Pith. sign in

REVIEW 2 major objections 4 minor 49 references

On the pointwise and sup-norm errors for local regression estimators

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Nearly isotropic local averaging cells are both necessary and sufficient for optimal nonparametric regression rates.

desk verdict A solid framework with a real, fixable gap: the CART algorithm can create leaves with fewer than m points, so Theorem 18's bound is unproven as written. read the letter →

arxiv 2507.07132 v1 pith:UI3CFXJ3 submitted 2025-07-08 math.ST stat.TH

Jérémy Bettinger , François Portier , Adrien Saumard This is my paper · ORCID
classification math.STstat.TH MSC 62G0862G2062C20
keywords localregressionmapsshaperegularityminimaxratessup-normconvergencenearestneighborsCARTtreesMondrian
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 asks when a local regression estimator—one that predicts at a point by averaging responses whose covariates fall in a small set around that point—attains the best possible statistical error for Lipschitz regression functions. The answer it defends is geometric: the averaging sets must be shape-regular, meaning their volume is comparable to the volume of a ball of the same diameter, and their Lebesgue volume must be calibrated to $n^{-d/(d+2)}$. Under that condition the pointwise and sup-norm errors are $(\log n/n)^{1/(d+2)}$ up to constants; without it, a single sufficiently varying Lipschitz function can force a strictly slower rate. The same condition organizes a wide family of concrete estimators: $k$-nearest neighbors, prototype rules, and a CART-like tree with shape-constrained splits all satisfy it, while uniform and centered random trees violate it and only get near-optimal rates. Mondrian trees satisfy it in probability and reach the minimax pointwise rate.

What carries the argument

The central object is the $\gamma$-shape-regular set: a measurable set $V$ satisfies $\mathrm{diam}(V)^d \le \gamma \lambda(V)$, meaning its volume is at least a fixed fraction of the ball with the same diameter—equivalently, the set is not too elongated. For hyper-rectangles this reduces to a bound on the ratio of longest to shortest side ($\beta$-shape regularity), which is what a tree-splitting rule can actually enforce. The paper couples this geometric condition with a VC-dimension control on the collection of possible averaging sets and a minimal-mass assumption on the covariate distribution, $P_X(V(x)) \ge \ell(x)\lambda(V(x))$, so that the variance term in the bias-variance decomposition is governed by Lebesgue volume rather than by the random number of points in a cell. The volume-diameter trade-off, optimized when $\lambda(V(x)) \asymp (\log((n+1)^v/\delta)/n)^{d/(d+2)}$, produces the rate $(\log n/n)^{1/(d+2)}$. The failure mode is also geometric: if the local cell is much longer than wide, a Lipschitz function that increases along all coordinates exposes the bias, and the variance cannot be reduced enough to compensate.

What would settle it

Simulate $d=2$ regression with $g(x,y)=x+y$, $X$ uniform on $[0,1]^2$, and compare two local-map estimators at the origin: one using cells that are $\gamma$-shape-regular with volume $n^{-1/2}$, the other using rectangles of the same volume but with one side of length $n^{-1/4}$ and the other $n^{-3/4}$, so that $\mathrm{diam}(V)^2/\lambda(V)\sim n^{1/2}$. If the elongated-cell estimator's pointwise or sup-norm error does not decay at least as fast as $(\log n/n)^{1/4}$, the necessity direction is confirmed; finding any non-shape-regular cell scheme that attains the minimax rate for every Lipschitz function would refute it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 12 together with Proposition 10: for Lipschitz regression functions on $\mathbb{R}^d$, a local map estimator attains the minimax pointwise and sup-norm rate $(\log n/n)^{1/(d+2)}$ up to constants if and only if its averaging sets are $\gamma$-shape-regular, $\mathrm{diam}(V)^d \le \gamma \lambda(V)$, with $\lambda(V)$ of order $n^{-d/(d+2)}$. The sufficiency direction follows from a bias-variance bound (Theorem 6) that replaces the empirical count in a local cell by its Lebesgue volume via a minimal-mass assumption on the covariate distribution, and then optimizes the volume-diameter trade-off. The necessity direction constructs a Lipschitz function whose partial derivatives are all one, $g(x)=\sum_k x_k$, and shows that any rectangular cell with $\mathrm{diam}(V)^d/\lambda(V)=\bar\gamma_n\to\infty$ yields expected squared error at least $C_d(\bar\gamma_n \sigma^2/n)^{1/(d+2)}$, so the optimal rate cannot hold. The paper therefore claims that shape regularity is a structural, distribution-free property that separates estimators achieving optimal pointwise and uniform behavior from those that do not.

Load-bearing premise

The load-bearing premise is the minimal-mass condition: every averaging set $V(x)$ must receive probability at least a fixed multiple of its Lebesgue volume under the covariate distribution. If a cell lands in a region where the covariate density is zero or extremely small, the variance term is no longer controlled by the cell's volume and the stated rates can fail.

Editorial extensions

If this is right

  • For $k$-nearest neighbors, the correct choice $k \asymp n^{2/(d+2)}$ yields the minimax sup-norm rate for every query point whenever a local density lower bound holds, extending earlier results that required a global density lower bound.
  • Proto-NN, whose convergence rates were previously open, attains the minimax rate up to logarithmic factors; OptiNet attains the same rate with substantially better probability guarantees.
  • A CART-like regression tree whose splits are restricted to $\beta$-shape-regular cuts and whose leaves hold at least $m$ points achieves the minimax sup-norm rate; in one dimension every cell is shape-regular, so failure of classical CART must come entirely from producing cells with too few points.
  • Uniform and centered random trees are not shape-regular on events of positive probability, which explains why they only reach near-optimal rates $n^{-1/(d+2)}e^{O(\sqrt{\log n\log\log n})}$ rather than the exact minimax rate.
  • Mondrian trees are shape-regular in probability and achieve the minimax pointwise rate, but with probability that decays polynomially rather than exponentially, because the random partition can occasionally create very small cells.

Reading between the lines

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

  • Beyond the paper's Lipschitz setting, the same shape-regularity criterion plausibly transfers to higher-order smoothness classes, where the optimal cell volume would scale as $n^{-d/(2s+d)}$ and an analogous necessary condition could be proved; the paper only treats the case $s=1$.
  • A practical diagnostic emerges: for any local averaging method, one can monitor $\mathrm{diam}(V(x))^d/\lambda(V(x))$ on the empirical cells and flag cells whose ratio grows with $n$; these are exactly the cells that spoil sup-norm performance, regardless of the split criterion.
  • Because the minimal-mass assumption is the only place the covariate density enters, methods that adaptively enlarge cells in low-density regions could maintain shape regularity and preserve the minimax rate without requiring a global lower bound on the density; testing this is a natural extension of the paper's framework.
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

2 major / 4 minor

Summary. The paper develops a unified theory of pointwise and sup-norm error bounds for local regression estimators indexed by VC classes of sets, centered on a new notion of shape-regular local maps. It proves a general deviation inequality (Theorem 4), introduces the minimal-mass assumption (X), shows that γ-shape-regularity suffices for near-minimax rates (Theorem 12) and that non-regular cells can be suboptimal (Proposition 10), and then applies the framework to k-NN, Proto-NN, OptiNet, a new CART-like tree algorithm (Algorithm 1), and purely random trees (uniform, centered, Mondrian). The proofs are detailed and the paper includes both positive and negative results, including new concentration bounds for Proto-NN and for CART-like trees.

Significance. If the main theorems are correct, the paper provides a coherent geometric condition—shape regularity—that unifies and extends known minimax guarantees for many local regression methods, and it supplies the first convergence-rate results for Proto-NN as well as near-optimal sup-norm guarantees for a data-dependent CART-type algorithm. The paper also contains explicit negative results for uniform and centered random trees, which are valuable for understanding when tree-based local estimators fail. The appendix is thorough, with self-contained proofs of the auxiliary concentration lemmas; the machine-checkable nature of the statement-level claims is a strength. However, the algorithmic guarantee for the CART-like tree in Section 5.3 has a genuine gap that affects a central advertised result.

major comments (2)
  1. [Section 5.3] The statement immediately after the definition of Sβ(V), 'when β ≥ 2, the Sβ(V) cannot be empty. Splitting the largest side in the middle is always in Sβ(V),' is false. For example, take d=2, V=[0,100]×[0,1], and β=2. Splitting the largest side in the middle produces children with side lengths 50 and 1, giving h+(child)/h−(child)=50, so neither child is 2-SR; indeed no split of this anisotropic cell can make both children 2-SR, since a split along the long side would require 100u≤2 and 100(1−u)≤2 simultaneously, which is impossible, and a split along the short side gives a child with largest side 100. This invalidates the claim used later in the proof of Theorem 18 that the β-SR constraint can never be a stopping criterion.
  2. [Section 5.3, Algorithm 1 and Theorem 18] Algorithm 1 does not enforce the condition nP_n(V)≥m for every leaf. In step (a), when the argmin over Sβ(V)∩Sm(V) has no solution, the fallback 'pick p as the largest side and u=1/2' is not checked against Sm(V). Concretely, with d=2, β=2, m=10, and a root cell [0,1]^2 containing 12 points at (0.1,0.1), 10 at (0.45,0.45), and 8 at (0.9,0.9), the only shape-regular split of the cube is the middle split, which puts 22 points in one child and 8 in the other, so it is not in Sm; the split u=0.3 along x is in Sm but not in Sβ. Thus Sβ∩Sm is empty, the fallback middle split is used, and the child containing 8 points is never split again because Sm=∅ there, producing a leaf with fewer than m points. Theorem 34, which is the basis of Theorem 18, requires every cell to satisfy nP_n(V)≥m for the variance bound 1/m, so the proof of Theorem 18 does not establish the claimed bound for the stated algorithm. The issue is fixable within the manuscript's scope, for example by treating 'Sβ(V)∩Sm(V)=∅' as a stopping condition rather than falling back to an unvetted split; this preserves nP_n(V)≥m because Sm(V)≠∅ implies the cell contains at least 2m points.
minor comments (4)
  1. [Section 6.4, Theorem 31 and its proof] The notation 'Λ ≍ n1/d+2' should read 'Λ ≍ n^{1/(d+2)}', and similarly in the proof 'C / n1/d+2' should be 'C n^{-1/(d+2)}'; as printed, the exponent is ambiguous and the final rate appears inverted.
  2. [Proof of Proposition 10] In the proof, the phrase 'where γ = γ1/d' is confusing: the symbol γ is introduced without a clear definition. It should be written as γ = \barγ^{1/d} so that the subsequent substitution γ^d = \barγ is transparent.
  3. [Theorem 2] Theorem 2 states the supremum over x∈R^d, while the local map V is only defined on SX; the statement should either restrict the supremum to x∈SX or extend V consistently to R^d.
  4. [Section 5.3, discussion after Algorithm 1] The sentence claiming that a leaf is kept 'if and only if the number of data points belonging to that cell is greater than m and strictly smaller than 2m' is not reflected in the pseudocode, because the fallback in step (a) can create a leaf with fewer than m points; this should be corrected after the algorithmic fix to the fallback rule.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: rates follow from explicit bias–variance bounds; shape regularity is an assumed geometric premise, not a fitted output.

full rationale

The derivation chain is self-contained. The upper-bound results (Theorems 4, 6, 12) are proved from the VC-type concentration inequality in Theorem 2 together with the Lipschitz bias bound; the minimal-mass assumption (X) is an explicit covariate-distribution hypothesis and is not defined in terms of the rate. Shape regularity, diam(V)^d ≤ γλ(V), is a geometric premise, and Theorem 12 merely balances the two terms of the Theorem 6 bound after imposing λ(V(x)) ≍ (log((n+1)^v/δ)/n)^{d/(d+2)}; this is an algebraic optimization of a displayed bound, not a quantity fitted to produce the conclusion. The necessity result, Proposition 10, is an independent lower bound for g(x)=Σx_k on a non-regular cell, obtained directly from variance and bias terms. No fitted parameter is renamed as a prediction. The only self-citation is Lemma 3 of Portier (2021), used to control the diameter of Voronoi cells in the Proto-NN and OptiNet bias terms; that lemma is a separate technical concentration bound on k-NN radii, not the paper’s central claim and not an unverified ansatz, so it does not make the argument circular. Separately, the fallback split in Algorithm 1 step (a) can leave a leaf with fewer than m points, so Theorem 18’s minimal-mass premise is not guaranteed by the implementation as written; this is a correctness gap, not a circularity, and does not change the score.

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

No free parameters are fitted to data; hyperparameters (k, m, beta, Lambda) are chosen as functions of n to balance variance and bias, and the rates are derived analytically. No new entities are postulated. The axioms are explicit modeling assumptions stated in the paper.

assumptions (7)
  • domain assumption Sub-Gaussian noise (E): E[exp(lambda epsilon)|X] <= exp(lambda^2 sigma^2 / 2).
    Used throughout for variance concentration; standard in nonparametric regression.
  • domain assumption IID sample (D): (X_i, Y_i) independent and identically distributed.
    Needed for all empirical process and concentration arguments.
  • domain assumption Lipschitz regression function (L).
    Defines the function class for which minimax rates are derived.
  • domain assumption Minimal mass assumption (X): P_X(V(x)) >= l(x) lambda(V(x)).
    Central to translate probability mass to Lebesgue volume; enters Theorem 6 and all sufficiency results.
  • standard math VC class and finite VC dimension for the local map.
    Sauer's lemma and the shattering coefficient are used in Theorem 2 and Theorem 4.
  • domain assumption Assumptions (XNN), (XTREE), (XZ), (DZ), (EZ) for specific estimators.
    Used to derive rates for k-NN, CART-like trees, Proto-NN, and OptiNet; they are technical versions of the minimal mass and iid assumptions.
  • standard math Borel-Cantelli lemma and Chernoff bounds for almost sure results on random trees.
    Used in Propositions 23 and 27 and Theorems 24, 28.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the pointwise and sup-norm errors for local regression estimators." pith.science (2026). https://pith.science/paper/UI3CFXJ3

@misc{pith2026250707132,
  author       = {Pith},
  title        = {Pith review of: On the pointwise and sup-norm errors for local regression estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UI3CFXJ3}},
  note         = {Machine review of arXiv:2507.07132}
}
read the original abstract

In this paper, we analyze the behavior of various non-parametric local regression estimators, i.e. estimators that are based on local averaging, for estimating a Lipschitz regression function at a fixed point, or in sup-norm. We first prove some deviation bounds for local estimators that can be indexed by a VC class of sets in the covariates space. We then introduce the general concept of shape-regular local maps, corresponding to the situation where the local averaging is done on sets which, in some sense, have ``almost isotropic'' shapes. On the one hand, we prove that, in general, shape-regularity is necessary to achieve the minimax rates of convergence. On the other hand, we prove that it is sufficient to ensure the optimal rates, up to some logarithmic factors. Next, we prove some deviation bounds for specific estimators, that are based on data-dependent local maps, such as nearest neighbors, their recent prototype variants, as well as a new algorithm, which is a modified and generalized version of CART, and that is minimax rate optimal in sup-norm. In particular, the latter algorithm is based on a random tree construction that depends on both the covariates and the response data. For each of the estimators, we provide insights on the shape-regularity of their respective local maps. Finally, we conclude the paper by establishing some probability bounds for local estimators based on purely random trees, such as centered, uniform or Mondrian trees. Again, we discuss the relations between the rates of the estimators and the shape-regularity of their local maps.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 44 canonical work pages

  1. [1]

    Analysis of purely random forests bias

    Sylvain Arlot and Robin Genuer. Analysis of purely random forests bias. arXiv preprint arXiv:1407.3939 , 2014

    arXiv 2014

  2. [2]

    T. W. Anderson. Some nonparametric multivariate procedures based on statistically equivalent blocks . Multivariate Analysis (P. R. Krishnaiah, ed), 5-27, Academic Press, New York., 1966

    work page 1966

  3. [3]

    A result of V apnik with applications

    Martin Anthony and John Shawe-Taylor. A result of V apnik with applications. Discrete Appl. Math. , 47(3):207--217, 1993

    work page 1993

  4. [4]

    Lectures on the nearest neighbor method

    G\' e rard Biau and Luc Devroye. Lectures on the nearest neighbor method . Springer Series in the Data Sciences. Springer, Cham, 2015

    work page 2015

  5. [5]

    Consistency of random forests and other averaging classifiers

    G \'e rard Biau, Luc Devroye, and G \'a bor Lugosi. Consistency of random forests and other averaging classifiers. Journal of Machine Learning Research , 9(66):2015--2033, 2008

    work page 2015

  6. [6]

    Classification and Regression Trees

    Leo Breiman, Jerome Friedman, Charles J Stone, and RA Olshen. Classification and Regression Trees . CRC Press, 1984

    work page 1984

  7. [7]

    Analysis of a random forests model

    G \'e rard Biau. Analysis of a random forests model. The Journal of Machine Learning Research , 13(1):1063--1095, 2012

    work page 2012

  8. [8]

    Concentration inequalities

    St\' e phane Boucheron, G\' a bor Lugosi, and Pascal Massart. Concentration inequalities. A nonasymptotic theory of independence . Oxford University Press, Oxford, 2013

    work page 2013

Show all 49 references
  1. [9]

    Some infinity theory for predictor ensembles

    Leo Breiman. Some infinity theory for predictor ensembles. Technical report, Citeseer, 2000

    2000

  2. [10]

    Random forests

    Leo Breiman. Random forests. Machine learning , 45:5--32, 2001

    2001

  3. [11]

    A random forest guided tour

    G \'e rard Biau and Erwan Scornet. A random forest guided tour. Test , 25:197--227, 2016

    2016

  4. [12]

    Nearest neighbor classificationin infinite dimension

    Fr \'e d \'e ric C \'e rou and Arnaud Guyader. Nearest neighbor classificationin infinite dimension. ESAIM: Probability and Statistics , 10:340--355, 2006

    2006

  5. [13]

    On the pointwise behavior of recursive partitioning and its implications for heterogeneous causal effect estimation

    Matias D Cattaneo, Jason M Klusowski, and Peter M Tian. On the pointwise behavior of recursive partitioning and its implications for heterogeneous causal effect estimation. arXiv preprint arXiv:2211.10805 , 2022

    2022 arXiv

  6. [14]

    Estimation by the nearest neighbor rule

    T Cover. Estimation by the nearest neighbor rule. IEEE Trans. Inform. Theory , 14(1):50--55, 1968

    1968

  7. [15]

    Asymptotic properties of high-dimensional random forests

    Chien-Ming Chi, Patrick Vossler, Yingying Fan, and Jinchi Lv. Asymptotic properties of high-dimensional random forests. The Annals of Statistics , 50(6):3415--3438, 2022

    2022

  8. [16]

    A probabilistic theory of pattern recognition , volume 31

    Luc Devroye, L \'a szl \'o Gy \"o rfi, and G \'a bor Lugosi. A probabilistic theory of pattern recognition , volume 31. Springer Science & Business Media, 1996

    1996

  9. [17]

    Uwe Einmahl and David M. Mason. An empirical process approach to the uniform consistency of kernel-type function estimators. J. Funct. Anal. , 13(1):1--37, 2000

    2000

  10. [18]

    Discriminatory analysis

    Evelyn Fix and Joseph Lawson Hodges. Discriminatory analysis. nonparametric discrimination: Consistency properties. Int. Stat. Rev. , 57(3):238--247, 1951

    1951

  11. [19]

    Rates of strong uniform consistency for multivariate kernel density estimators

    Evarist Gin \'e and Armelle Guillou. Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. Henri Poincar\' e Probab. Stat. , 38(6):907--921, 2002. En l'honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov

    2002

  12. [20]

    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

  13. [21]

    Classification in general finite dimensional spaces with the k -nearest neighbor rule

    S\' e bastien Gadat, Thierry Klein, and Cl\' e ment Marteau. Classification in general finite dimensional spaces with the k -nearest neighbor rule. Ann. Statist. , 44(3):982--1009, 2016

    2016

  14. [22]

    Near-optimal sample compression for nearest neighbors

    Lee-Ad Gottlieb, Aryeh Kontorovich, and Pinhas Nisnevitch. Near-optimal sample compression for nearest neighbors. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems , volume 27. Curran Associate...

    2014

  15. [23]

    Consistent nonparametric regression from recursive partitioning schemes

    Louis Gordon and Richard A Olshen. Consistent nonparametric regression from recursive partitioning schemes. Journal of Multivariate Analysis , 10(4):611--627, 1980

    1980

  16. [24]

    Universal consistency and rates of convergence of multiclass prototype algorithms in metric spaces

    L \'a szl \'o Gy \"o rfi and Roi Weiss. Universal consistency and rates of convergence of multiclass prototype algorithms in metric spaces. Journal of Machine Learning Research , 22(151):1--25, 2021

    2021

  17. [25]

    Universal bayes consistency in metric spaces

    Steve Hanneke, Aryeh Kontorovich, Sivan Sabato, and Roi Weiss. Universal bayes consistency in metric spaces. The Annals of Statistics , 49(4):2129--2150, 2021

    2021

  18. [26]

    A guided tour of chernoff bounds

    Torben Hagerup and Christine R \"u b. A guided tour of chernoff bounds. Information processing letters , 33(6):305--308, 1990

    1990

  19. [27]

    Non-asymptotic uniform rates of consistency for k - NN regression

    Heinrich Jiang. Non-asymptotic uniform rates of consistency for k - NN regression. In AAAI proceedings , volume 33, pages 3999--4006, 2019

    2019

  20. [28]

    Sharp analysis of a simple model for random forests

    Jason Klusowski. Sharp analysis of a simple model for random forests. In International Conference on Artificial Intelligence and Statistics , pages 757--765. PMLR, 2021

    2021

  21. [29]

    Nearest-neighbor sample compression: Efficiency, consistency, infinite dimensions

    Aryeh Kontorovich, Sivan Sabato, and Roi Weiss. Nearest-neighbor sample compression: Efficiency, consistency, infinite dimensions. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing...

    2017

  22. [30]

    On error and compression rates for prototype rules

    Omer Kerem and Roi Weiss. On error and compression rates for prototype rules. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 37, pages 8228--8236, 2023

    2023

  23. [31]

    Consistency of data of data-driven histogram methods for density estimation and classification

    Gabor Lugosi and Andrew Nobel. Consistency of data of data-driven histogram methods for density estimation and classification. The Annals of Statistics , 1996

    1996

  24. [32]

    Mondrian forests: Efficient online random forests

    Balaji Lakshminarayanan, Daniel M Roy, and Yee Whye Teh. Mondrian forests: Efficient online random forests. Advances in neural information processing systems , 27, 2014

    2014

  25. [33]

    Pattern classification and learning theory

    G \'a bor Lugosi. Pattern classification and learning theory. In Principles of nonparametric learning , pages 1--56. Springer, 2002

    2002

  26. [34]

    Universal consistency and minimax rates for online mondrian forests

    Jaouad Mourtada, Stéphane Gaïffas, and Erwan Scornet. Universal consistency and minimax rates for online mondrian forests. Advances in Neural Information Processing Systems 30 , 2017

    2017

  27. [35]

    Minimax optimal rates for mondrian trees and forests

    Jaouad Mourtada, Stéphane Gaïffas, and Erwan Scornet. Minimax optimal rates for mondrian trees and forests. Annals of Statistics , 2019

    2019

  28. [36]

    On the convergence of CART under sufficient impurity decrease condition

    Rahul Mazumder and Haoyue Wang. On the convergence of CART under sufficient impurity decrease condition. Advances in Neural Information Processing Systems , 36, 2024

    2024

  29. [37]

    On estimating regression

    Elizbar A Nadaraya. On estimating regression. Theory Probab. Appl. , 9(1):141--142, 1964

    1964

  30. [38]

    Histogram regression estimation using data-dependent partitions

    Andrew Nobel. Histogram regression estimation using data-dependent partitions. The Annals of Statistics , 1996

    1996

  31. [39]

    Nearest neighbor process: weak convergence and non-asymptotic bound

    Fran c ois Portier. Nearest neighbor process: weak convergence and non-asymptotic bound. arXiv preprint arXiv:2110.15083 , 2021

    2021 arXiv

  32. [40]

    The M ondrian process

    Daniel M Roy and Yee W Teh. The M ondrian process. In Advances in N eural I nformation P rocessing S ystems , pages 1377--1384, 2008

    2008

  33. [41]

    Consistency of random forests

    Erwan Scornet, G \'e rard Biau, and Jean-Philippe Vert. Consistency of random forests. Annals of Statistics , 43(4):1716--1741, 2015

    2015

  34. [42]

    Consistent nonparametric regression

    Charles J Stone. Consistent nonparametric regression. Ann. Statist. , 5(1):595--620, 1977

    1977

  35. [43]

    Optimal global rates of convergence for nonparametric regression

    Charles J Stone. Optimal global rates of convergence for nonparametric regression. The annals of statistics , pages 1040--1053, 1982

    1982

  36. [44]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to Nonparametric Estimation . Springer Publishing Company, Incorporated, 1st edition, 2008

    2008

  37. [45]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to nonparametric estimation . Springer Series in Statistics. Springer, 2009

    2009

  38. [46]

    V. N. Vapnik and A. Ya. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. In Measures of complexity , pages 11--30. Springer, Cham., 2015. Reprint of Theor. Probability Appl. 16 (1971), 264--280

    1971

  39. [47]

    Van Der Vaart and Jon A

    Aad W. Van Der Vaart and Jon A. Wellner. Weak Convergence and Empirical Processes. With Applications to Statistics . Springer Series in Statistics. Springer-Verlag, New York, 1996

    1996

  40. [48]

    Some special V apnik- C hervonenkis classes

    Roberta S Wenocur and Richard M Dudley. Some special V apnik- C hervonenkis classes. Discrete Math. , 33(3):313--318, 1981

    1981

  41. [49]

    Achieving the time of 1-nn, but the accuracy of k-nn

    Lirong Xue and Samory Kpotufe. Achieving the time of 1-nn, but the accuracy of k-nn. In International Conference on Artificial Intelligence and Statistics , pages 1628--1636. PMLR, 2018

    2018

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.