This paper investigates nonparametric quantile regression using recurrent neural networks (RNNs) and sparse recurrent neural networks (SRNNs) to approximate the conditional quantile function, which is assumed to follow a compositional hierarchical interaction model. We show that RNN- and SRNN-based estimators with rectified linear unit (ReLU) activation and appropriately designed architectures achieve the optimal nonparametric convergence rate, up to a logarithmic factor, under stationary, exponentially $\boldsymbol{\beta}$-mixing processes. To establish this result, we derive sharp approximation error bounds for functions in the hierarchical interaction model using RNNs and SRNNs, exploiting their close connection to sparse feedforward neural networks (SFNNs). Numerical experiments and an empirical study on the Dow Jones Industrial Average (DJIA) further support our theoretical findings.

The second-best congestion pricing (SBCP) problem is one of the most challenging problems in transportation because of its two-level hierarchical structure. In spite of various intriguing attempts at solving SBCP, existing solution methods are either heuristic without a convergence guarantee or suitable for solving SBCP on small networks only. In this paper, we first reveal some convexity-based structural properties of the marginal value function reformation of SBCP, and then, by effectively exploiting these structural properties, we propose two dedicated decomposition methods for solving SBCP on large-scale networks, which are different from existing methods in that they avoid linearizing nonconvex functions. We establish the convergence of the two decomposition methods under commonly used conditions and provide the maximum number of iterations for deriving an approximate stationary solution. The computational experiments based on a collection of real road networks show that in comparison with three existing popular methods, the two proposed methods are capable of solving SBCP on larger-scale networks, and for instances that can be solved by existing methods, the two proposed methods are substantially faster. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods and Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72032001, 72431007, 72131007, 72021002, and 12271161]. L. Guo was also supported by the Natural Science Foundation of Shanghai [Grant 22ZR1415900]. X. Wang was also supported by the Fundamental Research Funds for the Central Universities and CCF-DiDi GAIA Collaborative Research Funds for Young Scholars. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0144 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0144 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .