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Risk Bounds For Distributional Regression
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Derives non-asymptotic CRPS and worst-case MSE risk bounds for projection-based distributional regression under convex and non-convex constraints, applied to isotonic, trend filtering, and dense ReLU network estimators.
Problem
Estimating the vector of conditional CDF evaluations under structural constraints (monotonicity, bounded total variation, neural network composition) with uniform guarantees over thresholds t, measured by CRPS and worst-case MSE.
Approach
Reduce CRPS-based empirical risk minimization for distributional regression to per-threshold Euclidean projections of indicator vectors onto constraint sets K_t, and bound the risk via local Gaussian/entropy complexity of K-K, uniformly over t, extending to non-convex sets with an approximation-error term.
Key results
- Isotonic distributional regression achieves rate n^{-2/3} for average CRPS and worst-case MSE under convex monotonicity constraints.
- On the 2015 Chicago crime data, UnifDR (trend filtering) obtains CRPS 0.0976 vs 0.1223 for AddSS.
- On the 2015 Chicago crime data, UnifDR (dense ReLU) obtains CRPS 0.0811 vs best competitor DRF 0.0906 (reported in table).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use UnifDR Trend Filtering instead of AddSS | 2015 Chicago crime · 2015 Chicago crime | CRPS: 0.1223$\pm$0.0078 -> 0.0976$\pm$0.0017 | Helped |
| Use UnifDR Trend Filtering instead of AddSS | 2015 Chicago crime · 2015 Chicago crime | MSD: 0.2850$\pm$0.0391 -> 0.2509$\pm$0.0025 | Helped |
| Use UnifDR Dense ReLU networks instead of DRF | 2015 Chicago crime · 2015 Chicago crime | CRPS: $0.0906\pm 0.0018$ -> 0.0811$\pm$ 0.0018 | Helped |
| Use UnifDR Dense ReLU networks instead of DRF | 2015 Chicago crime · 2015 Chicago crime | MSD: $0.2477 \pm 0.0042$ -> 0.2133 $\pm$ 0.0031 | Helped |
Limitations
- Theory restricted to independent observations; dependent data left to future work.
- Isotonic and trend filtering theory is stated for univariate case (d=1) in the isotonic comparison.
- Experiments use only fixed small architectures and simulated/real data without released code.
Experimental setup
Datasets: 2015 Chicago crime dataset California housing 1990 simulated scenarios S1-S6
Benchmarks: 2015 Chicago crime dataset California housing 1990 ozone dataset
Metrics: CRPS MSD
Baselines: AddSS CART MARS Random Forests Distributional Random Forests Engression
Builds on: isotonic distributional regression trend filtering engression
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