# Conformalized Deep Splines for Optimal and Efficient Prediction Sets

arXiv: https://arxiv.org/abs/2311.00774 · HTML: https://mlsanity.com/paper/2311.00774/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 54 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 22 of 22)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Classification with Valid and Adaptive Coverage](https://mlsanity.com/paper/2006.02544/) (2020)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [Machine Learning for Survival Analysis: A Survey](https://mlsanity.com/paper/1708.04649/) (2017)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Nested conformal prediction and quantile out-of-bag ensemble methods](https://mlsanity.com/paper/1910.10562/) (2019)
- [A Conformal Prediction Approach to Explore Functional Data](https://mlsanity.com/paper/1302.6452/) (2013)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Selection by Prediction with Conformal p-values](https://mlsanity.com/paper/2210.01408/) (2022)
- [Flexible distribution-free conditional predictive bands using density estimators](https://mlsanity.com/paper/1910.05575/) (2019)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation](https://mlsanity.com/paper/1704.08095/) (2017)
- [Bayesian Optimization with Conformal Prediction Sets](https://mlsanity.com/paper/2210.12496/) (2022)
- [Bayes-optimal prediction with frequentist coverage control](https://mlsanity.com/paper/2105.14045/) (2021)
- [A three layer neural network can represent any multivariate function](https://mlsanity.com/paper/2012.03016/) (2020)
- [Deconvolutional Density Network: Modeling Free-Form Conditional Distributions](https://mlsanity.com/paper/2105.14367/) (2021)

## Built on by (most cited first, 5 of 5)

- [Probabilistic Conformal Prediction with Approximate Conditional Validity](https://mlsanity.com/paper/2407.01794/) (2024)
- [Conformal Generative Modeling with Improved Sample Efficiency through Sequential Greedy Filtering](https://mlsanity.com/paper/2410.01660/) (2024)
- [Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty](https://mlsanity.com/paper/2501.04757/) (2025)
- [Neural Optimal Transport Meets Multivariate Conformal Prediction](https://mlsanity.com/paper/2509.25444/) (2025)
- [Context-Aware Multi-Turn Visual-Textual Reasoning in LVLMs via Dynamic Memory and Adaptive Visual Guidance](https://mlsanity.com/paper/2509.05669/) (2025)

## Abstract (arXiv)

Uncertainty estimation is critical in high-stakes machine learning applications. One effective way to estimate uncertainty is conformal prediction, which can provide predictive inference with statistical coverage guarantees. We present a new conformal regression method, Spline Prediction Intervals via Conformal Estimation (SPICE), that estimates the conditional density using neural-network-parameterized splines. We prove universal approximation and optimality results for SPICE, which are empirically validated by our experiments. SPICE is compatible with two different efficient-to-compute conformal scores, one oracle-optimal for marginal coverage (SPICE-ND) and the other asymptotically optimal for conditional coverage (SPICE-HPD). Results on benchmark datasets demonstrate SPICE-ND models achieve the smallest average prediction set sizes, including average size reductions of nearly 50% for some datasets compared to the next best baseline. SPICE-HPD models achieve the best conditional coverage compared to baselines. The SPICE implementation is made available.