# Risk Bounds For Distributional Regression

arXiv: https://arxiv.org/abs/2505.09075 · HTML: https://mlsanity.com/paper/2505.09075/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 65 references that are in the map.
- First posted on arXiv 14 May 2025.

## Builds on (most cited first, 17 of 17)

- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Improving Regression Performance with Distributional Losses](https://mlsanity.com/paper/1806.04613/) (2018)
- [Engression: Extrapolation through the Lens of Distributional Regression](https://mlsanity.com/paper/2307.00835/) (2023)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Isotonic Distributional Regression](https://mlsanity.com/paper/1909.03725/) (2019)
- [Quantile regression with deep ReLU Networks: Estimators and minimax rates](https://mlsanity.com/paper/2010.08236/) (2020)
- [Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain](https://mlsanity.com/paper/1804.02921/) (2018)
- [Theoretical analysis of deep neural networks for temporally dependent observations](https://mlsanity.com/paper/2210.11530/) (2022)
- [Monotone Least Squares and Isotonic Quantiles](https://mlsanity.com/paper/1901.02398/) (2019)
- [New Risk Bounds for 2D Total Variation Denoising](https://mlsanity.com/paper/1902.01215/) (2019)
- [Non-parametric Quantile Regression via the K-NN Fused Lasso](https://mlsanity.com/paper/2012.01758/) (2020)
- [Dense ReLU Neural Networks for Temporal-spatial Model](https://mlsanity.com/paper/2411.09961/) (2024)
- [Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks](https://mlsanity.com/paper/2412.20355/) (2024)
- [Consistent estimation of distribution functions under increasing concave and convex stochastic ordering](https://mlsanity.com/paper/2105.03101/) (2021)
- [Spatio-temporal model via Locally Adaptive Regression Splines](https://mlsanity.com/paper/2308.16172/) (2023)

## Abstract (arXiv)

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.