the network f .

E.3 Proof of Theorem 3.2

Proof. For simplicity of notation, we don't explicitly write out the dependent on the inputs x , z , and write θ ( l ) ≜ θ ( l ) ( x , z ) , and ϕ ≜ ϕ ( x , z ) . We start the proof with the summation term on the R.H.S. of Eq. 7 in Lemma 3.1.

In step (a) above, we use the relation θ ( l ) = g ( θ ( l -1) ) , i.e., cos θ ( l ) = 1 -cos θ ( l -1) /π +sin θ ( l -1) /π ; in step (b), we used the fact that θ ( l ) < θ in (Theorem A.2) which stays small and used the Taylor expansion of tan . In step (c), we used the following lemma (proof is in Appendix G.7):

Lemma E.6. Given any inputs x , z such that θ in ( x , z ) ≪ 1 , for each l ∈ [ L ] , the l -embedding angle θ ( l ) ( x , z ) can be expressed as

By Lemma 3.1, there exists a constant c such that

When m> 16 π 2 c 2 L 2 θ 2 in cos 2 θ in , we have cos ϕ ( x , z ) < cos θ in ( x , z ) , namely, ϕ ( x , z ) > θ in ( x , z ) .

E.4 Proof of Theorem 3.5

Proof. We start the proof with an analysis of the embedding angles θ ( l ) in the infinite depth limit.

̸

First, by Lemma E.5 and Proposition A.1, we easily find that, for all input angle θ in = θ (0) = 0 , θ ( l ) is monotonically decreasing and converges to zero: lim l →∞ θ ( l ) → 0 . As the following analysis is independent of θ (0) , we will not explicitly write out the arguments x and z .

Now, we analyze its convergence rate, utilizing Eq.(57). As we are considering the infinite depth limit and θ ( l ) converges to zero, we can drop its o ( · ) term and rewrite Eq.(57) as:

Solving this differential equation, we get θ ( l ) = θ (0) / (1 + (3 π ) -1 θ (0) l ) , and

By Eq.(7), we get the following relation between ϕ L and ϕ L +1 :

Rearranging terms, we get

The right side of Eq.(39) converges to zero as L →∞ . Using Eq.(37) and lim L →∞ cos θ ( L +1) = 1 , we have

Hence, ϕ L converges to arccos 1 4 ≈ 75 . 5 ◦ , in the limit L →∞ .

E.5 Proof of Theorem 4.2

Proof. First of all, we note that in scenario (b), i.e., the network with all ReLU activation removed, the network simply becomes a linear neural network (while with the same trainable parameters W (1) as the ReLU network in scenario (a)). By the analysis in Section 2.1, we can easily see that the NTK matrix in scenario (b) is equivalent to the Gram matrix G , and κ b = κ 0 . Hence, whenever comparing the two scenarios, it suffices to compare the NTK K (and its condition number κ ) of ReLU network with the Gram matrix G (and its condition number κ 0 ).

We prove the theorem by induction.

Base case: ReLU neural network of depth L = 1 . First, consider the shallow ReLU neural network

where W are the trainable parameters.

The model gradient, for an arbitrary input x , can be written as

where δ ( x ) ∈ R 1 × m has the following expression

At initialization, W is a random matrix. Recall that the NTK K = FF T , where the gradient feature matrix F consist of the gradient feature vectors ∇ f ( x ) for all x for the dataset. Applying Lemma D.1 in the limit of m → ∞ , we have that each entry K ij is equivalent to ∑ k,l A ′ ikl A ′ jkl , with A ′ ikl := √ 2 X ik I { W l : X i : ≥ 0 } , where X ∈ R n × d is the matrix of input data. Then apply Lemma E.4, we immediately have that

Hence, we have that κ a < κ b .

In addition, note that this network has one hidden layer, and that the 'zero-hidden layer' network is just simply the linear model. For linear model, the NTK is simply the Gram matrix. Hence, for the base case, we have κ f 1 < κ f 2 = κ 0 , with network f 1 of depth 1 and network f 2 of depth 0 .

Induction hypothesis. Suppose that, for a ReLU network f L -1 of depth L -1 , its NTK condition number κ L -1 is strictly smaller than κ 0 .

Induction step. Now, let's consider the two ReLU networks f L of depth L and f L -1 . It is suffices to prove that κ L < κ L -1 . The model gradients, for any given input x , can be written as:

where

Note that the matrix W ( L ) has different dimensions for f L and f L -1 .

Using the same argument as in the base case, as well as applying Lemma D.1 when contracting the δ ( x ) 's, we directly obtain κ L < κ L -1 .

E.6 Proof of Theorem 4.3

Proof. First, consider the normalized NTK matrix 1 L +1 K . By Lemma 3.1, we have for it diagonal elements:

By Theorem 3.5, we have that, in the infinite depth limit, each of off-diagonal elements of the normalized NTK matrix converges to 1 4 . Namely,

where matrix J n has its all elements being ones. Therefore, lim L →∞ 1 L +1 K has one eigenvalue λ 1 = 1 + n 4 , and all remaining eigenvalues λ 2 = λ 3 = · · · = λ n = 3 4 . Then its condition number is κ = λ 1 λ n = 4+ n 3 .

F Relaxing the constraint on top layers

Theorem F.1. Consider a L -layer ReLU neural network f as defined in Eq.(1) in the infinite width limit m →∞ and at initialization. We compare the NTK condition numbers κ a and κ b of the two scenarios: (a) the network with the ReLU activation, and (b) the network with all the ReLU activation removed. Consider the dataset D = { ( x 1 , y 1 ) , ( x 2 , y 2 ) } with the input angle θ in between x 1 and x 2 small, θ in ≪ 1 . Then, the NTK condition number κ a < κ b . Moreover, for two ReLU neural networks f 1 of depth L 1 and f 2 of depth L 2 with L 1 > L 2 , we have κ f 1 < κ f 2 .

Proof. First, let's consider the scenario (a), i.e. the ReLU network. According to the definition of NTK and Lemma 3.1, the NTK matrix K for this dataset D = { ( x 1 , y 1 ) , ( x 2 , y 2 ) } is (NTK is normalized by the factor 1 / ( L +1) 2 ):

The eigenvalues of the NTK matrix K are given by

In the scenario (b), the ReLU activation is removed in the network, resulting in a linear neural network. In this case, the NTK is equivalent to the Gram matrix G , as given by Corollary 2.4. We have

and its eigenvalues as

̸

By Theorem 3.2, we have cos ϕ < cos θ in , when θ in ≪ 1 and θ in = 0 . Hence, we have the following relations

which immediately implies κ a < κ b .

When comparing ReLU networks with different depths, i.e., network f 1 with depth L 1 and network f 2 with depth L 2 with L 1 > L 2 , notice that in Eq.(45) the top eigenvalue λ 1 monotonically decreases in ϕ , and the bottom (smaller) eigenvalue λ 2 monotonically increases in ϕ . By the proof of Theorem 3.2, we know that the deeper ReLU network f 1 has a better separation than the shallower one f 2 , i.e., ϕ f 1 > ϕ f 2 . Hence, we get

Therefore, we obtain κ f 1 < κ f 2 . Namely the deeper ReLU network has a smaller NTK condition number.

G Technical proofs

G.1 Proof of Lemma D.1

Proof. We denote A ij as the ( i, j ) -th entry of the matrix A . Therefore, ( A T A ) ij = ∑ m k =1 A ki A kj . First we find the mean of each ( A T A ) ij . Since A ij are i.i.d. and has zero mean, we can easily see that for any index k ,

̸

Consequently,

That is E [ 1 m A T A ] = I d .

Now we consider the variance of each ( A T A ) ij . If i = j we can explicitly write,

̸

̸

In the case of i = j , then,

In the equality (a) above, we used the fact that A 2 ki ∼ χ 2 (1) . Therefore, lim m →∞ V ar ( 1 m ( A T A )) = 0 .

Now applying Chebyshev's inequality we get,

Obviously for any ϵ ≥ 0 as m →∞ , the R.H.S. goes to zero. Thus, 1 m A T A → I d × d , in probability.

G.2 Proof of Lemma E.1

Proof. Note that the random vector w is isotropically distributed and that only inner products w T v 1 and w T v 2 appear, hence we can assume without loss of generality that (if not, one can rotate the coordinate system to make it true):

In this setting, the only relevant parts of w are its first two scalar components w 1 and w 2 . Define ˜ w as

Then,

G.3 Proof of Lemma E.2

Proof. Note that the ReLU activation function σ ( z ) can be written as z I z ≥ 0 . We have,

Note that the random vector w is isotropically distributed and that only inner products w T v 1 and w T v 2 appear, hence we can assume without loss of generality that (if not, one can rotate the coordinate system to make it true):

In this setting, the only relevant parts of w are its first two scalar components w 1 and w 2 . Define ˜ w as

Then, in the limit of q →∞ ,

G.4 Proof of Lemma E.3

Proof.

In the step (a) above, we used the fact that U is independent of W , v 1 and v 2 . In the step (b) above, we applied Lemma E.1, and used the fact that E u ∼N (0 ,I s × s ) [ uu T ] = I s × s .

G.5 Proof of Lemma E.4

Proof. Starting from the definition of the smallest eigenvalue, we have that λ min ( B ′ ) satisfies

̸

̸

In the inequality ( a ) above, we made the following treatment: for each fixed l , we consider u i I { W l : A i : ≥ 0 } as the i -th component of a vector u ′ l ; by definition, the minimum eigenvalue of matrix B = AA T

̸

̸

moreover, this ≤ inequality becomes equality, if and only if all u ′ j are the same and equal to arg min u ′ =0 ( u ′ ) T G u ′ / ∥ u ′ ∥ 2 . It is easy to see, when the dataset is not degenerate, for different j , u ′ j are different, hence only the strict inequality < holds in step ( a ) .

̸

̸

Continuing from Eq.(51), we have

̸

Therefore, we have that λ min ( B ′ ) > λ min ( B ) .

As for the largest eigenvalue λ max ( B ′ ) , we can apply the same logic above for λ min ( K ) (except replacing the min operator by max and have < in step ( a ) ) to get λ max ( B ′ ) < λ max ( B ) .

G.6 Proof of Lemma E.5

Proof. Consider an arbitrary layer l ∈ [ L ] of the ReLU neural network f at initialization. Given two arbitrary network inputs x , z ∈ R d , the inputs to the l -th layer are α ( l -1) ( x )) and α ( l -1) ( z )) , respectively.

By definition, we have

with entries of W ( l ) being i.i.d. drawn from N (0 , 1) . Recall that, by definition, the angle between α ( l -1) ( x )) and α ( l -1) ( z )) is θ ( l -1) ( x , z ) . Applying Lemma E.2, we immediately have the inner product

In the special case of x = z , we have θ ( l -1) ( x , z ) = 0 , and obtain from the above equation that

Apply Eq.(55) back to Eq.(54), we also get

That is θ ( l ) ( x , z ) = g ( θ ( l -1) ( x , z )) . Recursively apply this relation, we obtain the desired result.

G.7 Proof of Lemma E.6

Proof. By Lemma E.5, we have that

Noting that the Taylor expansion of the cos function at zero is cos z = 1 -1 2 z 2 + o ( z 3 ) , one can easily check that, for all l ∈ [ L ] ,

̸

̸

Note that θ ( l ) ( x , z ) ≤ θ ( l -1) ( x , z ) = o (1 /L ) . Iteratively apply the above equation, one gets, for all l ∈ [ L ] , if θ (0) ( x , z ) = o (1 /L ) ,

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors investigate the advantages of using ReLU activation in the Neural Tangent Kernel (NTK) regime of fully connected neural networks. They demonstrate that ReLU improves data separation in the gradient\nfeature space, leading to better conditioning of the NTK compared to linear activations. This improvement becomes more pronounced as network depth increases. Their findings are supported by experiments on both synthetic and standard benchmark datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThe authors propose a new direction for investigation, focusing on characterizing the angles between data points in the NTK feature space. Additionally, they examine the spectral properties of the NTK, with particular attention to its condition number. Their theoretical insights are supported by a thorough experimental evaluation, showing agreement between empirical findings and theoretical predictions.\n\nWeaknesses\n\nThe main limitation of the paper lies in the narrow scope of the comparison. Specifically, the focus is on ReLU versus linear networks. The theoretical analysis emphasizes that ReLU networks yield greater angle\nseparation for similar data points in the NTK feature space. However, the impact of non-linear activations in neural networks is already well established, making this result less surprising. A more compelling \ndirection would have been to explore and compare a broader range of activation functions.\n\nIn this context, the setup of the trainable networks is rather restricted and lacks practical applicability. Specifically, in Theorem 2.2 (linear network), the feature space is attributed to the gradient at initialization.\nIn Theorem 4.2 (non-linear network), only the first layer is trainable, while all subsequent layers remain fixed.\n\nThroughout the manuscript, the intuition underlying the separation property is not clearly expressed. In fact, for similar data points, increased separation may be undesired, especially when the labels are the same. Yet, the manuscript does not explore how this separation behaviour is influenced by the labelling of the data.\n\nThe spectral properties of the NTK have been extensively studied in prior work. This manuscript focuses specifically on the condition number of the NTK, which is closely tied to its smallest eigenvalue. However, the comparison is again limited to the simplistic case of linear networks. It remains unclear what the practical implications of this comparison are. How does this insight benefit neural network training or inference? \n\nMoreover, while the manuscript claims that the condition number is small (line 57), it appears to be quite large when applied to larger datasets.\n\nProposition 4.1 has limited applicability, as it is demonstrated only in the context of small datasets where the number of samples n is less than the dimensionality d.\n\nQUESTIONS:\nSee above.\n\nLIMITATIONS:\n1. What are the practical implications of your theoretical findings?\n2. Could you clarify how the separation property of the data points relates to their corresponding labels?\n3. Figure 2 provides numerical evidence that the separation phenomenon also holds for various non-linear activation functions. However, the behaviour differs across ReLU², GeLU, and Tanh. Can your theoretical framework be extended to accommodate other non-linear activations and account for the observed differences?", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper reveals that non-linear activation functions not only enhance expressivity but also improve feature separation and NTK conditioning in wide neural networks. These effects are amplified with network depth, facilitating convergence of gradient-based methods. In contrast, linear networks lack these advantages, preserving the original data geometry and exhibiting poorly conditioned NTKs.  While the theoretical analysis focuses on ReLU, empirical results with other activations support the claims.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1 This paper is technically sound, The claims are well supported by theoretical analysis on ReLU and by empirical observations on other activations.\n\n2 The paper is well-written and clearly presented, making its theoretical insights and empirical findings easy to follow.\n\n3 The results are important for understanding the role of non-linear activations. The paper provides a comprehensive theoretical analysis demonstrating that the ReLU activation improves  feature separation and NTK conditioning. To the best of my knowledge, these findings are new.\n\nWeaknesses\n\n1 The title should be specific to ReLU activations as the main contribution of this paper is the theoretical findings, which is only conducted on ReLU.\n\n2 This paper demonstrates that as depth increases, the condition number of the NTK matrix with ReLU improves. However, many previous works [1] [2] show that as depth increases, the NTK degenerates. That is, the infinite depth limit of the NTK regime is trivial.\n\n3 To be honest, I acknowledge that the findings of this paper are novel, but not entirely surprising. While the theoretical analysis provides valuable insights, it is limited to ReLU activations. It seems that extending the results to other widely used activation functions may involve technical challenges. If the analysis could be extended to arbitrary non-linear activations, it could offer valuable guidance for the design of new activation functions.\n\n[1] Hayou S, Doucet A, Rousseau J. The curse of depth in kernel regime.\n\n[2] Bietti A, Bach F. Deep equals shallow for ReLU networks in kernel regimes.\n\nQUESTIONS:\nWhat are the specific technical challenges in extending the theoretical analysis to other activation functions? Is it because, for most non-linear activations, it is difficult to obtain a closed-form expression for the entries of the corresponding NTK? If so, have you considered analyzing the Hermite coefficients? Perhaps the higher-order coefficients could be bounded or even omitted.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

comments: SUMMARY:
This paper presents simple results on the tangent feature map of deep neural networks: (1) non-linear activations cause the angle in tangent feature space to be larger than input angles, and (2) this effect is amplified with depth. Importantly, this effect is not observed in linear networks. A natural consequence of this increased angle is improved conditioning of the NTK. This phenomenon is also empirically verified on datasets using finite width neural networks.

STRENGTHS AND WEAKNESSES:
Strengths:

1. The result is simple and presented in a straightforward and easy to follow manner.
2. It provides a novel perspective and concrete explanation (as far as this reviewer is aware) for the utility of non-linear activation functions. 
3. Theoretical results seem sound and are numerically verified as well.

Weaknesses:

1. The theory depends on infinite width of the network, and is restricted to ReLU activations.
2. The section on optimization acceleration could benefit from more quantitative analysis. For example, a quantitative convergence improvement as a function of depth, informed by the main result would be insightful. Especially if this quantitative analysis is corroborated by the increasing depth experiment that is presented. Currently, the faster convergence for higher depths is qualitative and does not build on the main result in a theoretical capacity.
3. Some of the writing could be slightly improved

QUESTIONS:
1. Can the authors comment on finite width effects?
2. Further, instead of increasing depth, could similar results be derived as a function of width for a fixed depth?

LIMITATIONS:
Yes, the authors have addressed limitations.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the effect of non-linear activation functions on the NTK and its associated feature map $x \\mapsto \\nabla_\\theta f_\\theta (x)$ in infinite-width neural networks. \nBased on the fact that feature norms are (virtually) unchanged by linear and ReLU networks, the authors focus on the study of the separation angle between data points and between their features. \n\nThey show that using linear activation functions results in a Gram matrix which is (virtually) unchanged across the hidden layers: the angles between data do not change and the associated NTK has a condition number that corresponds to the Gram matrix of the original data. Conversly, the use of activation functions increases the angles between data with small angles, and improves the NTK condition number. \n\nMore specifically, for ReLU networks, the paper provides a recursion relation for the angle between data points. This recursion allows the authors to study the infinite-width and infinite-depth networks, showing that: \n- All data pairs have an angle that converges to $\\sim 75,5$ degrees. \n- When the data lie on a sphere, the condition number will converge to $\\frac{n+4}{3}$. \n- The condition number decreases with the depth of the network. \n\nThe connection between the condition number and the training speed allows them to discuss implications for convergence speed.  Various experiments support these theoretical findings.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- The paper is clearly written and easy to follow, with an equality well-written and well-organized appendix. \n- It provides a fairly comprehensive study of the effect of ReLU activation function  on the NTK with no bias terms. A clear study of these neural network which lies at the boundary between the order and freeze phase was to my knowledge lacking. Its investigates both the separation angles and the features norms, provides recursive relations on angles, analyzes limits and variation of the condition number. The proofs feature interesting random matrices and geometric flavours, taking full advantage of the specific activation function they consider, the ReLU. I particularly like the result concerning the limiting angle,  and whose value is non-trivial (i.e. neither 0 nor 90 degrees).  I think this is one of the main new contributions of the paper, along with the decreasing property of the condition number and its explicit limiting value. The paper thus provides interesting quantitative results on top of the qualitative results. \n- Empicial results show that these theoretical insights can be extended to other activation functions. \n\n\n\nWeaknesses: \n- Link with previous literature is incomplete: \n    - Reference [1] shows that a transition occurs for the NTK between a “freeze” (order) phase, where the scaled NTK converges to a constant, and a “chaos” phase where the NTK looks more like a Dirac Kernel. It was shown that, in term of their characteristic value, ReLU networks without bias lie on the boundary between order and chaos. The current paper extends the study of this phenomena by looking specifically at what happens at the boundary, but does not explicitly mention this connection. \n    - I believe that the linear NTK has been studied before or is classical knowledge (it results directly from the recurrence relation in Theorem 1 of [2]). Check if it was not provided somewhere else.  \n    - The improved conditioning property (though without quantitative results provided in the paper) between linear and non-linear activation were already known. Specifically, the linear NTK has infinite condition number when the number of data points exceeds the dimension. Whereas, as shown in  Appendix A.4 [2], the NTK restricted to the sphere is positive-definite as soon as the activation function is non-polynomial. \n    - Could you discuss also relation with [3]. \n\n- It would be interesting to explicitly discuss the role of bias, as they can significantly affect the behaviour of the NTK. \n- The paper focuses exclusively on the speed of convergence, and does not discuss the generalization performance, despite a trade-off between these two effects. Empirical results exploring generalisation behaviour would be give valuable additional insights.\n\n\n\n[1] - Freeze and Chaos: NTK views on DNN Normalization, Checkerboard and Boundary Artifacts, Jacot, Gabriel, Ged, Hongler (https://proceedings.mlr.press/v190/jacot22a/jacot22a.pdf)\n\n[2] - Neural Tangent Kernel: Convergence and Generalization in Neural Networks, Jacot, Gabriel, Hongler, ArXiv Version https://arxiv.org/pdf/1806.07572\n\n[3] - Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension Karhadkar, Murray, Montúfar https://arxiv.org/abs/2405.14630\n\nQUESTIONS:\n- Fast convergence  can occur with a decrease of generalization; for instance, a NTK Gram matrix equal to the identity matrix would lead to fast convergence but no generalization capability. Could you discuss this trade-off explicitly? Specifically, how does generalization behave in Figure 4 (the experiments that illustrate the increase of training speed) ?\n- Relatedly, the experiments results provide an interesting fact that is not discussed in the paper: in the ReLU setting, angles converge to around 75.5 degrees, whereas with other activation functions used in Figure 2, it seems that the angles converges to around 90 degrees, and this could potentially reduce generalization properties. Could you provide the generalisation for these activation function in the setup of Figure 2 and discuss about this depending of what you obtain? Also can you report the limiting angles somewhere?  \n- In corollary 3.3, I suggest you could explicitly mention an insightfull consequence: $L$ should be of order $\\frac{1}{\\theta_{in}}$ for $\\Phi(x,z)$ to be non trivial. Do you agree with this statement? \n- Proof clarifications: \n    - Isn’t Lemma 4.1 a consequence of the Law of Large Numbers, given that each coordinates is the average of m independent variables? \n    - While the result is correct, and equation (40) captures the intuition, I believe it’s inaccurate to claim “solving the above equation (40), we found that …”. One should keep all approximations orders and prove that the l.h.s. is of order $\\frac{1}{L^2}$, thus implying that $1-4\\cos(\\Phi_L) \\to 0$ as $L\\to\\infty$. Do you agree ?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper discusses properties of neural tangent kernels and the effects of non-linear activation functions compared to linear activations. The field of discussion lies in the application to hyper-wide neural networks and examines the relation between network depth and width and the angular separation of internal representations in model gradient space. The key findings suggest that increasing the network depth significantly enhances the feature separation observed in wide models with non-linear activations. In particular, the authors show that in the infinite-width-then-depth limit, all data are equally separated with an angle ∼75.5 degrees.\n\nThe authors support their claims with detailed mathematical analysis under standard assumptions for wide network limits (e.g., random initialization, asymptotic regimes). They also validate these insights empirically using synthetic and real datasets like MNIST and LibriSpeech. Their findings offer a new perspective on why non-linear activations are crucial for effective learning in deep models.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper presents solid theoretical analysis with lemmas and theorems grounded in random matrix theory and kernel methods. Experimental results align well with theoretical predictions and are reproducible via detailed descriptions in Section 4.1.\nThe paper is well-written, logically structured, and accessible to readers familiar with NTK and wide network theory. The claims made in the abstract and introduction match the contributions in the body of the paper.\n\nThe insights into how non-linearities improve optimization dynamics via NTK conditioning and representation learning via feature separation are valuable for understanding modern current DL systems. While not  proposing a novel method, it offers interpretability tool and define specific numerical limits for parameter spaces.\n\nFocus solely on ReLU activations without considering a broad array of possible functions around may miss some insights.\n\nQUESTIONS:\nWhile ReLU is studied in depth, can the theoretical framework be extended to other activation functions and their derivatives? Would the conclusions (and numerical limits) still hold? If not, what aspects of the analysis break down?\n\nWhat are the practical takeaways for model design or training procedures, how your findings can be used in certain settings?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}