Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition

Ding-Xuan Zhou; Nathanael Tepakbong; Xiang Zhou

arxiv: 2505.08262 · v2 · submitted 2025-05-13 · 💻 cs.LG · math.ST· stat.TH

Super-fast Rates of Convergence for Neural Network Classifiers under the Hard Margin Condition

Nathanael Tepakbong , Xiang Zhou , Ding-Xuan Zhou This is my paper

Pith reviewed 2026-05-22 15:57 UTC · model grok-4.3

classification 💻 cs.LG math.STstat.TH

keywords deep neural networksbinary classificationhard margin conditionexcess risk boundsempirical risk minimizationfast convergence rateslow noise conditionactivation functions

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The pith

Deep neural networks achieve arbitrarily fast excess risk rates under the hard margin condition for many activations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that deep neural networks solving an empirical risk minimization problem with square loss and an ell_p weight penalty can reach excess risk that decays faster than any fixed power of the sample size when the data satisfies the hard margin condition. The result holds for a wide range of activations including ReLU, GELU, and SiLU as long as the Bayes regression function obeys a smoothness condition matched to the input distribution. A sympathetic reader would care because this implies that in cleanly separated classification problems the networks can approach optimal performance with far fewer examples than classical bounds suggest. The same fast rates appear under ordinary Tsybakov low-noise conditions with exponent near one, and the authors prove matching lower bounds showing the rates cannot be beaten for strong noise exponents.

Core claim

For DNNs solving the ERM problem with square loss and ell_p penalty, excess risk bounds of order O(n^{-α}) are achieved with α close to 1 under Tsybakov low-noise and with arbitrarily large α greater than 1 under the hard-margin condition, for a wide range of activations including ReLU, LeakyReLU, ELU, GELU, SiLU and others, provided the Bayes regression function η satisfies a distribution-adapted smoothness condition relative to the marginal distribution ρ_X; when the activation is tanh or sigmoid the rates follow from the standard assumption that η belongs to C^s; minimax lower bounds establish that these rates are optimal whenever the noise exponent q is at least 2.

What carries the argument

A novel decomposition of the excess risk for general ERM-based classifiers together with the distribution-adapted smoothness condition on the Bayes regression function η relative to ρ_X.

If this is right

Excess risk decays at rates O(n^{-α}) with α arbitrarily large under the hard-margin condition.
The same DNN training procedure yields rates near n^{-1} under Tsybakov low-noise conditions with any positive exponent.
The fast rates hold for many common activations including ReLU, GELU, SiLU, and Softplus.
For tanh and sigmoid activations the rates follow from the classical assumption that the regression function lies in C^s.
Minimax lower bounds match the upper bounds whenever the noise exponent q is at least 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The novel excess-risk decomposition may be reusable for analyzing convergence of other ERM classifiers beyond deep networks.
The distribution-adapted smoothness condition suggests DNNs can be applied in classification settings where standard uniform smoothness assumptions are too restrictive for most activations.
One could check whether relaxing the smoothness requirement further still permits the arbitrarily fast rates for networks of fixed depth.

Load-bearing premise

The Bayes regression function must satisfy a smoothness condition that is adapted to the specific marginal distribution of the inputs rather than a uniform smoothness requirement that works for every distribution.

What would settle it

Construct a data distribution satisfying the hard-margin condition but where the excess risk achieved by DNN ERM with ReLU activation remains bounded below by a positive constant times n to the power of minus some fixed number; such an example would show that arbitrarily large alpha cannot be attained.

Figures

Figures reproduced from arXiv: 2505.08262 by Ding-Xuan Zhou, Nathanael Tepakbong, Xiang Zhou.

**Figure 1.** Figure 1: Visualization of margin conditions (A1) and (A2) through histograms of values taken by η. Assumption (A2) was originally introduced in (Mammen and Tsybakov, 1999) as a characterization of classification problems for which the two classes are in some sense “separable", and has been repeatedly shown in the literature to lead to faster rates of convergence for various hypothesis classes. Consider the regulari… view at source ↗

**Figure 2.** Figure 2: Illustration of geometric margin conditions corresponding to Remarks [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: A toy dataset which satisfies the weak margin condition. By plotting the histogram of a DNN [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: Evolution of the testing error for a DNN trained on [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 5.** Figure 5: A toy dataset which satisfies the hard margin condition. By plotting the histogram of a DNN [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: Evolution of the testing error for a DNN trained on [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗

**Figure 7.** Figure 7: Interpolation of randomly selected images respectively in the “T-shirt" and “Pullover" class. [PITH_FULL_IMAGE:figures/full_fig_p044_7.png] view at source ↗

**Figure 8.** Figure 8: Histogram of predicted labels for images in the testing dataset: the histogram matches the function [PITH_FULL_IMAGE:figures/full_fig_p045_8.png] view at source ↗

**Figure 9.** Figure 9: Interpolation of randomly selected images respectively in the “Automobile" and “Truck" class. [PITH_FULL_IMAGE:figures/full_fig_p045_9.png] view at source ↗

**Figure 10.** Figure 10: Histogram of predicted labels for images in the testing dataset: the decay as [PITH_FULL_IMAGE:figures/full_fig_p046_10.png] view at source ↗

read the original abstract

We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) under Tsybakov's low-noise condition with exponent $q>0$, as well as its limit case $q=\infty$, which we refer to as the \emph{hard margin condition}. We demonstrate that, for a wide range of commonly used activation functions (including but not limited to ReLU, LeakyReLU, ELU, CELU, SELU, Softplus, GELU, SiLU, Swish, Mish, and Softmax), DNN solutions to the empirical risk minimization (ERM) problem with square loss surrogate and $\ell_p$ penalty on the weights $(0<p<\infty)$ can achieve excess risk bounds of order $\mathcal{O}\left(n^{-\alpha}\right)$ for $\alpha$ close to $1$ under the low-noise condition, and for arbitrarily large $\alpha>1$ under the hard-margin condition, provided that the Bayes regression function $\eta$ satisfies a \emph{distribution-adapted smoothness} condition relative to the marginal data distribution $\rho_{X}$. Furthermore, when the activation function is chosen as $\tanh$ or sigmoid, we show that the same rates follow from the standard assumption that $\eta\in \mathcal{C}^s$. Finally, we establish minimax lower bounds, showing that these rates cannot be improved upon whenever $q\ge2$. Our proof relies on a novel decomposition of the excess risk for general ERM-based classifiers which might be of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's new excess-risk decomposition lets DNNs hit arbitrarily fast rates under hard margin for many activations, but only if a distribution-adapted smoothness on η holds.

read the letter

The main thing to know is that the authors give a fresh decomposition of excess risk for general ERM classifiers and use it to prove that DNNs with square loss and ℓ_p penalties reach excess risk of order n^{-α} for any large α under the hard-margin case, covering activations like ReLU, GELU, and Swish via a distribution-adapted smoothness condition on the Bayes function η relative to ρ_X. For tanh and sigmoid they fall back to ordinary C^s smoothness and still get the rates. They also supply matching lower bounds once the noise exponent q is at least 2. That combination of a reusable decomposition plus the extension past smooth activations is the concrete advance over earlier fast-rate results for neural nets under Tsybakov noise. The lower bounds add weight because they show the upper bounds are sharp in the stated regime. The work is technically grounded in the sense that it separates approximation and estimation errors cleanly and ties the rates directly to the margin parameter without obvious circularity in the parameter choices. The abstract is explicit about the assumptions and the scope of activations, which helps. The soft spot is the distribution-adapted smoothness premise itself. It is doing the heavy lifting for the non-smooth activations, and it is not yet clear how often it holds for ordinary data distributions or whether it is materially weaker than classical smoothness in the regimes where the super-fast rates are claimed. If that condition ends up requiring roughly the same regularity as C^s in practice, the gain for ReLU-type nets shrinks. The proofs are asserted to exist but the derivation steps for the approximation error under the adapted condition would need checking to confirm the rates really go through without extra restrictions. This paper is for people who work on statistical rates for neural classifiers and low-noise conditions. A reader who already follows the literature on margin-based fast rates and DNN approximation will get the most out of it. It is worth sending to a serious referee because the claims are sharp, the lower bounds are present, and the decomposition looks reusable even if the smoothness assumption needs tightening in revision.

Referee Report

3 major / 2 minor

Summary. The paper establishes excess-risk bounds for DNN classifiers trained by ERM with square loss and ℓ_p weight penalty. Under Tsybakov's low-noise condition (exponent q>0) it obtains rates O(n^{-α}) with α close to 1; under the hard-margin case q=∞ it obtains arbitrarily large α>1. These rates hold for a wide family of activations (ReLU, LeakyReLU, GELU, etc.) provided the Bayes regressor η satisfies a distribution-adapted smoothness condition relative to the marginal ρ_X; for tanh/sigmoid the standard C^s assumption suffices. Matching minimax lower bounds are shown for q≥2. The argument rests on a novel excess-risk decomposition.

Significance. If the central claims are correct, the results would be significant: they deliver super-fast rates under the hard-margin condition for activations beyond the classical tanh/sigmoid setting, and the novel decomposition may be reusable. The matching lower bounds for q≥2 are a clear strength. The distribution-adapted smoothness assumption, however, is the load-bearing premise whose verification and comparison with C^s remain to be fully substantiated.

major comments (3)

[Definition of distribution-adapted smoothness] Definition of distribution-adapted smoothness (likely §2 or §3): the condition is invoked to obtain the approximation rates needed for general activations, yet the manuscript provides no explicit verification that it holds for standard data-generating processes (e.g., Gaussian mixtures or uniform on the sphere) nor demonstrates that it is strictly weaker than the classical C^s assumption in the regimes where super-fast rates are claimed. This premise directly controls both approximation and excess-risk bounds under q=∞.
[Excess-risk decomposition] Excess-risk decomposition (the novel step referenced in the abstract): the decomposition separates approximation, estimation, and noise terms; however, the passage from the adapted smoothness of η to the required DNN approximation rate for non-smooth activations is not shown in sufficient detail to confirm that the resulting α can be made arbitrarily large under hard margin.
[Minimax lower bounds] Minimax lower-bound construction (final section): the lower bounds match the upper bounds for q≥2, but it is unclear whether the adversarial distributions used in the lower-bound argument also satisfy the distribution-adapted smoothness condition required for the upper bounds; if they do not, the claimed optimality is not fully established.

minor comments (2)

Notation for the marginal ρ_X and the adapted smoothness modulus should be introduced once and used consistently; several passages reuse symbols without redefinition.
The statement that the rates hold 'for arbitrarily large α>1' under hard margin should be accompanied by an explicit dependence of α on the smoothness parameters and network depth/width.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points that we can address through clarifications, additional examples, and expanded proof details in a revised manuscript. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Definition of distribution-adapted smoothness] Definition of distribution-adapted smoothness (likely §2 or §3): the condition is invoked to obtain the approximation rates needed for general activations, yet the manuscript provides no explicit verification that it holds for standard data-generating processes (e.g., Gaussian mixtures or uniform on the sphere) nor demonstrates that it is strictly weaker than the classical C^s assumption in the regimes where super-fast rates are claimed. This premise directly controls both approximation and excess-risk bounds under q=∞.

Authors: We agree that concrete verification would improve the presentation. In the revision we will add a dedicated remark (or short subsection) in Section 2 that explicitly verifies the distribution-adapted smoothness condition for standard examples, including isotropic Gaussian mixtures and uniform distributions on the sphere, under mild regularity assumptions on the Bayes regressor η. We will also include a direct comparison with the classical C^s assumption, showing regimes (e.g., when ρ_X has regions of low density) in which the adapted condition is strictly weaker while still permitting arbitrarily large α under q=∞. revision: yes
Referee: [Excess-risk decomposition] Excess-risk decomposition (the novel step referenced in the abstract): the decomposition separates approximation, estimation, and noise terms; however, the passage from the adapted smoothness of η to the required DNN approximation rate for non-smooth activations is not shown in sufficient detail to confirm that the resulting α can be made arbitrarily large under hard margin.

Authors: We will expand the relevant proof section (currently Section 4) to include a more detailed, step-by-step derivation that explicitly connects the distribution-adapted smoothness of η to the DNN approximation error for non-smooth activations such as ReLU. The added steps will track the dependence on the smoothness parameters and demonstrate how α can be made arbitrarily large by increasing the smoothness index of η while respecting the hard-margin condition q=∞. revision: yes
Referee: [Minimax lower bounds] Minimax lower-bound construction (final section): the lower bounds match the upper bounds for q≥2, but it is unclear whether the adversarial distributions used in the lower-bound argument also satisfy the distribution-adapted smoothness condition required for the upper bounds; if they do not, the claimed optimality is not fully established.

Authors: We will revise the lower-bound section to clarify that the adversarial distributions can be constructed so that the associated regression function η satisfies the distribution-adapted smoothness condition while preserving the Tsybakov exponent q≥2 (or the hard-margin case). This is achieved by selecting sufficiently regular η that still induce the desired noise behavior; the revised argument will explicitly verify that the same class of distributions is used for both upper and lower bounds, thereby establishing matching minimax rates under the paper’s assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit assumptions and novel decomposition

full rationale

The paper derives excess-risk bounds from the low-noise/hard-margin conditions plus the stated distribution-adapted smoothness assumption on η (or C^s for tanh/sigmoid). A novel decomposition of excess risk for general ERM classifiers is introduced and used to control approximation and estimation errors. This decomposition supplies independent content rather than reducing the target rates to a fit or to a self-citation chain. The smoothness premise is an input assumption, not a quantity fitted to the excess-risk target or smuggled via prior self-citation. Minimax lower bounds are proved separately and do not rely on the upper-bound construction. No quoted equation or step equates the claimed rates to the inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the low-noise/hard-margin condition, the distribution-adapted smoothness of η, and standard approximation properties of DNNs with the listed activations. No free parameters are explicitly fitted in the abstract statement; the smoothness index and noise exponent q are treated as given.

axioms (2)

domain assumption Tsybakov's low-noise condition with exponent q>0 (and its limit q=∞ as hard margin)
Invoked throughout to control the excess risk in terms of the regression function deviation.
domain assumption Distribution-adapted smoothness of the Bayes regression function η relative to ρ_X
Required for the approximation error bounds when using general activations; stated as the key condition enabling the rates.

pith-pipeline@v0.9.0 · 5817 in / 1668 out tokens · 41676 ms · 2026-05-22T15:57:53.841244+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We demonstrate that ... DNN solutions to the ERM problem with square loss surrogate and ℓ_p penalty ... achieve excess risk bounds of order O(n^{-α}) ... provided that the Bayes regression function η satisfies a distribution-adapted smoothness condition relative to the marginal data distribution ρ_X.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Our proof relies on a novel decomposition of the excess risk for general ERM-based classifiers

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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work page 2026