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Approximating Full Conformal Prediction for Neural Network Regression with Gauss-Newton Influence
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Proposes ACP-GN, an approximate full conformal prediction method for neural network regression that trains once and uses Gauss-Newton influence plus network linearization to get closed-form prediction intervals without a label grid or sample splitting, and SCP-GN, a normalized split-CP variant.
Problem
Full conformal prediction is statistically efficient but requires retraining the network for every test point and candidate label, which is infeasible for deep neural network regression; split-CP avoids this but wastes data and yields non-adaptive intervals.
Approach
Approximate the retraining in full-CP with a single Gauss-Newton influence (Newton-step, rank-one) perturbation of the trained parameters, and linearize the network so the absolute residual is piecewise linear in the candidate label, allowing exact changepoint-based interval construction via conformalized ridge regression without a grid search.
Key results
- On yacht at 90% target coverage, ACP-GN gives average width 1.594 versus 2.553 for SCP while keeping coverage (87.36% reported, below nominal).
- On bounding box localization at 90% coverage, ACP-GN (split + refine) gives volume 0.472 (x10^-2) versus 1.166 for SCP and 1.098 for CRF, with comparable coverage (91.31%).
- ACP-GN miscovers on several larger datasets (e.g., facebook_2 and bike at higher target coverage), while the split + refine variant generally attains correct coverage.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ACP-GN instead of the conformal baselines | neural network regression on limited-data UCI datasets · yacht, boston, and energy | average prediction interval width | Helped |
| Use ACP-GN on larger datasets | neural network regression · UCI large datasets, especially facebook_2 at higher target coverage | average prediction interval width | Mixed |
| Use ACP-GN (split + refine) rather than ACP-GN without splitting and refinement (ablation) | neural network regression · UCI datasets | coverage validity | Helped |
| Use ACP-GN (split + refine) rather than ACP-GN without splitting and refinement (ablation) | neural network regression · UCI datasets | average prediction interval width | Hurt |
| Normalize split-CP residual scores using Gauss-Newton predictive variance (ablation) | SCP · UCI datasets and target coverage settings | average prediction interval width | Helped |
| Use studentized rather than add-one-in or leave-one-out nonconformity scores (ablation) | ACP-GN · neural network regression experiments | empirical performance | Helped |
| Use ACP-GN (split + refine) rather than ACP-GN (ablation) | bounding-box regression · Oxford-IIIT Pet bounding box localization | confidence-region volume | Helped |
| Use ACP-GN rather than conformal baselines | bounding-box regression · Oxford-IIIT Pet bounding box localization | confidence-region volume | Helped |
Surprising findings
- Sample splitting in the split + refine variant gave tighter bounding-box intervals than the full-data ACP-GN while matching baseline coverage.
- ACP-GN overcovers on the bounding box task, and all conformal methods overcover there even after Bonferroni correction.
Limitations
- Validity is not guaranteed because the retraining step is only locally approximated; miscoverage observed on several larger UCI datasets.
- Upfront Gauss-Newton Hessian inversion is cubic in parameter count, requiring KFAC or last-layer approximations for large networks.
Experimental setup
Datasets: yacht boston energy wine kin8nm power concrete bike community protein facebook_1 facebook_2
Benchmarks: yacht boston energy bike protein facebook_2 Oxford-IIIT Pet bounding box localization
Metrics: average prediction interval width average coverage confidence region volume
Baselines: Linearized Laplace split conformal prediction conformalized residual fitting conformalized quantile regression
Builds on: conformalized ridge regression Newton-step influence Linearized Laplace split conformal prediction conformalized quantile regression
Closest papers by meaning
- Self-Calibrating Conformal Prediction (NeurIPS 2024 · similarity 0.88)
- Verifiably Robust Conformal Prediction (NeurIPS 2024 · similarity 0.88)
- Optimal transport-based conformal prediction (ICML 2025 · similarity 0.87)
- Uncertainty Quantification for Deep Regression using Contextualised Normalizing Flows (NeurIPS 2025 · similarity 0.87)
- On Temperature Scaling and Conformal Prediction of Deep Classifiers (ICML 2025 · similarity 0.87)
- Class Adaptive Conformal Training (NeurIPS 2026 · similarity 0.87)
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