Fast Nonlinear Vector Quantile Regression

Quantile regression (QR) is a powerful tool for estimating one or more conditional quantiles of a target variable Y given explanatory features X. A limitation of QR is that it is only defined for scalar target variables, due to the formulation of its objective function, and since the notion of quantiles has no standard definition for multivariate distributions. Recently, vector quantile regression (VQR) was proposed as an extension of QR for vector-valued target variables, thanks to a meaningful generalization of the notion of quantiles to multivariate distributions via optimal transport. Despite its elegance, VQR is arguably not applicable in practice due to several limitations: (i) it assumes a linear model for the quantiles of the target Y given the features X; (ii) its exact formulation is intractable even for modestly-sized problems in terms of target dimensions, number of regressed quantile levels, or number of features, and its relaxed dual formulation may violate the monotonicity of the estimated quantiles; (iii) no fast or scalable solvers for VQR currently exist. In this work we fully address these limitations, namely: (i) We extend VQR to the non-linear case, showing substantial improvement over linear VQR; (ii) We propose vector monotone rearrangement, a method which ensures the quantile functions estimated by VQR are monotone functions; (iii) We provide fast, GPU-accelerated solvers for linear and nonlinear VQR which maintain a fixed memory footprint, and demonstrate that they scale to millions of samples and thousands of quantile levels; (iv) We release an optimized python package of our solvers as to widespread the use of VQR in real-world applications. Nonlinear VQR. To address the limitation of linear specification, in Section 4 we propose nonlinear vector quantile regression (NL-VQR). The key idea is fit a nonlinear embedding function of the input features jointly with the regression coefficients. This is made possible by leveraging the relaxed dual formulation and solver introduced in Section 3. We demonstrate, through synthetic and real-data experiments, that nonlinear VQR can model complex conditional quantile functions substantially better than linear VQR and separable QR approaches. Vector monotone rearrangement (VMR). In Section 5 we propose VMR, which resolves the co-monotonicity violations in estimated CVQFs. We solve an optimal transport problem to rearrange