A PyTorch Framework for Scalable Non-Crossing Quantile Regression

Quantile regression is fundamental to distributional modeling, yet independent estimation of multiple quantiles frequently produces crossing – where estimated quantile functions violate monotonicity, implying impossible negative probability densities. While Constrained Joint Quantile Regression (CJQR) elegantly enforces non-crossing by construction, existing formulations via Linear Programming exhibit $O((qn{)}^3)$ complexity, rendering them impractical for large-scale applications. We present the first scalable solution using PyTorch automatic differentiation: $\mathbf{\text{CJQR-ALM}}$, combining the $\mathbf{\text{Augmented}}\mathbf{\text{Lagrangian}}\mathbf{\text{Method}}$ with $\mathbf{\text{differentiable}}\mathbf{\text{pinball}}\mathbf{\text{loss}}$ and $\mathbf{\text{L-BFGS}}$ optimization. Our approach reduces computational complexity to $O(n)$, achieving near-zero crossing rates on datasets exceeding 70,000 observations within minutes. The differentiable formulation naturally extends to neural network architectures for non-linear conditional quantile estimation. Application to Student Growth Percentile calculations demonstrates practical utility for educational assessment, while simulation studies show negligible accuracy cost (RMSE increase $\approx 2.4$ points) relative to unconstrained estimation – a favorable trade-off for applications requiring valid probability statements across finance, healthcare, and engineering.