pith. sign in

arxiv: 1907.00275 · v1 · pith:BDR7PXKQnew · submitted 2019-06-29 · 💻 cs.LG · stat.ML

Efficient Regularized Piecewise-Linear Regression Trees

Leonidas Lefakis , Oleksandr Zadorozhnyi , Gilles Blanchard This is my paper

Pith reviewed 2026-05-25 12:30 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords regression treespiecewise-linear modelsregularizationgeneralization boundsRademacher complexityLASSOdecision treesGPU implementation
0
0 comments X

The pith

Regularized piecewise-linear regression trees with linear models at the leaves have new high-probability generalization bounds.

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

The paper establishes high-probability upper bounds on the generalization error of regression decision trees that use a regularized piecewise-linear node-splitting criterion together with regularized linear models at the leaves. These bounds are obtained through the Rademacher complexity framework and cover both LASSO-type and ℓ2 regularization, with further extension to general variable selection. The work also supplies an algorithmic implementation that renders the trees numerically stable and tractable on GPU hardware. Empirical comparisons on multiple datasets illustrate behavior relative to trees that use only piecewise-constant models at the leaves. The results supply theoretical support for preferring linear leaves under regularization when local linear trends are present in the data.

Core claim

The class of regression decision tree algorithms which employ a regularized piecewise-linear node-splitting criterion and have regularized linear models at the leaves admits new high-probability upper bounds for the generalization error based on the Rademacher complexity framework, including LASSO-type and ℓ2 regularization for the leaf models; the same class is shown to be numerically stable and tractable via a concrete algorithmic implementation and modern GPU technology.

What carries the argument

Rademacher complexity bounds applied to the class of regularized piecewise-linear regression trees that place regularized linear models at the leaves.

If this is right

  • High-probability generalization bounds hold for both LASSO-type and ℓ2 regularization of the leaf linear models.
  • The same bounds extend when a general variable selection procedure is added to the tree construction.
  • The proposed algorithmic implementation together with GPU acceleration renders the trees numerically stable and computationally practical.
  • Empirical performance on real datasets can be directly compared against piecewise-constant regression trees.

Where Pith is reading between the lines

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

  • The bounds offer a theoretical basis for choosing between different leaf regularizers when data exhibit local linear structure.
  • The GPU-oriented implementation suggests the method scales to data volumes where standard tree growing becomes prohibitive.
  • Variable selection inside the same framework may allow automatic feature pruning while preserving the generalization guarantee.

Load-bearing premise

The Rademacher complexity framework applies directly to yield valid high-probability generalization bounds for the specific class of regularized piecewise-linear regression trees with linear models at the leaves.

What would settle it

A concrete dataset, regularization strength, and sample size for which the observed test error exceeds the derived upper bound with frequency higher than the probability guarantee permits.

Figures

Figures reproduced from arXiv: 1907.00275 by Gilles Blanchard, Leonidas Lefakis, Oleksandr Zadorozhnyi.

Figure 1
Figure 1. Figure 1: Comparison of training times (in sec. ) for constructing a depth-10 tree on the various datasets. 6 Empirical Evaluation In this section we present an empirical evaluation of the generalization capabilities of the proposed PLRT algorithm. In particular, we show results on a number of different regression tasks comparing different regression tree algorithms, among them – PLRT trees that have been build usin… view at source ↗
Figure 2
Figure 2. Figure 2: The mean squared error of the 3 predictors for various tree depths. to γ ∈ {10.0, 1.0}, we provide a more detailed analysis of the effects of the γ parameter in the Appendix. The proposed method also outperforms the simpler CART method on all datasets; in fact in most cases CART significantly underperforms and its results are supressed in order to keep the plot compact. As highlighted by the results on the… view at source ↗
Figure 3
Figure 3. Figure 3: The mean squared error of the three speedup algorithms. As can be seen in most cases the difference in accuracy is negligible. Only in two of the Datasets (Energy, KDD2004) the approximation algorithms lead to poorer performance. Given the speedups they offer in computation time, we suggest that there is considerable merit in employing these algorithms despite the fact that they are not exact. B.3 Empirica… view at source ↗
Figure 4
Figure 4. Figure 4: The mean square error for various values of the l1-regularization parameter λ, for γ = 1.0. As can be seen, LASSO does not lead to improved performance on the datasets presented here. This may be related to the difference, when using LASSO, in regularization criteria in the nodes and the leaves. The tree itself is constructed by using an l2-regularization forcing the optimization algorithms in the prospect… view at source ↗
Figure 5
Figure 5. Figure 5: The mean squared error of the 3 predictors for various tree depths. C Numerical Stability As noted, one concern that can be raised regarding the proposed method is its numerical stability. the proposed method involves calculating, for each threshold, the inverse of a matrix of the form XT PX+λQ, where X is a N ×d matrix, meaning XT PX+λQ is the result of N rank-one updates to Q. We consider in the followin… view at source ↗
Figure 6
Figure 6. Figure 6: Numerical stability of calculating Λ = [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

We present a detailed analysis of the class of regression decision tree algorithms which employ a regulized piecewise-linear node-splitting criterion and have regularized linear models at the leaves. From a theoretic standpoint, based on Rademacher complexity framework, we present new high-probability upper bounds for the generalization error for the proposed classes of regularized regression decision tree algorithms, including LASSO-type, and $\ell_{2}$ regularization for linear models at the leaves. Theoretical result are further extended by considering a general type of variable selection procedure. Furthermore, in our work we demonstrate that the class of piecewise-linear regression trees is not only numerically stable but can be made tractable via an algorithmic implementation, presented herein, as well as with the help of modern GPU technology. Empirically, we present results on multiple datasets which highlight the strengths and potential pitfalls, of the proposed tree algorithms compared to baselines which grow trees based on piecewise constant models.

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 introduces a class of regression decision trees that use regularized piecewise-linear splitting criteria and regularized linear models (LASSO-type or ℓ₂) at the leaves. It derives new high-probability generalization bounds via the Rademacher complexity framework for these classes, extends the analysis to a general variable-selection procedure, describes a GPU-accelerated algorithmic implementation for tractability, and reports empirical comparisons against piecewise-constant tree baselines on multiple datasets.

Significance. If the generalization bounds are shown to be valid, the work would provide a useful theoretical characterization of regularized regression trees with linear leaves, complementing existing results on piecewise-constant trees. The algorithmic implementation and GPU scaling constitute a practical contribution, and the empirical section illustrates both strengths and limitations relative to standard baselines.

major comments (2)
  1. [theoretical analysis section] Theoretical analysis section: The high-probability bounds are obtained by applying the standard Rademacher complexity framework to the final hypothesis class of regularized piecewise-linear trees. Because the tree topology, partition, and variable selection are produced by a data-dependent regularized splitting rule, the derivation must include an explicit term (union bound, covering number over admissible trees, or structural-risk penalty) to account for this dependence; the provided description gives no indication that such a term is present.
  2. [theoretical analysis section] Theoretical analysis section, extension to general variable selection: The extension of the bounds to an arbitrary variable-selection procedure inherits the same issue; without a concrete argument showing that the selection mechanism can be absorbed into the fixed function class for Rademacher purposes, the claimed high-probability guarantee does not follow from the standard framework.
minor comments (2)
  1. [abstract] Abstract: the phrase 'theoretic standpoint' should read 'theoretical standpoint'.
  2. [empirical results] Empirical section: no information is supplied on the number of random seeds, error-bar computation, or data-exclusion rules used when reporting performance; this makes it difficult to assess the stability of the reported comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting an important subtlety in the theoretical analysis. We address the two major comments below.

read point-by-point responses

  1. Referee: [theoretical analysis section] Theoretical analysis section: The high-probability bounds are obtained by applying the standard Rademacher complexity framework to the final hypothesis class of regularized piecewise-linear trees. Because the tree topology, partition, and variable selection are produced by a data-dependent regularized splitting rule, the derivation must include an explicit term (union bound, covering number over admissible trees, or structural-risk penalty) to account for this dependence; the provided description gives no indication that such a term is present.

    Authors: We agree that the data-dependent selection of tree topology and partitions via the regularized splitting rule requires an explicit accounting that is not currently shown. Our derivation treated the hypothesis class as the collection of all admissible regularized piecewise-linear trees and bounded its Rademacher complexity directly, but this does not automatically control the complexity of the selection process itself. We will revise the theoretical analysis section to incorporate either a union bound over admissible tree structures or an additional structural-risk term that accounts for the dependence introduced by the splitting criterion. revision: yes

  2. Referee: [theoretical analysis section] Theoretical analysis section, extension to general variable selection: The extension of the bounds to an arbitrary variable-selection procedure inherits the same issue; without a concrete argument showing that the selection mechanism can be absorbed into the fixed function class for Rademacher purposes, the claimed high-probability guarantee does not follow from the standard framework.

    Authors: The same limitation applies to the extension. The current argument does not supply a concrete reduction showing that an arbitrary variable-selection procedure can be absorbed into a fixed function class without an additional complexity term. In the revised manuscript we will add an explicit argument (for example, by bounding the number of admissible selections or by treating the selection as part of a larger hypothesis class whose Rademacher complexity is controlled) so that the high-probability guarantee follows from the standard framework. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds derived from external Rademacher framework on the stated hypothesis class.

full rationale

The paper's central claim is new high-probability generalization bounds obtained by applying the Rademacher complexity framework to the class of regularized piecewise-linear regression trees. The abstract explicitly frames the result as an application of this external framework rather than any reduction to fitted quantities, self-defined parameters, or prior self-citations. No equations or steps in the provided text exhibit self-definitional, fitted-input, or self-citation load-bearing patterns. The derivation is therefore self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Rademacher complexity framework to the regularized tree class and on the existence of a tractable algorithmic implementation; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Rademacher complexity framework yields valid high-probability generalization bounds for regularized piecewise-linear regression trees with linear leaves
    The theoretical results are explicitly based on this framework as stated in the abstract.

pith-pipeline@v0.9.0 · 5692 in / 1237 out tokens · 35250 ms · 2026-05-25T12:30:14.517118+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

  1. [1]

    Annals of Statistics pp

    Bartlett, P., Bousquet, O., Mendelson, S.: Local rademacher complexities. Annals of Statistics pp. 1497–1537 (2005)

    work page 2005

  2. [2]

    Journal of Machine Learning Research pp

    Bartlett, P., Mendelson, S.: Rademacher and gaussian complexities: Risk bounds and results. Journal of Machine Learning Research pp. 463–482 (2002)

    work page 2002

  3. [3]

    Computing Science and Statistics (1994)

    Benett, K.: Global tree optimization. Computing Science and Statistics (1994)

    work page 1994

  4. [4]

    Chapman and Hall (1984)

    Breiman, L., Friedman, J., Olshen, A., Stone, C.: Classification and regression trees. Chapman and Hall (1984)

    work page 1984

  5. [5]

    CRC (1984)

    Breinman, L., Friedman, J., Olshen, R., Stone, C.: Classification and regression trees. CRC (1984)

    work page 1984

  6. [6]

    Statistica Sinica 4(1), 143–167 (1994)

    Chaudhuri, P., Huang, M.C., Loh, W.Y., Yao, R.: Piecewise-polynomial regression trees. Statistica Sinica 4(1), 143–167 (1994)

    work page 1994

  7. [7]

    Chipman, H.A., George, E.I., McCulloch, R.E.: Bart: Bayesian additive regression trees. Ann. Appl. Stat. 4(1), 266–298 (03 2010). https://doi.org/10.1214/09-AOAS285

    work page doi:10.1214/09-aoas285 2010

  8. [8]

    Jour- nal of Machine Learning Research 1, 143–160 (2001)

    Collobert, R., Bengio, S.: Svmtorch: Support vector machines for large-scale regression problems. Jour- nal of Machine Learning Research 1, 143–160 (2001)

    work page 2001

  9. [9]

    In: AAAI 2016 (2016)

    DeSalvo, G., Mohri, M.: Random composite forests. In: AAAI 2016 (2016)

    work page 2016

  10. [10]

    In: Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pp

    Dobra, A., Gehrke, J.: Secret: a scalable linear regression tree algorithm. In: Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 481–487 (2002)

    work page 2002

  11. [11]

    Annals of Statistics 25(5), 1870–1911 (1997)

    Donoho, D.: Cart and best ortho-basis: A connection. Annals of Statistics 25(5), 1870–1911 (1997)

    work page 1911

  12. [12]

    In: Lecture Notes in Statistics, pp

    Gey, S., Nedelec, E.: Risk bounds for cart regression trees. In: Lecture Notes in Statistics, pp. 369–379 (2003)

    work page 2003

  13. [13]

    In: Proceeding of ECAI’92

    Karalic, A.: Employing linear regression in regression tree leaves. In: Proceeding of ECAI’92. pp. 440– 441 (1992)

    work page 1992

  14. [14]

    IEEE Transactions on Infor- mation Theory 47(5), 1902–1914 (2001)

    Koltchinskii, V.: Rademacher penalties and structural risk minimization. IEEE Transactions on Infor- mation Theory 47(5), 1902–1914 (2001)

    work page 1902

  15. [15]

    In: The IEEE International Conference on Computer Vision (ICCV) (December 2015) 17

    Kontschieder, P., Fiterau, M., Criminisi, A., Rota Bulo, S.: Deep neural decision forests. In: The IEEE International Conference on Computer Vision (ICCV) (December 2015) 17

    work page 2015

  16. [16]

    ECML PKDD (2003)

    Landwehr, N., Hall, M., Eibe, F.: Logit model trees. ECML PKDD (2003)

    work page 2003

  17. [17]

    Lauer, P.: Error bounds for piecewise smooth and switching regression (2019), https://arxiv.org/abs/1707.07938

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  18. [18]

    Statistica Sinica 12(2), 361–386 (2002)

    Loh, W.Y.: Regression tress with unbiased variable selection and interaction detection. Statistica Sinica 12(2), 361–386 (2002)

    work page 2002

  19. [19]

    In: COLT 2000 (2000)

    Mansour, Y., Mcallster, D.: Generalization bounds of decision trees. In: COLT 2000 (2000)

    work page 2000

  20. [20]

    JLMR 13, 671–690 (2012)

    Maurer, A., Pontil, M.: Structured sparsity and generalization. JLMR 13, 671–690 (2012)

    work page 2012

  21. [21]

    Journal of Machine Learning Research:Workshop and Conference Proceedings 35, 1–21 (2014)

    Maurer, A., Pontil, M., Romera-Paredes, B.: An inequality with applications to structured sparsity and multitask dictionary learning. Journal of Machine Learning Research:Workshop and Conference Proceedings 35, 1–21 (2014)

    work page 2014

  22. [22]

    Natarajan, R., Pednault, E.: Segmented Regression Estimators for Massive Data Sets, pp. 566–582

    work page

  23. [23]

    In: NIPS’15 Proceedings of the 28th International Conference on Neural Information Processing Systems

    Norouzi, M., Collins, M., Johnson, M., Fleet, D., Kohli, P.: Efficient non-greedy optimization of decision trees. In: NIPS’15 Proceedings of the 28th International Conference on Neural Information Processing Systems. pp. 1729–1737 (2015)

    work page 2015

  24. [24]

    Machine Learning61(1), 5–48 (2005)

    Potts, D., Sammut, C.: Incremental learning of linear model trees. Machine Learning61(1), 5–48 (2005). https://doi.org/10.1007/s10994-005-1121-8

    work page doi:10.1007/s10994-005-1121-8 2005

  25. [25]

    In: Proc

    Quinlan, J.: Learning with continious classes. In: Proc. of Australian Joint Conf. on AI. pp. 343–348 (1992)

    work page 1992

  26. [26]

    Springer Science & Business Media (2008)

    Steinwart, I., Christmann, A.: Support Vector Machines. Springer Science & Business Media (2008)

    work page 2008

  27. [27]

    In: ICML-97

    Torgo, L.: Functional models for regression tree leaves. In: ICML-97. pp. 385–393 (1997)

    work page 1997

  28. [28]

    Torgo, L.: Computationally Efficient Linear Regression Trees, chap. I, pp. 409–417. Springer (2002)

    work page 2002

  29. [29]

    Foundations and Trends in Machine Learning 8, 1–230 (2015)

    Tropp, J.: An introduction to matrix concentration inequalities. Foundations and Trends in Machine Learning 8, 1–230 (2015)

    work page 2015

  30. [30]

    In: Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining

    Vogel, D., Asparouhov, O., Scheffer, T.: Scalable look-ahead linear regression trees. In: Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining. pp. 757–764 (2007)

    work page 2007

  31. [31]

    In: Proc

    Wang, Y., Witten, I.: Inducing model trees for continuos classes. In: Proc. of Poster Papers, European Conf. in Machine Learning (1997) 18 A Appendix. Proofs of the main results A.1 Generalization error bounds of the class of Piecewise Linear Regression tree models We remind the formal definition used for the class of piecewise-linear regression trees F. F...

    work page 1997

  32. [32]

    If for a given k , Lλ i,m≥ Lλ iT ,kT , we forgo calculating lλ j,m, rλ j,N−m,∀m, k≤ m≤ N− k

    ∀m, k≤ m≤ N− k we approximate Lλ i,m≈ lλ j,k + rλ j,N−k + (N− 2∗ k) ( min(lλ j,k− λ∥wj,k− w0∥2 Q, rλ j,N−k− λ∥wc j,N−k− wc 0∥2 Q) ) . If for a given k , Lλ i,m≥ Lλ iT ,kT , we forgo calculating lλ j,m, rλ j,N−m,∀m, k≤ m≤ N− k

    work page

  33. [33]

    If for a given i , Lλ i,m≥ Lλ iT ,kT , we forego calculating lλ j,m, rλ j,N−m,∀m, k≤ m≤ N− k

    ∀m, k≤ m≤ N− k we approximate Lλ i,m≈ lλ j,k + rλ j,N−k + (N− 2∗ k) ( max(lλ j,k− λ∥wj,k− w0∥2 Q, rλ j,N−k− λ∥wc j,N−k− wc 0∥2 Q) ) . If for a given i , Lλ i,m≥ Lλ iT ,kT , we forego calculating lλ j,m, rλ j,N−m,∀m, k≤ m≤ N− k. We set regularization parameterγ = 1 throughout these experiments and plot the loss for various tree depths on the various datase...

    work page 2048