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Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise
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Proposes Relaxed Quantile Regression (RQR), a direct interval-learning loss that drops the need to prespecify symmetric quantiles, with width (RQR-W) and conditional-coverage (RQR-O) regularizers, and shows narrower intervals than QR/SQR/IR on 12 benchmarks.
Problem
Quantile regression for prediction intervals requires prespecifying symmetric quantile levels, yielding wider-than-necessary intervals for asymmetric (skewed) noise, while learning all quantiles (SQR) is harder and interval-regression methods lack coverage guarantees.
Approach
A permutation-invariant loss (y-mu1)(y-mu2) weighted by (alpha - indicator of coverage) that directly learns an interval without fixing quantile levels, plus additive regularizers (squared width; Pearson correlation between width and miscoverage) to select preferable intervals among the many valid solutions.
Key results
- RQR-W obtains the desired coverage on 12/12 datasets with mean miscoverage 0.93% versus 1.45% for QR, and the narrowest intervals on 9/12 datasets versus 8/12 for QR.
- On synthetic truncated-Gaussian noise at alpha=0.8, RQR-W reduces width to 1.50 versus 1.61 for QR while keeping coverage near 0.81.
- RQR-O improves Pearson correlation over OQR by up to about +86% on power and +89% on kin8nm while keeping coverage near the target level.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RQR-W instead of QR | Prediction interval regression · 12-dataset benchmark | mean miscoverage (%): 1.45 -> 0.93 | Helped |
| Use RQR-W instead of QR | Prediction interval regression · 12-dataset benchmark | narrowest intervals: 8/12 -> 9/12 | Helped |
| Use RQR-W instead of QR | Prediction interval regression · Truncated Gaussian noise | Width: 1.61 (±0.007) -> 1.50 (±0.005) | Helped |
| Add the width regularizer to RQR (ablation) | RQR · Truncated Gaussian noise | Width: 1.72 (±0.010) -> 1.50 (±0.005) | Helped |
| Add the width regularizer to RQR (ablation) | RQR · Gaussian noise | Width: 0.26 (±0.002) -> 0.26 (±0.002) | No clear effect |
| Use RQR-W instead of QR | Prediction interval regression · Gaussian noise | Width: 0.26 (±0.003) -> 0.26 (±0.002) | No clear effect |
| Use RQR without a regularizer instead of QR (ablation) | Prediction interval regression · Truncated Gaussian noise | Width: 1.61 (±0.007) -> 1.72 (±0.010) | Hurt |
| Use RQR-O instead of OQR | Conditional-coverage interval regression · concrete | Pearson correlation improvement over OQR: +80.19 (0.72) | Helped |
| Use RQR-O instead of OQR | Conditional-coverage interval regression · yacht | Pearson correlation improvement over OQR: -45.80 (11.28) | Hurt |
| Pair the width-coverage independence regularizer with RQR rather than QR | Conditional-coverage interval regression · Benchmark datasets | Pearson correlation and HSIC | Mixed |
| Use RQR-W instead of QR | Prediction interval regression · 12-dataset benchmark | coverage obtained: 12/12 -> 12/12 | No clear effect |
Surprising findings
- RQR without regularization (unbiased, coverage-only) gave wider intervals than QR on the truncated Gaussian (1.72 vs 1.61), showing the coverage-only objective alone does not yield narrow intervals.
Limitations
- Width-minimizing regularization introduces a coverage bias that must be corrected by shifting the target coverage level.
- Conditional coverage gains via RQR-O come at the cost of wider intervals than strictly needed for marginal coverage.
- Empirical evaluation is limited to small tabular regression benchmarks with two-layer networks.
Experimental setup
Datasets: kin8nm concrete power wine yacht naval energy boston protein synthetic truncated Gaussian noise
Benchmarks: kin8nm concrete power wine yacht naval energy boston protein
Metrics: PICP coverage MPIW mean prediction interval width mean miscoverage Pearson correlation between width and miscoverage HSIC
Baselines: QR SQR-C SQR-N IR OQR
Builds on: quantile regression simultaneous quantile regression orthogonal quantile regression interval regression
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