在线训练非参数量化回归模型，无需存储历史数据。
Online Quantile Regression for Nonparametric Additive Models
- 基于投影函数梯度下降，用新损失函数处理在线学习。
- 每步计算复杂度仅O(J_t ln J_t)，预测只需O(J_t)时间。
- 理论证明达到最优收敛速率，适合大规模实时数据。
本文提出一种投影函数梯度下降算法（P-FGD），用于在在线场景下训练非参数加性量化回归模型。该算法将函数随机梯度下降框架扩展至pinball损失函数。P-FGD无需存储历史数据，每步计算复杂度为O(J_t ln J_t)，其中J_t表示当前基函数数量；在时间t进行量化函数预测时，仅需O(J_t)计算时间。相比常用的再生核希尔伯特空间（RKHS），P-FGD在在线学习中表现更优。通过一个新颖的希尔伯特空间投影恒等式，我们证明了所提出的在线量化函数估计器（P-FGD）达到了最小最大最优一致性速率O(t^{-2s/(2s+1)})，其中t为当前时间，s为量化函数的光滑度。此外，算法还推广至小批量学习情形。
原文摘要 · Abstract (English)
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.
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