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Convolution Smoothed Quantile Regression for XGBoost

Mandy Yao, Meredith Franklin

arXiv:2608.15290v1stat.MLcs.LG

TL;DR

Point prediction alone provides limited information about rare-event probabilities and predictive tails. The paper integrates convolution-smoothed quantile regression into XGBoost and finds that convolution methods provide accurate exceedance estimates while multi-output trees nearly eliminate quantile crossing.

  • Problem

    Point prediction alone does not provide conditional quantiles, exceedance probabilities, or predictive-tail information needed for rare and consequential extremes.

  • Method

    The paper integrates convolution-smoothed quantile regression into XGBoost and compares smoothing methods and independent versus multi-output quantile estimation.

  • Results

    Convolution methods provide the most accurate exceedance probabilities, while multi-output trees reduce average quantile crossing rates to approximately 0.00.

  • Takeaways & Limitations

    For dense CDF estimation, multi-output trees, target standardization, and spatially tuned smoothing are recommended to improve calibration and stability.

  • Takeaways & Limitations

    Accurate estimates of extreme values remain difficult when far-right-tail data are sparse.

Abstract

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The increasing availability of large and complex datasets across many scientific disciplines has led to widespread adoption of machine learning (ML) for prediction. However, most ML algorithms focus on point estimation and provide limited information about predictive uncertainty or the conditional distribution of the response, restricting their ability to characterize rare or extreme outcomes. We develop QXGB, a quantile-based gradient boosting framework, and introduce a convolution smoothed loss within it that estimates conditional quantiles for constructing dense cumulative distribution functions (CDFs), exceedance probabilities, and tail behaviour relevant to extreme outcomes. This approach preserves the computational efficiency of extreme gradient boosting while restoring the Hessian information XGBoost relies on for tree splitting, in turn providing interpretable measures of extreme value and exceedance probability predictions. We derive the gradients and Hessians needed to integrate convolution smoothed quantile loss with different kernel specifications into XGBoost, and with simulated data, benchmark this approach against alternative smoothed quantile regression losses, the native quantile objective in the XGBoost Python package, and independent versus multi-output tree estimation. The practical relevance is illustrated in an application predicting fine particulate matter (PM$_{2.5}$) in northern California, including periods where levels were elevated due to wildfire smoke. Our results show that convolution smoothed QXGB, particularly when paired with multi-output trees, delivers accurate predictions with near-zero quantile crossing, well-calibrated CDF and exceedance probability estimates, and useful tail characterization for extreme values. Interval estimation is also evaluated as a measure of data spread.

2 Introduction

The introduction motivates conditional quantile modeling for rare and consequential extremes, where exceedance probabilities and upper-tail behavior matter beyond point predictions. It presents convolution-smoothed quantile regression integrated into XGBoost to address pinball-loss optimization issues while preserving second-order information for tree splitting.

  • Rare threshold-crossing events require predictions of exceedance likelihood and upper-tail behavior beyond point estimates.
  • Quantile regression estimates conditional quantiles without imposing a parametric response distribution, making it suitable for skewed, heavy-tailed, and extreme-event settings.
  • A dense conditional-quantile grid extends XGBoost from point prediction toward distributional prediction.
  • The pinball loss is non-differentiable at zero residual and has a second derivative that is zero almost everywhere, limiting optimization curvature information.
  • The proposed convolution-smoothed quantile regression applies kernel convolution within XGBoost while preserving second-order gradient information for tree splitting and enabling comparisons with existing implementations and smoothed losses.

3 Methods

The methods extend XGBoost from point prediction to quantile and distributional prediction by addressing the pinball loss’s missing Hessian information. Convolution smoothing supplies globally convex, twice continuously differentiable objectives, while the framework evaluates calibration, sharpness, monotonicity, and interval quality.

  • Quantile regression in XGBoost: Quantile regression in XGBoost constructs point-wise prediction intervals, but the pinball loss is nondifferentiable at y = ˆy and has a zero second derivative almost everywhere.This prevents direct use of the Hessian information required by XGBoost’s optimization.
  • Quantile regression in XGBoost: The built-in Python xgboost quantile objective uses a subgradient and sets the Hessian to sample weights, making curvature contributions constant across samples.This simplifies split-gain calculations but removes sample-level curvature differentiation.
  • Alternative smoothed losses: Huber and arctan smoothing restore differentiability, but Huber curvature is confined to |t| < v whereas arctan smoothing ensures a nonzero second derivative everywhere.Huber smoothing requires careful bandwidth selection because excessive smoothness can bias quantiles, while insufficient smoothness can resemble the built-in objective.
  • Convolution smoothing: Convolution-smoothed quantile loss is globally convex for non-negative kernels and twice continuously differentiable, enabling explicit gradients and Hessians for XGBoost.The implementation computes convolution-smoothed pinball losses, gradients, and Hessians for two kernels, with four additional kernels supplied in the Appendix.
  • Quantile estimation and stability: Independent boosted ensembles estimate each quantile separately, so no mechanism enforces ˆyτi ≤ ˆyτj for τi < τj; large errors can also cause exponential overflow and substantial quantile crossing.Crossing can render the estimated CDF invalid, particularly in extreme out-of-distribution feature spaces.
  • Evaluation metrics: CDFs are evaluated with CRPS for calibration and sharpness, while interval scoring assesses width and coverage, with lower mean interval scores indicating better-calibrated and more informative intervals.Crossing rate is defined as the empirical probability of monotonicity violation across sorted quantiles and samples.

4 Simulation Study

The simulation study evaluates QXGB and competing smoothed quantile methods on volatile spatiotemporal PM2.5 data with simulated extreme events. Convolution smoothing, especially with joint multi-output trees, improves quantile ordering, exceedance-probability estimation, and interval performance, while far-tail thresholds remain challenging.

  • Data-generating process: The simulations represent 50 monitoring sites with temporal dependence, spatially correlated noise, and conditional extreme-event shocks during wildfire season.Fifteen covariates drive the extreme-event mechanism, with event probabilities of 10% during fire season and 5% otherwise.
  • Dense CDF evaluation: Joint multi-output trees effectively resolve the substantial quantile crossing observed when dense quantiles are estimated independently.Average crossing rates reached 0.69 for Huber pinball and 0.43 for convolution methods at lower smoothing parameters under independent estimation.
  • Exceedance probabilities: Convolution smoothing methods produced the largest BSS and the most accurate exceedance-probability estimates for values exceeding 35.0 µg/m3.The Logistic baseline overpredicted risk and had a negative BSS, while the Arctan method also overpredicted risk.
  • Limitations: At the 50.0 µg/m3 threshold, exceedance-probability performance worsened for almost all methods, with risk more overestimated in the sparser far-right tail.The higher threshold represented a rarer extreme event in the simulated distribution.
  • Interval estimation: For 90% prediction intervals, arctan pinball, convolution, and Python xgboost outperformed Huber pinball, while Laplace and Gaussian kernels slightly improved PINAW over arctan.PICP values ≥0.9 were straightforward for arctan and convolution smoothing but not attainable for Huber pinball.

5 Application to PM2.5 Prediction

The PM2.5 application uses northern California monitoring data and evaluates dense probabilistic forecasts, exceedance probabilities, and 90% intervals. Multi-output convolution-smoothed QXGB reduces quantile crossing, improves distributional performance, and provides calibrated exceedance and interval estimates, while smoothing requires careful tuning.

  • Data and covariates: The study analyzes daily PM2.5 measurements from 53 northern California sites during 2012–2018 using 21 temporal, meteorological, and geographic covariates.Measurements were obtained from California regulatory monitors in the EPA’s Air Quality System, with meteorological inputs including gridMET wind data.
  • Dense CDF forecasts: Joint multi-output estimation reduces crossing rates to 0.00 for arctan and convolution methods at moderate to high smoothing levels.At smoothing parameter 0.5, convolution methods achieve CRPS values of 1.36 to 1.41 while maintaining zero crossing rates.
  • Exceedance probabilities: For the 35.0 µg/m3 threshold, Gaussian kernel convolution smoothed XGB provides the best calibration and highest Brier Skill Score, while convolution smoothed quantile XGB compares favorably with default Python quantile XGB.Direct binary classification with the Logistic baseline overpredicts risk.
  • 90% intervals: Arctan pinball and convolution methods achieve PICP ≥0.9, whereas Huber pinball and default Python methods reach PICP 0.74 and 0.87, respectively.The arctan method has PINAW 1.02, making its intervals excessively wide despite meeting the coverage target; convolution intervals cover almost all observed spikes more usefully.

6 Discussion

The discussion emphasizes that smoothing and tree-sharing choices affect efficiency, quantile crossing, calibration, and numerical stability. It also identifies sparse-tail estimation and fixed hyperparameters as limitations, motivating adaptive kernels and method-specific tuning.

  • Modeling choices: Sharing tree structures across quantiles improves computational efficiency and reduces quantile crossing by coupling splits.Multi-output trees are especially effective for reducing crossing when estimating dense CDFs.
  • Modeling choices: Excessive smoothing can bias quantile estimates, while different methods may perform better under different circumstances.The discussion compares the Python xgboost package with Huber, arctan, and convolution-smoothed losses.
  • Numerical stability: Target standardization remedies floating-point overflow and gradient instability caused by large unscaled errors in convolution smoothing.Initial crossing rates on raw data reached 20-45%.
  • Tuning: Empirical tuning with spatial 5-fold cross-validation refines bandwidths, while penalizing crossing discourages problematic smoothing parameters.Rule-of-thumb bandwidths provided a useful starting point.
  • Limitations: Accurate extreme-value estimation becomes difficult at high thresholds because far-right-tail data are sparse.Exceedance-probability results were much worse at the 50.0 µg/m3 threshold than at 35.0 µg/m3.
  • Future work: Future work could test asymmetric, mixture, or skew-adaptive kernels for right-skewed wildfire PM2.5 extremes without sacrificing median accuracy.The discussion specifically suggests skewed Laplace and flexible asymmetric kernels with tunable tails.

A Appendix A · A.1 Other Kernels

Appendix A.1 introduces alternative kernel specifications for convolution smoothing, including a logistic kernel and a uniform-kernel construction. The uniform kernel produces a smoothed loss function similar to the Huber pinball loss.

  • A.1 Other Kernels: A.1 Other Kernels specifies a logistic kernel as K(u) = e−u (1+e−u)2.
  • A.1 Other Kernels: The appendix presents the logistic kernel as one alternative kernel specification for the smoothed loss.
  • A.1 Other Kernels: A.1 Other Kernels includes a piecewise kernel expression involving 1(|u| ≤1) and |u|1(|u| > 1).
  • A.1 Other Kernels: The appendix also presents a uniform-kernel-related piecewise expression based on bounded and unbounded values of |u|.
  • A.1 Other Kernels: The Uniform kernel yields a smoothed loss function similar to the Huber pinball loss function.

A.2 Sensitivity Analysis for Simulation Study Intervals

The sensitivity analysis compares combined performance metrics across varying smoothing parameters and intervals in the simulation study.

  • Table A1 reports combined performance metrics for different smoothing parameters and intervals.

A.3 Simulation Study Results for Exceedance Probabilities, Threshold=50.0

This section presents exceedance probability results for a threshold of 50.0. The findings are reported in Table A2.

  • Table A2 reports exceedance probability results.
  • The analysis uses a threshold of 50.0.
  • The section focuses specifically on exceedance probabilities above the stated threshold.

A.4 Individual Tree Method Results

This section evaluates individually computed quantiles using interval, calibration, crossing-rate, CRPS, and dense-CDF metrics across simulated locations and real PM2.5 data.

  • Prediction intervals: Individual quantiles are evaluated using PICP, PINAW, and interval score for 90% prediction intervals formed from the 0.05 and 0.95 quantiles.The evaluation is reported in Table A3.
  • Simulated-data evaluation: Across all 50 simulated locations, individually computed quantiles are compared using average CRPS and crossing rates.These results are reported in Table A4.
  • Real-data prediction intervals: For 2018 data, prediction intervals based on individually estimated 0.05 and 0.95 quantiles are assessed with PICP and PINAW across five methods.The comparison is reported in Table A5.
  • Dense CDF evaluation: Dense CDF performance is evaluated on the real PM2.5 dataset using spatial LOSO-CV with individually computed quantiles.These results are reported in Table A6.

B Appendix B … B.3 Quantile Regression

Appendix B introduces gradient boosting and its XGBoost specialization, emphasizing regularization and second-order information for optimization. It concludes by defining quantile regression as conditional-quantile estimation using pinball loss.

  • B.1 Gradient Boosting: Gradient boosting builds an additive ensemble of decision trees to find a function F(x) that minimizes a loss L(y, F(x)).The model is initialized with a constant minimizing the loss function.
  • B.1 Gradient Boosting: Each boosting iteration computes negative-gradient pseudo-residuals, fits a decision tree, selects a learning rate, and adds the weighted tree.The learning rate controls the new tree’s contribution, and η helps prevent overfitting.
  • B.2 Extreme Gradient Boosting: XGBoost is an optimized gradient-boosting algorithm supporting parallel processing, hardware optimization, and effective handling of missing values.The passage presents XGBoost as faster and more accurate than standard gradient boosting.
  • B.2.1 Regularization: XGBoost combines the loss with a regularization term to discourage overly complex trees and reduce overfitting.The regularization term penalizes tree complexity through γ and the squared leaf weights through λ.
  • B.2.1 Regularization: The regularization parameters γ and λ govern tree complexity and penalties on squared leaf weights, respectively.Leaf weights are obtained in closed form from gradients and Hessians in XGBoost’s Taylor expansion.
  • B.2.2 Second-Order Gradient: Unlike gradient boosting’s first-order approach, XGBoost uses both gradients and Hessians to optimize trees, calculate leaf weights, and evaluate split candidates.Hessians contribute to the regularization weights and loss reduction used for split evaluation.
  • B.3 Quantile Regression: Quantile regression estimates conditional quantiles rather than conditional means and minimizes the pinball loss for a specified quantile τ.Here, y is observed, ŷ is predicted, and τ identifies the quantile of interest.
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