Risk Bounds For Distributional Regression

2025-05-14

AI Summary

This paper investigates risk bounds for nonparametric distribution regression estimators, focusing on theoretical analysis of the Continuous Ranked Probability Score (CRPS) and in-domain worst-case mean squared error (MSE) under both convex and non-convex constraints. Methodologically, it extends isotonic and trend filtering to the distribution regression setting and integrates convex optimization, monotone regression, nonparametric inference, and neural network modeling. The key contributions are: (i) the first unified framework establishing general upper bounds on CRPS and worst-case MSE for both constraint classes; (ii) attainment of minimax-optimal convergence rates—matching those of mean estimation—under mild regularity conditions; and (iii) empirical validation via simulations and real-data experiments, confirming the tightness of the derived bounds and demonstrating significant improvements over baselines in both CRPS and MSE, with convergence behavior fully aligned with theoretical predictions.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Establishes risk bounds for nonparametric distributional regression estimators
• Derives convergence rates for isotonic and trend filtering regression
• Provides upper bounds for neural network-based distributional regression

Innovation

• Convex-constrained distributional regression for CRPS bounds
• Non-convex constraints applied to neural networks
• Isotonic and trend filtering regression convergence rates