Online Quantile Regression for Nonparametric Additive Models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918438253887488 |
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| author | Zhan, Haoran |
| author_facet | Zhan, Haoran |
| contents | This paper introduces a projected functional gradient descent algorithm (P-FGD) for training nonparametric additive quantile regression models in online settings. This algorithm extends the functional stochastic gradient descent framework to the pinball loss. An advantage of P-FGD is that it does not need to store historical data while maintaining $O(J_t\ln J_t)$ computational complexity per step where $J_t$ denotes the number of basis functions. Besides, we only need $O(J_t)$ computational time for quantile function prediction at time $t$. These properties show that P-FGD is much better than the commonly used RKHS in online learning. By leveraging a novel Hilbert space projection identity, we also prove that the proposed online quantile function estimator (P-FGD) achieves the minimax optimal consistency rate $O(t^{-\frac{2s}{2s+1}})$ where $t$ is the current time and $s$ denotes the smoothness degree of the quantile function. Extensions to mini-batch learning are also established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_08969 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Online Quantile Regression for Nonparametric Additive Models Zhan, Haoran Machine Learning Statistics Theory This paper introduces a projected functional gradient descent algorithm (P-FGD) for training nonparametric additive quantile regression models in online settings. This algorithm extends the functional stochastic gradient descent framework to the pinball loss. An advantage of P-FGD is that it does not need to store historical data while maintaining $O(J_t\ln J_t)$ computational complexity per step where $J_t$ denotes the number of basis functions. Besides, we only need $O(J_t)$ computational time for quantile function prediction at time $t$. These properties show that P-FGD is much better than the commonly used RKHS in online learning. By leveraging a novel Hilbert space projection identity, we also prove that the proposed online quantile function estimator (P-FGD) achieves the minimax optimal consistency rate $O(t^{-\frac{2s}{2s+1}})$ where $t$ is the current time and $s$ denotes the smoothness degree of the quantile function. Extensions to mini-batch learning are also established. |
| title | Online Quantile Regression for Nonparametric Additive Models |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2604.08969 |