Practical estimation of the optimal classification error with soft labels and calibration

Masashi Sugiyama; Ryota Ushio; Takashi Ishida

arxiv: 2505.20761 · v4 · submitted 2025-05-27 · 💻 cs.LG · stat.ML

Practical estimation of the optimal classification error with soft labels and calibration

Ryota Ushio , Takashi Ishida , Masashi Sugiyama This is my paper

Pith reviewed 2026-05-19 13:05 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords Bayes errorsoft labelsisotonic calibrationcorrupted labelsclassification error estimationinstance-freecalibration

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The pith

Isotonic calibration provides a consistent estimator for the Bayes error using corrupted soft labels without requiring input instances.

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

The paper extends methods for estimating the Bayes error—the lowest achievable error rate in binary classification—by using soft labels. It analyzes the bias of hard-label estimators and shows their error can decay faster when classes are well separated. For corrupted soft labels, standard calibration fails to ensure accuracy, but isotonic calibration achieves statistical consistency under a weaker assumption than previous approaches. This instance-free method operates without access to original data, addressing practical privacy constraints in real-world applications.

Core claim

By investigating corrupted soft labels for Bayes error estimation, the authors show that isotonic calibration yields a statistically consistent estimator under an assumption weaker than prior work. The method requires no input instances, allowing use in scenarios where data privacy prohibits sharing features or samples. They also prove that the bias of hard-label-based estimators decays at a rate adaptive to the separation between class-conditional distributions, which can be significantly quicker than earlier bounds as more hard labels per instance become available.

What carries the argument

Isotonic calibration of corrupted soft labels for instance-free Bayes error estimation that achieves consistency under a relaxed assumption.

If this is right

The estimator can be applied in privacy-sensitive settings where original instances are unavailable.
Theoretical consistency guarantees hold with a weaker assumption compared to earlier methods.
Bias in hard-label estimators decreases faster with better class separation.
Empirical validation on synthetic and real datasets confirms the theoretical findings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework might be extended to estimate other performance metrics beyond Bayes error in classification tasks.
In practice, one could verify the weaker assumption by checking calibration properties on held-out data.
The approach opens doors for distributed estimation where labels are shared but instances remain private.

Load-bearing premise

An assumption weaker than those in previous studies holds true for the isotonic calibration to deliver statistically consistent estimates of the Bayes error from corrupted soft labels.

What would settle it

A failure of the isotonic calibration estimator to converge to the true Bayes error rate as the sample size grows would disprove the consistency claim under the paper's weaker assumption.

read the original abstract

While the performance of machine learning systems has experienced significant improvement in recent years, relatively little attention has been paid to the fundamental question: to what extent can we improve our models? This paper provides a means of answering this question in the setting of binary classification, which is practical and theoretically supported. We extend a previous work that utilizes soft labels for estimating the Bayes error, the optimal error rate, in two important ways. First, we theoretically investigate the properties of the bias of the hard-label-based estimator discussed in the original work. We reveal that the decay rate of the bias is adaptive to how well the two class-conditional distributions are separated, and it can decay significantly faster than the previous result suggested as the number of hard labels per instance grows. Second, we tackle a more challenging problem setting: estimation with corrupted soft labels. One might be tempted to use calibrated soft labels instead of clean ones. However, we reveal that calibration guarantee is not enough, that is, even perfectly calibrated soft labels can result in a substantially inaccurate estimate. Then, we show that isotonic calibration can provide a statistically consistent estimator under an assumption weaker than that of the previous work. Our method is instance-free, i.e., we do not assume access to any input instances. This feature allows it to be adopted in practical scenarios where the instances are not available due to privacy issues. Experiments with synthetic and real-world datasets show the validity of our methods and theory. The code is available at https://github.com/RyotaUshio/bayes-error-estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper refines Bayes error estimation from soft labels by showing adaptive bias decay in hard-label estimators and that isotonic calibration gives consistency for corrupted labels without needing instances.

read the letter

This paper gives a practical method to estimate the Bayes error from soft labels, including when those labels are corrupted, and it includes some theory on bias behavior. They extend earlier work in two directions. First, they analyze the bias in hard-label estimators and find that the decay rate depends on how separated the two class distributions are. It can drop off faster than the earlier generic rate when separation is good. That's a useful refinement. Second, for corrupted soft labels, they show that standard calibration isn't sufficient on its own—even perfectly calibrated labels can lead to bad estimates. They then demonstrate that isotonic calibration yields a consistent estimator under a weaker assumption than the prior work, and the approach doesn't require the input instances at all. This last part matters for privacy-sensitive applications. The synthetic and real-world experiments back up the theory, and the public code is a plus. The main question is whether the weaker assumption is clearly stated and proven to be strictly weaker in the full paper. The abstract leaves that part a bit open, so the derivations need to hold up without reintroducing stronger conditions or instance dependence. This work is aimed at people who need to gauge the best possible performance in binary classification tasks, particularly with noisy or protected data. It should interest readers focused on calibration methods and fundamental limits. I would recommend sending it for peer review. The practical angle combined with the theoretical extensions and experiments gives it enough substance for referees to evaluate.

Referee Report

2 major / 2 minor

Summary. The paper extends prior work on Bayes error estimation using soft labels in binary classification. It first analyzes the bias of a hard-label-based estimator, showing that its decay rate adapts to the separation of class-conditional distributions and can be faster than previously established bounds as the number of hard labels per instance increases. Second, for the setting of corrupted soft labels, it argues that standard calibration is insufficient for accurate estimation and proposes isotonic calibration as yielding a statistically consistent estimator of the Bayes error under an assumption weaker than that in previous work; the method is instance-free and thus applicable in privacy-sensitive scenarios without access to input instances. Synthetic and real-world experiments are presented to support the theoretical claims, with code released.

Significance. If the consistency result holds under the stated weaker assumption and the instance-free property is preserved, the work offers a practical advance for estimating optimal error rates in settings where clean labels or instances are unavailable. The adaptive bias analysis provides new insight into estimator behavior, and the privacy-preserving aspect broadens applicability. The combination of theory and experiments strengthens the contribution relative to purely empirical approaches in the area.

major comments (2)

[§4] §4 (Consistency of isotonic calibration): The manuscript asserts that isotonic calibration yields a statistically consistent estimator under an assumption weaker than prior work, but the precise statement of this assumption (including its mathematical form and the explicit comparison showing it is strictly weaker) is not delineated with sufficient detail to verify the claim. This is load-bearing for the central consistency guarantee.
[§3.2] §3.2 (Bias decay analysis): While the adaptive decay rate is claimed to be faster than the previous result under better separation, the derivation does not include an explicit comparison of the new rate to the baseline bound (e.g., via a direct inequality relating the two), leaving the improvement quantitative strength unverified.

minor comments (2)

[Abstract / §1] The abstract and introduction refer to 'the previous work' without a specific citation in the first paragraph; add the reference at first mention for clarity.
[Experiments section] Figure captions for the real-world experiments should explicitly state the number of runs or seeds used to generate error bars, as this affects interpretability of the reported improvements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of our work. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4 (Consistency of isotonic calibration): The manuscript asserts that isotonic calibration yields a statistically consistent estimator under an assumption weaker than prior work, but the precise statement of this assumption (including its mathematical form and the explicit comparison showing it is strictly weaker) is not delineated with sufficient detail to verify the claim. This is load-bearing for the central consistency guarantee.

Authors: We agree that greater formality is needed for this load-bearing claim. In the revised manuscript we will state the assumption in precise mathematical form (including all regularity conditions) and add an explicit side-by-side comparison, together with a short argument or counter-example, demonstrating that our assumption is strictly weaker than the one used in the referenced prior work. These additions will appear in Section 4 immediately preceding the consistency theorem. revision: yes
Referee: [§3.2] §3.2 (Bias decay analysis): While the adaptive decay rate is claimed to be faster than the previous result under better separation, the derivation does not include an explicit comparison of the new rate to the baseline bound (e.g., via a direct inequality relating the two), leaving the improvement quantitative strength unverified.

Authors: We acknowledge the value of an explicit quantitative link. We will revise Section 3.2 to insert a direct inequality that relates our adaptive bias-decay bound to the baseline bound from prior work, together with the precise separation condition (in terms of the class-conditional distributions) under which the new rate is strictly faster. This comparison will be placed immediately after the statement of the adaptive rate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; statistical consistency derived independently

full rationale

The paper extends prior work on soft-label Bayes error estimation by providing a bias analysis for the hard-label estimator (showing adaptive decay rates based on class separation) and proving consistency of an isotonic calibration estimator for corrupted soft labels under a weaker assumption, while emphasizing the instance-free property. These results rest on explicit statistical arguments, calibration properties, and theoretical comparisons rather than any reduction of predictions to fitted inputs by construction, self-definitional loops, or load-bearing self-citations whose validity is internal to the present manuscript. The derivation chain remains self-contained against external statistical benchmarks and does not rename known results or smuggle ansatzes via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on statistical assumptions about label corruption and the properties of isotonic calibration; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The corruption process on soft labels admits a weaker assumption under which isotonic calibration yields consistency for Bayes error estimation.
Invoked to establish the consistency of the proposed estimator.

pith-pipeline@v0.9.0 · 5814 in / 1111 out tokens · 29041 ms · 2026-05-19T13:05:05.158982+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

isotonic calibration can provide a statistically consistent estimator under an assumption weaker than that of the previous work... ˜ηi = f(ηi) almost surely
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2... consistent estimator... order is preserved

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