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CONTRA: Conformal Prediction Region via Normalizing Flow Transformation
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Proposes CONTRA, a split-conformal method that uses the latent space of a conditional normalizing flow (RealNVP) with a Gaussian base to define a ball-shaped nonconformity region mapped back to the output space, plus ResCONTRA (flow on residuals of any point predictor) and MCQR (multi-target CQR boxes).
Problem
Constructing compact, coverage-guaranteed prediction regions for multi-dimensional outputs, since conformal prediction is mostly one-dimensional and multi-output methods restrict shapes to boxes or ellipsoids or yield disconnected unions of balls.
Approach
Use the latent space of a conditional normalizing flow with Gaussian base as the projection: nonconformity is the Euclidean norm of the latent code, so the conformal region is the image of a latent ball, giving a single connected, smooth, flexible region. ResCONTRA applies the same idea to residuals of any point predictor.
Key results
- On six real datasets, CONTRA attains the smallest volume in most multi-dimensional cases, e.g., 2D RF volume 5.29 vs 7.19 for PCP and 23.85 for RCP, and 4D RF volume 0.59 vs 1.13 for PCP.
- All methods reach near-nominal 90% coverage on most datasets; Energy shows slightly lower coverage (0.84-0.88).
- On synthetic mixture-Gaussian, CONTRA volume is 64.92 versus 71.32 for PCP and 109.51 for MCQR, with coverage 0.91.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CONTRA instead of PCP | 2D conformal regions · Mixture-Gaussian | volume: 71.32(0.845) -> 64.92(1.015) | Helped |
| Use CONTRA instead of PCP | 2D conformal regions · Spiral | volume: 23.22(1.154) -> 23.70(1.622) | Hurt |
| Use CONTRA instead of PCP | 2D conformal regions · Moon | volume: 1.66(0.015) -> 1.56(0.020) | Helped |
| Use CONTRA instead of PCP | 2D conformal regions · Ring | volume (× 10²): 1.97(0.030) -> 3.00(0.217) | Hurt |
| Use CONTRA instead of PCP | conformal regions · Bio | volume: 12.54(0.072) -> 13.23(0.383) | Hurt |
| Use CONTRA instead of PCP | Taxi conformal regions · Taxi | volume (× 10⁻³): 8.95(0.132) -> 8.71(0.290) | Helped |
| Use CONTRA instead of PCP | conformal regions · Energy | volume: 16.40(0.901) -> 18.24(1.269) | Hurt |
| Use CONTRA instead of PCP | conformal regions · 2D RF | volume: 7.19(0.164) -> 5.29(0.074) | Helped |
| Use CONTRA instead of RCP | conformal regions · SCM20D | volume (× 10⁴): 6.52(0.153) -> 6.30(0.120) | Helped |
| Use CONTRA instead of PCP | 4D conformal regions · 4D RF | volume (× 10²): 1.13 (0.057) -> 0.59(0.043) | Helped |
| Use CONTRA instead of shape-restricted methods | synthetic conformal regions · Mixture-Gaussian and Spiral | volume | Helped |
| Use CONTRA instead of competing methods | multi-dimensional conformal regions · real datasets | volume | Mixed |
Surprising findings
- PCP achieves the smallest volume on Bio and Energy and on synthetic spiral, slightly beating CONTRA in those cases.
- ResCONTRA is generally larger than CONTRA when the normalizing flow can learn the conditional density directly, since it uses only part of the training data.
Limitations
- Current version uses a Gaussian base distribution, limiting it to continuous outcomes.
- Performance depends on flow quality; underfitting or overfitting produce overly large or sensitive regions.
- Conditional coverage is not established; only marginal coverage guarantee is proven, with conditional analysis left as ongoing work.
Experimental setup
Datasets: NYC Taxi 2016 Bio (CASP 5-9 protein structure) Energy (UCI building energy) River flow (Mississippi network) 2D RF and 4D RF SCM20D (TAC SCM 2010) synthetic mixture-Gaussian, spiral, moon, ring
Benchmarks: Taxi Bio Energy 2D RF 4D RF SCM20D synthetic mixture-Gaussian synthetic spiral synthetic moon synthetic ring
Metrics: empirical coverage region volume
Baselines: PCP ST-DQR NLE RCP MCQR Dist-split_bon CQR_bon
Builds on: normalizing flows realNVP split conformal prediction conformalized quantile regression probabilistic conformal prediction
Closest papers by meaning
- Robust Conditional Conformal Prediction via Branched Normalizing Flow (NeurIPS 2026 · similarity 0.88)
- A Kernel Nonconformity Score for Multivariate Conformal Prediction (NeurIPS 2026 · similarity 0.87)
- A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression (ICML 2025 · similarity 0.87)
- Co-optimization for Adaptive Conformal Prediction (NeurIPS 2026 · similarity 0.86)
- Optimal transport-based conformal prediction (ICML 2025 · similarity 0.86)
- Kandinsky Conformal Prediction: Beyond Class- and Covariate-Conditional Coverage (ICML 2025 · similarity 0.86)
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