Online Quantile Regression for Nonparametric Additive Models

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(Jtln Jt) computational complexity per step where Jt denotes the number of basis functions. Besides, we only need O(Jt) 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^-frac2s2s+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.