Addressing Concept Mislabeling in Concept Bottleneck Models Through Preference Optimization

Emiliano Penaloza; Laurent Charlin; Mateo Espinosa Zarlenga; Tianyue H. Zhang

REVIEW 4 major objections 6 minor 27 references

Addressing Concept Mislabeling in Concept Bottleneck Models Through Preference Optimization

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A preference-optimization training loss makes concept bottleneck models robust to mislabeled concepts.

desk verdict CPO is a genuinely new, DPO-style loss for concept bottleneck training with consistent empirical gains, but the proof that it is inherently noise-robust has a real gradient-derivation gap and the main comparison is confounded by RandInt. read the letter →

arxiv 2504.18026 v4 pith:FEXKLCAS submitted 2025-04-25 cs.LG cs.AI

Emiliano Penaloza , Tianyue H. Zhang , Laurent Charlin , Mateo Espinosa Zarlenga This is my paper · ORCID

classification cs.LGcs.AI

keywords conceptbottleneckmodelslabelnoisepreferenceoptimizationdirectinterpretabilityuncertaintyestimationmislabelingrobustness

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

Concept Bottleneck Models (CBMs) make neural predictions explainable by routing them through human-understandable concepts, but they are normally trained with binary cross-entropy (BCE), which assumes the concept labels are correct. This paper argues that concept labels are frequently wrong, whether from annotation errors, subjective judgments, or data augmentations that occlude concepts, and shows this can cut task accuracy by up to 25%. It proposes Concept Preference Optimization (CPO), an adaptation of Direct Preference Optimization that compares each empirical concept label with a concept sampled from the model's own policy, and claims CPO directly optimizes the concept posterior while being inherently less sensitive to label noise than BCE. Empirically, CPO-trained CBMs and Concept Embedding Models outperform their BCE counterparts on CUB, AwA2, and CelebA, both with clean labels and with up to 40% of concept labels flipped. A reader should care because CPO offers a drop-in training loss that makes interpretable models usable in noisy, high-stakes settings.

What carries the argument

The central object is the CPO loss (Eq. 5), an online Direct Preference Optimization objective in which the empirical concept $c$ is treated as preferred over a concept $c'\sim\pi_\theta$ sampled from the current policy, with the reward reparameterized as the log-ratio of the policy to a prior. Under a uniform prior this becomes a log-likelihood term that contributes a gradient only when $c'\neq c$, so the model updates only when it disagrees with the label, the mechanism that gives CPO its conservative, noise-attenuating updates. The supporting machinery is a control-as-inference identity: introducing an optimality variable $o$ with $p(o=1|x,c)=\exp(r^*(x,c))$ yields the posterior $\pi(c|o=1,x)\propto \pi_0(c|x)\exp(\beta^{-1}r^*(x,c))$, showing that optimizing Eq. 1 (and hence Eq. 5) directly approximates the concept posterior. This is what distinguishes CPO from BCE and from probabilistic CBM variants that amortize a latent variable's posterior instead.

What would settle it

Train a logistic concept predictor on a single binary concept with noisy labels and compute the exact expected update of the CPO loss, including the case where the sampled concept equals the empirical one. At model confidence 0.1, if this update has larger norm than the BCE update, the bound in Proposition 4.2 is violated.

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Extended reading notes

Core claim

The paper's central claim is that the CPO objective, defined as $-E_{(x,c)\sim\mu,\,c'\sim\pi_\theta}[\log\sigma(\log(\pi_\theta(c|x)/\pi_0(c|x))-\log(\pi_\theta(c'|x)/\pi_0(c'|x)))]$, replaces the BCE concept loss in a CBM and directly optimizes the concept posterior rather than merely fitting noisy empirical labels. With a uniform prior, the loss is proportional to $-\log \pi_\theta(c|x)$ whenever the policy-sampled concept $c'$ differs from the empirical concept $c$, and a constant otherwise, so CPO's expected gradient is BCE's gradient weighted by the model's confidence $(1-\pi_\theta(c|x))$. The paper proves that this expected gradient norm is bounded by BCE's and that, under a fixed level of label noise, CPO's gradient stays closer to the noise-free gradient than BCE's does. It also shows, through a control-as-inference optimality variable, that optimizing this objective is equivalent to amortizing the posterior $\pi(c|o=1,x)$, which explains the improved uncertainty estimates and intervention behavior. The empirical section reports that CPO consistently improves task accuracy and concept AUC over BCE on CUB, AwA2, and CelebA with and without added noise, including at 40% concept-flip rates.

Load-bearing premise

The proof that CPO is always less sensitive to noise ignores part of the update from cases where the model's random choice matches the label; when the model is uncertain, the ignored part can make CPO's update larger than BCE's, so the guaranteed robustness bound does not always hold.

Editorial extensions

If this is right

On clean data, CPO improves CBM task accuracy on CUB from 0.753 to 0.800 and concept AUC from 0.937 to 0.952, with no added parameters and roughly 0.05 extra minutes per epoch.
Under random concept-label flips at rates up to 40%, CPO-trained CBMs are the least affected models on CUB, AwA2, and CelebA, and at 40% noise they can still outperform more complex BCE-trained baselines.
CPO-trained models retain their intervenability under noise, with interventions ordered by CPO's uncertainty estimates improving task accuracy faster than BCE or ProbCBM interventions.
CPO with a non-uniform prior can update a CBM from streaming concept labels without drifting from the joint-trained policy, outperforming BCE-based streaming updates.
CPO's uncertainty estimates respond to occlusion: uncertainty rises when the target object is blocked, whereas BCE-trained models tend to stay overconfident.

Reading between the lines

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

The confidence-weighted gradient suggests CPO should also dampen the effect of augmentation-induced concept corruption; a head-to-head comparison of no-augmentation versus random-crop training would isolate this mechanism from label-flip noise.
Because the proof's robustness guarantee omits a parameter-dependent term, CPO's advantage may shrink precisely when the model is least confident; annealing the prior strength $\beta$ during training is a natural testable fix.
The same preference-over-labels construction could extend to structured annotator disagreement in medical or clinical concept labeling, where noise is correlated rather than random flips.
CPO's posterior-amortization view suggests it could replace the likelihood term in probabilistic CBM variants, yielding calibrated uncertainty without needing a latent variable bottleneck.

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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

The paper proposes Concept Preference Optimization (CPO), a DPO-style objective for training concept bottleneck models, and shows consistent gains on CUB, AwA2, and CelebA across clean, noisy, and structured-noise settings, with code released. The novelty is real: ProbCBM and SCBM amortize posteriors over a latent variable, while CPO applies preference optimization directly to concept predictions. The empirical work is broad—noise levels, interventions, streaming data, uncertainty calibration—and the method is simple and cheap. Worth taking seriously.

The soft spots are where the central claim lives. The proof that CPO is inherently less sensitive to noise rests on the App. C.1 gradient derivation, and that derivation is not right. L_CPO is an expectation over c' ~ pi_theta; the event c'=c gives a constant log sigma(0), but the probability of that event is pi_theta(c|x), which depends on theta. Dropping it misses a term. For a single binary concept, the exact logit-space gradient at pi=0.1 is roughly -0.955, which is larger in magnitude than BCE's -0.9, directly contradicting Proposition 4.2. Theorem 4.3 inherits the problem, so the 'inherently less sensitive' conclusion is not established as written. That does not mean the method is bad—empirically it appears to work—but the theory needs either a corrected derivation or a weaker claim.

Second, the headline empirical comparison has a confound: CPO-trained models receive the RandInt augmentation (25% random concept interventions) while the plain CBM BCE baseline does not (App. A.1). That is a meaningful extra training signal, and some of the gain could come from it. The paper mentions this but never ablates it. A simple fix: run the BCE baseline with the same RandInt setting.

Third, Proposition 4.2 and Theorem 4.3 assume deterministic noise, while the experiments use random label flipping. That mismatch makes the theory support the empirical story less cleanly than the paper suggests.

The uncertainty analysis is the weakest part only in that it is mostly qualitative; the quantitative occlusion analysis shows CPO spreads uncertainty more evenly than BCE, but the aggregate differences are modest. The streaming-data experiment is exploratory but fine.

If I were advising the authors: correct or explicitly weaken the gradient theorem, add a BCE+RandInt ablation, and restate the noise theory for the actual noise model used. As it stands, the method is promising and the empirical pattern is consistent, but the central theoretical claim is unverified.

Referee Report

4 major / 6 minor

Summary. The paper introduces Concept Preference Optimization (CPO), a DPO-style loss for training Concept Bottleneck Models, and claims that CPO directly optimizes the concept posterior, is inherently less sensitive to concept-label noise than binary cross-entropy (BCE), and consistently outperforms BCE on CUB, AwA2, and CelebA with and without added noise. The theoretical component derives simplified forms of the CPO objective, a gradient bound relative to BCE, and a theorem about noise robustness; the empirical component compares CPO-trained CBMs and CEMs against BCE-trained baselines and ProbCBMs on task accuracy, concept AUC, intervention performance, uncertainty quantification, and noisy-label settings. The paper also includes a streaming-data experiment and releases code.

Significance. If the central claims were established, CPO would be a valuable, low-overhead drop-in loss for CBMs: it requires no extra parameters, adds modest compute, and appears to improve task accuracy and intervention behavior. The paper has concrete strengths: it ships code, evaluates across three datasets and several architectures, probes structured and uniform noise, includes an uncertainty study, and connects the loss to control-as-inference, which is a useful pedagogical framing. However, the central theoretical claim that CPO is 'inherently less sensitive' to concept noise rests on a gradient derivation that drops a theta-dependent term, and the headline empirical comparison is potentially confounded by the use of RandInt augmentation for CPO-trained models but not for the standard BCE baseline. The significance is therefore conditional on repairing the proof and deconfounding the experiments.

major comments (4)

[Sec. 4.2 / App. C.1, Eqs. (25)-(26), Prop. 4.2] This comment is complete.
[Prop. 4.1 / App. C.1, Eq. (6)] This comment is complete.
[Sec. 4.2 / App. C.2, Theorem 4.3] This comment is complete.
[App. A.1 / Table 1 / Figs. 5-6] This comment is complete.

minor comments (6)

[App. C.1, Eq. (11)] This comment is complete.
[App. H, first paragraph] This comment is complete.
[App. F.1] This comment is complete.
[Table 1] This comment is complete.
[App. A / App. I] This comment is complete.
[Throughout] This comment is complete.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: CPO is a direct DPO adaptation, and the posterior and noise-robustness claims rest on external derivations and empirical benchmarks rather than on fitted inputs or self-citation chains.

full rationale

The paper's central derivation chain is: Eq. (5) defines L_CPO by instantiating DPO's Eq. (4) with empirical concepts as preferred and policy-sampled concepts as dispreferred; Proposition 4.1 is an algebraic identity under a uniform prior; the gradient bound in Proposition 4.2 follows from the paper's own gradient formula; Theorem 4.3 applies that bound to noise; and the posterior interpretation invokes standard control-as-inference equivalences from Levine (2018), Haarnoja et al. (2017), and Korbak et al. (2022), none of which are self-citations. No fitted parameter is renamed as a prediction, no benchmark number is constructed from the paper's own fitted constants, and no uniqueness theorem from the authors' prior work is used to forbid alternatives. The apparent weakness in App. C.1—the c'=c event's probability pi_theta(c|x) is theta-dependent, so the exact expected gradient of Eq. (5) contains an additional term that can break Proposition 4.2 as stated—is a mathematical correctness issue in the derivation, not circularity: the theorem is not true by construction, it is simply unproven as written. The RandInt augmentation asymmetry between CPO and BCE baselines is an experimental confound, not a circular reduction. Therefore no circular step is established, and the score is 0.

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

The central results depend on the CPO loss with a uniform prior and on gradient derivations that assume conditional independence of concepts. No new physical entities are introduced; the main hand-set hyperparameters are beta=1 and the RandInt augmentation probability. The optimality variable o is a standard theoretical device, not a new postulated entity.

free parameters (4)

beta (KL regularization coefficient in CPO) = 1 (fixed, not tuned)
Controls deviation from the uniform prior in the DPO-style objective. Chosen by hand and held constant across all experiments; central to the loss definition.
lambda (concept loss weight) = tuned in {1,5,10}
Weights the concept loss in the joint CBM objective. Tuned on validation for all methods, so not unique to CPO, but it affects the reported comparisons.
RandInt intervention probability = 25% for CPO/CEM, 50% for ProbCBM; not used for CBM BCE
Random concept intervention applied during training. Applied to CPO-trained models and CEMs but not to the standard CBM BCE baseline, which confounds the comparison.
learning rate = tuned per dataset (e.g., {0.1,0.01} for CUB/AwA2)
Standard hyperparameter tuned for all baselines; not specific to CPO.

assumptions (5)

domain assumption The empirical concept label c is always preferred to a policy-sampled concept c' (c ≻ c'), even when c is noisy.
Core assumption of CPO in §4.1. It is weaker than assuming correctness, but noise robustness still relies on the gradient weight shrinking updates as policy confidence in a noisy c grows.
domain assumption Uniform prior pi_0(c|x) over concepts.
Used in Proposition 4.1 and all gradient derivations in App. C.1. The paper explores non-uniform priors in §5.4, so this is a design choice rather than a universal requirement.
domain assumption Conditional independence of concepts: c_i ⊥ c_j | x for i ≠ j.
Stated in App. C.1 to simplify gradient proofs; the paper claims it is not strictly necessary, but the main derivations use it.
domain assumption Optimality variable framing: p(o=1|x,c)=exp(r*(x,c)) with r* ∈ (-infinity, 0].
Used in §4.3 to equate the RL objective with amortized posterior inference. This is a standard maximum-entropy RL assumption, but r* is unknown and unmeasured.
domain assumption Task labels y are always correct; only concept labels may be noisy.
Assumed in §3, restricting the scope of the noise setting to concept mislabeling only.

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Cite this review

Pith. "Pith review of Addressing Concept Mislabeling in Concept Bottleneck Models Through Preference Optimization." pith.science (2026). https://pith.science/paper/FEXKLCAS

@misc{pith2026250418026,
  author       = {Pith},
  title        = {Pith review of: Addressing Concept Mislabeling in Concept Bottleneck Models Through Preference Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FEXKLCAS}},
  note         = {Machine review of arXiv:2504.18026}
}

read the original abstract

Concept Bottleneck Models (CBMs) propose to enhance the trustworthiness of AI systems by constraining their decisions on a set of human-understandable concepts. However, CBMs typically assume that datasets contain accurate concept labels-an assumption often violated in practice, which we show can significantly degrade performance (by 25% in some cases). To address this, we introduce the Concept Preference Optimization (CPO) objective, a new loss function based on Direct Preference Optimization, which effectively mitigates the negative impact of concept mislabeling on CBM performance. We provide an analysis of key properties of the CPO objective, showing it directly optimizes for the concept's posterior distribution, and contrast it against Binary Cross Entropy (BCE), demonstrating that CPO is inherently less sensitive to concept noise. We empirically confirm our analysis by finding that CPO consistently outperforms BCE on three real-world datasets, both with and without added label noise. We make our code available on Github.

Figures

Figures reproduced from arXiv: 2504.18026 by the authors.

**Figure 1.** Visualization of the behavior of LCPO and LBCE under different scenarios, where the concepts are sampled from the optimal policy π ∗ , the empirical data µ and the amortized sampling policy πθ. In detail the scenarios for the concepts are: Beak color where the not-optimal, but not worst concept (orange) is preferred over incorrectly sampled (red). Throat color where sampled and empirical data align as optimal (gree… view at source ↗

**Figure 2.** Comparison of graphical models for Bayesian CBMs. Dashed outlines indicate the variable over which an amortized posterior is taken. On the left (a), it can be seen that optimizing a CBM using LCPO directly approximates the posterior distribution of the concepts. On the right (b), it can be seen how other methods instead obtain the posterior over a latent variable z. may not accurately reflect their concept labels. I… view at source ↗

**Figure 4.** CUB Interventions without added label noise. In CUB and CelebA, LCPO models lead to the best intervention performance. While in AwA2, a substantial (∼8-15) number of interventions must be performed for LBCE-based models to outperform LCPO ones. 2022), which employ more expressive concept representations to increase model capacity, (3) Coop-CBMs (Sheth & Ebrahimi Kahou, 2024), which use an auxiliary head to improve… view at source ↗

Figures from the paper (12 more)

**Figure 5.** Figure 5: Performance metrics for all tasks across varying degrees of label noise. We find that across all noise levels, LCPO consistently alleviates performance loss. label noise probability p across {0.1, 0.2, 0.3, 0.4}. We observe that, under noisy label conditions, trainin…

**Figure 6.** Figure 6: Intervention performance under noisy settings. We find that LCPO is generally less affected by concept label noise in terms of interventionality. This finding is specifically relevant to CUB and CelebA where we find little to no performance degradation when compared to…

**Figure 7.** Figure 7: evaluates models using k% ∈ {50%, 100%} of total concepts. Curiously, we find that LCPO using a uniform prior performs worse when using more concept labels. We believe this is due to the policy drifting further from that of the initial checkpoint. We find that indeed b…

**Figure 8.** Figure 8: Intervention performance including Coop CBMs. C. Analysis While prior work has shown that under mild conditions, offline RL performs (equivalent CPO) better than likelihood-based training under noisy labels (Kumar et al., 2022; Rashidinejad et al., 2021), it still does…

**Figure 9.** Figure 9: Visualization of noisy (indicated by a −) and optimal (indicated with a *) gradients for ∇θLCPO and ∇LBCE. We can observe that even in low noise settings p = 0.1, LCPO better approximates its optimal gradient, with this difference growing as noise level increases espec…

**Figure 10.** Figure 10: Task Accuracy/Concept AUC vs the percentage of data the model has been trained on. We find that updating models initialized with a LBCE policy yields improved results for LBCE with detrimental ones for LCPO (A). In (B) we again visualize the result for updating the mo…

**Figure 11.** Figure 11: Intervention performance for all noise levels. We find that overall methods trained with LDPO yield better intervention performance under noise. These findings are specifically relevant to CUB and Celeb where we see all other methods are harshly impacted. We find the …

**Figure 12.** Figure 12: Performance metrics for all tasks across varying degrees of label noise. We find that across all noise levels, LCPO ourperforms all other models, including both joint and sequential ProbCBMs 21 [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]

**Figure 13.** Figure 13: Visualization of different images and concept uncertainty score. We display for each model the concept with the highest uncertainty alongside the label for that concept. For each model we display the uncertainty score alongside the probability of the concept being act…

**Figure 14.** Figure 14: Grid visualization of different images before and after applying a cropping and blocking augmentations. For each model we display the concept with the highest certainty score alongside the scores of other models. Overall we find that LCPO increases the confidence of t…

**Figure 15.** Figure 15: Distribution of changes in certainty (δaug) for correct concepts under increasing block sizes. Higher area under the curve above zero indicates a more appropriate increase in uncertainty. LCPO and LBCE respond more consistently to occlusion than ProbCBM, whose uncerta…

**Figure 16.** Figure 16: Per-concept AUC values under 4 × 4 blocking. LBCE concentrates uncertainty increases on a few concepts, while LCPO distributes uncertainty more evenly across concepts, suggesting better alignment with the effects of occlusion. by reducing visual access to relevant fea…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [4]

In our setup, the optimal values for γ were found to be γ= 5 for CUB,γ= 10 for AwA2, andγ= 0.01 for CelebA. We observed negligible differences between Coop-CBM and standard CBMs in terms of base and intervention performance (see Figure 8), except for in AwA2 where it improves intervention performance but outperforms ProbCBM at higher number of interventio...

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URL https: //arxiv.org/abs/1611.01144. Kaufmann, T., Weng, P., Bengs, V ., and H¨ullermeier, E. A survey of reinforcement learning from human feedback,

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URL https://arxiv.org/abs/ 2204.05618. Levine, S. Reinforcement learning and control as proba- bilistic inference: Tutorial and review,

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Performance metrics for all tasks across varying degrees of label noise. We find that across all noise levels,LCPO ourperforms all other models, including both joint and sequential ProbCBMs 21 Addressing Concept Mislabeling in Concept Bottleneck Models Through Preference Optimization G. Additional Analysis on Uncertainty In this Appendix, we provide a mor...

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