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Asymptotic Normality of Infinite Centered Random Forests -Application to Imbalanced Classification

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves a central limit theorem for infinite centered random forests with explicit rate and constant, then shows that after rebalancing and debiasing the estimator has smaller asymptotic variance as the minority class probability…

desk verdict The ICRF CLT with explicit constant is a real contribution, but the paper's headline variance-reduction claim for imbalanced classification is unproven because Theorem 3.5 leaves the i.i.d. framework without a valid substitute. read the letter →

arxiv 2506.08548 v1 pith:VW2BENBM submitted 2025-06-10 stat.ML cs.LG

Moria Mayala (LPSM (UMR_8001)) , Erwan Scornet (LPSM (UMR_8001)) , Charles Tillier (LMV) , Olivier Wintenberger (LPSM (UMR_8001)) This is my paper
classification stat.MLcs.LG MSC 60F0562G2062H30
keywords binaryclassificationimbalancedcenteredrandomforestsasymptoticnormalitycentrallimittheoremimportancesamplingvariancereductionU-statistics
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 establishes that infinite centered random forests, whose trees split at midpoints without using the data, are asymptotically normal at an explicit rate, and that the same holds after training on a rebalanced dataset. The debiased estimator, obtained by an importance-sampling correction, is centered at the true regression function rather than at the rebalanced regression function. In high-imbalance settings, the debiased estimator has asymptotic variance that is $O(p)$ times that of the forest trained on the original data, where $p$ is the minority class probability. The paper therefore gives a theoretical explanation for why rebalancing followed by debiasing can help predict rare events.

What carries the argument

The engine is the U-statistic representation of the infinite forest, which reduces the CLT to verifying that $nV^s_1 \to \infty$ and that the bias vanishes. Under the uniform covariate assumption the leaf probability is deterministic, $p_{k,U}(x)=2^{-k}$, which makes $V^s_1$ computable: it is asymptotically $C(d)2^k/(s^2 k^{(d-1)/2})\mu(x)(1-\mu(x))$, where $C(d)$ comes from a kernel equivalent $E[P(X_1 \in L_U(x)|X_1)^2] \sim C(d)/(2^k k^{(d-1)/2})$. The same variance analysis, extended to the rebalanced distribution through a density formula and the odds-ratio identity linking $\mu$ and $\mu'$, yields the CLT for the rebalanced forest and the variance comparison.

What would settle it

Simulate under (H0), (H1), (G1) with $d=2$ and hyperparameters satisfying (9), and estimate the empirical variance of $\sqrt{n k^{1/2}/2^k}(\widehat{\mu}^{\mathrm{ICRF}}_s(x)-\mu(x))$; if it does not converge to $C(2)\mu(x)(1-\mu(x))$, the CLT constant is wrong. Similarly, reweight one original sample to build IS-ICRF and check whether its variance ratio to ICRF actually decays like $p$; if not, the extension behind Theorem 3.5 fails.

Watch

Extended reading notes

Core claim

The central discovery is that the subsampling infinite centered random forest estimator $\widehat{\mu}^{\mathrm{ICRF}}_s(x)$ satisfies $\sqrt{n k^{(d-1)/2}/2^k}(\widehat{\mu}^{\mathrm{ICRF}}_s(x)-\mu(x)) \to N(0, C(d)\mu(x)(1-\mu(x)))$ under the paper's conditions (H0), (H1), (G1) and the rate condition (9), with the constant $C(d)$ given explicitly. Training on rebalanced data moves the centering to $\mu'(x)$, but the importance-sampling corrected IS-ICRF restores centering at $\mu(x)$. For small $p$, the asymptotic variance of IS-ICRF is only $O(p)$ times that of the original ICRF, so the debiased rebalanced forest is provably more stable in exactly the regime where the minority class is rare.

Load-bearing premise

The variance-reduction conclusion rests on treating the rebalanced training set as fresh draws from a reweighted distribution, while the practical construction reweights one original sample and creates repetitions, an extension the paper explicitly leaves unproved.

Editorial extensions

If this is right

  • Asymptotic confidence intervals for $\mu(x)$ can be built at the explicit rate $\sqrt{2^k/(n k^{(d-1)/2})}$.
  • The hyperparameter conditions in (9) give a concrete guide for choosing subsample size and tree depth in practice.
  • Training on a rebalanced dataset without correction is asymptotically biased; the debiasing step is necessary for valid inference about $\mu(x)$.
  • In high-imbalance settings, the debiased rebalanced forest reduces asymptotic variance by a factor $O(p)$ compared to the original-data forest.
  • Experiments suggest the same variance rates and variance reduction also hold for Breiman's random forests, not only centered forests.

Reading between the lines

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

  • The explicit constant $C(d)$ invites a finite-sample variance estimator for Breiman forests, since the experiments show matching slopes though not matching intercepts.
  • The i.i.d. assumption on rebalanced samples suggests the results cover undersampling without replacement but not oversampling or SMOTE-type duplicates; a CLT for resampling with replacement would be a natural extension.
  • One could optimize the choice of the rebalanced minority proportion $p'$ to minimize the variance ratio for a fixed $p$, connecting this analysis to cost-sensitive learning.
  • The variance-reduction mechanism may be generic: any base estimator whose CLT variance is of the form $\mathrm{const}\cdot \mu(x)(1-\mu(x))$ would inherit the same $O(p)$ reduction after this debiasing.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper studies the asymptotic normality of subsampling infinite centered random forests (ICRF), a purely random forest model in which splits occur at cell midpoints along uniformly chosen coordinates. The main technical results are a CLT for the ICRF around its own expectation (Theorem 2.4) and around the regression function μ(x) (Theorem 2.5), with an explicit limiting variance constant C(d) and explicit conditions linking the subsample size s, tree depth k, and sample size n. The paper then turns to imbalanced classification: it defines a rebalanced ICRF trained on an i.i.d. sample from a rebalanced distribution (X',Y'), introduces a debiased importance-sampling estimator IS-ICRF, proves CLTs for both (Theorem 3.2 and Corollary 3.4), and claims in Theorem 3.5 that for small minority probability p the IS-ICRF has asymptotic variance ratio O(p) relative to the ICRF, i.e., a variance reduction under high imbalance. Numerical experiments with Breiman's random forests illustrate the predicted variance decay rates and the variance reduction.

Significance. The ICRF portion of the paper is a solid contribution: Theorems 2.4–2.5 provide a U-statistic-style CLT with an explicit constant C(d), obtained through a detailed Hajek-projection variance computation (Proposition A.8) and a delicate kernel estimate (Proposition A.9). The paper supplies full proofs of the auxiliary lemmas and gives testable, explicit hyperparameter conditions, so the variance-rate predictions of Section 4 are falsifiable and indeed appear to agree qualitatively with simulations on Breiman's forests. The debiasing construction via the odds-ratio function g (Lemma B.2) and the Delta-method transfer (Corollary 3.4) are clean and potentially useful beyond this model. However, the headline application claim — the O(p) variance reduction of Theorem 3.5 — is not supported by the proofs in the setting in which it is stated, because the comparison relies on a reweighted sample that Section 3.4 itself concedes is outside the i.i.d. framework used to prove Corollary 3.4 and Proposition B.7. The ICRF CLTs are largely independent of this gap and should be assessed separately.

major comments (2)
  1. [Theorem 3.5 and Section 3.4] The stated comparison in Theorem 3.5 is not a consequence of the paper's theorems. Corollary 3.4 and Proposition B.7 are proved for an i.i.d. sample Z'1,...,Z'n' drawn from the rebalanced distribution (X',Y'), with CLT normalization sqrt(n'/(s^2 V'_{1,s})) in (21) and (30). Theorem 3.5 instead claims (36)–(37) with the same n in both normalizations, justified in Section 3.4 by assuming "both estimators are built using n observations" via reweighting; the same paragraph concedes that "this scenario departs from our theoretical analysis, as we do not allow repetitions in the observations." No theorem in the paper covers a reweighted (or otherwise dependent) sample, so (37) and (38) are unproven in the setting in which they are stated. This is not a harmless technicality: if one keeps the no-repetition i.i.d. construction, the rebalanced sample has size n' ≤ O(p n) (at most 2 p n when p' = 1/2), so the normalization in (21) carries a factor n'/n = Θ(p), and the variance ratio becomes (V'_{1,s} g'(μ')^2 / n') / (V_{1,s} / n) = O(p) · Θ(1/p) = Θ(1); the advertised O(p) reduction cancels. The variance-reduction claim therefore needs either a genuine CLT and variance analysis for the reweighted estimator, or a reformulated statement that does not assert the comparison in the reweighted setting.
  2. [Abstract, Section 1.2, and Theorem 2.4] The paper advertises "explicit convergence rates" and "exact rate of convergence" for the CLT. What Theorems 2.4–2.5 actually deliver is the magnitude of the sampling fluctuations, namely sqrt(2^k/(n k^{(d-1)/2})), or equivalently the reciprocal n^{1/2} k^{(d-1)/4}/2^{k/2}; this is the variance scale of the Hajek projection. The theorems do not provide any bound on the distance between the distribution of the normalized estimator and the Gaussian limit: no Berry–Esseen-type inequality is proved anywhere in the paper. The repeated phrasing "explicit convergence rate of the CLT" is therefore misleading and should be replaced by wording such as "explicit variance rate" unless a genuine distributional approximation bound is added.
minor comments (6)
  1. [Theorem 2.4 and Appendix A.4] The symbol N1 is overloaded: it denotes both the first component of the Gaussian vector and, in equation (83), the all-ones vector times the average N̄. The definition of Θ in (83) should use an explicitly defined mean vector to avoid confusion.
  2. [Lemma B.5, proof] The displayed Bayes computation for P(X ∈ L_U(x) | U, Y = 1) is garbled; a division by P(Y = 1 | U) appears to be missing. The final formula is correct, but the intermediate display should be corrected.
  3. [Assumption (H3)] Assumption (H3) postulates the existence of p'' such that the mixture p'' f_{X|Y=1} + (1-p'') f_{X|Y=0} is exactly uniform on [0,1]^d. This is a strong structural condition; no example or discussion of its plausibility is given, and its relationship to (H0) is not explained. Since Theorem 3.5 depends on it, a comment on when it holds (or a relaxation to bounded densities) would help the reader.
  4. [Section 3.1] The suggested construction "draw n' ≤ n1 samples from the original samples without replacement" produces an exchangeable sample, not an exactly i.i.d. sample from (X',Y'); a remark clarifying that the i.i.d. assumption in the theory is an idealization that is approximately valid when n' is small relative to n0 and n1 would be useful.
  5. [Definition 3.3 and Corollary 3.4] The IS-ICRF estimator (28) plugs in n0 and n1 for the unknown (1-p) and p, while the Delta-method proof of Corollary 3.4 treats p as known. A sentence noting that consistency of n1/n makes this substitution asymptotically negligible would remove a small gap.
  6. [Section 4, Figure 1] The claim that curves with the same β have the same slope rests on visual inspection; since the theory predicts slopes −(1−β) independent of α, reporting estimated slopes with standard errors would make the comparison quantitative.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the ICRF CLT rests on a self-cited general theorem whose conditions are verified independently in this paper, and Theorem 3.5's gap is an unproven extension, not a circularity.

full rationale

The central ICRF CLT (Theorems 2.4-2.5) is obtained by applying Theorem A.1, restated from Mayala et al. (2024) with substantial author overlap, and then verifying its conditions. This is a load-bearing self-citation, but the verification is performed in the present paper: Proposition A.8 computes V1^s ~ C(d)2^k/(s^2 k^{(d-1)/2}) mu(x)(1-mu(x)), Proposition A.9 (proved in the appendix, refining Arnould et al. 2023) supplies the kernel constant, and Proposition A.7 controls the bias. The CLT is therefore not assumed as its own conclusion. The debiased IS-ICRF CLT (Corollary 3.4) follows from the Delta method applied to the odds-ratio identity (Lemma B.2), and Theorem 3.5's O(p) variance reduction is derived from the explicit variance bounds of Proposition B.7 together with the explicit factor g'(mu'(x))^2; no fitted parameter is renamed as a prediction. The only load-bearing assumption not proved is in Theorem 3.5, where the comparison uses both estimators built from n reweighted observations, a scenario the paper itself flags as departing from the i.i.d. rebalanced-sample theory ('this scenario departs from our theoretical analysis, as we do not allow repetitions in the observations'). That is a correctness/extension gap rather than a circular reduction: the variance ratio is computed inside the i.i.d. rebalanced model, not derived from the reweighted estimator's own variance. No exhibited equation reduces to its own input, so the paper sits in the 0-2 no-significant-circularity band.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters appear in the theoretical results; all constants are derived. The main assumptions are distributional and smoothness conditions on the data, an imported U-statistic CLT from prior work, and a specialized condition for the variance comparison. The unproven reweighting scenario in Theorem 3.5 is a gap rather than an explicit axiom.

assumptions (6)
  • domain assumption (H0) X is uniformly distributed on [0,1]^d
    Used to obtain pk,U(x) = 2^{-k} and to derive the explicit constant C(d). Stated in Section 2.2 before Theorem 2.4.
  • domain assumption (H1) The regression function mu is L-Lipschitz with respect to the max norm
    Controls the bias of tree predictions inside cells and the bias-variance decomposition in Appendix A. Stated in Section 2.2.
  • domain assumption (G1) s/(k 2^k) -> infinity
    Ensures leaves contain many observations on average and controls variance of individual trees. Stated in Section 2.2.
  • standard math Theorem A.1, the U-statistic CLT from Mayala et al. (2024), is assumed without proof
    The present paper reduces its CLT proofs to verifying nV_1^s -> infinity and a bias condition, relying on this prior theorem from the same research group.
  • domain assumption Rebalanced samples Z'_i are i.i.d. from the distribution defined in (14)
    All rebalanced CLTs in Section 3 and Appendix B are derived under this i.i.d. assumption; the paper notes resampling with replicates (e.g., SMOTE) is excluded.
  • ad hoc to paper (H3) Existence of p'' making f_{X''} uniform, plus bounded Lipschitz conditional densities
    Introduced specifically for Theorem 3.5 to compare variances in a convenient mixture representation; it is not a standard condition in the random forest literature.

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Pith. "Pith review of Asymptotic Normality of Infinite Centered Random Forests -Application to Imbalanced Classification." pith.science (2026). https://pith.science/paper/VW2BENBM

@misc{pith2026250608548,
  author       = {Pith},
  title        = {Pith review of: Asymptotic Normality of Infinite Centered Random Forests -Application to Imbalanced Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VW2BENBM}},
  note         = {Machine review of arXiv:2506.08548}
}
read the original abstract

Many classification tasks involve imbalanced data, in which a class is largely underrepresented. Several techniques consists in creating a rebalanced dataset on which a classifier is trained. In this paper, we study theoretically such a procedure, when the classifier is a Centered Random Forests (CRF). We establish a Central Limit Theorem (CLT) on the infinite CRF with explicit rates and exact constant. We then prove that the CRF trained on the rebalanced dataset exhibits a bias, which can be removed with appropriate techniques. Based on an importance sampling (IS) approach, the resulting debiased estimator, called IS-ICRF, satisfies a CLT centered at the prediction function value. For high imbalance settings, we prove that the IS-ICRF estimator enjoys a variance reduction compared to the ICRF trained on the original data. Therefore, our theoretical analysis highlights the benefits of training random forests on a rebalanced dataset (followed by a debiasing procedure) compared to using the original data. Our theoretical results, especially the variance rates and the variance reduction, appear to be valid for Breiman's random forests in our experiments.

Figures

Figures reproduced from arXiv: 2506.08548 by the authors.

Figure 1
Figure 1. Variance of the CLT (second term in (39)) for the ICRF, as a function of the logarithm of the sample size, for different choices of parameters α, β (displayed in the legend as Subsampleα_Depthβ). bias rate and the variance rate. Note that a major difference between our theoreti￾cal analysis and our experimentation resides in forest algorithms: our theory is valid for centered random forests but experiments are perfo… view at source ↗
Figure 2
Figure 2. Bias of the CLT (first term in (39)) as a function of the logarithm of the sample size, for different choices of parameters α, β (displayed in the legend as Subsampleα_Depthβ). also observe that large values of β ∈ (0, 1) correspond to lower slopes as anticipated by theory. While the intercept cannot be visualised here, it appears that they are not in line with the theory we developed. Nevertheless, it is remarkable… view at source ↗
Figure 3
Figure 3. Histograms of predictions from different estimators in scenario [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Histograms of predictions from different estimators in scenario [PITH_FULL_IMAGE:figures/full_fig_p057_4.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.