{"id":"c03b3882-9055-4b4d-b6d2-38de6fd464e5","arxiv_id":"1908.01251","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A bootstrap procedure estimates the quantile of the regression ensemble convergence gap mse_t minus mse_infinity, with a non-asymptotic guarantee for the ideal functional and good empirical performance.","lead":"This paper develops a bootstrap method to tell how many trees are enough in bagging and random forests for regression, by estimating the gap between a finite ensemble and an infinite one. The method is cheap to run, comes with a non-asymptotic guarantee for an idealized version, and is tested on four real datasets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 guarantees coverage only for the theoretical bootstrap with the true functional ψ; Algorithm 1 as implemented uses estimated ψ̂ (equations 2.6 and 2.7), and no bound controls this extra error.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing concern: Theorem 3.1 covers only bootstrap samples built from the true functional ψ, while the implemented Algorithm 1 uses ψ̂ estimated from hold-out or oob samples, and no theorem accounts for this estimation error. My reading of the proof confirms that the theorem itself is internally sound: the decomposition of mse_t − mse∞ into a leading linear term plus a quadratic remainder, the Berry-Esseen approximation for both the original and bootstrap sums, and the variance-estimator concentration arguments in Appendices B and C are consistent and non-circular. The gap is external to the theorem, but it directly affects the method a practitioner would actually run. The paper is honest about the simplification in Section 3, yet neither the abstract nor the introduction qualifies the practical claim accordingly, and the numerical experiments do not isolate the true-ψ bootstrap from the estimated-ψ implementation. A secondary issue worth noting is that equation (4.3) defines τ_n(t0) as (1 − 1/n)^{n·t0}, which is a probability (approximately e^{−t0}) rather than an expected cardinality; the correct effective oob size is t0·(1 − 1/n)^n ≈ 0.37 t0. This appears to be a typographical or algebraic slip in the bias-correction discussion, and it does not affect Theorem 3.1, but it should be corrected. Because the central theoretical result is sound for the idealized bootstrap while the practical coverage guarantee is unproven, the appropriate verdict remains CONDITIONAL: the authors should either extend the theorem to the estimated-ψ setting or explicitly state in the abstract and introduction that the non-asymptotic guarantee applies only to the no-estimation-error version.","tokens_in":25402,"tokens_out":12469,"duration_ms":127403,"concrete_test":"Run a simulation with a synthetic test distribution ν so that ψ is known exactly. Fix n = 1000, t = 500, B = 50, α = 0.1, and choose k as in equation (3.6). Over many replications of D and ξ_t, estimate the conditional coverage of inequality (3.1) for three variants: (i) the theoretical bootstrap using the true ψ; (ii) Algorithm 1 with a hold-out ψ̂ of size m = n/5, as in Section 5.1; (iii) Algorithm 1 with the oob ψ̂. If variant (i) achieves coverage at or above 1 − α − c0δ_{t,k,B}(D) while variants (ii) or (iii) fall below it by more than Monte Carlo error, the estimated-ψ gap is load-bearing. Repeat with m = n/50 to stress small hold-out sizes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 3.1 and bound (3.5), is proved only for bootstrap samples mse*_t − mse_t defined in equation (2.5), which use the true functional ψ. The paper itself flags this in Section 3: it says the analysis focuses on bootstrap samples that are 'not affected by the extraneous error from estimating ψ.' Yet Algorithm 1 and all numerical experiments use ψ̂ estimated from hold-out samples (2.6) or out-of-bag samples (2.7). No theorem, lemma, or remark bounds the difference between the empirical quantile of ψ̂(Ṭ*_t) − ψ̂(Ṭ_t) and that of ψ(Ṭ*_t) − ψ(Ṭ_t). The experiments in Section 5.1 use a hold-out set of size only |H|/(|H|+|D|) ≈ 1/6 and an oob effective ensemble size of roughly 0.37 t0 for t0 = 500, so the estimation error is not a priori negligible at the operating point t0 = 500, B = 50. Thus the practical coverage guarantee advertised for the implemented method rests on an unproven premise: that the error from estimating ψ is small enough to leave the coverage bound (3.5) intact. This is a specific, identifiable gap rather than an inconsistency in the proof of Theorem 3.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a bootstrap method for deciding whether a randomized regression ensemble (bagging, random forests) is large enough. For a fixed training set D, it targets the (1−α)-quantile q_{1−α}(t) of the convergence gap mse_t − mse∞, where mse_t is the conditional test MSE of the averaged ensemble and mse∞ is its limit. Algorithm 1 resamples the t trained functions with replacement and uses the empirical quantile of ψ̂(T̅*_t) − ψ̂(T̅_t) as an estimator, with ψ estimated from hold-out or out-of-bag samples. Theorem 3.1 gives a non-asymptotic conditional coverage bound for the corresponding oracle bootstrap that uses the true functional ψ; Section 2.2 extends the idea to variable importance; Section 4 analyzes cost and proposes 1/√t extrapolation with an out-of-bag bias correction; Section 5 contains experiments on four datasets.","tokens_in":25653,"tokens_out":7975,"duration_ms":78780,"significance":"The oracle-bootstrap result is a genuine contribution if it stands: under assumptions A1–A2 it gives an explicit, non-asymptotic rate δ_{t,k,B}(D) → 0 in t and B, with only mild moment conditions, and the supplementary proof is detailed and uses standard tools (Talagrand/Rosenthal inequalities, DKW, Berry–Esseen). The CART moment bound in Eq. (3.3) and the cost comparison in Section 4.1 are also useful. However, the main theorem is proven only for bootstrap samples constructed from the unknown functional ψ, whereas Algorithm 1 and every numerical result use an estimated ψ̂; no result controls the estimation error. This gap is central to the paper's advertised practical claim and must be addressed in revision.","major_comments":[{"comment":"Theorem 3.1 is stated for q̂_{1−α}(t) defined as the empirical quantile of B bootstrap samples of the form (2.5), i.e. ψ(T̅*_t) − ψ(T̅_t) with the true functional ψ. The text at the beginning of Section 3 explicitly says the analysis focuses on bootstrap samples that are 'not affected by the extraneous error from estimating the functional ψ.' But Algorithm 1 and all experiments use z_{t,b} := ψ̂(T̅*_t) − ψ̂(T̅_t) with ψ̂ from (2.6) or (2.7). The manuscript contains no result bounding the difference between the empirical quantiles of these two bootstrap distributions, or the difference in coverage. At the numerical operating point (t0 = 500, B = 50, hold-out fraction ≈ 1/6, oob effective size ≈ 0.37 t0), the estimation error is not a priori negligible. Because the abstract and Section 1 present the bootstrap as a practical method, this is a load-bearing gap, not a cosmetic one.","section":"Section 3, Eq. (3.5); Section 2, Algorithm 1, Eqs. (2.5)–(2.7)"},{"comment":"The estimators plotted and recommended in Section 5 are the extrapolated oob estimator (4.4) and the hold-out estimator (4.2). Both rely on the heuristic q_{1−α}(t) ≈ κ/√t, and no theorem or quantitative bound is given for the error of this approximation or for the coverage of the extrapolated interval. Thus even a proof of Theorem 3.1 for the non-extrapolated ψ̂-based estimator would not justify the headline numerical claim that the method predicts convergence at t = 2,000 from t0 = 500. The paper should either prove, under stated assumptions, a bound on the extrapolation error, or explicitly mark the extrapolation as a heuristic and restrict the formal claims to the non-extrapolated estimate.","section":"Section 4.2, Eqs. (4.2) and (4.4); Section 5.2"}],"minor_comments":[{"comment":"There are small typos: 'The purpose of the current paper is develop a solution' should be 'is to develop a solution,' and 'has not beaddressed' in Section 1.2 should be 'has not been addressed.'","section":"Section 1"},{"comment":"The same symbol q̂_{1−α}(t) is used for the empirical quantile of the oracle bootstrap in Theorem 3.1 and for the output of Algorithm 1 with estimated ψ̂; these are different estimators and the notation should be distinguished.","section":"Section 3 and Section 2"},{"comment":"τ_n(t0) is defined as the expected cardinality of oob(X_j), but it is then called the 'effective' ensemble size; since the oob set is random, the text should make clear that q_{1−α}(τ_n(t0)) is an approximation based on an expectation.","section":"Section 4.2, Eq. (4.3)"},{"comment":"The 'true' quantile curves are approximations: they come from 1,000 runs, with mse_{2,000} used as a surrogate for mse∞. This approximation should be stated more prominently near the figures, not only in the experimental setup.","section":"Section 5.1"},{"comment":"No theoretical guarantee is stated for the variable-importance bootstrap; if this is meant as an empirical extension, the text should say so explicitly.","section":"Section 2.2, Algorithm 2"},{"comment":"The error bars in Figures 3–6 are 10th and 90th percentiles over runs; the text says they are not confidence intervals, but a brief explanation of what they do represent would improve clarity.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The main gap is well localized and, in principle, fixable by adding a result that controls the ψ estimation error under explicit assumptions on the hold-out/oob sizes, or by explicitly scoping the formal claims to the oracle bootstrap and reclassifying the numerical procedure as heuristic. The proof machinery appears sound, so I would not reject the paper; major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a competent, useful paper. It gives a bootstrap for deciding when a bagging/random-forest ensemble is large enough in regression, with a non-asymptotic guarantee, and it extends the same idea to variable importance. The proof is real: Banach-valued Rosenthal, Berry-Esseen, DKW, assembled cleanly. The rates are nearly parametric in t and B. That alone justifies a serious look.\n\nWhat's new: the regression analogue of Lopes's classification paper, but with a genuinely different argument and weaker assumptions. The uniform variable-importance convergence check is a nice addition and, as far as I know, not done elsewhere. The extrapolation with the oob bias correction is clever and works well in the experiments.\n\nThe soft spot, and it is the one worth naming: Theorem 3.1 covers bootstrap samples built from the true functional psi (equation 2.5). Algorithm 1 and all experiments use estimated psi from hold-out or oob samples (2.6, 2.7). The paper itself says in Section 3 that it excludes 'the extraneous error from estimating psi'. That is honest, but it leaves the implemented method's coverage guarantee resting on an unproven premise. The stress-test note is right that this is a specific named gap, not a hidden flaw. At the operating point t0=500, B=50, with a hold-out of about one sixth of the data or an oob effective size around 185 trees, the estimation error is not obviously negligible. The numerics suggest it doesn't matter much in practice, but the advertised guarantee for the algorithm as written is not the proved guarantee.\n\nMinor things: the variable importance algorithm has no theorem, only numerics; that's fine for a methods paper but should be flagged as such. The cost analysis assumes m=O(n); reasonable. Self-citation is appropriate here—companion paper, not a puff.\n\nBottom line: the theoretical core is sound, the gap is identifiable, and the practical value is clear. This deserves peer review. If I were the editor I'd send it out, with a gentle instruction to the authors to either extend the theorem to estimated psi or state prominently in the abstract that the guarantee is for the idealized bootstrap. Either way, the reader gets a useful tool and an honest statement of its limits.","headline":"A genuinely useful bootstrap for regression ensemble convergence, with a real theorem and a clearly named gap between the proved result and the implemented algorithm.","tokens_in":26190,"tokens_out":1772,"would_cite":true,"duration_ms":16945,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F40","65B05","68W20","60G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that resampling an ensemble's own trained functions yields a bootstrap quantile that serves as an approximately valid conditional upper bound on the gap between a finite ensemble's error and its infinite-ensemble…","keywords":["random forests","bagging","bootstrap","randomized algorithms","algorithmic convergence","ensemble size","mean-squared error","variable importance"],"falsifier":"Fix a training set $D$, run many independent ensembles on the same $D$, and compare the empirical frequency of $\\mathrm{mse}_t - \\mathrm{mse}_{\\infty} \\le \\hat q_{1-\\alpha}(t)$ with the predicted level $1-\\alpha - c_0\\delta_{t,k,B}(D)$, under all three implementations: the theoretical bootstrap using the exact functional $\\psi$, Algorithm 1 with a hold-out set of size $m$, and Algorithm 1 with out-of-bag predictions. If the theoretical version covers at the predicted level but the practical versions fall short precisely when $m$ or the effective out-of-bag size is small, the missing $\\widehat\\psi$-error bound is the cause; if the practical versions fail even with $\\psi$ known, the Gaussian and moment approximations in the proof are at fault.","tokens_in":25155,"feed_emoji":"🌳","tokens_out":20236,"duration_ms":165062,"temperature":0.7,"pith_summary":"The paper asks a practical question: when a random forest or bagged ensemble is trained on a fixed data set, how large must the ensemble be before adding more trees no longer matters? It claims that a bootstrap resampling of the ensemble's own trained functions answers this: the empirical quantile of the resampled mean-squared-error differences gives an approximate conditional upper bound on the gap between the current ensemble and the ideal infinite ensemble, holding with probability close to the nominal level given the data. The guarantee is non-asymptotic, with a coverage error that shrinks at nearly parametric rates as the ensemble size and the number of bootstrap replications grow. The same resampling idea extends to variable importance, where it certifies uniform convergence across all variables so that rankings are not distorted by algorithmic noise. Because the method works from the predicted values of a single run, checking convergence costs no more than training the ensemble itself when the number of bootstrap samples is moderate.","feed_headline":"Certify when a random forest is large enough","feed_subtitle":"Resampling within one run certifies the gap to the infinite-ensemble limit, shrinking at near-parametric rates.","key_machinery":"The identity $\\mathrm{mse}_t - \\mathrm{mse}_{\\infty} = \\psi(\\overline{T}_t) - \\psi(\\vartheta)$, where $\\psi(f) = \\int (y - f(x))^2 \\, d\\nu$ is the mean-squared-error functional, $\\overline{T}_t$ is the ensemble average, and $\\vartheta(x) = E[\\overline{T}_t(x) \\mid D]$ is its conditional expectation over algorithmic randomness. The identity converts the unobservable gap between a finite average and its limit into a difference of a smooth functional at two averages, so that resampling the trained functions $T_1,\\ldots,T_t$ with replacement and forming $\\mathrm{mse}^*_t - \\mathrm{mse}_t = \\psi(\\overline{T}^{\\,*}_t) - \\psi(\\overline{T}_t)$ produces bootstrap samples whose $(1-\\alpha)$-quantile mimics the target quantile. The load-bearing proof ingredients are a Rosenthal-type moment inequality for sums of independent Banach-valued random variables, which controls the residual $\\|\\overline{T}_t - \\vartheta\\|^2_{L^2}$; a Berry\\textendash Esseen (quantitative central limit) bound chaining the standardized pivot to a Gaussian; and the Dvoretzky\\textendash Kiefer\\textendash Wolfowitz inequality (the standard uniform deviation bound for empirical distributions) chaining the bootstrap empirical quantile to its conditional distribution.","core_discovery":"The central claim is Theorem 3.1. For any randomized regression ensemble satisfying the representation $T_i = \\phi(x; D, \\xi_i)$ with i.i.d. randomizing parameters $\\xi_i$ given the training data $D$, and under a mild moment assumption, the empirical $(1-\\alpha)$-quantile $\\hat q_{1-\\alpha}(t)$ of $B$ bootstrap samples of the form $\\mathrm{mse}^*_t - \\mathrm{mse}_t$ satisfies $$P\\big(\\mathrm{mse}_t - \\mathrm{mse}_{\\infty} \\le \\hat q_{1-\\$\\alpha$}(t) \\mid D\\big) \\ge 1 - \\$\\alpha$ - c_0\\delta_{t,k,B}(D),$$ where $\\delta_{t,k,B}(D) = \\frac{k^2}{\\sqrt{t}}\\left(\\frac{\\beta_{3k}(D)}{\\sigma(D)}\\right)^3 + e^{-k/2} + \\sqrt{\\frac{\\log B}{B}}$. In words, the bootstrap quantile is an approximately valid conditional upper confidence bound on the convergence gap between the finite ensemble and the ideal infinite ensemble, with an explicit non-asymptotic bound on the coverage error. The proof approximates $\\sqrt{t}(\\mathrm{mse}_t - \\mathrm{mse}_{\\infty})$ by a Gaussian with variance $\\sigma^2(D)$, approximates the bootstrap pivot by a Gaussian with the resampled variance, and shows the two variances are close; each step is controlled by a Banach-space moment inequality, a Berry\\textendash Esseen (quantitative central limit) bound, and the Dvoretzky\\textendash Kiefer\\textendash Wolfowitz inequality. When the base learner is CART (the standard greedy regression-tree algorithm) or any method whose predictions stay within the range of the training labels, choosing $k \\approx \\log t - 4\\log\\log t$ reduces the bound to $c(D)\\frac{\\log^2 t}{\\sqrt{t}} + \\sqrt{\\frac{\\log B}{B}}$, i.e., nearly parametric decay.","pith_inferences":["A natural next theorem, which the paper leaves open, would prove a matching guarantee for the estimated error functional, tracking the hold-out size or the out-of-bag effective size explicitly so that the coverage error picks up a term on the order of one over the square root of that size; the numerics suggest the out-of-bag version's larger effective sample is why it outperforms the small hold-ou","The one-sided quantile could support two-sided confidence statements about the infinite-ensemble error itself, since the same Gaussian approximation of the standardized gap that drives the proof should also yield intervals around the limit.","For data sets with very many variables, the worst-case importance deviation should grow like the square root of the log-variable-count over the ensemble size under the bootstrap; the method already handles this automatically, but the same machinery could predict how much of an observed spread in importance scores is purely algorithmic.","Because the method consumes only predicted values, it transfers to any randomized estimator with a convergent average, such as Bayesian additive regression trees, random rotations, or random projections, which the paper lists as within scope of its abstract representation but does not test."],"forward_implications":["A practitioner who runs bagging or random forests once on a fixed data set can compute the bootstrap quantile from the stored predicted values alone and obtain an upper bound, certified up to an explicit coverage error, on how far the finite ensemble's error is from the infinite-ensemble error at any chosen level.","Because the coverage error is non-asymptotic and shrinks at nearly parametric rates in both the ensemble size and the number of bootstrap replications, the guarantee has bite at finite ensemble sizes, not merely in the limit of infinitely many trees.","The extrapolation rule, based on the gap scaling as one over the square root of the ensemble size, lets a small initial ensemble predict the size needed to bring the gap within a chosen tolerance; when out-of-bag estimates are used, the effective ensemble size (the expected number of out-of-bag trees per training point) must be substituted, and the paper shows this bias correction is what makes th","Applied to variable importance, the same resampling certifies that every variable's importance score has converged uniformly, so that rankings compare variables fairly rather than reflecting algorithmic noise on slow-converging variables.","The computational cost of the convergence check is on the order of the number of bootstrap samples times the ensemble size times the validation size, which is no larger than the cost of training the trees when the number of bootstrap samples is modest; the variable-importance variant is independent of the sample size and highly scalable."],"supporting_citations":[{"why":"Supplies the Banach-space moment inequality (a Rosenthal-type bound) that controls the quadratic residual terms in the proof of Theorem 3.1.","marker":"(Talagrand, 1989)"},{"why":"Companion paper that introduced the bootstrap approach and extrapolation idea for classification ensembles, adapted here to regression.","marker":"(Lopes, 2019)"},{"why":"The main prior non-asymptotic bound on regression ensemble convergence, which this quantile-based guarantee complements.","marker":"(Scornet, 2016a)"},{"why":"Defines bagging and random forests, the ensemble methods whose convergence the paper measures.","marker":"(Breiman, 1996, 2001)"},{"why":"Origin of the Dvoretzky–Kiefer–Wolfowitz inequality used to control the bootstrap empirical quantile.","marker":"(Dvoretzky et al., 1956)"},{"why":"Gives the tight constant in the DKW inequality, yielding the explicit square-root-of-log-B-over-B term in the coverage error.","marker":"(Massart, 1990)"},{"why":"Companion moment inequalities for sums of independent Banach-valued random variables used alongside Talagrand's result.","marker":"(Johnson et al., 1985)"},{"why":"Supplementary hypercontraction moment inequalities cited with the Banach-space moment bounds.","marker":"(Kwapień et al., 1991)"},{"why":"Introduces CART; its bounded-prediction property justifies the simplified near-parametric error bound, and its cost model underlies the complexity comparison.","marker":"(Breiman et al., 1984)"},{"why":"The randomForest R package used for all numerical experiments on the four datasets.","marker":"(Liaw and Wiener, 2002)"}],"fun_headline_variants":["Bootstrap certifies when a random forest is big enough","A bootstrap bound for the infinite-ensemble gap","Random forest enough? A bootstrap check says","Near-optimal ensemble size via bootstrap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The coverage guarantee in Theorem 3.1 is proved only for bootstrap samples built from the true error functional $\\psi$, while the implemented Algorithm 1 and all experiments use an estimate $\\widehat\\psi$ from hold-out or out-of-bag samples, so the practical guarantee rests on the unproven premise that this estimation error is negligible, which can fail when the hold-out set is small or the effective out-of-bag size is small.","fun_headline_variants_meta":{"raw":{"variants":["Bootstrap certifies when a random forest is big enough","A bootstrap bound for the infinite-ensemble gap","Random forest enough? A bootstrap check says","Near-optimal ensemble size via bootstrap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1928,"prompt_tokens":1111,"completion_tokens":817,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":771}},"tokens_in":727,"tokens_out":817,"duration_ms":8841,"temperature":1.0,"reasoning_tokens":771,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:18:31.194474+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a training set $D$, run many independent ensembles on the same $D$, and compare the empirical frequency of $\\mathrm{mse}_t - \\mathrm{mse}_{\\infty} \\le \\hat q_{1-\\alpha}(t)$ with the predicted level $1-\\alpha - c_0\\delta_{t,k,B}(D)$, under all three implementations: the theoretical bootstrap using the exact functional $\\psi$, Algorithm 1 with a hold-out set of size $m$, and Algorithm 1 with out-of-bag predictions. If the theoretical version covers at the predicted level but the practical versions fall short precisely when $m$ or the effective out-of-bag size is small, the missing $\\widehat\\psi$-error bound is the cause; if the practical versions fail even with $\\psi$ known, the Gaussian and moment approximations in the proof are at fault.","supporting_citations":[{"cited_title":"(1989) Isoperimetry and integrability of the sum of independent B anach-space valued random variables","cited_arxiv_id":null,"evidence_quote":"Supplies the Banach-space moment inequality (a Rosenthal-type bound) that controls the quadratic residual terms in the proof of Theorem 3.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Companion paper that introduced the bootstrap approach and extrapolation idea for classification ensembles, adapted here to regression."},{"cited_title":"(1996) Bagging predictors","cited_arxiv_id":null,"evidence_quote":"Defines bagging and random forests, the ensemble methods whose convergence the paper measures."},{"cited_title":"and Wolfowitz, J","cited_arxiv_id":null,"evidence_quote":"Origin of the Dvoretzky–Kiefer–Wolfowitz inequality used to control the bootstrap empirical quantile."},{"cited_title":"(1990) The tight constant in the D voretzky- K iefer- W olfowitz inequality","cited_arxiv_id":null,"evidence_quote":"Gives the tight constant in the DKW inequality, yielding the explicit square-root-of-log-B-over-B term in the coverage error."},{"cited_title":"B., Schechtman, G","cited_arxiv_id":null,"evidence_quote":"Companion moment inequalities for sums of independent Banach-valued random variables used alongside Talagrand's result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces CART; its bounded-prediction property justifies the simplified near-parametric error bound, and its cost model underlies the complexity comparison."},{"cited_title":"and Wiener, M","cited_arxiv_id":null,"evidence_quote":"The randomForest R package used for all numerical experiments on the four datasets."}],"review_version":1}