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REVIEW 3 major objections 6 minor 58 references

Learning from Ambiguous Data with Hard Labels

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a class-wise positive-unlabeled risk estimator can train accurate classifiers from ambiguous data whose only labels are hard quantized labels.

desk verdict The QLL problem framing and the empirical results are worth a look, but the CPU estimator is not actually unbiased under the paper's own model, and the algorithm disagrees with the main equation. read the letter →

arxiv 2501.01844 v2 pith:TN75ZKY6 submitted 2025-01-03 cs.LG

Zeke Xie , Zheng He , Nan Lu , Lichen Bai , Bao Li , Shuo Yang , Mingming Sun , Ping Li This is my paper · ORCID
classification cs.LG
keywords quantizedlabellearningambiguousdatahardlabelspositive-unlabeledclass-wisePUrisksoft-labeldistributionquantizationmixupbenchmark
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

Real-world images are often genuinely ambiguous, but standard datasets assign each one a single hard label, and models trained on such labels become overconfident and generalize poorly. This paper argues that these labels are still usable if the hard label is treated as a sample from an underlying soft-label distribution and each class is learned as a positive-unlabeled problem. The proposed Class-wise Positive-Unlabeled (CPU) risk estimator trains a binary PU classifier per class and averages the risks, requiring no soft labels and no noise transition matrix. On mixed-image benchmarks built from CIFAR and animal-face images, CPU improves generalization over standard training and label-noise-robust baselines, with a mean accuracy gain of more than six points over the second-best method in the main comparison. If the paper is right, ambiguity-aware training can be done with the annotation budget we already have.

What carries the argument

The load-bearing object is the Class-wise Positive-Unlabeled (CPU) risk estimator. It converts multiclass training into $c$ binary PU problems: for class $j$, the positive set is the examples whose quantized label is $j$, and the unlabeled set is everything else; each binary risk uses the non-negative PU formula from nnPU, with the negative term clipped at zero, and the $c$ risks are averaged. The paper uses a stochastic Jensen-Shannon divergence as the loss inside the estimator, sampling the interpolation weight per iteration to balance optimization and robustness.

What would settle it

A direct test: on a two-class synthetic QLL dataset with known per-instance soft labels $s_j(x)$, compare the CPU risk estimate to the true expected risk computed by Monte Carlo; a systematic gap as the per-instance label probabilities vary would refute the unbiasedness claim. A second test: train CPU on quantized labels and compare with training on the underlying soft labels; if CPU cannot approach the soft-label model's accuracy, the claim that it recovers the information in ambiguous labels would be in doubt.

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Extended reading notes

Core claim

The central claim is that learning from ambiguous data with only quantized hard labels is tractable, and the way to make it tractable is to view every class as a separate positive-unlabeled problem. Under the label quantization hypothesis, an observed hard label $y$ is drawn from the instance's ground-truth soft-label distribution $P(y=k|x)=s_k$, so examples labeled $j$ are treated as positives for class $j$ and all other examples as unlabeled. The CPU estimator averages per-class non-negative PU risks, $\hat{R}_{pu}(g_j)=\pi_p^{(1)}\hat{R}_p^+(g_j)+\max\{\hat{R}_u^-(g_j)-\pi_p^{(2)}\hat{R}_p^-(g_j),0\}$, which prevents the model from memorizing biased labels. The paper reports that this simple estimator achieves the best accuracy across all four CIFAR-Q configurations and on the AFHQ-Q dataset, and that combining it with the FixMatch semi-supervised technique gives further gains.

Load-bearing premise

The load-bearing premise is that, for each class, the examples not carrying that label behave like a fixed blend of that class and everything else; when the label probability varies from image to image, that blend assumption is only approximate, and the estimator's unbiasedness is not guaranteed.

Editorial extensions

If this is right

  • If the CPU risk is correct, practitioners can train from ambiguous data using only the hard labels already being collected; no soft labels, label-distribution estimation, or noise transition matrix is needed.
  • The reported gains imply that per-class positive-unlabeled training is a stronger default than robust loss functions and complex noisy-label pipelines for quantized labels, and that adding one semi-supervised consistency term (FixMatch) makes it stronger still.
  • Because the risk is computed as an average of per-class binary risks with element-wise operations, it can drop into standard deep learning training loops with about the same cost as cross-entropy.
  • The mixing-based generation procedure gives future work a controlled benchmark: ambiguity level can be adjusted through the number of mixed instances and the patch masks, and quantized labels are sampled from known soft-label mixtures.

Reading between the lines

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

  • Editorial inference: the positive-versus-unlabeled decomposition may transfer to partial-label learning, where the candidate label set is the positive set and the rest are unlabeled; the paper does not test this.
  • Editorial inference: some of the gain may come from the max-clipping in the nnPU risk acting as a regularizer rather than from unbiased estimation; comparing CPU to label smoothing on the same ambiguous data would isolate the mechanism.
  • Editorial inference: on multi-annotator data, treating each annotator's vote as a quantized label and applying CPU should approach a soft-label oracle, a direct but untested consequence of the label quantization hypothesis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces Quantized Label Learning (QLL), a setting in which each instance has a ground-truth soft-label distribution and the observed hard label is sampled from that distribution. The authors propose a Class-wise Positive-Unlabeled (CPU) risk estimator, constructed by applying the non-negative PU (nnPU) risk per class, treating the instances labeled with a given class as positives and all other instances as unlabeled. They design mixing-based ambiguous-data benchmarks (CIFAR-10Q, CIFAR-100Q, and AFHQ-Q) and report accuracy gains over label-noise-robust baselines. The paper's main theoretical claim is that CPU is a generalized PU risk estimator that is unbiased under the QLL model, and its headline empirical claim is that CPU outperforms the second-best method by more than six points on average.

Significance. If the theoretical claim were valid, the paper would be a useful contribution: it proposes a new problem formulation (QLL) with practical relevance, a simple and efficient algorithm, and reproducible benchmark construction procedures. The empirical study is broad, covering CIFAR-10/100 and AFHQ with multiple mixing strategies and many baselines, and the reported gains are substantial. The hyperparameter robustness analysis is a positive feature. However, the theoretical justification as stated is invalid: the class-wise unlabeled set does not follow the mixture distribution required by the PU risk identity, so the unbiasedness claim does not hold. In addition, the implemented Algorithm 1 differs from the reported Eq. (5). The paper would need a corrected derivation or a reframing of CPU as a heuristic to support its central claims.

major comments (3)
  1. [Section III, Eqs. (2)-(4)] The derivation of the class-wise PU risk is invalid under QLL. In standard PU learning, the unlabeled data must be drawn from the mixture p_pu(x) = π_p p_p(x) + π_n p_n(x), and the positive set independently from p_p(x). In the class-wise construction described in Section III and Algorithm 1, the unlabeled set for class j is D_j^u = {x_i : y_i ≠ j}, so the unlabeled instances are drawn from p(x | y ≠ j). Under Definition 1, P(y = j | x) = s_j(x), so p(x | y ≠ j) = p(x)(1 - s_j(x)) / P(y ≠ j), which is a biased reweighting of p(x) and is not equal to the mixture π_p p_p(x) + π_n p_n(x). Consequently, Eq. (4) is not an unbiased estimator of any classification risk under QLL, and the nnPU max operation does not repair this bias. The authors should either provide a valid derivation under a different generative model or explicitly present CPU as a heuristic with empirical support.
  2. [Section III, Eq. (5) vs. Algorithm 1] The reported CPU risk and the implemented algorithm define different objectives. Eq. (5) states eR_pu(g_j) = π_p^(1) R_p^+(g_j) + max{ R_u^-(g_j) - π_p^(2) R_p^-(g_j), 0 }. Algorithm 1, lines 4-8, instead sets eR_pu(g_j) = -R_u^-(g_j) + π_p^(2) R_p^-(g_j) when the max term is negative, i.e., it uses the negative of the inside value and omits the positive-risk term π_p^(1) R_p^+(g_j). These are different losses with different gradients. Please specify which definition is the actual method and correct the inconsistency.
  3. [Section V-A and Appendix A] The class-prior hyperparameter π_p^(2) is set to m/c, where m is the number of mixed instances in the authors' benchmark generator. This uses privileged information about the synthetic data construction. The robustness analysis (Table VI and Fig. 3) shows non-negligible sensitivity to π_p^(2) (for example, CIFAR-10Q PatchMix best accuracies range from 71.57 to 72.69, and CIFAR-100Q Mixup from 51.33 to 51.85). The claim that CPU is robust to these hyperparameters should be qualified, and the method's performance on real-world ambiguous data without such privileged knowledge remains untested. Please provide a sensitivity analysis that does not assume knowledge of the generator.
minor comments (6)
  1. [Definition 1, Eq. (1)] Since s is a probability distribution that sums to 1, the denominator in Eq. (1) is redundant; please clarify whether s is assumed to be normalized.
  2. [Section III, paragraph on choosing class priors] The sentence 'the performance of the CPU is robust to π(1)_p and π(1)_p' contains a typo: the second prior should be π(2)_p.
  3. [Section III, Stochastic JS divergence] The text states that α ∈ (0, 0.5] but samples α from Beta(0.5, 0.5), whose support is (0, 1); please correct the range or the sampling distribution.
  4. [Algorithm 1, line 7] Line 7 uses π_p without the superscript (2); please align the notation with Eq. (5).
  5. [Tables I and II] The 'Mean' column appears between the dataset headers and the condition columns; it is unclear whether the mean is over the four mixing conditions or over repeated runs; please clarify the caption.
  6. [General] The paper does not provide a link to the implementation of CPU or the benchmark generator; for reproducibility, please release the code.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CPU is an openly adapted class-wise nnPU estimator evaluated on held-out test sets; the theoretical weakness is a correctness issue, not circularity.

full rationale

The paper's derivation chain is self-contained with respect to circularity. The CPU risk estimator (Eq. 4) is presented as a direct class-wise generalization of the external nnPU risk estimator of Kiryo et al. [20]; no new unbiasedness theorem is claimed, so there is no hidden step where a conclusion is encoded in the estimator by construction. The quantized-label benchmark is used only for evaluation, and the class-prior hyperparameter choice pi_p^(2)=m/c is disclosed explicitly ('we may choose the class-prior hyperparameters simply as pi_p^(1)=0.1 and pi_p^(2)=m/c without fine-tuning') and its sensitivity is tested (Fig. 3, Table VI). The headline accuracy comparison is an empirical result on held-out CIFAR and AFHQ test sets, not a quantity forced by a fitted constant or by a self-citation chain. Self-citations [13], [14], [16] are background on overfitting/noise memorization and are not load-bearing for the CPU derivation. The main weakness noted by a skeptical reader — that the standard PU identity in Eq. (2) requires unlabeled data drawn from a fixed mixture, whereas the class-wise complement distribution p(x|y!=j) is not that mixture under instance-dependent soft labels — is a validity/correctness concern about the estimator's unbiasedness, not a circularity of the kind defined in the rubric. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 3 assumptions · 1 invented entities

The central claim rests on the label quantization hypothesis, the validity of the PU mixture equation in a setting with instance-dependent labeling propensity, and several hand-chosen hyperparameters tied to the synthetic benchmark. The most fragile item is the PU mixture assumption, because it is imported from standard PU learning without proof and is violated by the paper's own Definition 1.

free parameters (4)
  • pi_p^(1) = 0.1
    Chosen by hand for all experiments as the positive-term class prior in Eq. (5); no estimation procedure is given.
  • pi_p^(2) = m/c, e.g., 0.2, 0.4, 0.02, 0.04
    Set from the number of mixed images m and the class count c; this quantity is only known because the benchmark is synthetic and generated by the authors.
  • alpha in Stochastic JS divergence = Sampled per iteration from Beta(0.5, 0.5)
    The final CPU method uses this stochastic interpolation loss; the distribution is chosen by hand and is not fitted to data.
  • lambda_mu for CPU (Semi) = Searched in {0.1, 0.5, 1, 5}
    Weight of the unlabeled FixMatch loss in the semi-supervised variant, selected per dataset.
assumptions (3)
  • domain assumption Label quantization hypothesis: human-annotated hard labels are sampled from a per-instance ground-truth soft-label distribution.
    Definition 1 postulates this mechanism. The paper calls it a hypothesis and provides no empirical validation against alternative models such as instance-dependent label noise.
  • ad hoc to paper Standard PU mixture and independence: unlabeled data are drawn from p_pu = pi_p p_p + pi_n p_n, and positive and unlabeled sets are sampled independently.
    Eq. (2) relies on this assumption. In QLL, the probability of being labeled positive for class j is s_j(x), which varies per instance, so the class-wise unlabeled set is not a fixed mixture.
  • domain assumption Averaging independent per-class binary PU risks yields a valid multi-class classifier.
    The paper averages class-wise risks but does not define the inference rule (e.g., argmax over g_j) or prove consistency of this one-vs-rest PU decomposition.
invented entities (1)
  • Quantized label (label quantization hypothesis)
    purpose: Formalizes observed hard labels as one sample from a ground-truth soft-label distribution, enabling the PU-based formulation.
    The paper defines it in Definition 1 and calls it a hypothesis; no falsifiable handle outside the paper is provided to test this label-generation model.

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Pith. "Pith review of Learning from Ambiguous Data with Hard Labels." pith.science (2026). https://pith.science/paper/TN75ZKY6

@misc{pith2026250101844,
  author       = {Pith},
  title        = {Pith review of: Learning from Ambiguous Data with Hard Labels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TN75ZKY6}},
  note         = {Machine review of arXiv:2501.01844}
}
read the original abstract

Real-world data often contains intrinsic ambiguity that the common single-hard-label annotation paradigm ignores. Standard training using ambiguous data with these hard labels may produce overly confident models and thus leading to poor generalization. In this paper, we propose a novel framework called Quantized Label Learning (QLL) to alleviate this issue. First, we formulate QLL as learning from (very) ambiguous data with hard labels: ideally, each ambiguous instance should be associated with a ground-truth soft-label distribution describing its corresponding probabilistic weight in each class, however, this is usually not accessible; in practice, we can only observe a quantized label, i.e., a hard label sampled (quantized) from the corresponding ground-truth soft-label distribution, of each instance, which can be seen as a biased approximation of the ground-truth soft-label. Second, we propose a Class-wise Positive-Unlabeled (CPU) risk estimator that allows us to train accurate classifiers from only ambiguous data with quantized labels. Third, to simulate ambiguous datasets with quantized labels in the real world, we design a mixing-based ambiguous data generation procedure for empirical evaluation. Experiments demonstrate that our CPU method can significantly improve model generalization performance and outperform the baselines.

Figures

Figures reproduced from arXiv: 2501.01844 by the authors.

Figure 1
Figure 1. Illustration of the CPU risk estimator. For a classification task with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Examples of synthetic AFHQ-Q training data with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Illustration of robustness to different π (1) p . The perfor￾mance of CPU is robust to various class priors. TABLE VI: Robustness analysis of π (2) p . CIFAR-10Q Mixup, k=2 Patchmix, k=4 π (2) p 0.15 0.2 0.25 0.35 0.4 0.45 Best test acc. 81.21 81.80 81.46 72.69 72.45 71.57 CIFAR-100Q Mixup, k=2 Patchmix, k=4 π (2) p 0.01 0.02 0.03 0.03 0.04 0.05 Best test acc. 51.54 51.85 51.33 39.73 40.25 39.94 but performs poorly … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The test curves of the baselines. Left: CIFAR-10Q, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.