Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A residual-based ReLU network estimator learns conditional variance at near-parametric rates and builds bootstrap confidence intervals with provable coverage.

desk verdict Solid variance estimation rates for ReLU networks; the bootstrap coverage theorem is not proven as written. read the letter →

arxiv 2412.20355 v2 pith:TRRMVKO6 submitted 2024-12-29 stat.ML cs.LG

Carlos Misael Madrid Padilla , Oscar Hernan Madrid Padilla , Yik Lun Kei , Zhi Zhang , Yanzhen Chen This is my paper · ORCID
classification stat.MLcs.LG MSC 62G0862G1568T07
keywords conditionalvarianceestimationReLUnetworksnonparametricregressionbootstrapconfidenceintervalssub-Exponentialnoiseheteroscedasticityuncertaintyquantificationdense
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that dense ReLU networks can do two things at once: estimate the conditional variance in a nonparametric regression and produce a confidence interval for the conditional mean with a finite-sample coverage guarantee. The method is the classic residual recipe, fit the mean with a ReLU network, square the residuals, and fit a second ReLU network to those squares. The main theoretical claims are a non-asymptotic $L^2$ error bound for the variance estimator, stated as the first of its kind for ReLU networks, and a bootstrap procedure whose interval covers $f^*(X)$ with probability at least $1-\alpha$. The noise assumption is relaxed to sub-Exponential errors, and simulations show the residual-based network beating random forests and MARS on variance estimation.

What carries the argument

The load-bearing object is the dense ReLU network class $F(L,\nu)$, fully connected feedforward networks with ReLU activation and all hidden layers of equal width. Three mechanisms carry the argument: the residual construction (estimate the conditional mean, square the residuals, and regress those squares to estimate the conditional variance), the approximation and entropy properties of deep ReLU networks over hierarchical composition classes, and a bootstrap correction that computes quantiles of prediction errors on a held-out validation set rather than quantiles of fitted values. The coverage proof combines concentration for retrained networks with DKW-type bounds on the empirical quantile adjustment.

What would settle it

Simulate a known pair $(f^*, g^*)$ in the hierarchical composition class, for example $f^*(x)=\sin(2\pi x_1)+x_2^2$ and $g^*(x)=\exp(-|x_1|-|x_2|)$, with sub-Exponential noise, then measure $\|\hat{g}-g^*\|^2$ on a large test set at $n=10^4,10^5,10^6$; if the error does not decay at the stated $\mathrm{poly}_1(\log n)(\phi_n+\psi_n+1/n)$ rate, Corollary 1 is falsified. For the confidence statement, fix $f^*$ and repeat the full bootstrap procedure many times, counting whether $f^*(X)$ falls in the interval; if the empirical coverage is systematically below $1-\alpha$ under the paper's assumptions, Theorem 5 is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that when the true conditional mean $f^*$ and conditional variance $g^*$ belong to a hierarchical composition class, the residual-based dense ReLU estimator $\hat{g}_{B_n}$ satisfies $\|g^* - \hat{g}_{B_n}\|^2_{L^2} = o_P(\mathrm{poly}_1(\log n)(\phi_n+\psi_n+1/n))$, and the bootstrap interval $[\frac{1}{eB}\sum_{j=B+1}^{B+eB} \hat{f}_{A_n}^{(j)}(X) \pm \Delta(\alpha)]$ covers $f^*(X)$ with probability at least $1-\alpha$. This is claimed as the first non-asymptotic conditional variance bound for ReLU networks and the first coverage-guaranteed bootstrap confidence interval in deep ReLU regression, with sub-Exponential noise allowed.

Load-bearing premise

The rates inherit from an external approximation theorem, cited in the paper, asserting that dense ReLU networks uniformly approximate functions in the hierarchical composition class at the advertised speed; if that sup-norm approximation only holds in $L^2$, the stated rate $r_n$ and the downstream variance and coverage results do not follow.

Editorial extensions

If this is right

  • Conditional variance estimation in heteroscedastic nonparametric regression becomes feasible with deep ReLU networks at non-asymptotic rates, beyond the usual kernel or local-polynomial settings.
  • The bootstrap confidence interval can be used when the regression function is a hierarchical composition and the noise is only known to be sub-Exponential, without a parametric error distribution.
  • The non-asymptotic bounds give concrete guidance for choosing the clipping level, retraining count, and interval width from the closed-form expressions in the theorems.
  • Because the general variance bound transfers to any mean estimator with a known $L^2$ rate, the same three-step framework applies to other function classes beyond ReLU networks.
  • In the homoscedastic special case the scalar variance estimator attains rate $\mathrm{poly}_1(\log n)(\phi_n+1/n)$, so constant-variance models are covered as a limit of the main result.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof strategy suggests the bounded-response assumption behind the coverage theorem could be replaced by a truncation argument for sub-Exponential $Y$, although the paper's stated result requires $P(|Y|\le A_n)=1$.
  • The held-out prediction-error bootstrap is a template that could be adapted to other deep uncertainty-quantification methods currently relying on heuristic ensembles; this adaptation is not part of the paper.
  • The rate in Corollary 1 implies a practical misspecification diagnostic: if a cross-validated residual-variance plot systematically disagrees with the fitted variance surface, the assumed hierarchical composition structure is likely wrong.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies nonparametric regression with dense ReLU networks under heteroscedastic noise. It proposes a residual-based estimator of the conditional variance function g*, derives nonasymptotic L2 error bounds for the conditional mean (Theorem 2) and for the conditional variance in heteroscedastic and homoscedastic settings (Theorems 3–4 and Corollaries 1–2), and then constructs a bootstrap confidence interval for the conditional mean f*(X) with a claimed finite-sample coverage guarantee (Theorem 5 and Corollary 3). The experimental section compares the proposed ReLU variance estimator with random forests and MARS on simulated and real data, and evaluates the coverage and length of the proposed confidence intervals.

Significance. If the results are correct, the paper would be a useful contribution to the theory of neural-network-based uncertainty quantification: it extends dense ReLU mean-estimation rates to sub-Exponential noise, gives the first nonasymptotic conditional-variance rates for ReLU networks under hierarchical composition assumptions, and proposes a bootstrap procedure with a theoretical coverage guarantee. The residual-based variance framework and the explicit nonasymptotic rates in Theorems 1–4 are coherent and follow standard empirical-process and approximation arguments. However, the central coverage claim for the confidence interval is not established as written: Theorem 5 contains an impossible hypothesis, and the proof uses a conditional-independence step that is invalid because it conditions on a common data-dependent event. In addition, the computable version of the interval in Corollary 3 rests on a strong high-probability condition whose derivation from the earlier rates is not supplied. These issues are load-bearing for the paper's headline confidence-interval contribution, so the manuscript needs substantial revision.

major comments (4)
  1. [Section 2.2 and Theorem 5] The hypothesis 'An > max{An, Bn}' is impossible, since the right-hand side contains An itself. This appears both in Section 2.2 ('we will assume that An is large enough such that An > max{An, Bn}') and in the statement of Theorem 5. Because this is an assumption of the main coverage theorem, Theorem 5 is vacuous as stated. The authors presumably intend a lower bound involving Bn and some norm of f*, but the condition must be corrected and the proof re-checked under the corrected statement.
  2. [Appendix E.1, Step 2] The proof conditions on the common event Ω1 = {‖fhat_An^{(j)} - f*‖_{L2}^2 ≤ a(α) for j = B+1,...,B+eB} and then writes Var(Σ_j Z_j | X, Ω1, D_{1:3}) = Σ_j Var(Z_j | X, Ω1, D_{1:3}), claiming the bootstrap fits are conditionally independent given X, Ω1, and D_{1:3}. This is not valid: the bootstrap fits are conditionally independent given the data D_{1:3} and X, but Ω1 is an intersection of events involving all j jointly, and the threshold a(α) is itself data-dependent through the validation set I4 and through ĝ^{(B+1)}. Conditioning on a common, non-product event destroys conditional independence, so the variance decomposition used to obtain the α/4 concentration bound does not follow. The subsequent bound P(Ω1^c) < 0.58α does not repair this step because the concentration inequality is applied conditionally on Ω1. Consequently, the proof of Theorem 5 and Corollary 3 does not currently establish the advertised coverage guarantee (35).
  3. [Corollary 3, condition (38)] The condition n^{q+1} P(max{‖g* - ĝ_An^{(B+1)}‖_{L2}^2, ‖f* - fhat_An^{(B+1)}‖_{L2}^2} > ε_n) → 0 with ε_n ≲ α^2/log^{2s} n is assumed rather than derived. The paper's earlier results, Theorem 2 and Corollary 1, provide only o_P rates of convergence; they do not imply the much stronger tail bound with polynomial factor n^{q+1}. Since Corollary 3 is what converts the oracle quantities a0 and b(α) in Theorem 5 into the computable choices used for the 'NN' method in Section 4.2, the coverage guarantee for the implemented procedure is not established. The authors should either prove (38) from the network approximation and empirical-process bounds or clearly state it as an additional unverified assumption and adjust the claims accordingly.
  4. [Appendix E, proof of Theorem 2] The proof relies on Theorem 3 of Kohler and Langer (2019) to assert the existence of a network fbar in F(L,ν) with uniform approximation error ‖fbar - f*‖∞ ≤ √ϕ_n. The manuscript does not state the hypotheses of that theorem or verify that the hierarchical composition class H(l1,P1) satisfies them. If the cited theorem only establishes approximation in L2(P) rather than in sup-norm, then the rate r_n in (25) does not follow, and the downstream variance rates in Corollary 1 inherit the gap. The authors should provide the precise statement of the external result and confirm that it yields the sup-norm approximation used here.
minor comments (5)
  1. [Theorem 3] The sentence 'so that ϕn is the approximating error' in the statement of Theorem 3 should refer to ψn, since the approximation is of g*.
  2. [Theorem 4] In the statement of Theorem 4, 'Bno(nb)' is a typo; it should read Bn = o(n^b).
  3. [Theorem 5, equation (34)] Equation (34) has a mismatched parenthesis: the expression should be 32[E((fhat_An^{(B+1)}(X) - Y)^2) - E(g*(X))]^{1/2} / (5α(1 - 0.58α)), not 32[E((fhat_An^{(B+1)}(X) - Y)^2 - E(g*(X))]^{1/2}.
  4. [Section 2.2, Step 5] The definition of a(α) in (14) uses a nested absolute value and parentheses that are difficult to parse; the authors should clarify the expression, especially which terms are inside the outer absolute value.
  5. [Abstract and Section 1.1] The claim that this is 'the first result for variance estimation using ReLU networks' should be qualified with respect to the specific nonasymptotic, dense-network setting considered here, since the prior literature includes various neural-network variance estimators without such theoretical guarantees.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's variance-estimation rates and bootstrap coverage proof are assembled from external approximation results and generic empirical-process bounds, not from the paper's own conclusions.

full rationale

The derivation chain is not circular in the sense defined here. The main variance-estimation result, Corollary 1, is obtained by inserting into Theorem 3 the mean-estimation rate rn from Theorem 2; Theorem 2 is in turn proven from the empirical-process Lemmas 3 through 6 and from the approximation theorem of Kohler and Langer (2019), which is external to this paper. No parameter is fitted to a subset of the data and then relabeled as a prediction: the bounds are stated for empirical-risk minimizers (7), (8), and (10) and are proved by standard oracle inequalities. The bootstrap confidence interval in Theorem 5 does not assume the coverage it proves; the half-width Delta(alpha) is built from a validation quantile a1(alpha) and concentration terms a2, a3, a0, and b(alpha), and the proof separately bounds the variance term, the bias term, and the failure event P(Omega_1^c). The self-citations (Padilla 2024, Padilla et al. 2022, Zhang et al. 2024) appear only in the literature review and are not load-bearing; no 'uniqueness theorem' from the authors is invoked to force a choice. The reliance on Kohler and Langer (2019, Theorem 3) for the existence of an approximating network is external (different authors) and would be a correctness risk if the cited theorem's assumptions are not met, but it is not circular. A separate concern is an apparent proof gap in Theorem 5, Appendix E.1 Step 2, where conditioning on the common event Omega_1 may break the conditional independence of the bootstrap draws, and there is the typo 'An > max{An, Bn}'; these are correctness issues rather than instances of circularity. Under the requested standard, no circular step is exhibited.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The central claims depend on standard empirical process tools, prior VC dimension and approximation results, and structural assumptions on f* and g*. No fitted parameters appear in the derivations; the only user-chosen constant is s in the correction terms. No new entities are postulated.

free parameters (1)
  • s = user-specified positive constant (e.g., s=2 in experiments)
    Controls the logarithm exponent in the deterministic correction terms a0=alpha/(100 log^s n) and b(alpha)=1/(100 log^s n); the theory allows any s>0 but the stated coverage guarantee depends on the assumed rate condition (38) with epsilon_n <= alpha^2/log^{2s} n.
assumptions (7)
  • standard math Empirical process tools: symmetrization, peeling, Dudley inequality, DKW inequality
    Used throughout the proofs in Section A; these are unproved background results relied upon for the deviation bounds and critical radius arguments.
  • standard math VC dimension bound for ReLU networks from Bartlett et al. (2019)
    Invoked in the proof of Theorem 2 to convert network complexity into an entropy bound. The paper cites the result but does not prove it.
  • domain assumption Sup-norm approximation of hierarchical composition functions by dense ReLU networks at rate sqrt(phi_n), from Kohler-Langer (2019, Theorem 3)
    Theorem 2's proof states 'from Theorem 3 in Kohler and Langer (2019), it follows that there exists f_bar in F(L,nu) with ||f_bar - f*||_inf <= sqrt(phi_n)'. This is load-bearing for the mean estimation rate and downstream results.
  • domain assumption f* in H(l1,P1) and g* in H(l2,P2) with max{p_max,K_max}=O(1)
    Theorem 2 and Corollary 1 require the true mean and variance functions to belong to hierarchical composition classes with bounded smoothness and input dimension.
  • domain assumption Error tail condition: P(||epsilon||_inf > U_n) -> 0
    Replaces the sub-Gaussian assumption; accommodates sub-Exponential and more general noise, used in (21) and throughout.
  • domain assumption Bounded response: P(|Y| <= A_n) = 1
    Assumed in Theorem 5 for the confidence interval coverage guarantee; the authors note it can be relaxed with more effort.
  • ad hoc to paper Condition (38) in Corollary 3: n^{q+1} P(max{||g*-g_hat||^2_L2, ||f*-f_hat||^2_L2} > epsilon_n) -> 0 with epsilon_n <= alpha^2/log^{2s} n
    This rate condition is assumed rather than derived from Theorem 2/Corollary 1 in the text; the deterministic choices of a0 and b(alpha) are proven valid only under this condition.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks." pith.science (2026). https://pith.science/paper/TRRMVKO6

@misc{pith2026241220355,
  author       = {Pith},
  title        = {Pith review of: Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRRMVKO6}},
  note         = {Machine review of arXiv:2412.20355}
}
read the original abstract

This paper addresses the problems of conditional variance estimation and confidence interval construction in nonparametric regression using dense networks with the Rectified Linear Unit (ReLU) activation function. We present a residual-based framework for conditional variance estimation, deriving nonasymptotic bounds for variance estimation under both heteroscedastic and homoscedastic settings. We relax the sub-Gaussian noise assumption, allowing the proposed bounds to accommodate sub-Exponential noise and beyond. Building on this, for a ReLU neural network estimator, we derive non-asymptotic bounds for both its conditional mean and variance estimation, representing the first result for variance estimation using ReLU networks. Furthermore, we develop a ReLU network based robust bootstrap procedure (Efron, 1992) for constructing confidence intervals for the true mean that comes with a theoretical guarantee on the coverage, providing a significant advancement in uncertainty quantification and the construction of reliable confidence intervals in deep learning settings.

Figures

Figures reproduced from arXiv: 2412.20355 by the authors.

Figure 1
Figure 1. The true variance function g ∗ (·) and the predicted ˆg(·) from the estimator based on the residuals. The rows from top to bottom refer to Scenario 1,2, and 3 with xi ∈ R 2 . For visualization purposes, regions with variance greater than 1 are colored in grey. errors in variance estimation are sub-Exponential, enabling us to validate our results beyond the sub-Gaussian setting. We examine five distinct scenarios bas… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Calibration Prediction Interval for Non-parametric Regression and Neural Networks

    stat.ME 2025-09 conditional novelty 5.0 of 10

    The authors construct prediction intervals by calibrating estimated conditional CDF values on a grid, and show asymptotic validity for DNN and kernel estimators, with a finite-sample coverage guarantee only under an o...

Reference graph

Works this paper leans on

66 extracted references · 53 canonical work pages · cited by 1 Pith paper

  1. [1]

    Robust variance estimation for the case-cohort design

    William E Barlow. Robust variance estimation for the case-cohort design. Biometrics, pages 1064--1072, 1994

    work page 1994

  2. [2]

    Nearly-tight vc-dimension and pseudodimension bounds for piecewise linear neural networks

    Peter L Bartlett, Nick Harvey, Christopher Liaw, and Abbas Mehrabian. Nearly-tight vc-dimension and pseudodimension bounds for piecewise linear neural networks. The Journal of Machine Learning Research, 20 0 (1): 0 2285--2301, 2019

    work page 2019

  3. [3]

    On deep learning as a remedy for the curse of dimensionality in nonparametric regression

    Benedikt Bauer and Michael Kohler. On deep learning as a remedy for the curse of dimensionality in nonparametric regression. 2019

    work page 2019

  4. [4]

    Random forests

    Leo Breiman. Random forests. Machine learning, 45 0 (1): 0 5--32, 2001

    2001

  5. [5]

    Confidence intervals for nonparametric regression

    Lawrence D Brown, Xin Fu, and Linda H Zhao. Confidence intervals for nonparametric regression. Journal of Nonparametric Statistics, 23 0 (1): 0 149--163, 2011

    work page 2011

  6. [6]

    Adaptive variance function estimation in heteroscedastic nonparametric regression

    T Tony Cai and Lie Wang. Adaptive variance function estimation in heteroscedastic nonparametric regression. The Annals of Statistics, 36 0 (5): 0 2025--2054, 2008

    work page 2025

  7. [7]

    Variance function estimation in multivariate nonparametric regression with fixed design

    T Tony Cai, Michael Levine, and Lie Wang. Variance function estimation in multivariate nonparametric regression with fixed design. Journal of Multivariate Analysis, 2009

    work page 2009

  8. [8]

    Adaptive confidence bands for nonparametric regression functions

    T Tony Cai, Mark Low, and Zongming Ma. Adaptive confidence bands for nonparametric regression functions. Journal of the American Statistical Association, 109 0 (507): 0 1054--1070, 2014

    work page 2014

Show all 66 references
  1. [9]

    Simultaneous confidence bands for nonparametric regression with functional data

    David A Degras. Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica, pages 1735--1765, 2011

    2011

  2. [10]

    Confidence bands in non-parametric errors-in-variables regression

    Aurore Delaigle, Peter Hall, and Farshid Jamshidi. Confidence bands in non-parametric errors-in-variables regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 77 0 (1): 0 149--169, 2015

    2015

  3. [11]

    Bootstrap methods: another look at the jackknife

    Bradley Efron. Bootstrap methods: another look at the jackknife. In Breakthroughs in statistics: Methodology and distribution, pages 569--593. Springer, 1992

    1992

  4. [12]

    An introduction to the bootstrap

    Bradley Efron and Robert J Tibshirani. An introduction to the bootstrap. Chapman and Hall/CRC, 1994

    1994

  5. [13]

    Confidence bands in nonparametric regression

    Randall L Eubank and Paul L Speckman. Confidence bands in nonparametric regression. Journal of the American Statistical Association, 88 0 (424): 0 1287--1301, 1993

    1993

  6. [14]

    Efficient estimation of conditional variance functions in stochastic regression

    Jianqing Fan and Qiwei Yao. Efficient estimation of conditional variance functions in stochastic regression. Biometrika, 85 0 (3): 0 645--660, 1998

    1998

  7. [15]

    Deep neural networks for estimation and inference

    Max H Farrell, Tengyuan Liang, and Sanjog Misra. Deep neural networks for estimation and inference. Econometrica, 89 0 (1): 0 181--213, 2021

    2021

  8. [16]

    Bootstrapping neural networks

    J \"u rgen Franke and Michael H Neumann. Bootstrapping neural networks. Neural computation, 12 0 (8): 0 1929--1949, 2000

    1929

  9. [17]

    Multivariate adaptive regression splines

    Jerome H Friedman. Multivariate adaptive regression splines. The Annals of Statistics, pages 1--67, 1991

    1991

  10. [18]

    Fast mars

    Jerome H Friedman. Fast mars. Technical report, Department of Statistics, Stanford University, 1993

    1993

  11. [19]

    Dropout as a bayesian approximation: Representing model uncertainty in deep learning

    Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pages 1050--1059. PMLR, 2016

    2016

  12. [20]

    Extremely randomized trees

    Pierre Geurts, Damien Ernst, and Louis Wehenkel. Extremely randomized trees. Machine learning, 63: 0 3--42, 2006

    2006

  13. [21]

    Variance reduction techniques for estimating value-at-risk

    Paul Glasserman, Philip Heidelberger, and Perwez Shahabuddin. Variance reduction techniques for estimating value-at-risk. Management Science, 46 0 (10): 0 1349--1364, 2000

    2000

  14. [22]

    Generative adversarial nets

    Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in neural information processing systems, 27, 2014

    2014

  15. [23]

    Long short-term memory

    Alex Graves and Alex Graves. Long short-term memory. Supervised sequence labelling with recurrent neural networks, pages 37--45, 2012

    2012

  16. [24]

    o rfi, Michael K \

    L \'a szl \'o Gy \"o rfi, Michael K \"o hler, Adam Krzy \.z ak, and Harro Walk. A distribution-free theory of nonparametric regression, volume 1. Springer, 2002

    2002

  17. [25]

    A distribution-free theory of nonparametric regression

    L \'a szl \'o Gy \"o rfi, Michael Kohler, Adam Krzyzak, and Harro Walk. A distribution-free theory of nonparametric regression. Springer Science & Business Media, 2006

    2006

  18. [26]

    On bootstrap confidence intervals in nonparametric regression

    Peter Hall. On bootstrap confidence intervals in nonparametric regression. The Annals of Statistics, pages 695--711, 1992

    1992

  19. [27]

    Variance function estimation in regression: the effect of estimating the mean

    Peter Hall and Raymond J Carroll. Variance function estimation in regression: the effect of estimating the mean. Journal of the Royal Statistical Society: Series B (Methodological), 51 0 (1): 0 3--14, 1989

    1989

  20. [28]

    A simple bootstrap method for constructing nonparametric confidence bands for functions

    Peter Hall and Joel Horowitz. A simple bootstrap method for constructing nonparametric confidence bands for functions. The Annals of Statistics, pages 1892--1921, 2013

    1921

  21. [29]

    Nonasymptotic bounds on the l 2 error of neural network regression estimates

    Michael Hamers and Michael Kohler. Nonasymptotic bounds on the l 2 error of neural network regression estimates. Annals of the Institute of Statistical Mathematics, 58: 0 131--151, 2006

    2006

  22. [30]

    Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands

    Wolfgang H \"a rdle and Adrian W Bowman. Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bands. Journal of the American Statistical Association, 83 0 (401): 0 102--110, 1988

    1988

  23. [31]

    Robust conditional variance estimation and value-at-risk

    Richard DF Harris and Cherif Guermat. Robust conditional variance estimation and value-at-risk. Available at SSRN 254569, 2000

    2000

  24. [32]

    Adaptive regression estimation with multilayer feedforward neural networks

    Michael Kohler and Adam Krzy \.z ak. Adaptive regression estimation with multilayer feedforward neural networks. Nonparametric Statistics, 17 0 (8): 0 891--913, 2005

    2005

  25. [33]

    On the rate of convergence of fully connected very deep neural network regression estimates

    Michael Kohler and Sophie Langer. On the rate of convergence of fully connected very deep neural network regression estimates. arXiv preprint arXiv:1908.11133, 2019

    1908 arXiv

  26. [34]

    Variance function estimation in high-dimensions

    Mladen Kolar and James Sharpnack. Variance function estimation in high-dimensions. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 875--882, 2012

    2012

  27. [35]

    Imagenet classification with deep convolutional neural networks

    Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. Communications of the ACM, 60 0 (6): 0 84--90, 2017

    2017

  28. [36]

    Simple and scalable predictive uncertainty estimation using deep ensembles

    Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017

    2017

  29. [37]

    Bootstrapping neural processes

    Juho Lee, Yoonho Lee, Jungtaek Kim, Eunho Yang, Sung Ju Hwang, and Yee Whye Teh. Bootstrapping neural processes. Advances in neural information processing systems, 33: 0 6606--6615, 2020

    2020

  30. [38]

    Theoretical analysis of deep neural networks for temporally dependent observations

    Mingliang Ma and Abolfazl Safikhani. Theoretical analysis of deep neural networks for temporally dependent observations. Advances in Neural Information Processing Systems, 35: 0 37324--37334, 2022

    2022

  31. [39]

    Locally apadtive regression splines

    Enno Mammen and Sara van de Geer. Locally apadtive regression splines. Annals of Statistics, 25 0 (1): 0 387--413, 1997

    1997

  32. [40]

    The tight constant in the dvoretzky-kiefer-wolfowitz inequality

    Pascal Massart. The tight constant in the dvoretzky-kiefer-wolfowitz inequality. The annals of Probability, pages 1269--1283, 1990

    1990

  33. [41]

    Convergence rates for single hidden layer feedforward networks

    Daniel F McCaffrey and A Ronald Gallant. Convergence rates for single hidden layer feedforward networks. Neural Networks, 7 0 (1): 0 147--158, 1994

    1994

  34. [42]

    earth: Multivariate Adaptive Regression Splines, 2024

    Stephen Milborrow, Trevor Hastie, and Rob Tibshirani. earth: Multivariate Adaptive Regression Splines, 2024. URL https://CRAN.R-project.org/package=earth. R package version 5.3.4

    2024

  35. [43]

    Simultaneous bootstrap confidence bands in nonparametric regression

    Michael H Neumann and J \"o rg Polzehl. Simultaneous bootstrap confidence bands in nonparametric regression. Journal of Nonparametric Statistics, 9 0 (4): 0 307--333, 1998

    1998

  36. [44]

    Why aren’t bootstrapped neural networks better? In 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Virtualonly Conference, 2020

    J Nixon, D Tran, and B Lakshminarayanan. Why aren’t bootstrapped neural networks better? In 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Virtualonly Conference, 2020

    2020

  37. [45]

    Assessing and improving neural network predictions by the bootstrap algorithm

    Gerhard Paass. Assessing and improving neural network predictions by the bootstrap algorithm. Advances in Neural Information Processing Systems, 5, 1992

    1992

  38. [46]

    Sparse spatial autoregressions

    R Kelley Pace and Ronald Barry. Sparse spatial autoregressions. Statistics & Probability Letters, 33 0 (3): 0 291--297, 1997

    1997

  39. [47]

    Variance estimation in graphs with the fused lasso

    Oscar Hernan Madrid Padilla. Variance estimation in graphs with the fused lasso. Journal of Machine Learning Research, 25 0 (250): 0 1--45, 2024

    2024

  40. [48]

    Risk bounds for quantile trend filtering

    Oscar Hernan Madrid Padilla and Sabyasachi Chatterjee. Risk bounds for quantile trend filtering. arXiv preprint arXiv:2007.07472, 2020

    2007 arXiv

  41. [49]

    Quantile regression with relu networks: Estimators and minimax rates

    Oscar Hernan Madrid Padilla, Wesley Tansey, and Yanzhen Chen. Quantile regression with relu networks: Estimators and minimax rates. Journal of Machine Learning Research, 23 0 (247): 0 1--42, 2022

    2022

  42. [50]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in P ython. Journal of Machine Learnin...

    2011

  43. [51]

    Convex regression with interpretable sharp partitions

    Ashley Petersen, Noah Simon, and Daniela Witten. Convex regression with interpretable sharp partitions. The Journal of Machine Learning Research, 17 0 (1): 0 3240--3270, 2016

    2016

  44. [52]

    On confidence bands for multivariate nonparametric regression

    Katharina Proksch. On confidence bands for multivariate nonparametric regression. Annals of the Institute of Statistical Mathematics, 68: 0 209--236, 2016

    2016

  45. [53]

    Nonlinear total variation based noise removal algorithms

    Leonid I Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: nonlinear phenomena, 60 0 (1-4): 0 259--268, 1992

    1992

  46. [54]

    Nonparametric regression using deep neural networks with relu activation function

    Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. 2020

    2020

  47. [55]

    Optimal estimation of variance in nonparametric regression with random design

    Yandi Shen, Chao Gao, Daniela Witten, and Fang Han. Optimal estimation of variance in nonparametric regression with random design. The Annals of Statistics, 48 0 (6): 0 3589--3618, 2020

    2020

  48. [56]

    Neural bootstrapper

    Minsuk Shin, Hyungjoo Cho, Hyun-seok Min, and Sungbin Lim. Neural bootstrapper. Advances in Neural Information Processing Systems, 34: 0 16596--16609, 2021

    2021

  49. [57]

    Deepface: Closing the gap to human-level performance in face verification

    Yaniv Taigman, Ming Yang, Marc'Aurelio Ranzato, and Lior Wolf. Deepface: Closing the gap to human-level performance in face verification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1701--1708, 2014

    2014

  50. [58]

    Tibshirani

    Ryan J. Tibshirani. Adaptive piecewise polynomial estimation via trend filtering. The Annals of Statistics, 42 0 (1): 0 285--323, 2014

    2014

  51. [59]

    Weak convergence

    Aad W Van Der Vaart and Jon A Wellner. Weak convergence. In Weak convergence and empirical processes, pages 16--28. Springer, 1996

    1996

  52. [60]

    Attention is all you need

    A Vaswani. Attention is all you need. NeurIPS, 2017

    2017

  53. [61]

    Methods to estimate the between-study variance and its uncertainty in meta-analysis

    Areti Angeliki Veroniki, Dan Jackson, Wolfgang Viechtbauer, Ralf Bender, Jack Bowden, Guido Knapp, Oliver Kuss, Julian PT Higgins, Dean Langan, and Georgia Salanti. Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research synthesis methods,...

    2016

  54. [62]

    High-dimensional statistics: A non-asymptotic viewpoint, volume 48

    Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint, volume 48. Cambridge University Press, 2019

    2019

  55. [63]

    Effect of mean on variance function estimation in nonparametric regression

    Lie Wang, Lawrence D Brown, T Tony Cai, and Michael Levine. Effect of mean on variance function estimation in nonparametric regression. The Annals of Statistics, 36 0 (2): 0 646--664, 2008

    2008

  56. [64]

    Some asymptotic results for learning in single hidden-layer feedforward network models

    Halbert White. Some asymptotic results for learning in single hidden-layer feedforward network models. Journal of the American Statistical association, 84 0 (408): 0 1003--1013, 1989

    1989

  57. [65]

    Bias-corrected confidence bands in nonparametric regression

    Yingcun Xia. Bias-corrected confidence bands in nonparametric regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60 0 (4): 0 797--811, 1998

    1998

  58. [66]

    Dense relu neural networks for temporal-spatial model

    Zhi Zhang, Carlos Misael Madrid Padilla, Xiaokai Luo, Oscar Hernan Madrid Padilla, and Daren Wang. Dense relu neural networks for temporal-spatial model. arXiv preprint arXiv:2411.09961, 2024

    2024 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.