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On Volume Minimization in Conformal Regression
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Analyzes split conformal regression as empirical volume minimization, bounds the excess volume loss, and proposes EffOrt (learn the base predictor by minimizing the empirical (1-alpha)-quantile of absolute error) and Ad-EffOrt (adaptive-length intervals), with synthetic-data experiments.
Problem
Split conformal regression produces prediction intervals that are valid but not necessarily of minimal length; the base predictor is usually fit by least squares, which is not volume-optimal, and the excess volume of split CP intervals has lacked finite-sample guarantees.
Approach
Show the calibration step is an empirical volume minimization, derive a finite-sample excess-volume bound, and modify the learning step to minimize the empirical (1-alpha)-quantile of absolute residuals (EffOrt), with an adaptive-width extension (Ad-EffOrt) that fits a pinball-loss quantile regressor for the width.
Key results
- On synthetic linear data, EffOrt yields shorter intervals than standard split CP in all four noise scenarios, with similar lengths under Gaussian noise.
- Ad-EffOrt gives marginally valid sets with average length smaller or similar to LW-CP and CQR, with much less variable sizes across repetitions.
- The main theorem gives a finite-sample excess-volume bound with two calibration terms (n_c) and one learning term phi(F, delta, n_l), holding with probability 1-2delta.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use EffOrt to minimize the empirical (1−α)-quantile of absolute error instead of fitting a standard or robust linear regressor | Split conformal regression · Synthetic linear regression, all four noise distributions | average interval length | Helped |
| Use EffOrt instead of split conformal with standard or robust linear regression | Split conformal regression · Synthetic linear regression, standard normal noise | interval length | No clear effect |
| Use EffOrt instead of split conformal with linear regression | Split conformal regression · Synthetic linear regression, Gaussian mixture noise | interval length | Helped |
| Use EffOrt instead of split conformal with robust linear regression | Split conformal regression · Synthetic linear regression, Pareto noise | interval length | Helped |
| Use EffOrt instead of standard or robust split conformal regression | Split conformal regression · Synthetic linear regression, Pareto-normal mixture noise | interval length | Helped |
| Use Ad-EffOrt instead of LW-CP or CQR | Adaptive split conformal regression · Synthetic heteroscedastic linear regression, all four noise distributions | average interval length | Mixed |
| Use Ad-EffOrt instead of LW-CP or CQR | Adaptive split conformal regression · Synthetic heteroscedastic linear regression, all four noise distributions | interval-length variability | Helped |
| Use Ad-EffOrt instead of CQR | Adaptive split conformal regression · Synthetic heteroscedastic linear regression, Pareto noise | average interval length | No clear effect |
Surprising findings
- Least-squares (MSE) base predictor is generally suboptimal for interval length; the QAE minimizer differs in asymmetric or heavy-tailed settings.
Limitations
- Ad-EffOrt lacks a theoretical guarantee analogous to the main excess-volume bound.
- Restricted to interval prediction sets; approximation error of this restriction is not studied.
- Empirical evaluation limited to synthetic data with linear models in the main text.
Experimental setup
Datasets: synthetic linear regression data with Gaussian, mixture, and Pareto noise
Benchmarks: synthetic heteroscedastic linear regression synthetic asymmetric and heavy-tailed linear regression
Metrics: marginal coverage average interval length
Baselines: split conformal prediction (least squares) split conformal prediction (Huber regression) locally-weighted conformal prediction conformalized quantile regression
Builds on: split conformal prediction conformalized quantile regression locally-weighted conformal inference minimum volume sets
Closest papers by meaning
- Volume Optimality in Conformal Prediction with Structured Prediction Sets (ICML 2025 · similarity 0.91)
- Co-optimization for Adaptive Conformal Prediction (NeurIPS 2026 · similarity 0.90)
- Non-exchangeable Conformal Prediction with Optimal Transport: Tackling Distribution Shift with Unlabeled Data (NeurIPS 2025 · similarity 0.88)
- False Coverage Proportion Control for Conformal Prediction (ICML 2025 · similarity 0.87)
- Improving the Statistical Efficiency of Cross-Conformal Prediction (ICML 2025 · similarity 0.87)
- Backward Conformal Prediction (NeurIPS 2025 · similarity 0.87)
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