Pith. sign in

REVIEW 3 major objections 3 minor 21 references

Can a Single Tree Outperform an Entire Forest?

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A single decision tree, trained end-to-end with gradient descent and linear leaves, can match—and on average beat—a random forest that pools hundreds of trees.

desk verdict The optimization method is solid and worth knowing, but the headline RF-beating claim fails on the paper's own statistics and on test-set-based regularization selection. read the letter →

arxiv 2411.17003 v1 pith:FCR5IW2N submitted 2024-11-26 cs.LG cs.AIstat.ML

Qiangqiang Mao , Yankai Cao This is my paper · ORCID
classification cs.LGcs.AIstat.ML
keywords obliqueregressiontreegradient-basedoptimizationscaledsigmoidapproximationrandomforestcomparisonhard-splitinferencesubtreepolishinterpretabilitydecisionvsensemble
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 overturn the usual assumption that a single decision tree is inherently less accurate than a random forest. It trains an oblique regression tree as one differentiable optimization problem, replacing non-differentiable split indicators with scaled sigmoid approximations and solving a sequence of increasingly sharp approximations. The result is GET, a hard-split tree with constant leaf predictions, and GET-Linear, whose leaves are linear regressions. Across 16 regression datasets, GET is statistically comparable to the classic random forest (0.17% lower test $R^2$), while GET-Linear outperforms the forest by an average of 2.03% (83.97% versus 81.94% test $R^2$). If the result holds, the accuracy cost of using one interpretable IF-THEN tree instead of a pool of hundreds of trees is much smaller than the field has assumed.

What carries the argument

The load-bearing object is the deterministic sample-route formulation coupled with the scaled sigmoid approximation. Each leaf assignment is the product $P_{i,t} = \prod_{j \in A^l_t} I_{i,j} \prod_{j \in A^r_t} (1 - I_{i,j})$ of branching-test indicators $I_{i,j} = \mathbf{1}(b_j - a_j^\top x_i > 0)$, and each indicator is replaced by the scaled sigmoid $\hat{I}_{i,j} = (1 + e^{-\alpha(b_j - a_j^\top x_i)})^{-1}$ so the whole tree is differentiable. Three mechanisms carry the argument: an iterative schedule that solves the loss for small $\alpha$ and warm-starts each larger-$\alpha$ problem, keeping the approximation sharp without destabilizing gradients; a subtree polish strategy that re-optimizes each branch node's subtree while holding the rest of the tree fixed, canceling approximation error that accumulates along deep paths; and hard-split inference with deterministically recomputed leaf values, which keeps the final model a genuine IF-THEN tree rather than a soft probabilistic one. Multi-start random initialization supplies diverse scale-factor schedules and better optima.

What would settle it

Re-run the paper's 16-dataset protocol with a random forest whose max_features and min_samples_leaf are tuned per dataset by cross-validation, and with GET-Linear's L1 strength selected on a validation split instead of by test-set improvement; if the forest then matches or beats GET-Linear in average test $R^2$ and in a paired t-test at the paper's own tolerance, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that an oblique regression tree trained end-to-end by gradient-based optimization can reach—and, with linear leaf predictions, exceed—the test accuracy of a classic random forest. The authors reformulate tree training as an unconstrained differentiable loss over the entire tree, routing each sample deterministically through products of branching-test indicators, and they approximate the indicators with scaled sigmoid functions so that gradients reach both split and leaf parameters. Because a single large scale factor makes the sigmoid nearly an indicator but starves the gradients, they solve a sequence of optimization problems with progressively larger scale factors, warm-starting each from the previous solution, and a subtree polish step re-optimizes each branch node's subtree while the rest of the tree stays fixed. At inference the tree keeps hard splits and recomputes leaf values deterministically as the mean (GET) or a fitted linear regression (GET-Linear) of the training samples that reach each leaf. Across 16 real-world regression datasets the authors report GET-Linear at an average test $R^2$ of 83.97% against 81.94% for a tuned random forest (a 2.03% edge), with GET at 81.77% and a paired t-test p-value of 0.877 for GET versus the forest.

Load-bearing premise

The 2.03-point edge reported for GET-Linear rests on the random forest being tuned only for tree count and depth while its other hyperparameters stay at library defaults, and on the L1 penalty strength being picked after observing a 0.73% test improvement rather than chosen on validation data.

Editorial extensions

If this is right

  • A single hard-split tree with roughly 100–300 times fewer parameters can match the test $R^2$ of a forest averaging 309 trees, so the interpretability-accuracy trade-off is looser than the prevailing mindset assumes.
  • GET-Linear's edge comes from its linear leaves: constant-leaf GET only reaches parity, so the practical win is specifically a piecewise-linear oblique tree.
  • Prediction is 24–30 times faster than the forest and parameter counts drop by factors of 119–324, which matters for the embedded and real-time control settings the paper motivates.
  • The parity is bought with training time: the optimized tree is thousands of times slower to train than the forest, though roughly 20 times faster than the prior best oblique-tree heuristic at depth 12.

Reading between the lines

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

  • Editorial extension: at the conventional 0.05 significance level the reported p = 0.127 for GET-Linear versus the forest would not reject equality; the paper's 'statistically significant' wording depends on its chosen tolerance of τ = 0.1.
  • Editorial extension: because the forest's max_features and min_samples_leaf are left at defaults, the 2.03% headline is best read as 'single tree versus a lightly tuned forest' until a fully tuned baseline is tested.
  • Editorial extension: the framework is demonstrated only for regression with an $R^2$ objective; a classification analogue that replaces the squared loss with cross-entropy over leaf distributions is a natural test the paper does not run.
  • Editorial extension: the authors name forests as future work; whether the iterative scaled-sigmoid and subtree-polish gains survive in an ensemble, where averaging already reduces variance, remains to be shown.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proposes a gradient-based entire-tree optimization framework for oblique regression trees, using a scaled sigmoid approximation of hard splits, an iterative annealing over the scale factor, and a subtree-polish strategy. The method is instantiated with constant leaf predictions (GET) and linear leaf predictions (GET-Linear). The central empirical claim is that, averaged over 16 UCI/OpenML regression datasets, GET-Linear outperforms a classic random forest by 2.03% in test R2, and that this difference is statistically significant, while GET is statistically comparable to random forest (Section 5.1, Table 1). The paper also reports that GET outperforms several decision-tree baselines, including CART, OC1, HHCART, SoftDT, GradTree, and ORT-LS (Table 2), and provides an analysis of training optimality (Table 3), an ablation of the proposed strategies (Table 4), and a comparison of parameter counts and prediction time (Table 5).

Significance. If the performance claim were fully supported, the result would be a meaningful challenge to the common belief that a single tree cannot reach random-forest-level test accuracy in regression, and it would strengthen the case for interpretable, lightweight alternatives in resource-constrained settings. The paper has clear strengths: the optimization reformulation is concisely presented, the algorithmic contribution (iterative scaled sigmoid plus subtree polishing) is explicit, the code is released, the experiments cover 16 datasets, and the ablation studies isolate the effect of the main components. However, the headline statistical-significance claim is not supported by the paper's own test, and the main accuracy gap relies on test-set-based selection of the L1 regularization strength. These issues are fixable but currently undermine the central conclusion as stated.

major comments (3)
  1. [Section 5.1, Figure 4] The claim that GET-Linear with p=0.127 is 'statistically superior' to random forest contradicts the paper's own pre-specified significance level. Section 5.1 sets the tolerance at τ=0.1 and then, when the paired t-test yields p=0.127, the text says 'if we accept a tolerance τ>0.127, we can reject the null hypothesis.' This is an ex post facto threshold change. Under the stated α=0.1, the null hypothesis is not rejected, so the bullet in Section 1 stating 'demonstrating a statistically significant difference' is not supported. The appropriate conclusion from p=0.127 is that the evidence for superiority is suggestive but not significant at the stated level; for a claim of comparability, a confidence interval or an equivalence test would be needed.
  2. [Appendix F.4, Table 10; Section 5.5] The 2.03% average gap in Table 1 is inflated by test-set-based hyperparameter selection. Appendix F.4 reports that GET-Linear's test accuracy improves from 83.24% to 83.97% when the L1 regularization strength λ is chosen from {0, 1e-5}, and the regularized result is used in Table 1. This is selection on the test set: without regularization the gap over RF is 1.30 percentage points, and no significance test is reported for that configuration. To make the central claim valid, the regularization strength should be selected on a validation fold, with the test set used only for final evaluation, and the result should be reported together with the corresponding significance test.
  3. [Section 5.1, first paragraph] The random forest baseline is only tuned over n_estimators and max_depth, while max_features and min_samples_leaf are left at sklearn defaults. The justification cites general robustness of defaults, but no dataset-specific evidence is given that these defaults are appropriate for the 16 datasets involved. Since the headline claim compares against this baseline, part of the 2.03% gap could reflect an under-tuned comparator. The comparison should either tune these hyperparameters (e.g., with the same cross-validation protocol used for depth) or provide sensitivity results showing that the conclusions are unchanged under reasonable variations of these defaults.
minor comments (3)
  1. [Section 4.2, Algorithm 1] The text contains a typo: 'all branch nondes' should read 'all branch nodes.' There are also minor grammatical issues in Section 4.3 ('once a branch node is optimally identified, we can sequentially polish its subtree') and in Section 5.1 ('GET-Linear significantly outperforms RF by 2.03%' overstates the statistical result).
  2. [Figure 4] The text labels in Figure 4 are rendered as unreadable Unicode escape sequences (e.g., 'p/uni...'), which makes the figure impossible to parse. The axis labels and p-values should be plain text.
  3. [Section 5.5, first paragraph] The phrase 'less than 20,000 samples' is repeated from the abstract but the numerical claim in Section 5.3 about training time uses averages; please clarify whether the training-time figures in Table 3 are averaged over the 16 datasets or over a subset, and specify the standard deviation or range.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the tree objective and held-out evaluation are self-contained; the only self-referential element is test-set-based regularization selection, which is a soundness caveat rather than circular reasoning.

full rationale

The paper's derivation chain is self-contained: the tree objective in Eq. (3) and the scaled-sigmoid approximation in Eq. (4) define a gradient-based optimization problem that does not encode the random-forest comparison or the claimed 2.03% advantage. GET and GET-Linear are evaluated on 25% held-out test data against an external Scikit-learn random forest baseline, so the central result is not constructed from the method's own outputs. There is no load-bearing self-citation: the references to Bertsimas & Dunn, Frosst & Hinton, Marton et al., and others are prior independent work, not a chain that forces the paper's conclusion. Two non-circular concerns remain. First, the L1 regularization strength for GET-Linear was selected by comparing testing accuracy between 0 and 1e-5 (Appendix F.4, Table 10: 83.24% to 83.97%), and the regularized value is used in Table 1; this is test-set-based model selection that can inflate the reported gap, but it is an evaluation-protocol flaw, not a derivation that reduces to a fitted constant. Second, the paper's own paired t-test gives p = 0.127 while Section 5.1 sets the significance tolerance at tau = 0.1, so the claim of a 'statistically significant difference' is inconsistent with the stated test; this is a statistical-reporting issue, not circularity. Accordingly, there are no circular steps to flag, and the circularity score is minimal.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim depends on several hand-set hyperparameters and on heuristics about the sigmoid approximation and hard-split leaf recomputation. No new physical or mathematical entities are introduced.

free parameters (5)
  • L1 regularization strength lambda for GET-Linear = between 0 and 1e-5
    Chosen after observing a 0.73% improvement in testing accuracy (Appendix F.4, Table 10), which is post hoc selection on the test set.
  • Scale factor sample range [alpha_min, alpha_max] = [5, 150]
    Set by hand in Appendix D as a compromise between approximation accuracy and gradient stability.
  • Number of sampled scale factors = 2
    The paper primarily samples two scale factors, one small in [5,25] and one large in [50,150], which affects approximation quality and computational cost.
  • Multi-start number Nstart = 10
    Set by hand to balance training optimality and compute time; larger values improve training accuracy but increase cost (Appendix D).
  • Epoch number Nepoch = 3000
    Set by hand; larger values generally improve training accuracy but increase cost (Appendix D).
assumptions (3)
  • domain assumption The scaled sigmoid S(x) = (1 + e^(-alpha*x))^(-1) approximates the indicator function well enough that minimizing the smoothed loss with increasing alpha approximates minimizing the true hard-split loss.
    Core to the method (Section 3.2, Eq. 4); the entire optimization relies on this approximation being valid for learning useful splits.
  • ad hoc to paper Deterministically recomputing leaf values K and h from hard-split routing after each optimization stage improves or maintains the training objective.
    Algorithm 1, Line 12 and Appendix C; this heuristic is not proven and the paper relies on it to bridge soft training and hard inference.
  • ad hoc to paper Gradient descent on the smoothed objective with warm-starting over increasing alpha avoids poor local optima.
    Section 4.1; no formal guarantee is provided, only empirical evidence.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Can a Single Tree Outperform an Entire Forest?." pith.science (2026). https://pith.science/paper/FCR5IW2N

@misc{pith2026241117003,
  author       = {Pith},
  title        = {Pith review of: Can a Single Tree Outperform an Entire Forest?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCR5IW2N}},
  note         = {Machine review of arXiv:2411.17003}
}
abstract

The prevailing mindset is that a single decision tree underperforms classic random forests in testing accuracy, despite its advantages in interpretability and lightweight structure. This study challenges such a mindset by significantly improving the testing accuracy of an oblique regression tree through our gradient-based entire tree optimization framework, making its performance comparable to the classic random forest. Our approach reformulates tree training as a differentiable unconstrained optimization task, employing a scaled sigmoid approximation strategy. To ameliorate numerical instability, we propose an algorithmic scheme that solves a sequence of increasingly accurate approximations. Additionally, a subtree polish strategy is implemented to reduce approximation errors accumulated across the tree. Extensive experiments on 16 datasets demonstrate that our optimized tree outperforms the classic random forest by an average of $2.03\%$ improvements in testing accuracy.

Figures

Figures reproduced from arXiv: 2411.17003 by the authors.

Figure 1
Figure 1. Example of deterministic sample routing 1 → 2 → 5. 3.2 LOSS FORMULATION AND DIFFERENTIABILITY Following the deterministic sample route, the training of oblique regression trees is subsequently reformulated as an unconstrained optimization problem. The objective function L is defined by L = Xn i=1 X t∈TL Pi,t yi − (k T t xi + ht) 2 . (3) Here, the variables A, b, K and h are implicitly expressed in terms of Ii,j and… view at source ↗
Figure 2
Figure 2. Forward and backward processes of loss calculation. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The illustrative example of our subtree polish strategy. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: T-statistic and p-value of the Paired T-test comparing [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The Paired T-test comparing our GET with various decision tree methods. 5.3 SUPERIOR TEST ACCURACY ANALYSIS: FROM TRAINING OPTIMALITY PERSPECTIVE To figure out the rationale behind the competitive testing accuracy of our optimized tree, we delve into the analysis of tr…
Figure 6
Figure 6. Figure 6: The trend of training optimality under different depth setting with different [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The trend of training optimality under different depth setting with various [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Disparity comparison of scaled sigmoid function over indicator function under varying [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 12 canonical work pages

  1. [7]

    Dheeru Dua and C

    doi: 10.1007/s10462-022-10275-5. Dheeru Dua and C. Graff. UCI machine learning repository.,

    doi:10.1007/s10462-022-10275-5

  2. [8]

    Manuel Fern ´andez-Delgado, Eva Cernadas, Sen ´en Barro, and Dinani Amorim

    doi: 10.3390/electronics10212640. Manuel Fern ´andez-Delgado, Eva Cernadas, Sen ´en Barro, and Dinani Amorim. Do we need hun- dreds of classifiers to solve real world classification problems? The journal of machine learning research, 15(1):3133–3181,

    work page doi:10.3390/electronics10212640

  3. [9]

    Sascha Marton, Stefan L ¨udtke, Christian Bartelt, and Heiner Stuckenschmidt

    doi: 10.3390/en15093413. Sascha Marton, Stefan L ¨udtke, Christian Bartelt, and Heiner Stuckenschmidt. Learning decision trees with gradient descent,

    work page doi:10.3390/en15093413

  4. [19]

    Joaquin Vanschoren, Jan N

    doi: 10.1109/FCCM.2012.47. Joaquin Vanschoren, Jan N. van Rijn, Bernd Bischl, and Luis Torgo. Openml: Networked science in machine learning. SIGKDD Explor. Newsl., 15(2):49–60,

    work page doi:10.1109/fccm.2012.47 2012

  5. [21]

    BM.Standard.E4.128

    The dataset size n and the number of features p are provided in the table. Typically, we allocated 75% of the samples for training purposes and the remaining 25% for test- ing. If an experiment requires cross validation for hyperparameters tuning like tree depth, we then 17 Table 7: Real-world datasets from UCI and OpenML Repository. Dataset Index Dataset...

    work page 2015

  6. [56]

    Accelerating a random forest classifier: Multi-core, gp-gpu, or fpga? In 2012 IEEE 20th International Symposium on Field- Programmable Custom Computing Machines, pp

    Brian Van Essen, Chris Macaraeg, Maya Gokhale, and Ryan Prenger. Accelerating a random forest classifier: Multi-core, gp-gpu, or fpga? In 2012 IEEE 20th International Symposium on Field- Programmable Custom Computing Machines, pp. 232–239,

    work page 2012

  7. [1994]

    12 Ramanathan Narayanan, Daniel Honbo, Gokhan Memik, Alok Choudhary, and Joseph Zambreno

    doi: 10.1613/jair.63. 12 Ramanathan Narayanan, Daniel Honbo, Gokhan Memik, Alok Choudhary, and Joseph Zambreno. An fpga implementation of decision tree classification. In Automation & Test in Europe Confer- ence & Exhibition 2007 Design, pp. 1–6,

    doi:10.1613/jair.63

  8. [1998]

    URL http://www

    World Scientific. URL http://www. researchgate.net/publication/2627892_Learning_With. Mark Segal and Yuanyuan Xiao. Multivariate random forests. WIREs Data Mining and Knowledge Discovery, 1(1):80–87,

    work page arXiv

Show all 21 references
  1. [2001]

    Leo Breiman, Jerome Friedman, Charles J

    doi: 10.1023/A: 1010933404324. Leo Breiman, Jerome Friedman, Charles J. Stone, and R. A. Olshen. Classification and Regression Trees. Taylor & Francis,

    doi

  2. [2002]

    Dimitris Bertsimas and Jack Dunn

    doi: 10.1109/TAC.2002.805688. Dimitris Bertsimas and Jack Dunn. Optimal classification trees. Machine Learning, 106(7):1039– 1082,

    2002

  3. [2006]

    ISBN 978- 3-540-49058-6

    Springer. ISBN 978- 3-540-49058-6. doi: 10.1007/11925231

    doi

  4. [2007]

    Thais Mayumi Oshiro, Pedro Santoro Perez, and Jos ´e Augusto Baranauskas

    doi: 10.1109/DATE.2007.364589. Thais Mayumi Oshiro, Pedro Santoro Perez, and Jos ´e Augusto Baranauskas. How many trees in a random forest? In Machine Learning and Data Mining in Pattern Recognition , pp. 154–168, Berlin, Heidelberg,

    2007

  5. [2011]

    doi: 10.1002/widm.12. David J. Sheskin. Handbook of Parametric and Nonparametric Statistical Procedures, Fifth Edition. CRC Press,

    doi

  6. [2012]

    ISBN 978-3-642-31537-4

    Springer. ISBN 978-3-642-31537-4. doi: 10.1007/978-3-642-31537-4

    doi

  7. [2014]

    URL https://dl.acm.org/doi/10.1145/2641190.2641198

    doi: 10.1145/2641190.2641198. URL https://dl.acm.org/doi/10.1145/2641190.2641198. Alvin Wan, Lisa Dunlap, Daniel Ho, Jihan Yin, Scott Lee, Henry Jin, Suzanne Petryk, Sarah Adel Bargal, and Joseph E. Gonzalez. Nbdt: Neural-backed decision trees,

  8. [2017]

    Rico Blaser and Piotr Fryzlewicz

    doi: 10.1007/s10994-017-5633-9. Rico Blaser and Piotr Fryzlewicz. Random rotation ensembles. Journal of Machine Learning Re- search, 17(4):1–26,

    doi

  9. [2019]

    URL https://onlinelibrary.wiley.com/doi/abs/10

    doi: 10.1002/widm.1301. URL https://onlinelibrary.wiley.com/doi/abs/10. 1002/widm.1301. J. R. Quinlan. Learning with continuous classes. In Proceedings Australian Joint Conference on Artificial Intelligence., pp. 343–348, Singapore,

    doi

  10. [2020]

    ISBN 978-1-4398-5804-2. Peter J. Tan and David L. Dowe. Decision forests with oblique decision trees. In MICAI 2006: Advances in Artificial Intelligence, pp. 593–603, Berlin, Heidelberg,

    2006

  11. [2021]

    doi: 10.3390/electronics10030314. A. Bemporad, F. Borrelli, and M. Morari. Model predictive control based on linear programming - the explicit solution. IEEE Transactions on Automatic Control, 47(12):1974–1985,

    1974 doi

  12. [2022]

    Adri´an Alcolea and Javier Resano

    URL http://arxiv.org/abs/2103.15965. Adri´an Alcolea and Javier Resano. Fpga accelerator for gradient boosting decision trees. Electron- ics, 10(3):314,

    arXiv

  13. [2023]

    Leo Breiman

    doi: 10.1007/s10601-023-09367-y. Leo Breiman. Random forests. Machine Learning , 45(1):5–32,

    doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.