Asymptotic Theory and Phase Transitions for Variable Importance in Quantile Regression Forests
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Nakamura, Tomoshige Shiraishi, Hiroshi |
| author_facet | Nakamura, Tomoshige Shiraishi, Hiroshi |
| contents | Quantile Regression Forests (QRF) are widely used for non-parametric conditional quantile estimation, yet statistical inference for variable importance measures remains challenging due to the non-smoothness of the loss function and the complex bias-variance trade-off. In this paper, we develop a asymptotic theory for variable importance defined as the difference in pinball loss risks. We first establish the asymptotic normality of the QRF estimator by handling the non-differentiable pinball loss via Knight's identity. Second, we uncover a "phase transition" phenomenon governed by the subsampling rate $β$ (where $s \asymp n^β$). We prove that in the bias-dominated regime ($β\ge 1/2$), which corresponds to large subsample sizes typically favored in practice to maximize predictive accuracy, standard inference breaks down as the estimator converges to a deterministic bias constant rather than a zero-mean normal distribution. Finally, we derive the explicit analytic form of this asymptotic bias and discuss the theoretical feasibility of restoring valid inference via analytic bias correction. Our results highlight a fundamental trade-off between predictive performance and inferential validity, providing a theoretical foundation for understanding the intrinsic limitations of random forest inference in high-dimensional settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Theory and Phase Transitions for Variable Importance in Quantile Regression Forests Nakamura, Tomoshige Shiraishi, Hiroshi Machine Learning Statistics Theory Quantile Regression Forests (QRF) are widely used for non-parametric conditional quantile estimation, yet statistical inference for variable importance measures remains challenging due to the non-smoothness of the loss function and the complex bias-variance trade-off. In this paper, we develop a asymptotic theory for variable importance defined as the difference in pinball loss risks. We first establish the asymptotic normality of the QRF estimator by handling the non-differentiable pinball loss via Knight's identity. Second, we uncover a "phase transition" phenomenon governed by the subsampling rate $β$ (where $s \asymp n^β$). We prove that in the bias-dominated regime ($β\ge 1/2$), which corresponds to large subsample sizes typically favored in practice to maximize predictive accuracy, standard inference breaks down as the estimator converges to a deterministic bias constant rather than a zero-mean normal distribution. Finally, we derive the explicit analytic form of this asymptotic bias and discuss the theoretical feasibility of restoring valid inference via analytic bias correction. Our results highlight a fundamental trade-off between predictive performance and inferential validity, providing a theoretical foundation for understanding the intrinsic limitations of random forest inference in high-dimensional settings. |
| title | Asymptotic Theory and Phase Transitions for Variable Importance in Quantile Regression Forests |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2511.23212 |