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Optimal transport-based conformal prediction
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Proposes OT-CP, a conformal prediction framework using Monge-Kantorovich vector ranks and quantiles to build flexible, possibly non-convex prediction regions from multivariate nonconformity scores, with finite-sample coverage and an adaptive variant (OT-CP+) with asymptotic conditional coverage.
Problem
Standard conformal prediction uses scalar nonconformity scores, which fail to exploit the geometry of multivariate outputs in multi-output regression and multiclass classification, and existing multivariate methods impose predefined convex shapes.
Approach
Apply Monge-Kantorovich (optimal transport) vector ranks and quantiles to multivariate scores computed on a split calibration set, so prediction regions take data-driven, potentially non-convex shapes with distribution-free finite-sample coverage; an adaptive variant uses conditional MK ranks from k-nearest neighbours for asymptotic conditional coverage.
Key results
- On simulated multi-output regression with alpha=0.9, OT-CP gives smaller, more precise prediction regions than ellipsoidal and hyperrectangle baselines while all reach the target coverage.
- OT-CP+ on real Mulan datasets keeps worst-set coverage close to the 0.9 target across datasets, whereas the adaptive ellipsoidal approach shows greater variability, at the price of larger set sizes and runtimes.
- On MNIST and Fashion-MNIST, OT-CP achieves label-conditional coverage on par with APS while keeping the efficiency and informativeness of IP and MS scores.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use OT-CP with multivariate residual scores instead of ellipsoidal or hyperrectangle prediction regions | Multi-output regression · Simulated regression | prediction-region efficiency (volume) | Helped |
| Use OT-CP with multivariate residual scores instead of ellipsoidal or hyperrectangle prediction regions | Multi-output regression · Multi-output regression | prediction-region conservativeness | Mixed |
| Add local k-nearest-neighbor conditional MK ranks | OT-CP for multi-output regression · Simulated regression with input-dependent residual variance | conditional coverage | Helped |
| Add local k-nearest-neighbor conditional MK ranks | OT-CP for multi-output regression · Mulan real datasets | worst-set coverage | Helped |
| Use local k-nearest-neighbor conditional MK ranks instead of the adaptive ellipsoidal approach | Adaptive multi-output conformal regression · Mulan real datasets | worst-set coverage | Helped |
| Add local k-nearest-neighbor conditional MK ranks | OT-CP for multi-output regression · Mulan real datasets | prediction-region size | Hurt |
| Add local k-nearest-neighbor conditional MK ranks | OT-CP for multi-output regression · Mulan real datasets | computational time | Hurt |
| Use the multivariate absolute-error score with OT-CP instead of IP or MS scores | Multiclass classification · Simulated Gaussian-mixture classification | conditional coverage | Helped |
| Use the multivariate absolute-error score with OT-CP instead of IP or MS scores | Multiclass classification · MNIST and Fashion-MNIST | label-wise conditional coverage | Helped |
| Use the multivariate absolute-error score with OT-CP instead of IP or MS scores | Multiclass classification · MNIST and Fashion-MNIST | efficiency and informativeness | No clear effect |
Surprising findings
- In multi-output regression OT-CP can be more conservative than the ellipsoidal approach, as the authors acknowledge.
- OT-CP+ tends to overcover on the smallest datasets (atp1d and jura, around 350 samples).
Limitations
- Multivariate OT rank computation has O(n^3) cost, more expensive than univariate conformal methods.
- Multi-output regression sets can be more conservative than ellipsoidal ones.
- Finite-sample coverage has an upper slack of n_ties/(n_2+1) due to ties; asymptotic conditional coverage only holds under density regularity assumptions.
Experimental setup
Datasets: MNIST Fashion-MNIST Mulan datasets (atp1d, jura, among others) simulated Gaussian mixture regression data simulated Gaussian mixture classification data
Benchmarks: MNIST Fashion-MNIST Mulan regression datasets simulated Gaussian mixture data
Metrics: marginal coverage worst-set coverage worst-slab coverage (WSC) volume average set size singleton fraction
Baselines: ellipsoidal conformal regression hyperrectangle conformal regression with Bonferroni inverse probability score margin score adaptive prediction sets (APS) local ellipsoids (kNN ellipsoidal CP)
Builds on: split conformal prediction Monge-Kantorovich quantiles conformalized quantile regression adaptive prediction sets (APS)
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