Locally Optimized Random Forests

Kimberly Kaufeld; Lucas Mentch; Mary Frances Dorn; Tim Coleman

REVIEW 3 major objections 6 minor 32 references

Locally Optimized Random Forests

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Weighting random-forest splits and predictions by the estimated test-to-training likelihood ratio makes the forest target the shifted test distribution.

desk verdict A genuinely useful and reproducible weighting scheme for random forests under covariate shift, held back by a consistency proof that doesn't cover the implemented weights and by empirical claims that run ahead of the tables. read the letter →

arxiv 1908.09967 v1 pith:7UK5ZU6T submitted 2019-08-27 stat.ML cs.LGstat.ME

Tim Coleman , Kimberly Kaufeld , Mary Frances Dorn , Lucas Mentch This is my paper · ORCID

classification stat.MLcs.LGstat.ME MSC 62G0862G0962H30

keywords randomforestsimportancesamplingcovariateshiftlikelihoodratioestimationuLSIFhurricanepoweroutageforecastingquantileregressionout-of-bagerror

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

Standard supervised learning validates models on a test set drawn from the same distribution as training, but real problems often involve a different target distribution. This paper proposes Locally Optimized Random Forests, which reweight both the split-selection and prediction stages of every tree by an estimate of the likelihood ratio $dP_2^*/dP_1^*$ between the test and training covariate distributions, so the forest behaves as if trained on the target distribution. The motivating application is forecasting county-level power outages during hurricanes, where rare, severe storms are exactly the cases standard validation handles worst. The paper claims the weighted out-of-bag error converges in probability to the generalization error under the test distribution, giving a model-selection criterion that does not need labeled test data. If right, the method offers a data-driven way to adapt a broadly used predictive model to extreme events.

What carries the argument

The load-bearing object is the weighted node distribution $\tilde P_{A,w}$, which places weight $w_i$ on observation $(X_i,Y_i)$ inside node $A$ and renormalizes. All tree computations, the split-quality functional $T_{j,z}(\tilde P_{A,w})$ and the prediction functional $T_1(\tilde P_{A^*(x),w})$, are evaluated against this distribution, so importance weighting enters the structure of the tree, not just the bootstrap resampling. The weights come from unconstrained least-squares importance fitting (uLSIF), which models the ratio as a nonnegative linear combination of Gaussian kernels $\sum_k \alpha_k K_\sigma(X, x_k)$, regularized by an $\ell^1$ penalty and smoothed by selecting a power $\lambda$ that fixes the effective sample size at a pre-chosen $n_0 = 0.75n$. The effective sample size is $n_{\mathrm{eff}} = (\sum_i w_i)^2 / \sum_i w_i^2$, so fixing it at $n_0$ prevents a few observations from dominating.

What would settle it

On data where $P_1$ and $P_2$ are known, compare a forest weighted by the true oracle ratio $dP_2^*/dP_1^*$ with one weighted by the uLSIF estimates across increasing shift strength, for example the Dirichlet simulation with $\lambda$ from 1 to 1.5. If the oracle-weighted forest tracks the $P_2$ generalization error while the estimated-weight forest diverges from it, the reweighting idea is sound and the fragility is in the weight estimator; if the oracle-weighted forest also fails, the covariate-shift assumption $P(Y\mid X)$ unchanged is the point of failure.

Watch

Extended reading notes

Core claim

Under the covariate-shift model $P_i(X,Y)=P(Y\mid X)P_i^*(X)$, the paper's central claim is that replacing the empirical node distribution inside a random forest with the weighted distribution $\tilde P_{A,w} = \sum_i w_i \mathbb{I}(X_i\in A)\delta_{(X_i,Y_i)} / \sum_j w_j \mathbb{I}(X_j\in A)$, where $w_i \propto \hat{\ell}(X_i)$ estimates $dP_2^*/dP_1^*$, makes both the split criterion $L(j,z)$ and the terminal-node prediction $T(x;D)$ approximate the target distribution $P_2$ instead of the training distribution $P_1$. The weights are learned from unlabeled test covariates, making the procedure semi-supervised. Proposition 1 states that the weighted out-of-bag error converges in probability to the $P_2$ generalization error, so tuning hyperparameters by minimizing weighted out-of-bag error selects a model for the target distribution. The paper also extends the weighting to quantile regression forests for prediction intervals and supplies a quantile-regression imputation step that preserves the training covariate distribution.

Load-bearing premise

The benefit of the method depends on the estimated weights being close to the true likelihood ratio $dP_2^*/dP_1^*$; Proposition 1 assumes the sup-norm estimation error tends to zero, but the paper does not verify this on the hurricane data and its simulations show the weighted forest's advantage erodes and can reverse as the shift grows.

Editorial extensions

If this is right

Weighted out-of-bag error can be used to tune $mtry$ and other hyperparameters when the test distribution differs from training, without labeled test data (Proposition 1).
The weighting extends to quantile regression forests, giving prediction intervals that stay near nominal coverage while growing narrower under covariate shift.
In the simulations, the weighted forest improves RMSE and score over an unweighted forest for moderate shifts, with the largest gains in Models 1 and 2.
Gains shrink as the shift grows: both weighted and unweighted forests degrade with increasing $\lambda$, and the weighted forest can underperform on large shifts.
The quantile-regression imputation procedure generates missing covariates from the full conditional distribution, preserving the training covariate distribution so the density-ratio weights remain meaningful.

Reading between the lines

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

Because the weighted node distribution is defined without assuming a particular splitter, the same two-stage recipe could be tested in gradient-boosted trees or other recursive ensemble methods, an extension the paper does not explore.
A practical diagnostic would be to compare the effective sample size after uLSIF with the effective sample size of the oracle weights; when these diverge, the weight estimator is the fragile component rather than the forest.
The proof of Proposition 1 assumes nonnegative responses and bounded fourth moments, so applying the method to zero-inflated or censored outage counts is a natural stress test.
Weighted out-of-bag error could be paired with conformal prediction under covariate shift to produce calibrated intervals for extreme storms, a neighbouring development the paper does not develop.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Tom — quick take on arXiv:1908.09967. The core idea is worth having: instead of retraining a forest for each test point as in case-specific random forests, they estimate the density ratio once via uLSIF and feed those weights into split selection, node prediction, quantile forest weighting, and a weighted OOB error. That combination is new to me, and the paper ships R code and data, so the empirical pieces are checkable. Under moderate covariate shift the weighted forest does reduce RMSE in the simulations, and the oracle-versus-learned weight comparison in Figure 3 is a nice sanity check.

The soft spots are mostly in the theory. Proposition 1 assumes the estimated weights are proportional to the true likelihood ratio up to a vanishing error term, but the implemented method replaces the weights with w^lambda, choosing lambda so the effective sample size hits 0.75n. For lambda != 1, w^lambda is not proportional to the target ratio, so the theorem's hypothesis isn't satisfied by the algorithm being evaluated. The proof also has an internal slip: after showing S2,N -> 0 it concludes Var(S2,N) -> 1 (surely meant 0), and the variance bound on S1,n implicitly needs a bounded likelihood ratio or a finite P1-moment of L, which is not stated. These are fixable, but as written the tuning rationale is not established.

The simulations are supportive but the text overstates them. No error bars are shown over the 150 runs, and the advantage shrinks and reverses at larger shift. In the hurricane application, the weighted method improves the score for Irma, Matthew, Nate, and Sandy, but is worse for Harvey and Arthur — and the text says Harvey improves, which Table 3 contradicts. Coverage also tends to be lower for the weighted forest, and that deserves a more careful discussion than a brief dismissal.

Who gets value: anyone working on covariate shift with tree ensembles, and forecasters chasing extreme-event performance. The paper deserves a serious referee. I'd recommend major revision: fix the proposition's hypotheses and proof, add error bars or per-run summaries, and correct the summary of the real-data results. The method itself is sensible and the reproducibility is a real plus.

Referee Report

3 major / 6 minor

Summary. The paper proposes a weighted random forest procedure for prediction under covariate shift. The method estimates the likelihood ratio dP2*/dP1* between the test and training covariate distributions using uLSIF (Kanamori et al., 2009), then uses these weights both in the split-selection criterion and in the prediction stage of each tree, with a weighted quantile regression forest variant for prediction intervals. To stabilize the weights, Section 3.2.3 replaces w(X) with w(X)^lambda so that the effective sample size equals a pre-specified n0, and Section 3.3 introduces a weighted out-of-bag error for model tuning. Proposition 1 claims that this weighted OOB error converges in probability to the P2 generalization error. The paper also proposes a quantile-regression-based missing-data imputation scheme. The method is evaluated on five synthetic data-generating models with varying shift severity, comparing weighted and unweighted forests, and on a hurricane power-outage forecasting application with six held-out storms.

Significance. If the claimed properties hold, the paper makes a practically useful contribution: it extends random forests to covariate shift with a relatively simple modification, and the hurricane outage application is a credible motivating problem. A notable strength is the design of the simulation in Section 4.1, which compares the learned uLSIF weights against oracle weights and thereby isolates the reweighting mechanism from density-ratio estimation error. The paper also provides a theoretical target, Proposition 1, for justifying weighted OOB model selection, and it makes concrete, falsifiable predictions about performance as a function of shift magnitude. However, the theoretical support as currently stated does not cover the regularized weights actually used in the experiments, and the empirical sections lack uncertainty quantification. With those issues addressed, the paper would be a solid contribution to the covariate-shift and random-forest literatures.

major comments (3)

[Section 3.2.3, Section 3.3, Appendix B] Proposition 1 does not apply to the weighting scheme actually implemented and evaluated. The proposition assumes w_N(x)=c dP2*/dP1*(x)+epsilon_N(x), but Section 3.2.3 replaces the estimated weights with w(X_i)^lambda, choosing lambda so that the effective sample size equals n0=0.75n; Table 3 reports lambda values of 0.0084, 0.1305, and 0.3602, for which the weight vector is not proportional to the likelihood ratio. Consequently, the tuning rationale in Section 3.3 -- that minimizing the weighted OOB error should select a good model for P2 -- is not supported by Proposition 1 for the regularized estimator used in the simulations and the hurricane study. Please either extend the consistency result to ESS-regularized weights or explicitly restrict the claim to the un-regularized estimator.
[Appendix B] The proof of Proposition 1 has gaps that need repair. The variance bound for S_{1,n} requires a boundedness or finite-moment condition on the likelihood ratio, e.g., sup_x dP2*/dP1*(x)<infinity or a finite second moment under P1; the displayed step bounding E_{P1}[L(X)(mB-Y)^4] by max(E_P[W^4], E_P[Z^4]) is not justified as written because the change of measure is not controlled. In addition, after arguing that S_{2,N} -> 0, the text concludes Var(S2,N) -> 1; the limit should be 0. These issues matter because Proposition 1 is the stated theoretical basis for using the weighted OOB error in model selection.
[Section 4.3, Figure 4, Tables A2-A6] The simulation results are reported without any measure of variability. Figure 4 and Tables A2-A6 show only point estimates averaged over the 150 runs, so the reader cannot assess whether differences such as Model 1 at lambda=1.5 (weighted RMSE 1.214 vs unweighted 1.404) are meaningful. The same tables also show that the weighted forest loses its advantage at high shift levels in Models 3 and 4, e.g., Table A4 at lambda=1.357 and higher, and the coverage plots show weighted intervals falling below the 0.80 nominal level in that regime. Please add standard errors, intervals, or statistical comparisons, and discuss explicitly the range of shift magnitudes in which the method is beneficial.

minor comments (6)

[Proposition 1, Section 3.3] The notation in Proposition 1 is inconsistent: it samples (X_i,Y_i)_{i=1}^{n+m} but then defines N, n, and m=sum Z_i without clearly relating n and m to N. Please clarify the sample-size notation.
[Section 5, Table 3] The hurricane results are mixed: the weighted model improves the score for Harvey, Irma, and Matthew, but for Sandy, Nate, and Arthur the weighted model is worse on most reported metrics, including score. The text should state this balance explicitly and discuss possible reasons, since the hurricane application is central to the paper's motivation.
[Section 3.2.3] The statement that n0 is chosen in (1,n) conflicts with the later choice n0=0.75n; please specify whether n0 is an integer or a proportion and how the search for lambda is implemented.
[Section 3.4, Equation (3.6)] Equation (3.6) is hard to parse: the displayed convergence appears to reverse the roles of the estimator and the limiting CDF. Rewriting this step would improve clarity.
[Table 3] The lambda column is reported for unweighted rows as well as weighted rows; since unweighted forests do not use lambda, this is confusing and should be clarified in the table caption or by leaving those entries blank.
[Figure 1 caption] The caption says "Fitted vs Predicted" but the figure appears to plot observed versus predicted values; please correct the wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the weighted OOB guarantee is a standard importance-sampling consistency result under an explicit weight-convergence hypothesis; the main caveats are unverified assumptions and proof gaps, not circular reasoning.

full rationale

Proposition 1 (Section 3.3, Appendix B) is the only candidate for a circular step, but on inspection it is not one. The weighted OOB metric in Eq. (3.5) is explicitly a self-normalized importance-sampling estimator, and its hypothesis w_N(x) = c dP2*/dP1* + epsilon_N(x) is an external consistency assumption about the uLSIF weight estimates, not a restatement of the conclusion theta_P2. The target theta_P2 is defined independently as the P2 expectation of the limiting OOB error, so the proposition's conclusion does not reduce to its input by construction; it is the standard importance-sampling identity, with a variance argument supplied in the proof. The proof does contain validity gaps — for example, the implemented ESS-regularized weights w(x)^lambda with lambda != 1 are not proportional to the likelihood ratio, so Proposition 1's hypothesis is not satisfied by the evaluated method, and the appendix contains an apparent slip in concluding Var(S2,N) -> 1 after showing S2,N -> 0. These are correctness and assumption gaps, not circularity. Likewise, the weights are estimated from unlabeled test covariates rather than test responses, and the simulations include oracle-weight comparisons that benchmark the weighting itself; the observed degradation under large shift (Figure 4, Section 6) is an empirical limitation, not a definitional identity. The self-citations (Mentch and Hooker; Coleman et al.; Peng et al.) appear only in related-work and simulation-design contexts and carry no load-bearing weight. No fitted parameter is renamed as a prediction, and no self-citation chain forces the paper's conclusions.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central method depends on the covariate shift assumption, mutual absolute continuity, and the unverified assumption that the estimated weights converge to the true likelihood ratio. The chosen effective sample size and the uLSIF hyperparameters are additional tuning inputs. No new physical or statistical entities are introduced.

free parameters (3)

n0, minimum effective sample size target = 0.75n
Used in Section 3.2.3 to select the weight smoothing exponent lambda by solving n_eff = n0. Chosen by the authors as a heuristic, not by cross-validation, and directly controls how aggressively weights are shrunk toward uniform.
uLSIF bandwidth sigma = not reported, selected by leave-one-out cross-validation
Gaussian kernel bandwidth in the density ratio model l(X) = sum_k alpha_k K_sigma(X, x_k); affects the smoothness of the estimated ratio.
uLSIF regularization parameter lambda = not reported, selected by leave-one-out cross-validation
L1 penalty coefficient in objective (3.3); affects the sparsity and stability of the estimated density-ratio weights.

assumptions (6)

domain assumption Covariate shift model: P1(X,Y)=P(Y|X)P1*(X) and P2(X,Y)=P(Y|X)P2*(X), so the conditional response distribution is identical across domains.
Equation (1.1) in Section 1. The entire reweighting scheme is invalid if P(Y|X) differs between training and test.
domain assumption P1* and P2* are mutually absolutely continuous with matching support.
Stated in Section 1: for all measurable A, P1*(A)>0 iff P2*(A)>0. Needed for the likelihood ratio l(X)=dP2*/dP1* to be well-defined everywhere.
ad hoc to paper Estimated weights are consistently proportional to the true likelihood ratio: wN(x)=c dP2*/dP1*(x)+epsilon_N(x) with sup|epsilon_N| -> 0.
Assumed in Proposition 1 (Section 3.3 and Appendix B). The paper does not prove uLSIF attains this in the high-dimensional setting, and the probabilistic classification alternative is shown to be unstable in Figure 2.
domain assumption Weak correlation of squared OOB residuals: rho*_n = max corr[(mB_i(X_i)-Y_i)^2, (mB_j(X_j)-Y_j)^2] -> 0.
Proposition 1 assumption, needed for the variance bound in the proof in Appendix B.
domain assumption Bounded fourth moments of Y and of tree predictions, with Y_i >= 0 almost surely.
Proposition 1 assumes sup_x E(Y^4|X=x) < K and positive responses; used for the variance arguments in Appendix B.
domain assumption Meinshausen quantile regression forests provide consistent conditional CDF estimators.
Section 3.4 relies on assumption (A2), motivated by Meinshausen (2006), to justify the quantile-forest imputation scheme and its claimed consistency.

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Cite this review

Pith. "Pith review of Locally Optimized Random Forests." pith.science (2026). https://pith.science/paper/7UK5ZU6T

@misc{pith2026190809967,
  author       = {Pith},
  title        = {Pith review of: Locally Optimized Random Forests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UK5ZU6T}},
  note         = {Machine review of arXiv:1908.09967}
}

abstract

Standard supervised learning procedures are validated against a test set that is assumed to have come from the same distribution as the training data. However, in many problems, the test data may have come from a different distribution. We consider the case of having many labeled observations from one distribution, $P_1$, and making predictions at unlabeled points that come from $P_2$. We combine the high predictive accuracy of random forests (Breiman, 2001) with an importance sampling scheme, where the splits and predictions of the base-trees are done in a weighted manner, which we call Locally Optimized Random Forests. These weights correspond to a non-parametric estimate of the likelihood ratio between the training and test distributions. To estimate these ratios with an unlabeled test set, we make the covariate shift assumption, where the differences in distribution are only a function of the training distributions (Shimodaira, 2000.) This methodology is motivated by the problem of forecasting power outages during hurricanes. The extreme nature of the most devastating hurricanes means that typical validation set ups will overly favor less extreme storms. Our method provides a data-driven means of adapting a machine learning method to deal with extreme events.

Figures

Figures reproduced from arXiv: 1908.09967 by the authors.

**Figure 1.** Fitted vs Predicted for each storm-holdout model. Blue line represents perfect prediction, and grey bars represent 80% prediction intervals 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Comparison of estimated density ratios between an inverted random forest classifier and the uLSIF method of Kanamori et al. (2009). In this example, P ∗ 1 (X) = N (0, 2.5 2 ) and P ∗ 2 (X) = N (0.5, 0.952 ), and models were learned with n = 1500 examples from each. In the above example, the RF attained RMSE of 0.355 while the uLSIF method attained an RMSE of 0.139. where λ is a tuning parameter, and the first term u… view at source ↗

**Figure 3.** Top: Fitted functions according to the three tested models, along with an overlay of the training points. Center: The training and test densities used. Bottom: Estimated density ratio terms and true density ratio terms. the densities are (higher λ leads to higher discrepancy). In these simulations, we use λ ∈ {1, 1.07, 1.14, 1.21, 1.29, 1.36, 1.43, 1.5} - noting that λ = 1 is the case where P1 = P2. Note that P2 con… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Results for the Score (top), RMSE (center), and Coverage probabilities (bottom) from the simulation study from subsection 4.2. The dashed line in the bottom indicates the nominal coverage level, 0.80. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: Out of bag error versus holdout RMSE. Top: Results for the unweighted forest. Bottom: Results for the weighted forest. 6 Conclusion We sought to modify the random forest algorithm to account for distributional changes between test and training sets, which often arise i…

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

F., Candes, E

Barber, R. F., Candes, E. J., Ramdas, A., and Tibshirani, R. J. (2019). Conformal prediction under covariate shift. arXiv preprint arXiv:1904.06019

arXiv 2019

[2] [2]

Breiman, L. (2001). Random forests. Machine learning , 45(1):5--32

2001

[3] [3]

P., Latto, A

Cangialosi, J. P., Latto, A. S., and Berg, R. (2018). Hurricane irma. In National Hurricane Center Tropical Cyclone Report

work page 2018

[4] [4]

Chen, S. (2014). Imputation of missing values using quantile regression . PhD thesis, Iowa State University

work page 2014

[5] [5]

Coleman, T., Peng, W., and Mentch, L. (2019). Scalable and efficient hypothesis testing with random forests. arXiv preprint arXiv:1904.07830

work page Pith review arXiv 2019

[6] [6]

Friedman, J., Hastie, T., and Tibshirani, R. (2001). The elements of statistical learning , volume 1. Springer series in statistics New York, NY, USA:

work page 2001

[7] [7]

Friedman, J. H. (1991). Multivariate adaptive regression splines. The annals of statistics , pages 1--67

work page 1991

[8] [8]

Guikema, S. D. and Quiring, S. M. (2012). Hybrid data mining-regression for infrastructure risk assessment based on zero-inflated data. Reliability Engineering & System Safety , 99:178--182

work page 2012