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arxiv: 2606.05599 · v1 · pith:QAEBYUCHnew · submitted 2026-06-04 · 💻 cs.LG · math.ST· stat.ME· stat.ML· stat.TH

Mitigating the Curse of Dimensionality in Uniform Convergence of Deep Neural Networks via Smooth Activations

Yizhe Ding , Runze Li , Jia Liu , Lingzhou Xue This is my paper

Pith reviewed 2026-06-28 02:37 UTC · model grok-4.3

classification 💻 cs.LG math.STstat.MEstat.MLstat.TH
keywords deep neural networksuniform convergencecurse of dimensionalitysmooth activationsnonparametric regressionpseudo-dimension boundsHölder norms
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The pith

Smooth DNNs mitigate the curse of dimensionality in uniform convergence by exploiting low-dimensional hierarchical structures.

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

Standard ReLU networks achieve minimax-optimal L2 rates but suffer the curse of dimensionality in uniform convergence, as established by a new lower bound. The paper analyzes DNNs with smooth activations to derive non-asymptotic uniform rates that adapt to the target's low-dimensional hierarchical composition structure. Novel pseudo-dimension bounds, approximation guarantees, and Hölder-norm bounds support the analysis for both feedforward and residual architectures. These rates apply across Huber, least-squares, quantile, and logistic regression. A reader would care because uniform guarantees matter for downstream tasks that demand worst-case reliability rather than average-case performance.

Core claim

Smoothly activated deep neural networks encompassing both feedforward and residual structures achieve non-asymptotic uniform convergence rates across multiple statistical contexts by deriving novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds, which allow them to adaptively exploit the low-dimensional hierarchical composition structure of the target function and thereby mitigate the curse of dimensionality in uniform convergence.

What carries the argument

Smoothly activated DNN approximators that adaptively exploit the low-dimensional hierarchical composition structure of the target function

If this is right

  • Non-asymptotic uniform convergence rates hold for Huber, least-squares, quantile, and logistic regression.
  • Smooth DNNs provide a theoretically grounded alternative to ReLU networks for tasks requiring uniform guarantees.
  • The derived rates apply to both feedforward and residual network structures.
  • Simulation studies and real-world applications confirm the theoretical uniform rates.

Where Pith is reading between the lines

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

  • The same smooth-activation approach could extend to other regression or classification settings that require uniform control.
  • Practitioners facing high-dimensional data with suspected compositional structure might switch to smooth activations for improved worst-case reliability.
  • The framework invites comparison with other nonparametric estimators that also assume hierarchical low-dimensional structure.

Load-bearing premise

The target function possesses a low-dimensional hierarchical composition structure that the smooth DNN approximators can exploit.

What would settle it

A demonstration that smooth DNN uniform convergence rates still scale exponentially with ambient dimension when the target function lacks low-dimensional hierarchical composition structure.

Figures

Figures reproduced from arXiv: 2606.05599 by Jia Liu, Lingzhou Xue, Runze Li, Yizhe Ding.

Figure 1
Figure 1. Figure 1: ReLU and its non-C ∞ variants (left); and its C ∞ variants (right). differentiable function. The explicit functional forms are given by: SiLU(x) =x · exp(x) 1 + exp(x) ; (5) GELU(x) =x · 1 2  1 + tanhp 2/π (x + 0.044715 x 3 )   ; (6) Mish(x) =x · tanh log(1 + exp(x)) . (7) Here we use the hyperbolic tangent version of the GELU activation, and its associated func￾tion ψ approximates the distribution fun… view at source ↗
Figure 2
Figure 2. Figure 2: Architectures of an FNN (left) and residual blocks with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Side-by-side boxplots of the L 2 (P) estimation errors. 0.25 0.5 1.0 2.0 4.0 0 2 4 6 8 10 L error L error under noise=t2 0.25 0.5 1.0 2.0 4.0 0 2 4 6 8 10 L error L error under noise=t4 n=512 ReLU n=512 SiLU FNN n=512 SiLU RN n=1024 ReLU n=1024 SiLU FNN n=1024 SiLU RN n=2048 ReLU n=2048 SiLU FNN n=2048 SiLU RN [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Side-by-side boxplots of the L ∞ estimation errors. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Overall counterfactual difference by SiLU ResNets in ozone concentration. [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
read the original abstract

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and H\"older-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves a lower bound showing that least-squares ReLU DNN estimators suffer the curse of dimensionality in uniform norm. It then derives novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for smoothly activated feedforward and residual DNNs. These are used to obtain non-asymptotic uniform convergence rates for Huber, least-squares, quantile, and logistic regression estimators that mitigate the curse by exploiting the low-dimensional hierarchical composition structure of the target function. The claims are supported by simulations and a real-data example.

Significance. If the derivations hold, the work supplies a concrete theoretical distinction between ReLU and smooth activations for uniform-norm guarantees, which is relevant for downstream tasks needing worst-case reliability. The explicit lower bound paired with matching upper bounds under the structural assumption, together with the multi-context regression results, strengthens the contribution over purely approximation-theoretic comparisons.

minor comments (3)
  1. [Abstract] Abstract: the claim of 'novel pseudo-dimension bounds' would be strengthened by a brief comparison sentence to the best existing ReLU pseudo-dimension results.
  2. [Section 5] The non-asymptotic rates are stated to hold 'across multiple statistical contexts'; a short table summarizing the precise rate exponents and the role of the smoothness parameter for each context (Huber, quantile, etc.) would improve readability.
  3. [Section 6] The simulation section reports empirical uniform errors but does not state the number of Monte Carlo repetitions or whether error bars reflect variability over random seeds; adding this information would make the numerical support more reproducible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so we have no point-by-point responses to provide at this stage. We will make the minor revisions as appropriate in the next version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives uniform convergence rates for smooth DNNs via pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm controls, then invokes the low-dimensional hierarchical composition structure as an explicit modeling assumption to obtain dimension-free rates. These steps rest on standard tools from statistical learning theory and approximation theory without any reduction of a claimed prediction to a fitted parameter, self-definitional loop, or load-bearing self-citation chain. The structural assumption is stated as the mechanism for mitigation and is consistent with external nonparametric benchmarks; no equation or result is shown to equal its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on standard mathematical assumptions about Holder norms, pseudo-dimensions, and the existence of low-dimensional hierarchical structure in the target function.

axioms (2)
  • domain assumption Target functions admit low-dimensional hierarchical composition structure
    Invoked to obtain adaptive uniform rates that mitigate curse of dimensionality
  • domain assumption Smooth activations satisfy required differentiability for Holder-norm bounds
    Used in approximation guarantees for feedforward and residual networks

pith-pipeline@v0.9.1-grok · 5766 in / 1251 out tokens · 34714 ms · 2026-06-28T02:37:09.137583+00:00 · methodology

discussion (0)

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Reference graph

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