# Direct Prediction Set Minimization via Bilevel Conformal Classifier Training

arXiv: https://arxiv.org/abs/2506.06599 · HTML: https://mlsanity.com/paper/2506.06599/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 76 of its 107 references that are in the map.
- First posted on arXiv 7 Jun 2025.

## Builds on (most cited first, 30 of 76)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Mask R-CNN](https://mlsanity.com/paper/1703.06870/) (2017)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (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)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Uncertainty Sets for Image Classifiers using Conformal Prediction](https://mlsanity.com/paper/2009.14193/) (2020)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Predictive inference with the jackknife+](https://mlsanity.com/paper/1905.02928/) (2019)
- [Coresets via Bilevel Optimization for Continual Learning and Streaming](https://mlsanity.com/paper/2006.03875/) (2020)
- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
- [Conformal Prediction With Conditional Guarantees](https://mlsanity.com/paper/2305.12616/) (2023)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Ask Me Anything: Dynamic Memory Networks for Natural Language Processing](https://mlsanity.com/paper/1506.07285/) (2015)
- [Localized Conformal Prediction: A Generalized Inference Framework for Conformal Prediction](https://mlsanity.com/paper/2106.08460/) (2021)
- [Learning Optimal Conformal Classifiers](https://mlsanity.com/paper/2110.09192/) (2021)
- [Class-Conditional Conformal Prediction with Many Classes](https://mlsanity.com/paper/2306.09335/) (2023)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum](https://mlsanity.com/paper/2102.07367/) (2021)
- [Provably Faster Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2106.04692/) (2021)
- [BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach](https://mlsanity.com/paper/2209.08709/) (2022)

## Built on by (most cited first, 8 of 8)

- [ConformalSAM: Unlocking the Potential of Foundational Segmentation Models in Semi-Supervised Semantic Segmentation with Conformal Prediction](https://mlsanity.com/paper/2507.15803/) (2025)
- [Singleton-Optimized Conformal Prediction](https://mlsanity.com/paper/2509.24095/) (2025)
- [Provably Minimum-Length Conformal Prediction Sets for Ordinal Classification](https://mlsanity.com/paper/2511.16845/) (2025)
- [Cost-Sensitive Conformal Training with Provably Controllable Learning Bounds](https://mlsanity.com/paper/2511.17861/) (2025)
- [Clinician-in-the-Loop Smart Home System to Detect Urinary Tract Infection Flare-Ups via Uncertainty-Aware Decision Support](https://mlsanity.com/paper/2511.18334/) (2025)
- [Class Adaptive Conformal Training](https://mlsanity.com/paper/2601.09522/) (2026)
- [Conformal Margin Risk Minimization: An Envelope Framework for Robust Learning under Label Noise](https://mlsanity.com/paper/2604.06468/) (2026)
- [Conformal Risk Minimization for Semi-Supervised Domain Adaptation via Optimal Transport](https://mlsanity.com/paper/2608.23153/) (2026)

## Abstract (arXiv)

Conformal prediction (CP) is a promising uncertainty quantification framework which works as a wrapper around a black-box classifier to construct prediction sets (i.e., subset of candidate classes) with provable guarantees. However, standard calibration methods for CP tend to produce large prediction sets which makes them less useful in practice. This paper considers the problem of integrating conformal principles into the training process of deep classifiers to directly minimize the size of prediction sets. We formulate conformal training as a bilevel optimization problem and propose the em Direct Prediction Set Minimization (DPSM) algorithm to solve it. The key insight behind DPSM is to minimize a measure of the prediction set size (upper level) that is conditioned on the learned quantile of conformity scores (lower level). We analyze that DPSM has a learning bound of O(1/√n) (with n training samples), while prior conformal training methods based on stochastic approximation for the quantile has a bound of Omega(1/s) (with batch size s and typically s ll √n). Experiments on various benchmark datasets and deep models show that DPSM significantly outperforms the best prior conformal training baseline with 20.46%downarrow in the prediction set size and validates our theory.