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Estimation of the Label-Noise Transition Matrix with Performance Guarantees via Selective Classification

Xabier de Juan, Santiago Mazuelas, Yilun Zhu, Clayton Scott

arXiv:2609.39829v1stat.MLcs.LG

TL;DR

Estimating label-noise transition matrices matters because large-scale labeling can be costly, while existing estimators rely on fragile class-posterior estimates and lack finite-sample guarantees. The paper uses one-sided selective classification to estimate the matrix, establishing finite-sample guarantees and reporting lower experimental error rates than posterior-based methods.

  • Problem

    Transition-matrix estimators rely on fragile pointwise class-posterior estimates, while the literature lacks finite-sample performance guarantees for these estimators.

    Source: §1 · p. 1

  • Method

    The method estimates each transition-matrix column with a selection function that minimizes false discovery rate subject to a minimum coverage constraint.

    Source: §1 · p. 2

  • Results

    The estimator achieves the parametric convergence rate up to a small bias and avoids the curse of dimensionality; experiments report lower error rates than posterior-based methods.

    Source: §1 · p. 2, §4.3 · p. 9

  • Takeaways & Limitations

    The methodology can help reduce label noise's impact on supervised learning and make model deployment more cost-efficient.

    Source: §5 · p. 10

    The anchor-based estimator remains vulnerable to the curse of dimensionality because it relies on accurate class-posterior estimates.

    Source: §3 · p. 5

Abstract

from arXiv · show

Modern machine learning depends heavily on massive datasets, but obtaining high-quality annotations at scale is often expensive. As a result, learning from noisily-labeled data has become common, making accurate estimation of the label-noise transition matrix crucial. However, existing transition matrix estimators rely on the fragile estimation of class-posteriors and do not provide finite-sample performance guarantees. In this work, we propose a novel methodology to estimate the transition matrix based on one-sided selective classification. This approach bypasses class-posterior estimation, provides finite-sample performance guarantees, and leverages flexible learning methods for binary classification. Moreover, we introduce effective algorithms to implement the proposed methodology and provide their refined finite-sample performance bounds.

1 Introduction

High-quality labels are costly at scale, and label-noise transition matrices are important for understanding and addressing noisy labels. Existing estimators depend on fragile class-posterior estimates and lack finite-sample guarantees; this work proposes a selective-classification approach to address those gaps.

1 Introduction

  • The transition matrix captures label-flip probabilities, and accurate estimates support applications including loss correction, prediction sets, fairness, and noisy-label diffusion models.High-quality annotations at scale are often cost-prohibitive, contributing to the use of less accurate labeling procedures.Source: §1 · p. 1
  • Existing estimators rely on fragile pointwise class-posterior estimates, while finite-sample guarantees are missing for transition-matrix estimation.Even one poor posterior estimate can cause a large matrix-estimation error, and related mixture-proportion methods have practical restrictions.Source: §1 · p. 1
  • The method estimates each transition-matrix column through one-sided selective classification, minimizing false discovery rate subject to minimum coverage.Its estimator has finite-sample error bounds and achieves the parametric convergence rate up to a small bias, while avoiding the curse of dimensionality.Source: §1 · p. 2
  • The proposed methodology also has computationally tractable algorithms based on general binary-classification methods, with refined finite-sample performance bounds.The algorithms are theoretically analyzed and admit bounds analogous to classical classification generalization bounds.Source: §1 · p. 2

2 Preliminaries

The preliminaries define the transition matrix under class-dependent label noise and describe assumptions and existing estimation approaches. Related methods include anchor-point estimators, posterior-based approaches, and mixture-proportion estimators, each with stated limitations or scope constraints.

2.1 Label-noise transition matrix

  • The transition-matrix entry T_i,j is the probability that true label j is observed as noisy label i; the matrix is assumed row-diagonally dominant.The goal is to estimate T from i.i.d. noisy samples, under class-dependent noise: the noisy label depends on the clean label, not directly on features.Source: §2.1 · p. 2

2.2 Related work

  • Anchor-based estimation assumes each class has a pure anchor instance, then estimates noisy class-posteriors and uses their classwise maxima to estimate transition-matrix columns.The anchor-point assumption requires an instance with class-posterior probability one for every class.Source: §2.2 · p. 2
  • Later estimators remove explicit anchor search but still depend on accurate pointwise class-posteriors, which are difficult to estimate reliably, especially in higher dimensions.A separate approach bypasses posterior estimation but uses assumptions the authors describe as unlikely in practice, including a uniqueness condition that is often unmet.Source: §2.2 · p. 3
  • Mixture-proportion estimators offer finite-sample guarantees at a parametric rate up to small bias, but some require exhaustive search with loose VC bounds and others are binary-only.These methods estimate inverse flip-rates rather than the transition-matrix entries directly.Source: §2.2 · p. 3
  • Selective classification learns a classifier and reject option under a minimum-coverage constraint; its one-sided form selects a high-purity subset for a target class.The one-sided objective minimizes false discovery rate for the target class.Source: §2.2 · p. 3

3 Transition matrix estimation via one-sided selective classification

The section estimates each transition-matrix column through one-sided selective classification rather than pointwise class-posterior estimation. It provides finite-sample error guarantees and explains how coverage, false discovery, and matrix conditioning affect the bound.

3 Transition matrix estimation via one-sided selective classification

  • For each class, the method minimizes false discovery rate subject to minimum coverage, then estimates that transition-matrix column from accepted noisy-label frequencies.The empirical accepted-label probability approximates the corresponding transition-matrix entry when the selection function has low false discovery rate.Source: §3 · pp. 3–4
  • The guarantee also covers settings without anchor points, where a positive limiting value of R(j)_γ represents unavoidable bias.The method remains effective when R(j)_γ is small.Source: §3 · p. 4
  • Decreasing minimum coverage can reduce false discovery but accepts fewer samples, increasing variance; coverage therefore controls the bias-variance trade-off.The passages state that choosing coverage above ω(1/n) lets both the false-discovery and variance terms decrease with sample size.Source: §3 · p. 5
  • Compared with the cited mixture-proportion methods, the approach covers multiclass transition matrices and permits any one-sided selective-classification method.The cited mixture-proportion guarantees are described as limited to binary settings and specific learning methods.Source: §3 · p. 5
  • Theorem 1 provides the first finite-sample performance guarantees for transition-matrix estimators, with a bias-variance decomposition of estimation error.The bound includes bias from the selection function’s false discovery rate and a condition number tied to the transition matrix’s diagonal margin.Source: §3 · pp. 4–5
  • Unlike anchor-based estimation, the method’s suboptimality term is excess false discovery rate rather than pointwise class-posterior error.The passages describe that excess false discovery rate as small under realistic assumptions, while anchor-based estimates can inherit the curse of dimensionality.Source: §3 · p. 5

4 Tractable algorithms with finite-sample performance guarantees

The section gives two selective-classification algorithms for transition-matrix estimation with finite-sample guarantees, then compares their errors with existing estimators and an oracle.

4.1 Threshold selection approach

  • Threshold selection learns a class-specific score, then chooses a threshold on held-out data to minimize false discoveries subject to minimum coverage.The estimated transition-matrix entries are computed from accepted instances in the second sample split.Source: §4.1 · p. 6
  • The threshold method’s error depends on ranking excess risk; an order-preserving score gives zero ranking excess risk, and ranking can be easier than accurate posterior estimation.The analysis also relates ranking excess risk to the training-sample size.Source: §4.1 · pp. 6–7

4.2 Cost-sensitive risk minimization approach

  • Cost-sensitive risk minimization learns a class-specific accept/reject classifier, choosing its rejection cost to minimize false discoveries while meeting minimum coverage.It estimates transition-matrix entries from accepted instances in the second sample split.Source: §4.2 · p. 7
  • The cost-sensitive method’s error is controlled by classification excess risk, a milder requirement than the threshold method’s ranking excess risk; calibrated convex-surrogate learning also yields guarantees.For hinge loss, the excess-risk bound scales linearly; the cited analysis notes a square-root dependence for another surrogate.Source: §4.2 · pp. 7–9

4.3 Experimental results

  • Both proposed algorithms achieve significantly lower error than methods relying on pointwise class-posterior estimates; their errors decrease with sample size like the oracle, while the anchor-based method plateaus.Figure 1 reports MAE as training-sample size increases.Source: §4.3 · p. 9, Figure 1 · p. 10

5 Conclusion

The conclusion presents one-sided selective classification as a way to estimate the label-noise transition matrix without pointwise class-posterior estimation. It also identifies limits in scaling with class count and under class imbalance.

5 Conclusion

  • The method estimates the transition matrix by learning selection functions that minimize false discovery rate subject to minimum coverage, with finite-sample guarantees and flexible binary-classification algorithms.The conclusion states that the approach avoids the curse of dimensionality, unlike existing estimators.Source: §5 · p. 10
  • Using labels from inexpensive sources may reduce label-noise impacts on supervised learning and make deployment more cost-efficient.This is presented as a potential benefit of the methodology.Source: §5 · p. 10
  • The approach scales poorly with class count because it solves separate binary problems per class, and class imbalance may reduce selection-function precision for minority classes.The conclusion notes that the per-class problems are parallelizable and that large training samples are typically available in the stated setting.Source: §5 · p. 10

A Proofs

The proof section develops an error bound using conditional label probabilities and decomposes transition-matrix estimation error into separate terms.

A Proofs

  • Lemma 1 lower-bounds the true class probability within a set using the corresponding noisy-label probability and transition-matrix entries.The displayed bound subtracts the largest off-diagonal entry in the relevant row.Source: A Proofs · p. 14
  • The transition-entry estimation error is split by the triangle inequality into deviation from a conditional noisy-label probability and its estimation error.The proof then applies Hölder’s inequality to the first term.Source: A Proofs · p. 14
  • The proof relates the distance from a class-specific basis vector to the conditional class probability, defining the conditional label vector’s components explicitly.Its stated norm equals twice one minus the probability of the target class within the set.Source: A Proofs · p. 14

A.1 Proof of Theorem 1

These appendices prove the finite-sample bounds for the main estimator and its threshold-selection analysis, using conditional probability bounds and concentration arguments.

A.1 Proof of Theorem 1

  • For Theorem 1, accepted instances define a coverage-constrained set; conditioning on the accepted-sample count yields a binomial estimate to which a Chernoff bound is applied.The proof then uses a union bound over classes.Source: §A.1 · p. 15

A.2 Proof of Proposition 1

  • The proof of Proposition 1 applies Lemma 1 to singleton sets and the selected instance.Source: §A.2 · p. 15

A.3 Proof of Theorem 2

  • Theorem 2’s proof uses a coverage-constrained family and bounds the suboptimality gap; uniform convergence of empirical conditional measures controls the estimate on the second sample split.The proof also takes a union bound over classes.Source: §A.3 · pp. 15–16

A.4 Proof of Theorem 3

The proof bounds the transition-probability estimate using accepted samples, then combines bounds over a grid and controls the resulting suboptimality gap.

A.4 Proof of Theorem 3

  • Conditioning on the number of accepted samples makes the estimate binomial, enabling a conditional Chernoff bound.The proof then uses the law of total probability over possible accepted-sample counts.Source: §A.4 · p. 17
  • A union bound over a grid of size ⌊1/ε⌋ yields the stated result with probability at least 1 − δ.The probability statement is over the samples indexed by S2.Source: §A.4 · p. 17
  • The proof obtains the result by bounding the suboptimality gap against the constrained selection-function objective.The proof defines the feasible class using selection functions that accept with probability at least γ.Source: §A.4 · pp. 16–17

A.5 Proof of Theorem 4

The proof of Theorem 4 derives a bound by analyzing excess conditional risks and relating the classifier’s decisions to a transformed conditional probability.

A.5 Proof of Theorem 4

  • The proof derives the Theorem 4 bound from Theorem 3 and establishes an intermediate bound.The supplied passage does not specify the intermediate bound’s full expression.Source: §A.5 · p. 17
  • The proof defines excess conditional risk as the conditional risk minus its minimum over the two actions.It identifies accept and reject with +1 and −1 when defining the conditional loss.Source: §A.5 · p. 18
  • The classifier proof splits on whether its sign agrees with the sign of the transformed conditional probability.In the agreement case, the conditional excess risk is zero, and ψΦ(0)=0 is used in the bound.Source: §A.5 · pp. 18–19

B Analysis of the effect of the number of accepted samples

The accepted-sample analysis identifies a plateau where estimates are stable and close to the true transition probability, while very low or excessive acceptance can worsen estimation.

B Analysis of the effect of the number of accepted samples

  • The empirical estimate has three regimes: high-variance fluctuations, a stable plateau near Tj,j, then a sharp drop when acceptance extends beyond the plateau.Beyond the plateau, selection functions must accept samples with labels y ≠ j.Source: B Analysis of the effect of the number… · p. 19
  • With fewer instances, the plateau is shorter and noisier, leading to higher estimation errors.The figure captions report plateau values close to Tj,j; Figure 3 specifies Tj,j = 0.8 under uniform noise.Source: B Analysis of the effect of the number… · p. 19, Figure 3 · p. 20
  • Choosing the highest accepted-sample count within the plateau keeps variance low while bias remains controlled.The practical choice of N+ balances having enough accepted instances to reduce variance against maintaining a low false discovery rate.Source: B Analysis of the effect of the number… · pp. 19–20

C Implementation details and additional experimental results

  • The appendix also describes implementation details and reports a table comparing the proposed algorithms with the anchor-based method.The supplied passage does not state the table’s outcome.Source: C Implementation details and… · p. 21

C.1 Implementation details

The implementation details specify benchmark datasets, synthetic noise settings, and model choices for the proposed algorithms and comparison methods.

C.1 Implementation details

  • The experiments use four publicly available datasets: Letter, Satellite, MNIST, and CIFAR-10.Their listed sizes range from 6,435 Satellite samples to 70,000 MNIST samples.Source: §C.1 · p. 21
  • Binary classifiers are random forests for Letter and Satellite, an MLP for MNIST, and logistic regression on pretrained ConvNeXt features for CIFAR-10.The MLP has hidden layers of sizes 128 and 64; the random forests use 200 maximum splits and minimum leaf size 5.Source: §C.1 · p. 21
  • Noise is generated synthetically under uniform-noise and label-flip regimes.The implementation follows standard practices cited as [16].Source: §C.1 · p. 21
  • The proposed algorithms split samples into two equal sets, while the anchor-based comparison uses the 97th percentile of the noisy class-posterior.The percentile choice follows the original work [8].Source: §C.1 · p. 21
  • The oracle jointly estimates classifier parameters and the transition matrix by maximum likelihood using clean-data network representations.Its estimate is constrained to be column-stochastic and initialized at the ground-truth transition matrix.Source: §C.1 · p. 21
  • Reported results average 50 trials for the main sample-size experiments, 100 repetitions in Appendix B, and 5 or 20 repetitions for Appendix C.2 tables.The Appendix B plots show standard-deviation shading.Source: §C.1 · p. 22

C.2 Additional experimental results

Across three noise regimes, the proposed methods outperform the anchor-based baseline on almost all datasets; under random asymmetric noise, at least one proposed algorithm outperforms it on every dataset. The additional experiments also examine noise severity, dimensionality, and runtime.

C.2 Additional experimental results

  • The cross-dataset comparison uses neural networks, random forests, and logistic regression, with a strong neural network for high-dimensional CIFAR-10.Tables 1–3 report transition-matrix MAE under 0.2-uniform, 0.45-flip, and random asymmetric uniform noise, respectively.Source: §C.2 · p. 22, Table 1 · p. 22, Table 2 · p. 22
  • Across three noise regimes, the proposed methods outperform the anchor-based baseline on almost all datasets; under random asymmetric noise, at least one proposed algorithm outperforms it on every dataset.The results cover 0.2-uniform, 0.45-flip, and random asymmetric uniform noise.Source: §C.2 · p. 22, Table 1 · p. 22, Table 2 · p. 22
  • Under 0.45-flip noise, all evaluated methods have higher absolute error than under 0.2-uniform noise, while Algorithm 2 performs well with neural networks on CIFAR-10 and MNIST.The authors relate the increase in error to the more challenging noise setting and its larger theoretical constant C_T.Source: §C.2 · p. 22, Table 2 · p. 22, Table 1 · p. 22
  • As dimension increases from 100 to 1,000, the anchor-based method’s error grows, while Algorithms 1 and 2 remain essentially flat.The experiment uses PCA-reduced ConvNeXt CIFAR-10 features, logistic regression, uniform 0.2-noise, and n = 5,000; Theorem 1’s bound has no explicit dependence on dimension.Source: §C.2 · p. 23, Figure 5 · p. 23
  • Algorithm 1’s runtime is comparable to the anchor-based method, while Algorithm 2 takes at most 43 times as long and runs in at most a few minutes.Algorithm 2 searches 20 cost values per class; the authors attribute some overhead beyond the 20× grid size to implementation and note warm-starting could reduce it.Source: §C.2 · p. 23, Figure 6 · p. 23

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