CMQNet: Continuous and Monotone Quantile Networks

Deep quantile regression has emerged as a powerful paradigm for characterizing conditional distributions and quantifying uncertainty. However, existing neural network approaches are often limited to estimating quantiles at discrete grids, resulting in discontinuity with respect to the quantile level and failing to structurally guarantee the non-crossing property. To address these challenges, we propose the Continuous and Monotone Quantile Network (CMQNet). Leveraging the fundamental theorem of calculus for absolutely continuous functions, CMQNet models the conditional quantile function as the summation of a baseline term and the integral of a non-negative derivative with respect to the quantile index. By modeling this derivative using a neural network with strictly positive activation functions, our framework naturally ensures both monotonicity and continuity by construction, thereby eliminating quantile crossing and admitting a well-defined conditional density. Furthermore, we introduce an efficient training strategy based on stochastic quantile sampling, allowing the model to learn the global quantile process beyond fixed grids. Extensive experiments on simulated datasets and real-world benchmarks demonstrate that CMQNet significantly outperforms state-of-the-art methods in terms of estimation accuracy and stability. The code can be found at https://anonymous.4open.science/r/CMQNet-C420/.