Title: Optimal Rates  for Generalization of Gradient Descent for Deep ReLU Classification
Venue: NeurIPS 2025
Accepted: yes
Keywords: statistical learning theory

generlization analysis

SGD

neural tangent kernel

optimal rates
Abstract: Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of $O(1/\sqrt{n})$, or focus on networks with smooth activation functions, incurring exponential dependence on network depth $L$. In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off optimization and generalization errors, achieving only polynomial dependence on depth. Specifically, under the assumption that the data are NTK separable from the margin $\gamma$, we prove an excess risk rate  of $\widetilde{O}(L^4 (1 + \gamma L^2) / (n \gamma^2))$, which aligns with the optimal SVM-type rate $\widetilde{O}(1 / (n \gamma^2))$ up to depth-dependent factors.
A key technical contribution is our novel control of activation patterns near a reference model, enabling a sharper Rademacher complexity bound for deep ReLU networks trained with gradient descent.

Optimal Rates for Generalization of Gradient Descent for Deep ReLU Classification

Abstract

Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of O (1 / √ n ) , or focus on networks with smooth activation functions, incurring exponential dependence on network depth L . In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off optimization and generalization errors, achieving only polynomial dependence on depth. Specifically, under the assumption that the data are NTK separable from the margin γ , we prove an excess risk rate of ˜ O ( L 6 / ( nγ 2 )) , which aligns with the optimal SVM-type rate ˜ O (1 / ( nγ 2 )) up to depth-dependent factors. A key technical contribution is our novel control of activation patterns near a reference model, enabling a sharper Rademacher complexity bound for deep ReLU networks trained with gradient descent.

1 Introduction

Deep neural networks trained through first-order optimization methods have achieved remarkable empirical success in diverse domains [Krizhevsky et al., 2012]. Despite their widespread adoption, a rigorous theoretical understanding of their optimization dynamics and generalization behavior remains incomplete, particularly for ReLU networks. The inherent challenges arise from the nonsmoothness and non-convexity of the loss landscape induced by ReLU activations and network architectures, which complicate the classical analysis. Intriguingly, empirical evidence demonstrates that over-parameterized models often achieve zero training error but still generalize well even in the absence of explicit regularization [Zhang et al., 2016]. This phenomenon has spurred significant theoretical research to understand its underlying mechanisms.

A prominent line of research uses the neural tangent kernel (NTK) framework to analyze neural network training [Jacot et al., 2018]. In the infinite-width limit, gradient descent (GD) dynamics can be characterized by functions in the NTK's reproducing kernel Hilbert space (RKHS), with convergence guarantees established for both shallow and deep networks [Du et al., 2019b,a]. These results demonstrate that GD converges to global minima within a local neighborhood of initialization,

provided that the network width is sufficiently large. In particular, the appealing work [Arora et al., 2019a] showed that, if the network width m = ˜ O (poly( n, 1 /λ 0 , L )) , then the generalization error is of the order L √ y ⊤ ( H ∞ ) -1 y n , where H ∞ denotes the NTK gram matrix over the training data and λ 0 = λ min ( H ∞ ) . However, the assumption λ 0 > 0 is a strong assumption because it tends to zero if the size of the training data tends to infinity as shown by Su and Yang [2019]. Ji and Telgarsky [2020] achieved logarithmic width requirements for NTK-separable data with a margin γ . They derived the risk bound of order 1 / ( γ 2 √ n ) for two-layer ReLU networks using Rademacher complexity. The recent work Chen et al. [2021] extended their results from shallow to deep neural networks, the authors considered the NTK feature learning and proved the bound of ˜ O ( e O ( L ) γ √ m n ) . Recently, Lei et al. [2026] derived the bound of 1 / ( γ 2 n ) for two-layer ReLU networks. However, all the above bounds explicit suboptimal 1 √ n dependence on the sample size n or only focus on shallow networks.

Complementing the NTK framework, another line of research employs algorithmic stability to analyze neural networks. In particular, Liu et al. [2020] demonstrated that the Hessian spectral norm scales with width m as ˜ O ( 1 √ m ) , providing the theoretical basis for subsequent studies on generalization in overparameterized networks [Richards and Kuzborskij, 2021, Lei et al., 2022, Taheri and Thrampoulidis, 2024, Taheri et al., 2025]. The work [Taheri and Thrampoulidis, 2024] achieves the bound of ˜ O (1 /nγ 2 ) for shallow neural networks, which is almost optimal, as illustrated by Shamir [2021], Schliserman and Koren [2023]. More recently, Taheri et al. [2025] extended this line of work to deep networks, obtaining a generalization bound of ˜ O ( e O ( L ) / ( nγ 2 )) . However, this approach often assume smooth activation functions and can not apply to the non-smooth ReLU networks. In summary, these works either consider smooth neural networks or develop generalization bounds with exponential dependency on L . These limitations motivate two fundamental questions:

Can we develop optimal risk bounds of 1 / ( γ 2 n ) for deep ReLU networks through refined Rademacher complexity analysis? Furthermore, is it possible to replace the exponential dependence on L with poly( L ) by using neural networks with width as a polylogarithmic function of n ?

In this paper, we provide affirmative answers to both questions. Our main contributions are listed below.

We prove that gradient decent with step size η and T iterations achieves the convergence rate of F S ( W ) / ( ηT ) , where F S ( W ) = 3 ηT L S ( W ) + ∥ W -W (0) ∥ 2 F , W is a reference model, W (0) is the initialization point and L S ( · ) is the training error. This indicates that all iterates remain within a neighborhood of W . By refining the analysis of ReLU activation patterns, we reduce the overparameterization requirement by a factor of L 6 as compared to Chen et al. [2021] (see Remark 1 ).

We establish a population risk bound of ˜ O ( L 4 F ( W ) /n ) , where F ( W ) extends the empirical counterpart F S ( W ) to the population setting. Our analysis introduces two key technical contributions. First, we derive sharper Rademacher complexity bounds for the hypothesis class encompassing all gradient descent iterates. A central innovation is our representation of the

complexity via products of sparse matrices, whose norms are tightly controlled using optimizationinformed estimates (see Remark 2). Second, by leveraging the covering number techniques, we prove that ReLU networks remain ˜ O ( L 2 ) -Lipschitz continuous in a neighborhood around the initialization-a substantial improvement over previous exponential bounds [Xu and Zhu, 2024, Taheri et al., 2025] (see Remark 3).

For NTK separable data with a margin γ , we show that neural networks with m = ˜ O ( L 16 log( n/δ )(log n ) 8 /γ 8 ) can achieve ˜ O ( L 6 / ( γ 2 n )) risk. This improves on Chen et al. [2021]'s ˜ O ( e O ( L ) γ -1 √ m/n ) by simultaneously (a) removing the exponential depth dependence, (b) eliminating the √ m width factor and (c) achieving the optimal dependence on sample complexity (see Table 1 for a comparison with existing work).

2 Related Works

2.1 Optimization

The foundational work of Jacot et al. [2018] introduced the Neural Tangent Kernel (NTK) framework, demonstrating that in the infinite-width limit, neural networks behave as linear models with a fixed tangent kernel [Liu et al., 2020, Lee et al., 2019]. This lazy training regime [Chizat et al., 2019], where parameters remain close to initialization, enables gradient descent to converge to global optima near initialization [Du et al., 2019a, Arora et al., 2019a]. These analyses showed that the training dynamics can be governed by the NTK Gram matrix, which leads to substantial overparameterization ( m ≳ n 6 /λ 4 0 ). This was later improved by Oymak and Soltanolkotabi [2020]. They showed that if the square-root of the number of the network parameters exceeds the size of the training data, randomly initialized gradient descent converges at a geometric rate to a nearby global optima. The work (Ji and Telgarsky [2020]) achieved polylogarithmic width requirements for logistic loss by leveraging the 1-homogeneity of two-layer ReLU networks. However, it should be mentioned that this special property does not hold for deep networks. The NTK framework was extended to deep architectures by Arora et al. [2019b] for CNNs and by Du et al. [2019b] for ResNets using the last-layer NTK. Xu and Zhu [2024] pointed out that such a characterization is loose, only capturing the contribution from the last layer. They further gave the uniform convergence of all layers as m →∞ and convergence guarantee for stochastic gradient descent (SGD) in streaming data setting. Allen-Zhu et al. [2019] showed that the optimization landscape is almost-convex and semi-smooth, based on which they proved that SGD can find global minima. Cao and Gu [2019] introduced the neural tangent random feature and showed the convergence of SGD under the overparameterized assumption m ≳ n 7 .

2.2 Generalization

The NTK framework has yielded generalization bounds scaling as √ y ⊤ ( H ∞ ) -1 y /n [Arora et al., 2019a, Cao and Gu, 2019]. This data-dependent complexity measure helps to distinguish between random labels and true labels. Li and Liang [2018] showed that SGD trained networks can achieve small test error on specific structured data. A very popular approach to studying the generalization of neural networks is via the uniform convergence, which analyzes generalization gaps in a hypothesis space using tools such as Rademacher complexity or covering numbers [Neyshabur et al., 2015, Bartlett et al., 2017, Golowich et al., 2018, Liu et al., 2024]. However, this could lead to vacuous generalization bound in some cases [Nagarajan and Kolter, 2019]. Moreover, these bounds typically exhibit exponential dependence on depth L , thus often leading to loose bounds [Chen et al., 2021]. This capacity-based method usually results in the generalization rate of the order ˜ O (1 / √ n ) . Recent work has also exploited stability arguments for generalization guarantees [Richards and Kuzborskij, 2021, Lei et al., 2022, Taheri and Thrampoulidis, 2024, Deora et al., 2023, Taheri et al., 2025]. The main idea of algorithmic stability is to study how the perturbation of training samples would affect the output of an algorithm [Rogers and Wagner, 1978]. The connection to generalization bound was established in Bousquet and Elisseeff [2002]. Hardt et al. [2016] gave the stability analysis of SGD for Lipschitz, smooth and convex problems. Lei and Ying [2020] further studied SGD under much wilder assumptions. Liu et al. [2020] identified weak convexity of neural networks, enabling stability analyses with polynomial width requirements for quadratic loss [Richards and Kuzborskij, 2021, Lei et al., 2022]. Moreover, Taheri and Thrampoulidis [2024], Taheri et al. [2025] obtained generalization bounds of order ˜ O (1 /n ) by using a generalized local quasi-convexity property for

sufficiently parameterized networks. However, these methods depend on smooth activations, and whether similar or even better bound can be established for deep ReLU networks is still unknown. The recent work derived excess risk bounds of order ˜ O (1 /n ) for shallow ReLU networks [Lei et al., 2026].

3 Preliminaries

Notation Throughout the paper, we denote a ≲ b if there exists a constant c > 0 such that a ≤ cb , and denote a ≍ b if both a ≲ b and b ≲ a hold. We use the standard notation O ( · ) , Ω( · ) and use ˜ O ( · ) , ˜ Ω( · ) to hide polylogarithmic factors. Denote by I {·} the indicator function (i.e., taking the value 1 if the argument holds true, and 0 otherwise). We use N ( µ, σ 2 ) to denote the Gaussian distribution of mean µ and variance σ 2 . For a positive integer n , we denote [ n ] := { 1 , . . . , n } . For a vector x ∈ R d , we use ∥ x ∥ 2 to denote its Euclidean norm. For a matrix A ∈ R m × n , we denote ∥ A ∥ 2 and ∥ A ∥ F the corresponding spectral norm and Frobenius norm respectively. The (2 , 1) -norm of A is defined as ∥ A ∥ 2 , 1 = ∑ n j =1 ∥ A : j ∥ 2 . Let ⟨· , ·⟩ be the inner product of a vector or a matrix, i.e., for any matrices A,B ∈ R m × n , we have ∥ A ∥ 2 F = tr ( A ⊤ A ) and ⟨ A,B ⟩ = tr ( A ⊤ B ) . Let L ∈ N , A = ( A 1 , . . . , A L ) and B = ( B 1 , . . . , B L ) be two collections of arbitrary matrices such that A i and B i have the same size for all i ∈ [ L ] . We define ⟨ A , B ⟩ = ∑ L i =1 tr ( A ⊤ i B i ) . Denote ∥ A ∥ 2 , ∞ = max l ∥ A l ∥ 2 . For a matrix W , we define ( w r ) ⊤ the r -th row of W . Denote ∥ · ∥ 0 the l 0 -norm which is the number of nonzero entries of a matrix or a vector. We denote C ≥ 1 as an absolute value, which may differ from line to line.

Let X ⊆ R d be the input space, Y = { 1 , -1 } be the output space, and Z = X × Y . Let ρ be a probability measure defined on Z . Let S = { z i = ( x i , y i ) : i = 1 , . . . , n } be a training dataset drawn from ρ . Let W := R m × d × ( R m × m ) L -1 be the parameter space. W 1 ∈ R m × d and W l ∈ R m × m for l = 2 , . . . , L is the weight of the l -th hidden layer. W = ( W 1 , . . . , W L ) ∈ W denotes the collection of weight matrices for all layers. Let a = ( a 1 , . . . , a m ) ⊤ ∈ R m be the weight vector of the output layer and σ ( · ) = max {· , 0 } denote the ReLU activation function. For x ∈ X , we consider the L -layer deep ReLU neural networks with width m as follows,

Given an input x ∈ X and parameter matrix W = ( W 1 , · · · , W L ) of an L -layer ReLU network f W ( x ) . We denote the output of the l -th layer by h l ( x ) = √ 2 m σ ( W l h l -1 ( x )) with h 0 ( x ) = x . Then f W ( x ) = a ⊤ h L ( x ) . We define B R ( W ) = { ˜ W ∈ W : max l ∥ W -˜ W l ∥ 2 ≤ R } . The performance of the network f W ( x ) is measured by the following empirical risk L S ( W ) and population risk L ( W ) , respectively:

where we use logistic loss ℓ ( z ) := log(1 + exp( -z )) throughout this paper. We further assume the following symmetric initialization [Nitanda and Suzuki, 2020, Kuzborskij and Szepesvári, 2023, Xu and Zhu, 2024]:

Assumption 1 (Symmetric initialization) . Without loss of generality, we assume the network width m is even, and a r + m 2 = -a r ∈ {-1 , +1 } for 1 ≤ r ≤ m/ 2 . W (0) ∈ W satisfies

We remark that this initialization is for theoretical simplicity, using general initialization techniques will not affect the main results. We fix the output weights { a r } and use Gradient Descent (GD) to train the weight matrix W [Ji and Telgarsky, 2020, Arora et al., 2019a, Zou et al., 2018].

Definition 1 (Gradient Descent) . GD updates { W ( k ) } by

where η > 0 is the step size.

Note that in each layer we employ √ 2 /m as the regularization factor instead of the conventional √ 1 /m [Ji and Telgarsky, 2020], which is due to E x ∼N (0 , 1) σ 2 ( x ) = 1 / 2 for our activation function σ ( · ) . This scaling matches both the theoretical framework of Du et al. [2019a] and the initialization scheme of [He et al., 2015] (where weights w l r ∼ N (0 , 2 /m ) ). As will be shown later (Appendix A), this regularization ensures stable gradient propagation and maintains consistent variance across layers.

The following assumption is standard in the literature [Cao and Gu, 2019, Ji and Telgarsky, 2020, Chen et al., 2021].

Assumption 2. We assume X = S d -1 be the sphere.

Throughout the paper, we assume that Assumptions 1 and 2 always hold true.

Error decomposition In this work, we analyze the performance of gradient descent through the lens of population risk. To facilitate this analysis, we decompose the population risk L ( W ( T )) as follows

where the first term captures the generalization gap, quantifying the network's performance on unseen data. The second term represents the optimization error, which reflects GD's ability to find global minima. We will use tools in the optimization theory to study the optimization error [Ji and Telgarsky, 2020, Schliserman and Koren, 2022], and Rademacher complexity to control the generalization gap [Mohri et al., 2018].

4 Main Results

In this section, we present the main results. In Section 4.1, we show the optimization analysis of gradient descent. In Section 4.2, we use Rademacher complexity to control the generalization gap. In Section 4.3, we apply our generalization results to NTK-separable data with a margin γ .

4.1 Optimization Analysis

We introduce the following notations for a reference model W

Without loss of generality, we assume F S ( W ) ≥ 1 .

Theorem 1. Let Assumptions 1, 2 hold. If m ≳ L 16 (log m ) 4 log( nL/δ ) F 4 S ( W ) , η ≤ min { 4 / (5 L ) , 1 / (20 L ˜ F S ( W )) } , then with probability at least 1 -δ , for all t ≤ T we have

Remark 1 . Our theorem shows that the convergence rate is bounded by the optimization error of a reference model, implying that any low-loss reference point guarantees good convergence. While prior works relied on NTK-induced solutions [Richards and Kuzborskij, 2021, Arora et al., 2019a], we prove that there exists a reference model near initialization under the milder Assumption 3. Furthermore, our analysis implies that all training iterates remain within a neighborhood of the reference point, and thus near initialization, aligned with previous observations but without studying the kernel or the corresponding Gram matrix directly [Du et al., 2019a,b].

Here we provide the proof sketch and compare it with previous works. The starting point is to show deep ReLU networks admit almost convexity ( Lemma 19 ):

for W , W ∈ B R ( W (0)) .

Then we can show all the iterates remain in B 2 √ F S ( W ) ( W (0)) and the following inequality holds (Lemma 21),

Telescoping gives the theorem. Chen et al. [2021] introduce the following neural tangent random feature (NTRF) function class:

They show that gradient descent achieves a training loss of at most 3 ϵ NTRF , where ϵ NTRF denotes the minimal loss over the NTRF function class (see Theorem 3.3 therein). In contrast, our approach directly analyzes the GD iterates and shows that the existence of a nearby reference point with small training error is sufficient to ensure convergence. For a fair comparison, under Assumption 3, both analyses yield an optimization error of ˜ O (1 /T ) . However, our method significantly relaxes the overparameterization requirement, improving the width dependence by a factor of L 6 (see Remark 5).

4.2 Generalization Analysis

We use Rademacher complexity to study the generalization gap, which measures the ability of a function class to correlate random noises.

Definition 2 (Rademacher complexity) . Let F be a class of real-valued functions over a space X , S 1 = { x 1 , · · · , x n } ⊂ X . We define the following empirical Rademacher complexity as

where ϵ = ( ϵ i ) i ∈ [ n ] ∼ {± 1 } n are independent Rademacher variables, i.e., taking values in {± 1 } with the same probability.

We further define the following worst-case Rademacher complexity,

We define G = sup z ℓ ( yf W ( x )) , and

We consider the following function space

where the parameter space is defined as

Here we use F ( W ) instead of F S ( W ) to get a data-independent hypothesis space. We will show F ( W ) is an upper bound of F S ( W ) with high probability. According to Theorem 1, all the iterations fall into W 1 with high probability. We use the following lemma to relate the generalization gap of smooth loss function with Rademacher complexity.

Lemma 1 (Srebro et al. [2010]) . Let G ′ = sup z, W ∈W 1 ℓ ( yf W ( x )) . For any 0 < δ < 1 , we have with probability at least 1 -δ/ 2 over S , for any W ∈ W 1 ,

Now we need to control R S 1 ,n ( F ) and G ′ . As will be shown in Lemma 22, with high probability there holds

To estimate G ′ , we employ covering numbers to derive a uniform upper bound of f W ( x ) -f W ( x ) . Then we use the smoothness of ℓ to show that for all G ′ -2 G ≲ L 4 log mF ( W ) . Plugging these bounds into Lemma 1 gives the generalization gap. Combined with Theorem 1, we derive the following excess risk error. The full proofs are provided in Appendix C.

Theorem 2. Let Assumptions 1, 2 hold. If m ≳ L 16 d (log m ) 5 log( nL/δ ) F 4 S ( W ) , η ≤ min { 4 / (5 L ) , 1 / (20 L ˜ F S ( W )) } , ηT ≍ n , then with probability at least 1 -δ , we have

Remark 2 (Improved Rademacher complexity) . In previous work [Chen et al., 2021], they derived the bound of ˜ O ( 4 L L √ mF ( W ) /n ) . Specifically, they use the generalization analysis in Bartlett et al. [2017], which requires to estimate the following term

Note that ∏ L l =1 ∥ W l ∥ 2 could lead to exponential dependence on the depth. Moreover, ∥ ( W l ) ⊤ -( W l ) ⊤ ∥ 2 , 1 ≤ √ m ∥ W l -W l ∥ F , inducing an explicit √ m term. However, we reduce the dependence of L to polynomial ( L 2 ). Furthermore, we sharpen the dependence on width from √ m to logarithmic terms. The key idea is to introduce the following expression

where ̂ G l L, 0 ( x i ) is a matrix defined in (36), h l 0 is the output of l -th layer of f W (0) . We further show that ̂ G l L, 0 ( x i ) is of the order ˜ O ( L/ √ m ) with high probability, from which we can derive R S 1 ,n ( F ) = ˜ O ( L 2 / √ n ) . We will show that this improved Rademacher complexity is crucial to get almost optimal risk bounds in NTK separable data.

Remark 3 (Analysis of Lipschitzness) . To bound the term G ′ = sup z, W ∈W 1 ℓ ( yf W ( x )) , we analyze the difference f W ( x ) -f W ( x ) . Since both W , W ∈ B R ( W (0)) for some R , we only need to study the local variation f W ( x ) -f W (0) ( x ) . This approach necessitates characterizing the uniform behavior of deep networks in B R ( W (0)) , specifically establishing control over their Lipschitz constants near initialization. Existing works usually lead to an exponential dependence on L [Xu and Zhu, 2024, Taheri et al., 2025], thus resulting in a e O ( L ) term in the generalization bound. In particular, Lemma F.3 and F.5 in Liu et al. [2020] pointed out that ∥ h l ( x ) ∥ ≤ C L , ∥ ∂f W ( x ) /∂h l ( x ) ∥ 2 ≤ C L -l +1 √ m . Based on these observations, Taheri et al. [2025] showed that

On the other hand, to analyze the output difference near initialization, we observe that

reducing our task to bounding the hidden layer perturbation. Previous approaches, including Xu and Zhu [2024] and Du et al. [2019b], employ a recursive estimation:

where in the second inequality they used ∥ h l 0 ( x ) ∥ 2 ≤ C L and ∥ W l (0) ∥ 2 ≲ √ m . Although this method provides a straightforward bound, it leads to an exponential dependence on depth L due to the recursive nature of the estimation.

In contrast to previous work, we develop the covering-number strategy to avoid the exponential dependence on depth. Specifically, we first show that for any finite set of size N : K = { x 1 , · · · , x N } , if m = ˜ Ω( L 10 log( N ) R 2 ) , then ∥ h l ( x i ) -h l 0 ( x i ) ∥ 2 = ˜ O ( L 2 R √ m ) holds for i ∈ [ N ] , l ∈ [ L ] (Lemma 15). We further take a 1 / ( C L √ m ) -covering D = { x j : j = 1 , . . . , | D |} of the input space. Recall that the input space X = S d -1 , it is well known from Corollary 4.2.13 in Vershynin [2018] that the number of 1 / ( C L √ m ) -covering is given by | D | ≤ (1 + 2 C L √ m ) d . Applying Lemma 15 to D derives that if m = ˜ Ω( L 10 log( | D | ) R 2 ) , then

Note that although the covering number could be exponential in L , we only require logarithm of it, thus leading to polynomial dependence. For any input x , we use the closest cover point x j ∈ D to approximate ∥ h l ( x ) -h l ( x j ) ∥ 2 , ∥ h l 0 ( x ) -h l 0 ( x j ) ∥ 2 . Combining these yields the key technical lemma (Lemma 16):

This implies that the network is ˜ O ( L 2 ) -Lipschitz near initialization. More details can be found in Lemma 16 and its proof.

4.3 Optimal rates on NTK-separable data

In this section, we apply our general analysis to NTK-separable data [Ji and Telgarsky, 2020, Nitanda et al., 2020, Chen et al., 2021, Taheri and Thrampoulidis, 2024, Deora et al., 2023], and obtain the optimal rates.

Assumption 3. There exists γ > 0 and a collection of matrices W ∗ = { W 1 ∗ , · · · , W L ∗ } satisfying ∑ L l =1 ∥ W l ∗ ∥ 2 F = 1 , such that

This means that the dataset is separable by the NTK feature at initialization with a margin γ . Nitanda et al. [2020] pointed out that this assumption is weaker than positive eigenvalues of NTK Gram matrix, which has been widely used in the literature [Du et al., 2019b,a, Arora et al., 2019a]. With the above assumption, we have the following optimal risk bound on NTK separable data. The proof is given in Appendix D.

Theorem 3. Let Assumptions 1, 2, 3 hold. If m ≳ L 16 d (log m ) 5 log( nL/δ )(log T ) 8 /γ 8 , η ≤ 4 / (5 L ) , ηT ≍ n , then with probability at least 1 -δ , we have

Remark 4 (Proof sketch) . To apply the result in Theorem 2, we need to estimate F ( W ) , for which it suffices to bound L ( W ) and G = sup z ℓ ( yf W ( x )) . For the first part, we control it by L S ( W ) using Bernstein inequality (Eq.(14)). Let W = W (0) + 2 log T W ∗ /γ , plugging into (4) obtains ℓ ( y i f W ( x i )) ≤ 1 /T and further L S ( W ) ≤ 1 /T , implying F S ( W ) = ˜ O (1 /γ 2 ) . In order to control ℓ ( yf W ( x )) , we leverage the ˜ O ( L 2 ) -Lipschitzness of f W ( x ) . Indeed, for any x ∈ X , there holds

Remark 5 (Discussion on optimization error) . Under Assumption 3, Theorem 3.3 and Proposition 4.2 of Chen et al. [2021] show that when the network width satisfies m = ˜ Ω( L 22 /γ 8 ) , the training error is of the order ˜ O (1 /T ) . We achieve the same guarantee under a significantly milder width condition of m = ˜ Ω( L 16 /γ 8 ) . This improvement is enabled by two key technical advances: a sharper bound for (4), and a tighter estimate of the iterate distance ∥ W ( t ) -W ∥ F . Specifically, we improve the bound in (4) by a factor of L 1 / 3 , and show that ∥ W ( t ) -W ∥ F = ˜ O (1 /γ ) , improving upon the previous ˜ O ( √ L/γ ) bound. Together, these refinements reduce the required network width by a factor of L 6 .

Remark 6 (Comparison) . Under similar assumptions, Ji and Telgarsky [2020] derived the bound ˜ O ( 1 γ 2 √ n ) for shallow networks, which was recently improved to ˜ O ( 1 γ 2 n ) based on an improved control of the Rademacher complexity [Lei et al., 2026]. For deep ReLU networks, Chen et al. [2021] developed the bound of the order ˜ O ( e O ( L ) γ √ m n ) via Rademacher complexity [Bartlett et al., 2017], suffering from exponential depth dependence, explicit width requirement and suboptimal √ 1 /n scaling. Taheri et al. [2025] improved the result to ˜ O ( e O ( L ) γ 2 n ) for deep networks. The dependence on n, γ is optimal up to a logarithmic factor [Shamir, 2021, Schliserman and Koren, 2023]. However, their results require smooth activations and exponential width in L . Our rate is almost-optimal and enjoys polynomial dependence over the network depth. Furthermore, our bound holds under the overparameterization ˜ Ω(1 /γ 8 ) , matching the requirement in Ji and Telgarsky [2020], Chen et al. [2021]. This is much better than 1 /γ 6 L +4 in Taheri et al. [2025]. To the best of our knowledge, these are the best generalization bound and width condition for deep neural networks.

5 Experiments

In this section, we make some experimental verifications to support our theoretical analysis. Our excess risk analysis in Theorem 3 imposes an NTK separability assumption, which has been validated in the literature. For example, [Ji and Telgarsky, 2020] demonstrates that Assumption 3 holds for a noisy 2-XOR distribution, where the dataset is structured as follows:

Here, the factor 1 √ d -1 ensures that ∥ x ∥ 2 = 1 , × above denotes the Cartesian product, and the label y only depends on the first two coordinates of the input x . As shown in Ji and Telgarsky [2020], this dataset satisfies Assumption 3 with 1 /γ = O ( d ) , which implies that our excess risk bound in Theorem 3 becomes O ( d 2 /n ) for this dataset. We conducted numerical experiments and observed that the test error decays linearly with d 2 /n . The population loss for the test error is computed over all 2 d points in the distribution.

Settings We train two-layer ReLU networks by gradient descent on noisy 2-XOR data. We fix the width m = 128 , T = 500 , η = 0 . 1 . We have conducted two experiments. With a fixed dimension d = 6 , we vary the sample size n . The results are presented in Figure 1a. With a fixed sample size n = 64 , we vary the dimension d and the corresponding table is provided in Figure 1b.

(a) Test error for different

n

Test Error vs d²/n

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(b) Test error for different

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Test Error vs d²/n

Linear Fit: y = 0.1648x + -0.0241

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3.5

d°/n

In both experiments, we observe that the test error is of the order d 2 /n (approximately 0 . 15 d 2 /n ). This shows the consistency between our excess risk bounds in Theorem 3 and experimental results. We conducted the experiments on Google Colab. A simple demonstration reproducing our numerical experiments is available as a Google Colab notebook at: https://github.com/YuanfanLi2233/ nips2025-optimal .

6 Conclusion and Future Work

In this paper, we present optimization and generalization analysis of gradient descent-trained deep ReLU networks for classification tasks. We explore the optimization error of F S ( W ) / ( ηT ) under a milder overparameterization requirement than before. We establish sharper bound of Rademacher complexity and Lipschtiz constant for neural networks. This helps to derive generalization bound of order ˜ O ( F ( W ) /n ) . For NTK-separable data with a margin γ , our methods lead to the optimal rate of ˜ O (1 / ( nγ 2 )) . We improve the existing analysis and require less overparameterization than previous works.

There remain several interesting questions for future works. First, it is an interesting question to extend our methods to SGD. Second, while we establish polynomial Lipschitz constants near initialization, investigating whether similar bounds hold far from initialization would deepen our theoretical understanding. Finally, we only consider fully-connected neural networks. It is interesting to study the generalization analysis of networks with other architectures, such as CNNs and Resnets [Du et al., 2019b].

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A Technical Lemmas

We define the diagonal sign matrix Σ l ( x ) with l ∈ [ L ] by

Then the deep ReLU network has the following matrix product representation:

together with the presentation of h l ( x ) :

We further define G l l ( x ) = √ 2 /m Σ l ( x ) and

from which we can rewrite f W ( x ) as

Hence, for l ∈ [ L ] , we have

Similarly, we define

We denote Σ l 0 ( x ) , h l 0 ( x ) , G a b, 0 ( x ) , H a b, 0 ( x ) as (10), (12), (13) and (14) with W = W (0) .

A.1 Properties of the Initialization

Given a set of N points on the sphere K = { x 1 , · · · , x N } . We provide general results for any finite set K , then it can be applied to specific choices of K , for example, the training dataset S 1 = { x 1 , · · · , x n } .

Lemma 2 (Theorem 4.4.5 in Vershynin [2018]) . With probability at least 1 -L exp( -Cm ) over the random choice of W (0) , there exists an absolute constant c 0 > 1 such that for any l ∈ [ L ] , there holds

For a sub-exponential random variable X , its sub-exponential norm is defined as follows:

X -E X is sub-exponential too, satisfying

If Y is a sub-gaussian random variable, we define the sub-gaussian norm of Y by

Suppose Y ∼ N (0 , r 2 ) , then σ ( Y ) is also sub-guassian and we have

We have the following lemma:

Lemma 3 (Lemma 2.7.6 in Vershynin [2018]) . A random variable X is sub-gaussian if and only if X 2 is sub-exponential. Moreover,

Now we introduce Bernstein inequality with respect to ∥ · ∥ ϕ 1 ,

Lemma 4 (Theorem 2.8.2 in Vershynin [2018]) . Let X 1 , · · · , X m be independent, mean zero, subexponential random variables, and d = ( d 1 , · · · , d m ) ∈ R m , K ≥ max r ∥ X r ∥ ϕ 1 . Then for every t ≥ 0 , we have

for some absolute constant c .

We introduce the following technical lemma related to the conditional expectation of Gaussian indicator function.

̸

Lemma 5. Suppose w is a m -dim Gaussian random vector with distribution N (0 , I ) . Let c = 0 , b be two given vectors of m -dim. Then we have the following property

Proof. Let u = ⟨ w , c ⟩ , v = ⟨ w , b ⟩ . Then u ∼ N (0 , ∥ c ∥ 2 2 ) , v ∼ N (0 , ∥ b ∥ 2 2 ) . We decompose v into a component dependent on u and an independent residual z :

where z ∼ N ( 0 , ∥ b ∥ 2 2 -⟨ c , b ⟩ 2 ∥ c ∥ 2 2 ) is independent of u . Hence, we have

where the second equality is due to E [ I { u ≥ 0 } u 2 ] = ∥ c ∥ 2 2 2 and the independence of u , z . The third equality follows from the distribution of z . The proof is completed.

The following lemma studies the output of initialization at each layer.

Lemma 6. For any δ > 0 , if m ≳ L 2 log( NL/δ ) , then with probability at least 1 -δ ,

Proof. This result directly follows Corollary A.2 in Zou et al. [2018], and we give the proof here for completeness. Note that for 1 ≤ i ≤ N, 1 ≤ l ≤ L ,

Condition on h l -1 0 ( x i ) , we have ⟨ w l j (0) , h l -1 0 ( x i ) ⟩ ∼ N (0 , ∥ h l -1 0 ( x i ) ∥ 2 2 ) , hence

By (16) and Lemma 3, we have

where the last inequality is due to (17). Let X r = 2 σ 2 ( ⟨ w l r (0) , h l -1 0 ( x i ) ⟩ ) - ∥ h l -1 0 ( x i ) ∥ 2 2 , d r = 1 /m,K = C ∥ ∥ h l -1 0 ( x i ) ∥ ∥ 2 2 and apply Lemma 4. We have for any 0 ≤ t ≤ 1 ,

Taking union bounds over i, l , there holds for any 1 ≤ i ≤ N and 1 ≤ l ≤ L ,

Since m ≳ L 2 log( NL/δ ) , let t = √ log( NL/δ ) /m , we have with probability at least 1 -δ , there holds

Now we show the following inequality holds with probability at least 1 -δ ,

When l = 0 , it is true. If (20) holds for l ∈ [ L -1] , then ∥ h l 0 ( x i ) ∥ 2 2 ≤ 4 / 3 . Combined with (19), we have with probability at least 1 -δ ,

Hence, (20) holds for all i ∈ [ N ] , l ∈ [ L ] , which implies the lemma.

Lemma 7. For any x ∈ X , we have

Proof. Note the r -th row of Σ L 0 ( x ) W L (0) is I {⟨ w L r (0) , h L -1 0 ( x ) ≥ 0 } w L r (0) . Since a r = -a r + m 2 and w L r (0) = w L r + m 2 (0) for all r ∈ [ m 2 ] , we have

It then follows that for all l ∈ [ L -1] ,

Hence, the proof is completed.

Lemma 8. Suppose m ≳ L 2 log( NL/δ ) , then with probability at least 1 -δ , for all i ∈ [ N ] ,

Proof. By Hoeffding inequality, condition on h L -1 0 ( x i ) , with probability at least 1 -δ , there holds

Combined with Lemma 6, we have

The proof is completed.

Let Σ 1 , Σ 2 ∈ R m × m be two diagonal matrices with entries in { 0 , 1 } .

Lemma 9. Suppose m ≳ L 2 log( NL/δ ) , s ≲ m/ ( L 2 log m ) , then with probability at least 1 -δ we have for all i ∈ [ N ] , 2 ≤ a ≤ b ≤ L ,

Proof. We need to prove that for any v ∈ S m -1 , there holds

Note that ∥ Σ 1 v ∥ 0 ≤ s and ∥ Σ 1 v ∥ 2 ≤ 1 . Let P = { v ∈ S m -1 : ∥ v ∥ 0 ≤ s } . We only need to prove that the following inequality holds with probability at least 1 -δ :

Let S be a subspace of S m -1 that has at most s non-zero coordinates. For such a subspace, we choose a 1 / 2 -cover of it and denote this cover by Q . By Lemma 4.2.13 in Vershynin [2018],

The number of such subspaces is M = ( m s ) . We denote all subspaces by S 1 , · · · , S M , and the corresponding covers Q 1 , · · · , Q M . Let ⋃ Q = { v 1 , · · · , v M ′ } with M ′ ≤ ( m s ) 5 s . We first prove that (23) is true for all v j , then it holds simultaneously for all elements in P .

For a unit vector v , we define

and v a -1 ( x i ) = v . Note that condition on h l -1 0 ( x i ) , v l -1 ( x i ) , we take expectation over w l r (0) , applying Lemma 5 implies that

It then follows that

Similar to the proof of Lemma 6, for every v j , 1 ≤ j ≤ M ′ , we apply Lemma 4 to get

Taking the union bounds for all j, l, i yields

where we have used ( m s ) ≤ ( em/s ) s ≤ ( em ) s in the second inequality and the last inequality is due to m ≳ L 2 log( NL/δ ) , s ≲ m/ ( L 2 log m ) . Hence, we have with probability at least 1 -δ ,

For any unit vector v with ∥ v ∥ 0 ≤ s , consider the subspace S containing it and the corresponding 1 / 2 - cover Q . There exists a unit vector v j , j ∈ [ M ′ ] , ∥ v -v j ∥ 0 ≤ s and ∥ v -v j ∥ 2 ≤ 1 / 2 . Thus,

Taking sup to the both sides yields

which implies sup v ∈P ∥ H a b, 0 ( x i ) v ∥ 2 ≲ 1 . Hence (23) holds and the proof is completed.

Remark 7 . Our proofs are inspired by Lemma A.9 in Zou et al. [2018], which establishes the estimates under the condition s ≳ log( NL/δ ) . However, we eliminate this assumption via a more refined analysis.

Lemma 10. Suppose m ≳ L 2 log( NL/δ ) , then with probability at least 1 -δ , we have for all i ∈ [ N ] , 2 ≤ a ≤ b ≤ L ,

Although the left-hand side of above inequality could be the production of L terms, it is bounded by ˜ O ( L ) . This lemma shows that the introduction of ReLU activation can avoid exponential explosion.

Proof. For any unit vector v , we decompose it as v = v 1 + · · · + v q , where v j , j ∈ [ q ] are all s -sparse vectors on different coordinates. Therefore,

Here we choose s ≍ m/ ( L 2 log m ) , then q ≲ m/s ≲ L 2 log m . Applying Lemma 9, we have

where we have used Cauchy-Schwartz's inequality in the second inequality. Hence,

The proof is completed.

From the above lemma, we know that if m ≳ L 2 log( NL/δ ) log m , then

Remark 8 . In Lemma 10, we introduce a useful technique that decomposes the unit vector into sparse components. This approach reduces the covering number from 5 m to 5 s ( m s ) , making it easier for high-probability bounds to hold. A related method appears in Lemma 7.3 of Allen-Zhu et al. [2019], but their width exhibits polynomial dependence on n . In contrast, our analysis achieves polylogarithmic width, substantially relaxing the overparameterization requirement.

Using similar techniques we can obtain the following lemma:

Lemma 11. Suppose m ≳ L 2 log( NL/δ ) , then with probability at least 1 -δ , for all i ∈ [ N ] , 2 ≤ a ≤ b ≤ L, ∥ Σ 1 ∥ 0 , ∥ Σ 2 ∥ 0 ≤ s ≲ m/ ( L 2 log m ) ,

Proof. If s = 0 , the above inequality becomes 0 ≲ 1 /L , which holds true. Now we assume s ≥ 1 . Similar to Lemma 9, we only need to prove that for any s -sparse unit vector v there holds

We use the same notation as in Lemma 9, it then follows that for all j ∈ [ M ′ ] , a ≤ l ≤ b, i ∈ [ N ] ,

For a fixed Σ 1 , we assume Σ 1 = diag { d 1 , · · · , d m } with d r ∈ { 0 , 1 } , ∑ r d r ≤ s, r ∈ [ m ] . We have

Condition on v b -1 j ( x i ) , there holds

Let X r = 2 m ( ⟨ w b r (0) , v b -1 j ( x i ) ⟩ ) 2 -2 m ∥ v b -1 j ( x i ) ∥ 2 2 and d = ( d 1 , · · · , d m ) . Then X r are mean-zero sub-exponential random variables, following similar discussions in Lemma 6, we have ∥ X r ∥ ϕ 1 ≤ C m ∥ v b -1 j ( x i ) ∥ 2 2 . Moreover, ∥ d ∥ 2 2 = ∑ m r =1 d 2 r = ∑ m r =1 d r ≤ s, ∥ d ∥ ∞ = 1 . Applying Lemma 4, we have

Choosing t = 1 /L 2 and note that s ≤ m/ ( L 2 log m ) , we have

where the first inequality is due to (26) and ∑ m r =1 d r ≤ s , the last inequality results from (27).

Taking union bounds over all x i , Σ 1 , v j , l , and note that

We have with probability at least 1 -δ ,

Hence, we have

Following same techniques of using 1 / 2 -cover in the proof of Lemma 9, we can prove (25), and then complete the proof of the lemma.

Additionally, apply similar methods in Lemma 10 by decomposing the unit vector into s -sparse vectors, we have

Remark 9 . To analyze the influence of the sparse matrix Σ 1 on (25), we propose a key technical improvement: instead of resorting to covering number arguments as in Zou et al. [2018], Allen-Zhu et al. [2019], we leverage a weighted Bernstein inequality. Particularly, existing methods require taking another union bound over both all s -sparse subspaces and their covers (of size ∼ ( m s ) 9 s ), whereas our analysis only needs to union over the sparse subspaces themselves (of cardinality ( m s ) ). Our method directly demonstrates that sparsity inherently lowers computational costs by avoiding the need for dense covers. The simplicity of our technique also underscores the intrinsic benefits of sparse structures in optimization.

A.2 Properties of Perturbation Terms

Recall that

For any l ∈ [ L ] , let ̂ W l and the diagonal matrix ̂ Σ l ( x ) be the matrices with the same size of W l (0) and Σ l 0 ( x ) , respectively. Define

Lemma 12. Let ̂ G a b ( x ) with 1 ≤ a ≤ b ≤ L be the matrix defined in (29) . Assume max l ∈ [ L ] ∥ ̂ W l ∥ 2 ≤ R ≲ √ m/ ( L 2 √ log m ) , m ≳ L 2 log( NL/δ ) and ̂ Σ l ( x i ) , ̂ Σ l ( x i ) + Σ l 0 ( x i ) ∈ [ -1 , 1] m × m , ∥ ̂ Σ l ( x i ) ∥ 0 ≤ s ≲ m/ ( L 2 log m ) for all i ∈ [ N ] , l ∈ [ L ] . Then, with probability at least 1 -δ for all 1 ≤ a ≤ b ≤ L, i ∈ [ N ] , there holds

Proof. The proof is similar to that of Lemma 8.6 in Allen-Zhu et al. [2019], the differences lie in the dependence of L . We first prove that for any 1 ≤ a ≤ b ≤ L ,

̸

We define a diagonal matrix ( ̂ Σ l 1 ( x i )) k,k = I { ̂ Σ l ( x i ) k,k = 0 } , and ∥ ̂ Σ l 1 ( x i ) ∥ 0 ≤ s . Therefore, ̂ Σ l ( x i ) = ̂ Σ l 1 ( x i ) ̂ Σ l ( x i ) ̂ Σ l 1 ( x i ) . We decompose the left term of (30) into 2 b -a terms and control them respectively. Each matrix can be written as (ignoring the superscripts and x i ).

Then, with probability at least 1 -δ , there holds:

By Lemma 9, ∥ ∥ ∥ Σ 0 √ 2 m W (0) · · · ̂ Σ 1 ∥ ∥ ∥ 2 ≲ 1 .

By Lemma 11, ∥ ∥ ∥ ̂ Σ 1 √ 2 m W (0) · · · √ 2 m W (0) ̂ Σ 1 ∥ ∥ ∥ 2 ≲ 1 /L .

By (28), ∥ ∥ ∥ ̂ Σ 1 √ 2 m W (0) · · · Σ 0 √ 2 m W (0) ∥ ∥ ∥ 2 ≲ √ log m .

When there is no ̂ Σ , by Lemma 10, ∥ ∥ ∥ Σ 0 √ 2 m W (0) · · · Σ 0 √ 2 m W (0) ∥ ∥ ∥ 2 ≲ L √ log m .

Combined with these results, counting the number of ̂ Σ , we obtain

where in the second inequality we have used ( b -a j ) ≤ ( e ( b -a ) /j ) j ≤ ( eL/j ) j , the last inequality is due to ∑ L j =1 ( e/j ) j converges and it is bounded by a constant. Now we have proved (30).

Denote Σ ′ = Σ 0 + ̂ Σ , through similar expansion, Σ ′ √ 2 m ( W (0) + ̂ W ) · · · (Σ ′ ) √ 2 m ( W (0) + ̂ W )

is the sum of following terms

Since ∥ Σ ′ ∥ 2 ≲ 1 , using Eq. (30), we have

Note that max l ∈ [ L ] ∥ ̂ W l ∥ 2 ≤ R ≲ √ m/ ( L 2 √ log m ) , then by counting the number of ̂ W , we have

The proof is completed.

Denote ˜ Σ( x ) , ˜ h l ( x ) , ˜ G a b ( x ) as (10), (12),(13) when W = ˜ W .

Lemma 13 (Claim 11.2 and Proposition 11.3 in Allen-Zhu et al. [2019]) . For any W , ˜ W ∈ B R ( W (0)) . There exists a series of diagonal matrices { (Σ ′′ ) l ∈ R m × m } l ∈ [ L ] with entries in [ -1 , 1] such that for any l ∈ [ L ] , there holds

The above lemma shows that the difference of ReLU networks can be expressed explicitly as the operations of matrices. The main idea is to show that σ ( a ) -σ ( b ) = ( I [ a ≥ 0] -ξ )( a -b ) for ξ ∈ [ -1 , 1] . Now we introduce the following Bernstein inequality under bounded distributions.

Lemma 14 (Theorem 2.8.4 in Vershynin [2018]) . Let X 1 , · · · , X N be independent, mean-zero random variables, such that | X i | ≤ K for all i . Then for every t ≥ 0 , we have

where λ 2 = ∑ N i =1 E X 2 i is the sum of the variance.

The following lemma shows that under overparameterized setting, the outputs and activation patterns for deep relu networks near initialization do not change much.

Lemma 15. Suppose m ≳ L 10 log( NL/δ )(log m ) 4 R 2 . Then with probability at least 1 -δ , for any W ∈ B R ( W (0)) , i ∈ [ N ] and l ∈ [ L ] , there holds

Proof. We prove these two inequalities by induction. Note that (31) holds for l = 0 . Now we suppose (31) holds for l -1 . Let κ > 0 be a constant. For i ∈ [ N ] and l ∈ [ L ] , we define A l ( x i ) = { r ∈ [ m ] : I {⟨ w l r , h l -1 ( x i ) ⟩ ≥ 0 } ̸ = I {⟨ w l r (0) , h l -1 0 ( x i ) ⟩ ≥ 0 }} , then ∥ Σ l ( x i ) -Σ l 0 ( x i ) ∥ 0 = | A l ( x i ) | . Furthermore, we decompose A l ( x i ) into two parts based on the behavior of w l r (0) :

A l 1 ( x i ) = { r ∈ A l ( x i ) : |⟨ w l r (0) , h l -1 0 ( x i ) ⟩| ≤ κ } and A l 2 ( x i ) = { r ∈ A l ( x i ) : |⟨ w l r (0) , h l -1 0 ( x i ) ⟩| > κ } . We will control | A l 1 ( x i ) | and | A l 2 ( x i ) | respectively.

For r ∈ [ m ] , we define F l r,i = I {|⟨ w l r (0) , h l -1 0 ( x i ) ⟩| ≤ κ } . From Lemma 6 we know that 2 / 3 ≤ ∥ h l -1 0 ( x i ) ∥ 2 2 ≤ 4 / 3 . Condition on h l -1 0 ( x i ) , ⟨ w l r (0) , h l -1 0 ( x i ) ⟩ ∼ N (0 , ∥ h l -1 0 ( x i ) ∥ 2 2 ) , then we have

Then by Lemma 14, choose K = 1 , t = mCκ,λ 2 ≤ mCκ , it then follows that

Hence, taking union bounds over l, i , with probability at least 1 -CnL exp( -mκ ) , there holds for all i, l ,

For r ∈ A l 2 ( x i ) , since I {⟨ w l r , h l -1 ( x i ) ⟩ ≥ 0 } ̸ = I {⟨ w l r (0) , h l -1 0 ( x i ) ⟩ ≥ 0 } , we have ( ⟨ w l r , h l -1 ( x i ) ⟩ - ⟨ w l r (0) , h l -1 0 ( x i ) ⟩ ) 2 ≥ |⟨ w l r (0) , h l -1 0 ( x i ) ⟩| 2 > κ 2 .

We deduce that

By assumption ∥ h l -1 ( x i ) -h l -1 0 ( x i ) ∥ 2 ≲ L 2 R √ log m/m and Lemma 6, we get

Combined with (33), we have

From (32) and (34) we know that

where in the last inequality we choose κ = L 4 / 3 (log m ) 1 / 3 R 2 / 3 m -1 / 3 . Hence, due to the overparameterization of m , we have with probability at least 1 -δ , for i ∈ [ N ] ,

Applying Lemma 13, we have

where ̂ G k l, 0 ( x i ) is defined as

It then follows that

Our overparameterization requirement implies that R ≲ √ m/ ( L 2 √ log m ) . Hence, by Lemma 12, we have

where the last inequality results from Lemma 6. As a result, (31) holds for l . We have completed the proof of the lemma.

The above lemma and Lemma 6 imply that with probability at least 1 -δ , for all l ∈ [ L ] , i ∈ [ N ] ,

Remark 10 . Although our approach shares similarities with Lemma B.3 in Zou et al. [2018], our analysis relaxes the required conditions. Specifically, we only require R/ √ m = ˜ O ( L -5 ) , whereas their result demands the stricter scaling R/ √ m = ˜ O ( L -11 ) . Furthermore, compared to Lemma 8.2 in Allen-Zhu et al. [2019], they derive the bound ∥ h l ( x i ) -h l 0 ( x i ) ∥ 2 ≲ RL 5 / 2 √ log m/ √ m , which is worse than our result by a factor of √ L .

The following lemma shows the uniform concentration property of deep ReLU networks, which is crucial in the generalization analysis.

Lemma 16. Let R ≥ 1 be a constant. Assume m ≳ L 11 d (log m ) 5 log( L/δ ) R 2 . Then with probability at least 1 -δ , for W ∈ B R ( W (0)) , l ∈ [ L ] , we have

Proof. We consider the 1 / ( C L √ m ) -cover of S d -1 and denote it by D = { x 1 , · · · , x | D | } . By Lemma 4.2.13 in Vershynin [2018],

Therefore,

Note that Lemma 15 holds for any finite set K = { x 1 , · · · , x N } . Letting K = D , we obtain that if m ≳ L 11 d (log m ) 5 log( L/δ ) R 2 ≳ L 10 log( | D | L/δ )(log m ) 4 R 2 , then

For any x ∈ X , there exists x j ∈ D with ∥ x -x j ∥ 2 ≤ 1 / ( C L √ m ) . It then follows that

where the first inequality is due to σ ( · ) is 1 -Lipschitz. In the second inequality we have used ∥ W l ∥ 2 ≤ ∥ W l (0) ∥ 2 + R ≲ √ m due to Lemma 2. Similarly, we derive that

Therefore, combined with (40), we have

where the last inequality results from R ≥ 1 .

The proof is completed.

Remark 11 . This lemma is a property of deep ReLU networks near initialization that does not depend on the training data. Compared to prior work, while Allen-Zhu et al. [2019], Zou et al. [2018] only establishes bounds for the training data, we prove the uniform convergence over the entire input space. Previous work on uniform concentration demonstrated that sup x ∈X ∥ h l ( x ) -h l 0 ( x ) ∥ 2 ≲ C L R/ √ m [Xu and Zhu, 2024]. We present a significant improvement, reducing the dependence on L from exponential to polynomial.

In the following part, we apply previous technical lemmas to K = S 1 and get properties of deep neural networks over the training dataset.

Lemma 17. Suppose m ≳ L 10 log( nL/δ )(log m ) 4 R 2 . Then with probability at least 1 -δ for all W ∈ B R ( W (0)) , l ∈ [ L ] , i ∈ [ n ]

Proof. For the case l = L ,

where the last inequality is due to Lemma 15.

Now we suppose l < L , then

where the last equality is according to Lemma 7. Applying Lemma 12 and Lemma 15, there holds

This implies that

where we have used (41) in the third inequality. The proof is completed.

Lemma 18. Assume m ≳ L 10 log( nL/δ )(log m ) 4 R 2 . Then with probability at least 1 -δ , for any W ∈ B R ( W (0)) , i ∈ [ n ] and l ∈ [ L ] , there holds

Proof. Since ∥ xy ⊤ ∥ F = ∥ x ∥ 2 ∥ y ∥ 2 for two vectors x, y , we have

Using (38) and Lemma 17, we have

Applying Lemma 15 and (24), we obtain

It then follows that

The proof is completed.

B Proofs for Optimization

Lemma 19. Suppose m ≳ L 10 log( nL/δ )(log m ) 4 R 2 , then with probability at least 1 -δ , for i ∈ [ n ] , ˜ W , W ∈ B R ( W (0)) , we have

This lemma shows that deep ReLU networks near initialization are almost linear.

Proof. Note that

Since f W ( x i ) = a ⊤ h L ( x i ) , applying Lemma 13, we obtain

the last inequality is due to Lemma 15. We further let

Following the proof of Lemma 17, we have

which implies that

Hence,

where in the last inequality we have used (38). The proof is completed.

The following lemma shows that L S is almost convex near initialization. It becomes more convex as the width grows.

Lemma 20. Suppose m ≳ L 10 log( nL/δ )(log m ) 4 R 2 , then with probability at least 1 -δ , we have for ˜ W , W ∈ B R ( W (0)) ,

Proof. Since ℓ is 1 / 4 -smooth, it enjoys the co-coercivity, i.e., ℓ ( a ) ≥ ℓ ( b ) + ( a -b ) ℓ ′ ( b ) + 2( ℓ ′ ( a ) -ℓ ′ ( b )) 2 , which implies that

We combine the above inequality with Lemma 19 and obtain

The following lemma shows how the distance between gradient descent iterators and the reference model would change after a single gradient descent.

Lemma 21. Suppose m ≳ L 10 log( nL/δ )(log m ) 4 R 2 . Then with probability at least 1 -δ , for η ≤ 4 / (5 L ) and ˜ W , W ∈ B R ( W (0)) ,

Proof. By Lemma 8 and Lemma 18 we know that ∥ ∥ ∥ ∂f W ( x i ) ∂ W l ∥ ∥ ∥ F ≤ 2 , hence ∥ ∥ ∥ ∂f W ( x i ) ∂ W ∥ ∥ ∥ F ≤ 2 √ L , which implies that

where we have used the standard inequality ( a + b ) 2 ≤ 5 a 2 +5 b 2 / 4 . Then by applying Lemma 20 we have

the proof is completed.

Proof of Theorem 1. We prove it through induction. It holds for t = 0 . Suppose it holds for k = 0 , · · · , t -1 , then we have,

plugging R = 2 √ F S ( W ) and m into Lemma 21, we have

Telescoping and note that ˜ F S ( W ) ≤ L S ( W ) , we obtain

which implies

≤ ∥

W

(0)

-

W

∥

2

F

+(2 + 20

ηL

˜

F

S

(

W

Hence, when m ≳ F 4 S ( W )(log m ) 4 L 16 , η ≤ 1 / (20 L ˜ F S ( W )) , there holds

))

ηT

L

S

(

W

)

.

This implies that

Therefore, the induction holds for t , the proof is completed.

C Proofs for Generalization

From (35), we know that there exists ̂ G l L, 0 ( x i ) , 1 ≤ l ≤ L, i ∈ [ n ] , such that

Let E 1 be the event (w.r.t. W (0) ) that ∥ ̂ G l L, 0 ( x i ) ∥ 2 ≤ CL √ log m/m for all W ∈ W 1 . Let E 2 be the event such that ∥ h l -1 0 ( x i ) ∥ 2 2 ≤ 4 / 3 , 1 ≤ l ≤ L, i ∈ [ n ] . Then we have the following bound on Rademacher complexity:

Lemma 22. Let F and W 1 be defined in (7) and (8) , respectively. If the events E 1 , E 2 hold, then

where

where the last inequality is due to Cauchy-Schwartz inequality. Since the event E 1 holds, we have for all W ∈ W 1 , 1 ≤ k ≤ l ≤ L, i ∈ [ n ] ,

where we have used ∥ AB ∥ F ≤ ∥ A ∥ 2 ∥ B ∥ F for two matrices A,B . Therefore,

For l ∈ [ L ] , let ( g l 1 ) ⊤ , · · · , ( g l m ) ⊤ be the rows of matrix ̂ G l L, 0 (˜ x i )( W l -W l (0)) , we obtain

Therefore,

This implies that

The last inequality is due to the event E 2 . Combined with (44), we have

The proof is completed.

Now we provide the proof for Theorem 2

Proof of Theorem 2. We first control G ′ = sup z, W ∈W 1 ℓ ( yf W ( x )) . We denote ¯ h l ( x ) as the output of l -th layer of the network f W ( x ) . Then f W ( x ) = a ⊤ ¯ h L ( x ) . By the definition of F ( W ) , we have max l ∥ W l -W l (0) ∥ 2 ≤ ∥ W -W (0) ∥ F ≤ √ F ( W ) . For W ∈ W 1 , there holds ∥ W l -W l (0) ∥ 2 ≤ ∥ W l -W l (0) ∥ 2 + ∥ W l -W l ∥ 2 ≤ 2 √ F ( W ) for all l ∈ [ L ] . By the overparameterization of m and Lemma 16,

Since logistic loss ℓ is 1 / 4 -smooth, the following property holds,

|

ℓ

′

(

x

)

√

It then follows that for any W ∈ W 1 , z ∈ Z ,

where we have used ab ≤ 2 a 2 + b 2 / 8 in the second inequality. Hence, G ′ ≤ 2 G + CL 4 log mF ( W ) . According to Lemma 14, we have with probability at least 1 -δ ,

It then follows that

ℓ

(

x

)

/

2

,

x

∈

R

.

which implies that F S ( W ) ≤ F ( W ) . Combined with Theorem 1, we know that with probability at least 1 -δ , W ( t ) ∈ W 1 . It means that all the iterates are in the hypothsesis space. Furthermore, events E 1 , E 2 hold due to Lemma 6 and (37) in Lemma 15. Hence, by Lemma 1 and Lemma 22, there holds

As a result,

from which we derive

where we have used ηT ≍ n . The proof is completed.

D Proofs on NTK separability

Proof of Theorem 3. We show that there exists W with small F ( W ) for NTK separable data. Let W = W (0) + λ W ∗ . Choose λ = 2log T/γ . Applying Lemma 19 and letting R = λ , we know that if m ≳ L 16 d (log m ) 5 log( nL/δ )(log T ) 2 /γ 8 , then with probability at least 1 -δ , for all i ∈ [ n ] , there holds

where we have used f W (0) ( x ) = 0 for any x ∈ X due to Lemma 7. Therefore, by Assumption 3, we have

As a result,

where we have used log(1 + x ) ≤ x . For any x ∈ X , Lemma 16 implies that

Hence,

where the last inequality is due to log(1+ t ) ≤ log(2 t ) ≤ 2 log( t ) for t ≥ 2 . Note that if x 2 ≤ αx + β , then x 2 ≤ α 2 +2 β . Combined with (45), it then follows that (let x = √ L ( W ) )

Thus, we have

Applying Theorem 2 and note that ˜ F S ( W ) ≤ L S ( W ) ≤ 1 T , there holds

The proof is completed.

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Peer reviews:
comments: SUMMARY:
Improved bounds on the excess risk rate for deep ReLU networks.

STRENGTHS AND WEAKNESSES:
Strengths. Improved bounds wrt those known that are polynomial in depth.
Weaknesses. No attempt at an experimental confirmation, but they are on time to present it for the rebuttal.

QUESTIONS:
- gradient decent -> gradient descent, line 57

LIMITATIONS:
The limitations are the assumptions of their theorems, which are clearly stated.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-09

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the convergence and generalization analysis of deep fully connected neural networks. Specifically, this paper proves that GD achieves a sublinear convergence rate in the order of $1/T$. Furthermore, the derived generalization bound exhibits an optimal rate of $1/n$ with respect to the sample size $n$. Additionally, the dependence on $L$ has been improved from an exponential to a polynomial order by establishing a tighter Rademacher complexity bound.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: \n\n1. A tighter and more practically relevant generalization bound.\n\n2. A thorough comparison with existing works.\n\nWeakness:\n\nOverall, I feel that while this paper shows improvements over prior work, it focuses on a less meaningful direction. The insights gained from this improvement are either unclear or do not align well with what is observed in practice. As a result, I am not convinced that this setting is appropriate for studying the generalization error of deep neural networks.\n\n1. The assumption that the data is NTK-separable with a margin $\\gamma$ lacks explanation in practical contexts. Specifically, what kinds of data will succeed or fail in this assumption? In addition, this assumption is extremely strong in the sense that it requires the network to be initialized in such a way that the data, when processed through this predefined model, is already linearly separable.\n\n2. Compared with prior work, the main difference I observe is that the generalization error bound has been improved from an exponential dependence to a polynomial dependence. However, the key insight remains the same as in previous studies: increasing the number of layers leads to worse generalization. I do not see any fundamentally new insights presented in this paper. \n\n3. The theoretical insights presented in this paper do not adequately explain the current success and widespread effectiveness of deep learning models that rely on large and deep architectures.\n\n4. The paper does not include any experiments to validate the theoretical results it presents.\n\nQUESTIONS:\nSince the major contribution of this paper is the improvement in the dependence of the generalization bound with respect to $L$, I have several questions:\n  \n1. What are the key insights that this paper aims to convey through this improved bound?\n    \n2. If Assumption 3 holds for real-world data, what would be the practical difference between using a deep neural network versus a shallow one? For example, how does $L$ relate to $\\gamma$ in this context?\n\n3. Could you specify lines that reflect \"novel control of activation patterns near a reference model\"?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-25"}

comments: SUMMARY:
The authors establish a convergence rate for Gradient Descent applied on deep neural network with rail activations that is polynomial in the depth of the network $L$.

STRENGTHS AND WEAKNESSES:
**Strengths**

1. The paper is well written and easy to follow.
2. The authors present their proof technique and the relation to prior work in a very clear manner.
3. The result is significant and improves the exponential bound given in previous work.

**Weaknesses**

1. My main concern is about the correctness of Lemma 16, which is a central component of the paper. While Lemma 15 is proved only for the data points \$x\_i\$ in the training set, Lemma 16 applies it to points in the covering set \$D\$, which includes points outside the training set. Why is this use of Lemma 15 justified? Could the authors elaborate on this point? If I’m mistaken, I will revise my score accordingly. Another reason this transition may not be valid is that the network’s weights depends on the training set. Therefore, a bound that holds for the training set may not necessarily apply to points outside it.

2. Additionally, I am uncertain about the correctness of the proof of Lemma 15. In the inductive argument showing that $A_l \leq O(B_l) $, it is crucial that the constant hidden in the big-$O$ notation remains uniform across all induction steps. Otherwise, for example, if the constant grows multiplicatively at each step, the resulting bound would be exponential in $ l $. Could you clarify why this issue does not arise in the proof of Lemma 15?


3. A minor concern: The authors obtain a bound only in the case where \$m \geq d\$.

QUESTIONS:
Why is it necessary to assume that $R \geq 1$?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the generalization error of $L$-th layer ReLU neural networks trained in the NTK regime. On the data that is $\\gamma$-separated by the neural tangent kernel, the authors provide the excess risk of order $O(L^4(1+\\gamma L^2)/\\gamma^2 n)$, which is faster than the well-known suboptimal one $O(1/\\sqrt{n})$ and only depends polynomially on the network depth $L$. They also provide a global convergence guarantee for the network using gradient descent and the NTK framework. In the analysis, they carefully evaluate the dependence on $L$ for the convergence analysis and Rademacher complexity bound, which enables them to obtain a sharper bound.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n- As the authors mentioned in the paper, much of the existing literature on the generalization error bound of deep neural networks exponentially depends on the network depth. This work mitigates this dependence to the polynomial one, which is a significant refinement. \n- This paper is easy to follow, with detailed explanations of theorems and concise proof overviews.\n\n**Weakness**\n- The primary concern is that this paper mainly focuses on NTK-separable data, which imposes a substantial restriction on the data structure. It appears that several refinements, such as the polynomial dependence on the network depth, do not require the separability assumption. Therefore, it would be more plausible to compare the results obtained in this work with other NTK-based results that do not focus on the separability condition.\n\nQUESTIONS:\n- How is the initialization scheme defined in Assumption 1 crucial for obtaining the sharper bound? For example, while [Chen et al., 2021] considers the standard Gaussian initialization for the weights, can the same bound be derived for such settings?\n- As the authors also mentioned as a future work, extending the results to stochastic optimization, which is covered by [Chent et al., 2021], will be a significant topic. What is the difficulty of such an extension?\n- Another question is about the extension to other models. For example, how is the ReLU activation essential for obtaining the results, and is it possible to extend the results to other activations? Moreover, I am curious about the applicability of this approach to different models, such as ResNet and CNN.\n\nLIMITATIONS:\nThe authors properly address the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}