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A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression
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Unified comparative study of nine multi-output conformal methods within one framework, plus two new generalized conformity-score classes (CDF-based and latent-based) that give asymptotic conditional coverage with exact finite-sample marginal coverage, evaluated on 13 tabular datasets.
Problem
Conformal prediction for multi-output regression is underexplored; existing methods either ignore output dependencies, are computationally costly (sampling or density-based), or fail to achieve asymptotic conditional coverage.
Approach
Generalize univariate HPD-split to any multivariate conformity score via the CDF of that score conditional on x (CDF-based scores, giving C-HDR and C-PCP), and define a latent-space score via an invertible conditional generative model (L-CP), avoiding directional quantile regression and grid construction.
Key results
- C-PCP, L-CP and C-HDR obtain the best conditional coverage on most datasets, while HD-PCP, STDQR, PCP and DR-CP are the least conditionally calibrated.
- C-HDR yields the smallest median region size among methods with asymptotic conditional coverage; DR-CP gives the smallest mean region size.
- Sampling-based methods (PCP, C-PCP, C-HDR, HD-PCP, STDQR) typically require 100 to 200 times more computation time than M-CP, L-CP and DR-CP.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use C-PCP, which adapts PCP's ball radius using a conditional empirical CDF | PCP · 13 tabular datasets with MQF² | conditional coverage | Helped |
| Use L-CP's latent-space conformity score | multi-output conformal regression · 13 tabular datasets with MQF² | conditional coverage | Helped |
| Use C-HDR instead of methods with less conditionally calibrated regions | multi-output conformal regression · 13 tabular datasets with MQF² | conditional coverage | Helped |
| Use C-HDR rather than C-PCP or L-CP | asymptotically conditionally valid methods · 13 tabular datasets with MQF² | median region size | Helped |
| Use C-PCP or L-CP rather than C-HDR | asymptotically conditionally valid methods · 13 tabular datasets with MQF² | median region size | Hurt |
| Use DR-CP rather than other methods that do not achieve asymptotic conditional coverage | multi-output conformal regression · 13 tabular datasets with MQF² | median region size | Helped |
| Use M-CP or CopulaCPTS rather than methods with more flexible region shapes | multi-output conformal regression · 13 tabular datasets with MQF² | median region size | Hurt |
| Use PCP rather than HD-PCP or STDQR | sample-based multi-output conformal regression · 13 tabular datasets with MQF² | median region size | Hurt |
| Use M-CP or CopulaCPTS rather than sampling-based methods | multi-output conformal regression · 13 tabular datasets with MQF² | computation time | Helped |
| Use sampling-based methods rather than M-CP, CopulaCPTS, L-CP, or DR-CP | multi-output conformal regression · 13 tabular datasets with MQF² | computation time: 100 to 200 times more | Hurt |
| Use C-PCP rather than PCP's fixed-radius region | PCP on the NYC Taxi example with low uncertainty · NYC Taxi, low-uncertainty example | region size: 8.67 compared to 8.2 | Helped |
Surprising findings
- M-CP attains intermediate conditional coverage and performs slightly better than CopulaCPTS, despite not achieving asymptotic conditional coverage in general.
Limitations
- Asymptotic conditional coverage relies on strong assumptions such as oracle base predictors and infinite calibration/sample sizes.
- Sampling-based methods are computationally expensive and sensitive to sampling randomness; PCP-style regions are limited in volume.
Experimental setup
Datasets: 13 tabular multi-output datasets (7,207 to 50,000 instances) CIFAR-10
Benchmarks: 13 tabular multi-output datasets CIFAR-10 image regression NYC Taxi
Metrics: worst-slab coverage (WSC) conditional expected calibration error (CEC) marginal coverage region size computation time
Baselines: M-CP CopulaCPTS DR-CP C-HDR PCP HD-PCP STDQR CP2-PCP
Builds on: HPD-split probabilistic conformal prediction distributional conformal prediction conformalized quantile regression CopulaCPTS
Closest papers by meaning
- Rectifying Conformity Scores for Better Conditional Coverage (ICML 2025 · similarity 0.90)
- Conformal Prediction via Regression-as-Classification (ICLR 2024 · similarity 0.89)
- A Kernel Nonconformity Score for Multivariate Conformal Prediction (NeurIPS 2026 · similarity 0.89)
- Generative Conformal Prediction with Optimized Coverage Allocation (NeurIPS 2026 · similarity 0.89)
- Kandinsky Conformal Prediction: Beyond Class- and Covariate-Conditional Coverage (ICML 2025 · similarity 0.88)
- Optimal transport-based conformal prediction (ICML 2025 · similarity 0.88)
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