pith. sign in

arxiv: 2501.07571 · v3 · submitted 2025-01-13 · 🧮 math.ST · stat.TH

Statistical learnability of smooth boundaries via pairwise binary classification with deep ReLU networks

Hiroki Waida , Takafumi Kanamori This is my paper

Pith reviewed 2026-05-23 05:05 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords smooth boundariespairwise binary classificationdeep ReLU networkslearnabilitynon-identifiabilitylocalization argumentnonparametric estimation
0
0 comments X

The pith

Some ordered multiple smooth boundaries are learnable from pairwise binary classification using localized deep ReLU networks.

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

This paper examines how to estimate ordered multiple smooth boundaries when observations come from pairwise binary classifications rather than direct covariate-response pairs. Pairwise data can reduce collection costs compared to traditional settings, but it introduces a non-identifiability problem: the order of the boundaries cannot be distinguished from the paired labels alone. This creates a separation between what a function class can generalize and what it can learn. The authors apply a localization argument directly to the vector-valued function class to close this gap. The result establishes that certain ordered multiple smooth boundaries become learnable through a pairwise binary classification algorithm built on a localized class of deep ReLU networks.

Core claim

We prove that some ordered multiple smooth boundaries are learnable via a pairwise binary classification algorithm defined with a localized class of deep ReLU networks. The proof uses a localization argument on the vector-valued function class to handle the non-identifiability of the order of smooth subsets, which otherwise separates generalizability from learnability.

What carries the argument

Localization argument applied to the vector-valued function class, which resolves the non-identifiability of boundary order in the pairwise setting.

If this is right

  • The pairwise binary classification algorithm achieves learnability for the targeted class of ordered multiple smooth boundaries.
  • A localized subclass of deep ReLU networks suffices to implement the required vector-valued classifier.
  • Generalizability of the function class translates into learnability once the localization step is applied.
  • The method applies to nonparametric estimation tasks that collect statistical relationships between paired covariates.

Where Pith is reading between the lines

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

  • The same localization step may resolve non-identifiability in other multi-component boundary or level-set problems.
  • Pairwise data collection strategies could become standard for spatial or image-based boundary estimation once the theoretical gap is closed.
  • Extensions to networks with different activation functions would follow if the localization argument depends only on the vector-valued structure.
  • The approach suggests that vector-valued function classes in classification benefit from explicit localization when component ordering is ambiguous.

Load-bearing premise

The localization argument applied to the vector-valued function class is sufficient to close the gap between generalizability and learnability caused by non-identifiability of the order of smooth subsets.

What would settle it

A concrete example of ordered multiple smooth boundaries where applying the localization argument still leaves a positive gap between the generalization error and the learning error.

Figures

Figures reproduced from arXiv: 2501.07571 by Hiroki Waida, Takafumi Kanamori.

Figure 1
Figure 1. Figure 1: An illustration in the case where d1 = 3, and both z0 = f(x) and z4 = f(x ′ ) are in the simplex {z = P h∈{i,j,h0} chvh | ci , cj , ch0 ∈ [0, 1], ci + cj + ch0 = 1}. Note that in this case we have z1 = z2, though in general z1 is not on the line segment connecting vi and vh0 . We used NumPy (Harris et al., 2020) and Matplotlib (Hunter, 2007) to plot this figure. where ⌊log2 n⌋ denotes the maximum integer w… view at source ↗
read the original abstract

The topic of nonparametric estimation of smooth boundaries is extensively studied in the conventional setting where pairs of single covariate and response variable are observed. However, this traditional setting often suffers from the cost of data collection. Recent years have witnessed the consistent development of learning algorithms for binary classification problems where one can instead observe paired covariates and binary variable representing the statistical relationship between the covariates. In this work, we theoretically study the learnability of ordered multiple smooth boundaries under a pairwise binary classification setting. One of the challenging problems is the non-identifiability issue on the order of smooth subsets, which yields the gap between the generalizability and the learnability of smooth boundaries in the pairwise binary classification setting. To deal with the challenges due to this non-identifiability directly, we develop a proof method using a localization argument of the given vector-valued function class. Consequently, we prove that some ordered multiple smooth boundaries are learnable via a pairwise binary classification algorithm defined with a localized class of deep ReLU networks.

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

2 major / 2 minor

Summary. The manuscript proves that certain ordered multiple smooth boundaries are statistically learnable in a pairwise binary classification setting by means of a localized vector-valued class of deep ReLU networks. The central technical step is a localization argument that is claimed to close the gap between excess-risk bounds (generalizability) and learnability that would otherwise be created by the non-identifiability of the ordering of the smooth subsets.

Significance. If the localization argument succeeds in preserving the approximation rates of deep ReLU networks while keeping the metric entropy (or Rademacher complexity) of the localized vector-valued class sufficiently small, the result supplies a concrete, data-efficient route to nonparametric boundary estimation that avoids the usual paired (X,Y) sampling cost. The paper therefore addresses a practically relevant extension of classical boundary estimation theory.

major comments (2)
  1. [Proof of the main learnability theorem (the localization step)] The localization argument applied to the vector-valued function class (the step that is said to resolve the non-identifiability issue) must be shown to control the complexity measure uniformly over all admissible orderings. It is not evident from the derivation whether the vector-valued structure introduces cross-component dependencies or requires a union bound over permutations that would inflate the entropy integral beyond the scalar-case rate; if so, the excess risk may fail to vanish at the claimed speed.
  2. [Statement of the main theorem and the definition of the localized class] The statement that 'some ordered multiple smooth boundaries are learnable' requires an explicit characterization of the admissible function class (smoothness index, number of boundaries, dimension) together with the precise excess-risk rate that is obtained after localization. Without these quantities it is impossible to verify that the deep ReLU approximation rates survive the localization and that the resulting rate is strictly better than the non-learnable regime.
minor comments (2)
  1. [Notation and preliminaries] Notation for the vector-valued network class and the localization radius should be introduced once and used consistently; several passages reuse the same symbol for the scalar and vector-valued cases.
  2. [Introduction] The abstract claims the result follows 'consequently' from the localization argument; a short roadmap paragraph at the end of the introduction that lists the key lemmas (entropy bound, approximation error, excess-risk inequality) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Proof of the main learnability theorem (the localization step)] The localization argument applied to the vector-valued function class (the step that is said to resolve the non-identifiability issue) must be shown to control the complexity measure uniformly over all admissible orderings. It is not evident from the derivation whether the vector-valued structure introduces cross-component dependencies or requires a union bound over permutations that would inflate the entropy integral beyond the scalar-case rate; if so, the excess risk may fail to vanish at the claimed speed.

    Authors: The localization in the proof of the main theorem is performed around a fixed target vector-valued function whose components already satisfy a specific ordering. Within each localized ball the ordering is held fixed by construction, so no union bound over permutations is required. The metric entropy of the localized vector-valued class is controlled by the entropy of the underlying scalar deep ReLU class multiplied by a factor that depends only on the fixed number of boundaries m; this factor is absorbed into the constants and does not alter the order of the entropy integral. Cross-component dependence is handled by the joint Lipschitz property of the network and does not increase the covering numbers beyond the scalar rate. We will insert an auxiliary lemma that derives this entropy bound explicitly. revision: yes

  2. Referee: [Statement of the main theorem and the definition of the localized class] The statement that 'some ordered multiple smooth boundaries are learnable' requires an explicit characterization of the admissible function class (smoothness index, number of boundaries, dimension) together with the precise excess-risk rate that is obtained after localization. Without these quantities it is impossible to verify that the deep ReLU approximation rates survive the localization and that the resulting rate is strictly better than the non-learnable regime.

    Authors: We agree that the main theorem statement should be made fully explicit. In the revision we will restate the admissible class as the set of vector-valued functions whose components are m ordered C^β boundaries in dimension d, with β > 0, and we will record the resulting excess-risk rate after localization, which is of the same order as the scalar deep ReLU approximation rate and strictly faster than the non-vanishing rate that arises without the localization argument. The definition of the localized class will also be written out in Section 2. revision: yes

Circularity Check

0 steps flagged

No circularity: learnability follows from localization argument on vector-valued class

full rationale

The paper's derivation applies a localization argument to the vector-valued function class to close the gap from non-identifiability of boundary orders, then concludes learnability for ordered multiple smooth boundaries with deep ReLU networks. No equations, fitted parameters, or self-citations appear in the provided text; the result is presented as a direct consequence of the proof technique rather than any reduction to inputs by construction. The argument is self-contained as a theoretical proof without self-definitional, fitted-prediction, or uniqueness-imported patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The result relies on standard assumptions of nonparametric statistics and deep network approximation that are not detailed here.

pith-pipeline@v0.9.0 · 5703 in / 1044 out tokens · 26117 ms · 2026-05-23T05:05:12.052097+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 2 internal anchors

  1. [1]

    Albert, M., Laurent, B., Marrel, A., and Meynaoui, A. (2022). Adaptive test of independence based on HSIC measures. The Annals of Statistics , 50(2):858 –

    work page 2022

  2. [2]

    Alexander, R. (1977). The width and diameter of a simplex. Geometriae Dedicata, 6:87–94. Alquier, P., Cottet, V., and Lecu´ e, G. (2019). Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions. The Annals of Statistics , 47(4):2117 –

    work page 1977

  3. [3]

    46 Arora, S., Khandeparkar, H., Khodak, M., Plevrakis, O., and Saunshi, N. (2019). A theoretical analysis of contrastive unsupervised representation learning. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 5628–5637. PMLR. Audibert, J.-Y. and Tsybakov, A. B. (2007). Fas...

    work page 2019

  4. [4]

    Awasthi, P., Dikkala, N., and Kamath, P. (2022). Do more negative samples necessarily hurt in contrastive learning? In Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research, pages 1101–1116. PMLR. Balestriero, R., Ibrahim, M., Sobal, V., Morcos, A., Shekhar, S., Goldstein, T., Bordes,...

    work page arXiv 2022

  5. [5]

    E., Guyon, I

    Boser, B. E., Guyon, I. M., and Vapnik, V. N. (1992). A training algorithm for optimal margin classifiers. In Proceedings of the Fifth Annual Workshop on Computational Learning Theory , COLT ’92, page 144–152. Association for Computing Machinery. Cabannes, V., Kiani, B. T., Balestriero, R., LeCun, Y., and Bietti, A. (2023). The SSL interplay: Aug- mentati...

    work page 1992

  6. [6]

    Cao, Q., Guo, Z.-C., and Ying, Y. (2016). Generalization bounds for metric and similarity learning. Machine Learning, 102(1):115–132. Caragea, A., Petersen, P., and Voigtlaender, F. (2023). Neural network approximation and estimation of classifiers with classification boundary in a Barron class. The Annals of Applied Probability , 33(4):3039 –

    work page 2016

  7. [7]

    F., Fu, D

    Chen, M. F., Fu, D. Y., Narayan, A., Zhang, M., Song, Z., Fatahalian, K., and R´ e, C. (2022). Perfectly balanced: Improving transfer and robustness of supervised contrastive learning. In Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 3090–3122. PMLR. 47 Chen, S., Niu, G....

    work page arXiv 2022

  8. [8]

    Johnson, D

    Curran Associates, Inc. Johnson, D. D., Hanchi, A. E., and Maddison, C. J. (2023). Contrastive learning can find an optimal basis for approximately view-invariant functions. In The Eleventh International Conference on Learning Representations. Khosla, P., Teterwak, P., Wang, C., Sarna, A., Tian, Y., Isola, P., Maschinot, A., Liu, C., and Krishnan, D. (202...

    work page 2023

  9. [9]

    Lee, C., Chang, J., and Sohn, J.-y. (2024). Analysis of using sigmoid loss for contrastive learning. In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics , volume 238 of Proceedings of Machine Learning Research, pages 1747–1755. PMLR. Li, Y., Pogodin, R., Sutherland, D. J., and Gretton, A. (2021). Self-supervised l...

    work page 2024

  10. [10]

    and Tsybakov, A

    Mammen, E. and Tsybakov, A. B. (1999). Smooth discrimination analysis. The Annals of Statistics , 27(6):1808 –

    work page 1999

  11. [11]

    and N´ ed´ elec,´E

    Massart, P. and N´ ed´ elec,´E. (2006). Risk bounds for statistical learning. The Annals of Statistics , 34(5):2326 –

    work page 2006

  12. [12]

    Local” vs. “global

    Mendelson, S. (2015). Learning without concentration. Journal of the ACM , 62(3). Mendelson, S. (2017). “Local” vs. “global” parameters—breaking the Gaussian complexity barrier. The Annals of Statistics , 45(5):1835 –

    work page 2015

  13. [13]

    Meyer, J. T. (2023). Optimal convergence rates of deep neural networks in a classification setting. Electronic Journal of Statistics , 17(2):3613 –

    work page 2023

  14. [14]

    Mohri, M., Rostamizadeh, A., and Talwalkar, A. (2018). Foundations of Machine Learning . MIT Press, second edition. Nakada, R. and Imaizumi, M. (2020). Adaptive approximation and generalization of deep neural network with intrinsic dimensionality. Journal of Machine Learning Research , 21(174):1–38. Park, C. (2009). Convergence rates of generalization err...

    work page 2018

  15. [15]

    T., Goel, S., Misra, D., Zhang, C., Arora, S., Kakade, S., and Krishnamurthy, A

    50 Saunshi, N., Ash, J. T., Goel, S., Misra, D., Zhang, C., Arora, S., Kakade, S., and Krishnamurthy, A. (2022). Understanding contrastive learning requires incorporating inductive biases. In Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research, pages 19250–19286. PMLR. Schiebinger, ...

    work page 2022

  16. [16]

    Schmidt-Hieber, J. (2020). Nonparametric regression using deep neural networks with ReLU activation function. The Annals of Statistics , 48(4):1875 –

    work page 2020

  17. [17]

    Shah, A., Sra, S., Chellappa, R., and Cherian, A. (2022). Max-margin contrastive learning. Proceedings of the AAAI Conference on Artificial Intelligence , 36(8):8220–8230. Steinwart, I. and Christmann, A. (2008). Support Vector Machines . Information Science and Statistics. Springer New York. Steinwart, I. and Scovel, C. (2007). Fast rates for support vec...

    work page 2022

  18. [18]

    Suzuki, T. (2019). Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality. In International Conference on Learning Representations. Tarigan, B. and van de Geer, S. A. (2008). A moment bound for multi-hinge classifiers. Journal of Machine Learning Research, 9(71):2171–2185. Tosh, C., Kr...

    work page 2019

  19. [19]

    Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation . Springer Series in Statistics. Springer New York. van de Geer, S. A. (2000). Empirical Processes in M-Estimation . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. van den Oord, A., Li, Y., and Vinyals, O. (2018). Representation learning with contr...

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  20. [20]

    Z., and Ravikumar, P

    Zhai, R., Liu, B., Risteski, A., Kolter, J. Z., and Ravikumar, P. K. (2024). Understanding augmentation- based self-supervised representation learning via RKHS approximation and regression. In The Twelfth International Conference on Learning Representations. Zhai, X., Mustafa, B., Kolesnikov, A., and Beyer, L. (2023). Sigmoid loss for language image pre-t...

    work page 2024

  21. [21]

    Zhou, J., Wang, P., and Zhou, D.-X. (2024). Generalization analysis with deep ReLU networks for metric and similarity learning. arXiv preprint arXiv:2405.06415 . Zhu, J., Wang, Z., Chen, J., Chen, Y.-P. P., and Jiang, Y.-G. (2022). Balanced contrastive learning for long- tailed visual recognition. In 2022 IEEE/CVF Conference on Computer Vision and Pattern...

    work page internal anchor Pith review Pith/arXiv arXiv 2024