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Conformal Prediction under Lévy-Prokhorov Distribution Shifts: Robustness to Local and Global Perturbations
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Models distribution shift as Lévy-Prokhorov ambiguity sets combining local (Wasserstein-infinity) and global (TV) perturbations, propagates them to one-dimensional score distributions, and derives closed-form worst-case quantile/coverage for robust split conformal prediction.
Problem
Conformal prediction loses finite-sample coverage under distribution shifts that combine local input perturbations and global population/label shifts, and existing ambiguity-set methods rely on absolute continuity or likelihood ratios.
Approach
Use LP ambiguity sets (a union of TV balls around W-infinity balls) around the training distribution; show LP shifts propagate through a Lipschitz score into LP shifts on scores, yielding closed-form worst-case quantile Quant(beta+rho)+eps and worst-case coverage F(q-eps)-rho, and a robust conformal interval with explicit (eps, rho) dependence.
Key results
- On MNIST and ImageNet under label corruption plus input noise, standard split CP fails to keep coverage while LP-based methods maintain valid coverage at alpha=0.1 across three shift levels.
- RSCP keeps valid coverage but yields very large prediction sets, especially on ImageNet.
- On iWildCam, pairs of (eps, rho) to the right of the 90% coverage contour give valid coverage on the out-of-distribution test set; the data-driven estimate performs close to the grid-optimal pair.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use LP_ε instead of standard split conformal | MNIST and ImageNet · MNIST and ImageNet under simulated local noise and label corruption | coverage | Helped |
| estimate ε and ρ from samples instead of using the fixed Lipschitz-based ε | LP robust conformal prediction · MNIST and ImageNet under simulated shifts | prediction set size | No clear effect |
| use LP robust conformal prediction instead of the other non-RSCP baselines | MNIST and ImageNet · MNIST and ImageNet under simulated local noise and label corruption | coverage | Helped |
| use LP robust conformal prediction instead of RSCP | MNIST and ImageNet · MNIST and ImageNet under simulated shifts | prediction set size | Helped |
| estimate (ε, ρ) from data rather than select them by exhaustive grid search | LP robust conformal prediction · iWildCam out-of-distribution test set | prediction set size | No clear effect |
Surprising findings
- Standard parameter choice rho=p and eps=ku with a global Lipschitz constant is overly conservative, so a fixed k=2 is used instead.
Limitations
- Results rely on score-space modeling; the Lipschitz constant k is fixed heuristically rather than estimated rigorously.
- Estimating (eps, rho) requires test samples, and the paper notes standard CP would be preferable when test data are available.
- Coverage guarantee degrades by rho in the bound and the method is not extended to sequential settings.
Experimental setup
Datasets: MNIST ImageNet iWildCam
Benchmarks: MNIST ImageNet iWildCam
Metrics: empirical coverage average prediction set size
Baselines: standard split conformal prediction chi^2-divergence robust conformal prediction weighted conformal prediction under covariate shift randomly smoothed conformal prediction (RSCP) fine-grained conformal prediction (FG-CP)
Builds on: conformal prediction f-divergence robust conformal prediction weighted conformal prediction randomly smoothed conformal prediction
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