No Free Lunch in Few-Step Flow Matching for Conformal Prediction

Ziyun Li, Henrik Boström
Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, PMLR 329:1078-1083, 2026.

Abstract

Conformal prediction (CP) turns a black-box predictor into prediction sets with finite-sample marginal coverage (Vovk et al., 2005). When p(y | x) is multimodal, a single interval either overcovers the gaps between modes or misses them; probabilistic conformal prediction (PCP; Wang et al., 2023) instead draws K samples from a conditional generative model and returns the union of balls of calibrated radius around them, tracing the support rather than summarizing it. Flow matching (FM; Lipman et al., 2023; Tong et al., 2024) is a natural generator, and recent work builds conformal procedures directly on flows (Li and Boström, 2025; Fang et al., 2025, 2026; Lee et al., 2025). All of this work assumes an exact sampler. FM sampling instead solves an ODE numerically, one evaluation of the learned velocity field per solver step, so its cost is the number of function evaluations (NFE), which the FM literature invests heavily in cutting (Liu et al., 2023; Song et al., 2023). Does that endanger the guarantee? No: for any fixed sampler, calibration and test scores stay exchangeable, so PCP retains coverage at any NFE, down to a single Euler step (Proposition 1, proved in Appendix B). However, there is no free lunch. (i) Coarse sampling is safe but not free. Set volume grows as the budget shrinks, by up to 1.51$\times$ at a single step, and across our four datasets the cost is ordered by $\hat{q}$/$\sigma$, the calibrated radius over the sampler’s conditional spread (Section 2), computable at full budget before any sweep. Where the ratio is large the K balls already coincide and the set is one ball around a point prediction, so few-step sampling is free exactly where sample-based CP was buying nothing over an interval method. (ii) Budget mismatch silently breaks coverage. Exchangeability requires the same sampler at calibration and deployment, an easily violated and previously unexamined condition: calibrating at 50 steps and deploying at 1 drops coverage from 90% to 48%, and to 11% on our hardest task, with no error signal. The rule: sample cheaply if you like, but calibrate with the sampler you deploy.

Cite this Paper

BibTeX
@InProceedings{pmlr-v329-li26b, title = {No Free Lunch in Few-Step Flow Matching for Conformal Prediction}, author = {Li, Ziyun and Bostr{\"o}m, Henrik}, booktitle = {Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications}, pages = {1078--1083}, year = {2026}, editor = {Ahlberg, Ernst and Johansson, Ulf and Boström, Henrik and Carlevaro, Alberto and Hallberg Szabadváry, Johan and Carlsson, Lars}, volume = {329}, series = {Proceedings of Machine Learning Research}, month = {02--04 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v329/main/assets/li26b/li26b.pdf}, url = {https://proceedings.mlr.press/v329/li26b.html}, abstract = {Conformal prediction (CP) turns a black-box predictor into prediction sets with finite-sample marginal coverage (Vovk et al., 2005). When p(y | x) is multimodal, a single interval either overcovers the gaps between modes or misses them; probabilistic conformal prediction (PCP; Wang et al., 2023) instead draws K samples from a conditional generative model and returns the union of balls of calibrated radius around them, tracing the support rather than summarizing it. Flow matching (FM; Lipman et al., 2023; Tong et al., 2024) is a natural generator, and recent work builds conformal procedures directly on flows (Li and Boström, 2025; Fang et al., 2025, 2026; Lee et al., 2025). All of this work assumes an exact sampler. FM sampling instead solves an ODE numerically, one evaluation of the learned velocity field per solver step, so its cost is the number of function evaluations (NFE), which the FM literature invests heavily in cutting (Liu et al., 2023; Song et al., 2023). Does that endanger the guarantee? No: for any fixed sampler, calibration and test scores stay exchangeable, so PCP retains coverage at any NFE, down to a single Euler step (Proposition 1, proved in Appendix B). However, there is no free lunch. (i) Coarse sampling is safe but not free. Set volume grows as the budget shrinks, by up to 1.51$\times$ at a single step, and across our four datasets the cost is ordered by $\hat{q}$/$\sigma$, the calibrated radius over the sampler’s conditional spread (Section 2), computable at full budget before any sweep. Where the ratio is large the K balls already coincide and the set is one ball around a point prediction, so few-step sampling is free exactly where sample-based CP was buying nothing over an interval method. (ii) Budget mismatch silently breaks coverage. Exchangeability requires the same sampler at calibration and deployment, an easily violated and previously unexamined condition: calibrating at 50 steps and deploying at 1 drops coverage from 90% to 48%, and to 11% on our hardest task, with no error signal. The rule: sample cheaply if you like, but calibrate with the sampler you deploy.} }
Endnote
%0 Conference Paper %T No Free Lunch in Few-Step Flow Matching for Conformal Prediction %A Ziyun Li %A Henrik Boström %B Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications %C Proceedings of Machine Learning Research %D 2026 %E Ernst Ahlberg %E Ulf Johansson %E Henrik Boström %E Alberto Carlevaro %E Johan Hallberg Szabadváry %E Lars Carlsson %F pmlr-v329-li26b %I PMLR %P 1078--1083 %U https://proceedings.mlr.press/v329/li26b.html %V 329 %X Conformal prediction (CP) turns a black-box predictor into prediction sets with finite-sample marginal coverage (Vovk et al., 2005). When p(y | x) is multimodal, a single interval either overcovers the gaps between modes or misses them; probabilistic conformal prediction (PCP; Wang et al., 2023) instead draws K samples from a conditional generative model and returns the union of balls of calibrated radius around them, tracing the support rather than summarizing it. Flow matching (FM; Lipman et al., 2023; Tong et al., 2024) is a natural generator, and recent work builds conformal procedures directly on flows (Li and Boström, 2025; Fang et al., 2025, 2026; Lee et al., 2025). All of this work assumes an exact sampler. FM sampling instead solves an ODE numerically, one evaluation of the learned velocity field per solver step, so its cost is the number of function evaluations (NFE), which the FM literature invests heavily in cutting (Liu et al., 2023; Song et al., 2023). Does that endanger the guarantee? No: for any fixed sampler, calibration and test scores stay exchangeable, so PCP retains coverage at any NFE, down to a single Euler step (Proposition 1, proved in Appendix B). However, there is no free lunch. (i) Coarse sampling is safe but not free. Set volume grows as the budget shrinks, by up to 1.51$\times$ at a single step, and across our four datasets the cost is ordered by $\hat{q}$/$\sigma$, the calibrated radius over the sampler’s conditional spread (Section 2), computable at full budget before any sweep. Where the ratio is large the K balls already coincide and the set is one ball around a point prediction, so few-step sampling is free exactly where sample-based CP was buying nothing over an interval method. (ii) Budget mismatch silently breaks coverage. Exchangeability requires the same sampler at calibration and deployment, an easily violated and previously unexamined condition: calibrating at 50 steps and deploying at 1 drops coverage from 90% to 48%, and to 11% on our hardest task, with no error signal. The rule: sample cheaply if you like, but calibrate with the sampler you deploy.
APA
Li, Z. & Boström, H.. (2026). No Free Lunch in Few-Step Flow Matching for Conformal Prediction. Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, in Proceedings of Machine Learning Research 329:1078-1083 Available from https://proceedings.mlr.press/v329/li26b.html.

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