Variance-Reduced Conformal Prediction via Shrinkage for Long-Tailed Recognition

Class-conditional conformal prediction is difficult when some classes have few calibration examples. In this regime, class-specific quantile estimates can vary substantially across calibration splits, which leads to large prediction sets. We propose **SARC-CP** (Shrinkage-Adjusted Rank-Calibrated Conformal Prediction). SARC-CP shrinks each class-specific quantile toward a global quantile using a sample-size-dependent weight, $q_y^{\ast }=(n_y{q}_y+\kappa q_{\mathrm{global}})/(n_y+\kappa )$. Classes with fewer calibration examples receive more shrinkage, while classes with more calibration data remain closer to their class-specific estimates. We derive a bias-variance condition under which the shrunk estimator has lower mean-squared error than the plug-in quantile. We then combine this estimator with rank-calibrated conformal prediction (RC3P), adding one hyperparameter and no model retraining. Across CIFAR-10/100-LT, ImageNet-LT, and iNaturalist-2018, SARC-CP reduces average prediction set size at comparable class-coverage levels. Relative to RC3P, the reduction reaches 19% on CIFAR-100-LT, 55-65% on ImageNet-LT, and about 80% on iNaturalist-2018. The largest gains occur in the large-class, low-calibration regimes. Code: https://anonymous.4open.science/r/sarc-cp-shrinkage-0CE7/README.md