Efficient Conformal Prediction for Regression Models under Label Noise

Jacob Goldberger; Tom Tirer; Yahav Cohen

REVIEW 3 major objections 5 minor 31 references

Efficient Conformal Prediction for Regression Models under Label Noise

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A deconvolution identity lets conformal prediction recover near-oracle confidence intervals even when the calibration labels are corrupted by Gaussian noise.

desk verdict A practical deconvolution method for regression CP with noisy labels that works when the noise is homoscedastic and independent of X; the coverage is conditional on that assumption, which is plausible but unvalidated in the motivating medical setting. read the letter →

arxiv 2509.15120 v2 pith:TKGDMAJI submitted 2025-09-18 cs.LG

Yahav Cohen , Jacob Goldberger , Tom Tirer This is my paper

classification cs.LG MSC 62G1562G0568T07

keywords conformalpredictionlabelnoiseregressiondeconvolutionmedicalimaginguncertaintyquantificationcoverageguarantee

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes the first conformal prediction (CP) method for regression whose calibration labels are noisy, targeting high-stakes settings like medical imaging. The central idea is to estimate what the CP threshold would have been on clean labels, instead of accepting the inflated intervals that result from applying CP directly to noisy labels. The authors derive an identity relating the noisy-label score distribution to the clean one through a convolution with the noise kernel, then invert it numerically and iteratively shrink the threshold until the estimated clean coverage hits 1−α. On two medical age-regression datasets with additive Gaussian label noise, the method attains near-oracle interval length and coverage, significantly beating the naive noisy-CP baseline. The paper also gives a practical estimator for the noise variance when no clean calibration labels exist.

What carries the argument

The load-bearing object is the deconvolution identity of Equation (2): M_q^n = M_q^c * k, where M_q^c(ℓ, y) = P(ℓ ∈ C_q(X) | Y = y) p_Y(y) and k is the noise kernel. Because the clean coverage probability is the trace of M_q^c, the algorithm reconstructs M_q^c from the noisy-calibration empirical M_q^n by convex deconvolution, then shrinks the threshold q until the estimated clean coverage drops to 1−α.

What would settle it

Construct a regression dataset where the label noise variance is a function of X (e.g., high noise only for out-of-distribution or low-quality inputs), run the method with a constant σ, and check whether the empirical coverage on clean test labels drops below 1−α; if it does, the independence assumption has been violated.

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Extended reading notes

Core claim

The paper establishes that the joint distribution M_q^n(ℓ, ỹ) = P(ℓ ∈ C_q(X) | Ỹ = ỹ) p_Ỹ(ỹ) computed from noisy labels is the convolution of the corresponding clean-label distribution M_q^c(ℓ, y) with the label-noise kernel k(·; σ²). This identity lets the clean coverage probability P(Y ∈ C_q(X)) be recovered by deconvolving M_q^n along the label axis and integrating its diagonal, without ever seeing clean calibration labels. The practical algorithm discretizes the label range, solves a convex deconvolution for each bin, and searches for the largest q whose estimated clean coverage still exceeds 1−α; the result is a threshold close to the oracle clean-label threshold, producing short interv

Load-bearing premise

The entire recovery relies on the assumption that, given the true label, the noisy label is independent of the input X; if mislabeling is more likely for certain images, the deconvolution identity no longer holds and the corrected threshold has no validity.

Editorial extensions

If this is right

A practitioner who knows or can estimate the noise kernel can calibrate a conformal regressor on noisy labels and get intervals essentially as short as if the calibration set were clean.
The method's coverage remains close to the nominal 1−α even when the assumed noise variance is moderately misspecified; overestimating σ makes intervals slightly smaller and underestimating makes them slightly larger.
When label noise is present in both training and calibration, the threshold can still be corrected using an estimated noise level, and the empirical estimator finds that level well in the tested settings.
The approach transfers the standard classification-side idea of noise-robust CP to continuous labels, where the main obstacle is the infinite label space and the solution is binning plus per-bin deconvolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: if label noise depends on the input (e.g., ambiguous images are mislabeled more often), Assumption A.1 fails and the deconvolved threshold will not be calibrated; a natural fix would be to model the noise scale as a function of X and deconvolve conditional on X.
Editorial extension: the noise-variance estimator in Section 3.3 relies on the existence of easy samples with near-zero aleatoric variance; on datasets where every sample has substantial irreducible uncertainty, the estimator will overestimate σ and produce slightly short intervals.
Editorial extension: the same deconvolution framework should extend to non-Gaussian noise kernels (e.g., Laplace or heteroscedastic Gaussian) as long as the kernel is known, since Equation (2) only requires a translation-invariant kernel.
Editorial extension: the iterative threshold search can be replaced by a one-step quantile estimate on the deconvolved score distribution, which may yield improved finite-sample stability; the paper does not explore this.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know: this paper gives the first adaptive conformal prediction method for regression when the calibration labels are noisy. The core idea is to recover the noise-free conformal score distribution by deconvolving the noisy score distribution with the known noise kernel. The derivation of Eq. (2) is clean under Assumptions A.1 and A.2, and the experiments on BoneAge and Chest X-Ray show intervals close to oracle length with coverage near 90% when sigma is set correctly. The paper is worth a serious look.

What's actually new: prior work on noisy labels for CP is mostly classification (Sesia et al., Penso et al., Clarkson et al.). The regression alternative "noisy CP" (Einbinder et al.) simply applies standard CP on noisy labels, which gives over-coverage and wider intervals. This paper is the first to attempt to recover the clean threshold via deconvolution. The practical implementation with discretization and regularized deconvolution is a real engineering contribution, and the authors are careful about masking empty bins and using finite-sample corrections.

Soft spots:

1. The load-bearing Assumption A.1 is that given the true label Y, the noisy label tildeY and the input X are independent. For medical imaging, this is often false: ambiguous images get mislabeled more often, and the noise variance likely varies across X. If A.1 fails, Eq. (2) is not a convolution and the deconvolved threshold has no coverage meaning. The paper acknowledges A.1 is common but provides no evidence it holds in their data. All experiments use synthetic homoscedastic Gaussian noise independent of X, so the central claim is only demonstrated under the assumption, not for the motivating application.

2. Section 3.3's estimator for sigma^2 relies on the existence of "easy" samples with u^2≈0. If none exist, sigma is overestimated, and Table 1 shows that overestimating sigma from 0.2 to 0.25 drops coverage to 88.5% on Chest X-Ray—below the 90% target. The paper calls this "robustness," but the degradation is in the unsafe direction for high-stakes use.

3. There is no finite-sample coverage guarantee for the practical algorithm. The population-level identity is exact only with infinite data and known sigma. The paper is honest about this, but readers should not mistake the empirical coverage for a guarantee.

Overall, the paper is a solid, useful contribution to the conformal prediction literature, with a clear mathematical core and honest reporting. The central limitation is the unvalidated assumption of instance-independent label noise, which matters most in the medical imaging scenarios they cite. That is a reason to require targeted experiments or a discussion of when A.1 approximately holds, not a reason to reject the paper.

Recommendation: send it to peer review. The method is new, the write-up is clear, and the flaw is an explicit assumption that the authors can be asked to address. I'd cite it in my own work.

Referee Report

3 major / 5 minor

Summary. The paper proposes a conformal prediction (CP) method for regression when the calibration labels are noisy. Under Assumptions A.1 (conditional independence of X and the noisy label given the clean label) and A.2 (Gaussian additive noise), it derives a convolution identity (Eq. (2)) relating the noisy-label joint score distribution M_n^q to the clean-label distribution M_c^q. Algorithm 1 exploits this identity: starting from the noisy CP threshold, it iteratively decreases q, estimates M_n^q from noisy calibration data, deconvolves to recover M_c^q, and estimates the clean-label coverage as the trace of M_c^q. The practical implementation discretizes the label space, solves a regularized least-squares deconvolution, and estimates the noise variance via the smallest 1% of model predictive variances. Experiments on two medical imaging datasets (Chest X-Ray and BoneAge) with synthetic Gaussian label noise show that the method achieves coverage and interval lengths close to the oracle clean-label CP and clearly outperforms the 'noisy CP' baseline.

Significance. If the method works as claimed, it addresses a genuine gap: extending conformal prediction to regression with noisy labels, where the existing baseline (noisy CP) is valid but inefficient. The convolution identity in Eq. (2) is clean and provides a principled population-level account of how label noise inflates the CP threshold. The paper includes an oracle baseline, a noisy baseline, and a sensitivity analysis over the noise variance. However, the contribution is conditional on strong assumptions, and the practical algorithm has no finite-sample coverage guarantee. The experiments only consider homoscedastic Gaussian noise added independently of the input, which sidesteps the most likely failure mode in the motivating medical-imaging setting. Thus the significance is moderate: it is a useful proof-of-concept with a sound population-level derivation, but it is not yet a robust CP method with the usual coverage guarantees.

major comments (3)

[§3.1, Eq. (2)] Assumption A.1 is load-bearing: the fourth equality in (2) drops the conditioning on X. If the label-noise mechanism is instance-dependent (e.g., ambiguous scans are mislabeled more often), M_n^q is not the convolution of M_c^q with the homoscedastic kernel k(·;σ^2), and the deconvolved threshold is not the clean-label threshold. All experiments use synthetic Gaussian noise added independently of X, so the central claim is validated only under exactly the assumption that is most questionable in the motivating medical-imaging setting. Please either relax/robustify the method or clearly scope the claim and include experiments with X-dependent noise to assess the failure mode.
[§3.3, Table 1] The estimator for σ^2 relies on the heuristic existence of 'easy' samples with u^2≈0. When this fails, σ is overestimated, and Table 1 shows that overestimating σ from 0.2 to 0.25 lowers coverage on Chest X-Ray to 88.52% (below the 90% target). Since the algorithm's stopping rule uses σ as an input, this is not a benign tuning issue. The paper should either provide a principled estimator, quantify sensitivity across datasets, or explicitly state the easy-sample condition and verify it in the experiments.
[§3.2, Algorithm 1] The practical procedure has no finite-sample coverage guarantee. Steps involve binning with δ_y=0.01, masking empty bins, L2 regularization with λ=0.01, and stopping based on an estimated coverage P ℓ M^c_q[ℓ,ℓ] δ_y. None of the approximation errors are controlled, so the method can under-cover for finite n (as in Table 1). This is a departure from the usual CP guarantee, and the paper should state this limitation explicitly, e.g., by calling the target 'approximate coverage' or providing an error analysis for the discretized deconvolution.

minor comments (5)

[§3.2] The notation ℓ and ˜y is used both for continuous values and for discretized bin indices; this should be clarified to avoid confusion with the population-level definitions in Eq. (2).
[Algorithm 1] Step 1 says 'For ℓ,˜y ∈ Y' but the algorithm operates on discretized bins; please make the discretization explicit in the pseudocode.
[§4.1, Table 1] The statement 'vice versa for a larger σ' is a trend rather than a strict monotonic relation in Table 1 (e.g., BoneAge with σ=0.25 has smaller length but also slightly lower coverage than the oracle). Consider phrasing it as a general trend.
[References] Reference [21] appears to have incomplete author information ('Safwan S. Halabi, William Prevedello, Michael Kalpathy-Cramer, and et al.'); please correct the list.
[General] No code or reproducibility details are provided. Adding a link to an implementation would strengthen the practical contribution.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the derivation chain; Eq. (2) is a genuine convolution identity, and self-citations are supporting, not load-bearing.

full rationale

The paper's central derivation is Eq. (2): M_n^q(ell, tilde y) = (M_c^q(ell, ·) * k(·; sigma^2))(tilde y), obtained from the definitions of M_n and M_c via Bayes rule and Assumptions A.1/A.2. This is a population-level identity, not a restatement of the target result. The algorithm then deconvolves M_n to estimate M_c and chooses q by root-finding on estimated coverage; no clean calibration labels are used to fit q, so the estimated threshold is not a fitted input renamed as a prediction. The score function s=|y-hat y|/hat u is adopted from the authors' prior work [20], and Assumption A.1 is cited to works including [17,18] by the same group, but these citations support design choices or explicit assumptions rather than the conclusion that the deconvolved threshold has target coverage. Section 3.3's sigma estimator is a heuristic ('easy samples') explicitly flagged for future work; an inaccurate sigma would bias coverage (as Table 1 shows), but that is an estimation weakness, not a circular construction. Hence no circular step is present; the main risk is the unvalidated instance-independent noise assumption (A.1) for medical imaging, which is a correctness concern, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or conceptual entities; the deconvolved M^c and the noise-free threshold q are derived quantities. The main commitments are the conditional independence assumption, the known-Gaussian-noise assumption, and the heuristic uncertainty-based variance estimator.

free parameters (4)

lambda (deconvolution regularization) = 0.01
Chosen by hand for the convex optimization in Step 2; affects the deconvolved M^c and hence the estimated coverage.
delta_y (label discretization width) = 0.01
Chosen for binning continuous labels; granularity controls discretization error.
delta_q (threshold decrement) = 0.05
Step size in Algorithm 1 search for q; biases the returned threshold upward by up to delta_q.
lowest 1% fraction for sigma^2 estimate = 1%
Section 3.3 uses the average of the lowest 1% of uncertainty values to estimate noise variance; the percentile choice is arbitrary and affects the noise kernel used in deconvolution.

assumptions (5)

domain assumption Given Y, the clean label, the noise Ỹ and input X are independent (Assumption A.1).
Used in Eq. (2) to replace P(ℓ in C_q(X) | Ỹ=ỹ, Y=y) with P(ℓ in C_q(X) | Y=y). Likely violated when label noise correlates with image difficulty.
domain assumption Noise follows a known kernel k(ỹ|y)=k(ỹ-y; sigma^2), specifically additive Gaussian (Assumption A.2).
Needed for the convolution identity and deconvolution; experiments inject exactly this noise, so behavior under other noise models is untested.
ad hoc to paper Deconvolution of M^n with the Gaussian kernel recovers M^c, and the regularized finite-sample solution is accurate.
Algorithm 1 Step 2 assumes identifiability and stability of the deconvolution problem; no error bounds are provided. Masking and lambda=0.01 affect the result.
ad hoc to paper The model's uncertainty head estimates u^2(X)+sigma^2 when trained on noisy labels, and there exist 'easy' samples with u^2≈0.
Section 3.3 uses the lowest 1% of uncertainty values to estimate sigma^2. This assumes the NLL-trained variance head is calibrated and easy samples exist, which is not proven.
standard math Calibration and test samples are exchangeable.
Inherited from standard conformal prediction (Theorem 1); needed for the baseline and for the method's calibration usage.

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Cite this review

Pith. "Pith review of Efficient Conformal Prediction for Regression Models under Label Noise." pith.science (2026). https://pith.science/paper/TKGDMAJI

@misc{pith2026250915120,
  author       = {Pith},
  title        = {Pith review of: Efficient Conformal Prediction for Regression Models under Label Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKGDMAJI}},
  note         = {Machine review of arXiv:2509.15120}
}

read the original abstract

In high-stakes scenarios, such as medical imaging applications, it is critical to equip the predictions of a regression model with reliable confidence intervals. Recently, Conformal Prediction (CP) has emerged as a powerful statistical framework that, based on a labeled calibration set, generates intervals that include the true labels with a pre-specified probability. In this paper, we address the problem of applying CP for regression models when the calibration set contains noisy labels. We begin by establishing a mathematically grounded procedure for estimating the noise-free CP threshold. Then, we turn it into a practical algorithm that overcomes the challenges arising from the continuous nature of the regression problem. We evaluate the proposed method on two medical imaging regression datasets with Gaussian label noise. Our method significantly outperforms the existing alternative, achieving performance close to the clean-label setting.

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Reference graph

Works this paper leans on

31 extracted references · 1 linked inside Pith

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INTRODUCTION Regression deep neural networks (DNNs) play a pivotal role in modern machine learning, serving as the backbone of systems that predict continuous values from complex, high- dimensional data. They are widely applied in diverse do- mains: estimating patients’ anatomical metrics from medical images, forecasting energy consumption and financial t...

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BACKGROUND Let(X, Y)denote a sample and its label, distributed overX × Y. For a regression task with scalar label, we haveY=R. Consider a DNN that for an input samplex∈ Xoutputs a predictionˆy(x)∈ Y, a calibration set of labeled samples {xi, yi}n i=1, and a predefinedα∈(0,1). Conformal Prediction (CP) establishes a decision rule for generating confidence ...
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noisy CP

CONFORMAL PREDICTION FOR REGRESSION WITH LABEL NOISE The goal of this paper is to design a CP method that satisfies the pre-defined coverage requirement while providing confi- dence intervals that are much shorter than those of the “noisy CP”. To this end, we will base our method on estimating the CP threshold of the noise-free case. 3.1. Estimating the n...
[7]

EXPERIMENTS In this section, we evaluate the usefulness of our approach in reducing the confidence intervals while preserving coverage. Datasets.We use two medical imaging datasets.BoneAge [21]: hand CT age regression from the RSNA pediatric bone age dataset.Chest X-Ray[22]: large-scale collection of chest radiographs from the NIH, commonly used for disea...
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We developed an iterative deconvolution-based procedure for estimating the noise-free CP threshold, based on a solid mathematical derivation

CONCLUSION We addressed the problem of applying CP for regression DNNs with noisy labels in the calibration set. We developed an iterative deconvolution-based procedure for estimating the noise-free CP threshold, based on a solid mathematical derivation. We evaluated the method on two medical imaging regression datasets with Gaussian label noise and showe...

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

prediction set

INTRODUCTION Regression deep neural networks (DNNs) play a pivotal role in modern machine learning, serving as the backbone of systems that predict continuous values from complex, high- dimensional data. They are widely applied in diverse do- mains: estimating patients’ anatomical metrics from medical images, forecasting energy consumption and financial t...

arXiv 2025

[2] [2]

For a regression task with scalar label, we haveY=R

BACKGROUND Let(X, Y)denote a sample and its label, distributed overX × Y. For a regression task with scalar label, we haveY=R. Consider a DNN that for an input samplex∈ Xoutputs a predictionˆy(x)∈ Y, a calibration set of labeled samples {xi, yi}n i=1, and a predefinedα∈(0,1). Conformal Prediction (CP) establishes a decision rule for generating confidence ...

[3] [3]

A higher score should encode a lower level of agreement betweenxandy

Define a heuristic score functions(x, y)∈Rbased on some output of the model. A higher score should encode a lower level of agreement betweenxandy

[4] [4]

, s(xn, yn)}

Calibration phase: Computeˆqas the⌈(n+ 1)(1−α)⌉/n quantile of the scores{s(x 1, y1), . . . , s(xn, yn)}

[5] [5]

noisy CP

Deployment phase: Useˆqto create the prediction set for a new samplex n+1:C(x n+1) ={y:s(x n+1, y)≤ˆq}. In the regression case, whereY=R, essentially we get the confidence intervalC(x n+1)⊂R. CP methods possess the following coverage guarantee. Theorem 1 (Theorem 1 in [7])Suppose that{(X i, Yi)}n i=1 and(X n+1, Yn+1)are i.i.d. Defineˆqas in step 2 above a...

[6] [6]

noisy CP

CONFORMAL PREDICTION FOR REGRESSION WITH LABEL NOISE The goal of this paper is to design a CP method that satisfies the pre-defined coverage requirement while providing confi- dence intervals that are much shorter than those of the “noisy CP”. To this end, we will base our method on estimating the CP threshold of the noise-free case. 3.1. Estimating the n...

[7] [7]

EXPERIMENTS In this section, we evaluate the usefulness of our approach in reducing the confidence intervals while preserving coverage. Datasets.We use two medical imaging datasets.BoneAge [21]: hand CT age regression from the RSNA pediatric bone age dataset.Chest X-Ray[22]: large-scale collection of chest radiographs from the NIH, commonly used for disea...

[8] [8]

We developed an iterative deconvolution-based procedure for estimating the noise-free CP threshold, based on a solid mathematical derivation

CONCLUSION We addressed the problem of applying CP for regression DNNs with noisy labels in the calibration set. We developed an iterative deconvolution-based procedure for estimating the noise-free CP threshold, based on a solid mathematical derivation. We evaluated the method on two medical imaging regression datasets with Gaussian label noise and showe...