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 $O(J_t\ln J_t)$ 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.

Consistency for Large Neural Networks

Neural networks have shown remarkable success, especially in overparameterized or "large" models. Despite increasing empirical evidence and intuitive understanding, a formal mathematical justification for the behavior of such models, particularly regarding overfitting, remains incomplete. In this paper, we prove that the Mean Integrated Squared Error (MISE) of neural networks with either $L^1$ or $L^2$ penalty decreases after a certain model size threshold, provided that the sample size is sufficiently large, and achieves nearly the minimax optimality in the Barron space. These results challenge conventional statistical modeling frameworks and broadens recent findings on the double descent phenomenon in neural networks. Our theoretical results also extend to deep learning models with ReLU activation functions.