Making Reliable and Flexible Decisions in Long-tailed Classification

Bolian Li; Ruqi Zhang

REVIEW 2 major objections 5 minor 19 references

Making Reliable and Flexible Decisions in Long-tailed Classification

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A single decision-theoretic objective lets long-tailed classifiers be trained for the errors that actually matter, with no loss in accuracy or calibration.

desk verdict A practical cost-sensitive long-tailed method with strong empirical results, but the optimality claim rests on swapping log E g for E log g; that gap is fixable. read the letter →

arxiv 2501.14090 v1 pith:H2TXSNH3 submitted 2025-01-23 cs.LG stat.ML

Bolian Li , Ruqi Zhang This is my paper · ORCID

classification cs.LGstat.ML

keywords long-tailedclassificationBayesiandecisiontheorycost-sensitivelearningutilitymatrixfalseheadratevariationalinferenceparticle-basedensemblesuncertaintyquantification

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

The paper tries to show that long-tailed classifiers can be trained to make decisions that are optimal for the user's actual error costs, not just to maximize accuracy. It builds a single objective from Bayesian decision theory—the integrated-gain lower bound—in which a utility matrix assigns penalties to specific mispredictions and an importance weight $1/f(n_y)$ corrects for class imbalance. The paper argues that this objective is a principled alternative to re-weighting and logit-adjustment losses, and that the accompanying test-time rule is the right way to use the learned ensemble. If the claim holds, practitioners can directly suppress high-risk errors such as calling a tail-class patient healthy while preserving or improving standard accuracy and calibration.

What carries the argument

The load-bearing object is the integrated gain $G(d) = \mathbb{E}_{(x_1,y_1),\dots,(x_N,y_N)\sim p_{\mathrm{test}}(x,y)}\mathbb{E}_{\theta\sim p(\theta|D)}\prod_i g(d_i|x_i,\theta)$, where the per-input decision gain is $g(d|x,\theta)=\prod_{y'} p(y'|x,\theta)^{U_{y',d}}$. This fuses the task-specific utility matrix $U$, the class-imbalance correction $p_{\mathrm{test}}(y)/p_{\mathrm{train}}(y)\propto 1/f(n_y)$, and a posterior over models into one quantity. The paper converts this quantity into the tractable lower bound $L(q,d=Y)$ of Eq. 10 using Jensen's inequality, importance sampling, and a particle approximation $q(\theta)=\sum_j w_j\delta(\theta-\theta_j)$, with a repulsive-force term from the KL divergence that keeps particles diverse. The same machinery supplies the test-time decision rule in Eq. 16.

What would settle it

Train RF-DLC on a long-tailed benchmark, then evaluate it on a held-out test set with a known non-uniform class distribution and compare against a version trained with the correct $p_{\mathrm{test}}(y)/p_{\mathrm{train}}(y)$ ratio; if the decision gain or accuracy is worse, the uniform-test assumption is the cause. A second check is to compare per-class feature statistics between training and test images for the same label, which would directly test the intra-class consistency assumption.

Watch

Extended reading notes

Core claim

The central claim is that optimal decision-making on long-tailed data can be written as one objective: maximize a variational lower bound on the integrated gain, the expected utility-weighted predictive likelihood under the test distribution. The training loss is $L(q,d=Y) = \sum_i \mathbb{E}_{\theta\sim q(\theta)}[\frac{1}{f(n_{y_i})}(\sum_{y'} U_{y',y_i}\log p(y'|x_i,\theta) + \log p(y_i|x_i,\theta))] - \mathrm{KL}(q(\theta)\|p(\theta)) + C$, with $q(\theta)$ represented by an ensemble of particles. At test time, the optimal decision for input $x^*$ is $d^* = \arg\max_d \sum_j \sum_{y'} U_{y',d} \log p(y'|x^*,\theta_j)$. The experiments report that a tail-sensitive utility lowers false-head errors on CIFAR10-LT, CIFAR100-LT, and ImageNet-LT, that a one-hot utility preserves or improves overall and tail accuracy, and that the resulting Bayesian predictive distribution improves calibration.

Load-bearing premise

The whole weighting scheme reduces to $1/f(n_y)$ only because the paper assumes the test set is class-uniform and that, within each class, test inputs match training inputs; if either fails, the objective reweights the wrong quantity.

Editorial extensions

If this is right

A user can encode any asymmetric penalty in the utility matrix, and the same training and inference equations apply without changing the model family.
With a tail-sensitive utility, false-head errors drop on CIFAR10-LT, CIFAR100-LT, and ImageNet-LT across tail regions of 25%, 50%, and 75%.
With a one-hot utility, the framework reduces to a class-weighted Bayesian ensemble, and the reported overall and tail accuracies are at or above the compared long-tailed baselines.
The Bayesian predictive distribution from the particle ensemble gives better AUC and lower ECE than maximum softmax and evidential uncertainty baselines.
The ablation recommends the linear form $f(n_y)=n_y$, since larger growth rates between head and tail weights improve tail and overall accuracy.

Reading between the lines

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

If the test set's class distribution is known and non-uniform, the same derivation suggests replacing $1/f(n_y)$ with $p_{\mathrm{test}}(y)/p_{\mathrm{train}}(y)$; the paper notes this as a limitation but does not test it.
The inference-time rule in Eq. 16 improves FHR when applied to other baselines, so part of the reported gain may come from decision-time re-scoring rather than from the new training objective alone.
The objective is a lower bound on the integrated gain, and the gap from the one-sample Monte Carlo approximation and Jensen's inequality is not measured; a tighter or multi-sample estimate could be tested.
A regression analogue could be built by replacing the categorical decision gain with a continuous utility function, a direction the paper lists as future work.

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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 thing to know: this paper gives you a workable recipe for injecting asymmetric misprediction costs into long-tailed classifiers, and it ships a new metric (False Head Rate) that practitioners will likely borrow. The experiments are consistent and reasonably thorough: three datasets, ablations on the utility matrix, the reweighting form, particle count, repulsive force, and an ensemble comparison that partially addresses the unfair main-table setup. If you work on decision-aware or cost-sensitive classification, this is worth a read.

What is genuinely new is the specific combination: product-form decision gain, class-count reweighting, particle variational inference, and test-time utility weighting. The derivation in Eq. 10 is a legitimate ELBO-style bound, not a curve fit to the metric. The authors also state their distributional assumptions (Assumption 1) and list limitations honestly, including the uniform-test restriction.

The soft spots are real but mostly addressable. The load-bearing one is internal and was not in the reader's take. The test-time rule in Eq. 16 is not the Bayes-optimal rule claimed. The true posterior expected gain under the particle approximation is log E_q g = log (1/M) sum_j exp(sum_{y'} U_{y',d} log p(y'|x, theta_j)), but Eq. 16 maximizes E_q log g = (1/M) sum_j sum_{y'} U_{y',d} log p(y'|x, theta_j). Jensen's inequality says these differ whenever particles disagree, and the gap persists even if q equals the exact posterior. So the central optimality claim is not established. Fixing this requires either justifying the swap as an approximation or running experiments with the exact rule, which is a concrete revision. Second, the implemented objective (Eq. 24) includes an alpha rescaling and an annealed repulsive force (Eq. 23) that are not in the derived Eq. 10; these extra knobs are practical but should be acknowledged as modifications. Third, the closest recent cost-aware long-tailed baseline (He et al. 2024) is cited but not compared, though Table 14 does include Elkan-style baselines.

Who is this for? Researchers or engineers who want a flexible, empirically solid way to reduce tail-to-head errors without trashing accuracy. The theory section overclaims, but the method and metric are useful. I would send this to peer review and ask for a revision that fixes Eq. 16 or weakens the claim, reports exact-rule results, and adds the missing baseline. With those changes, it becomes a solid paper.

Referee Report

2 major / 5 minor

Summary. The paper proposes RF-DLC, a Bayesian decision-theoretic framework for long-tailed classification. It defines an integrated gain that combines a task utility matrix with the training-data class distribution, derives a lower bound (Eq. 10) using importance sampling under Assumption 1, approximates the posterior with particles (Eq. 13) plus repulsive regularization (Eq. 15), and makes test decisions via Eq. (16). Experiments on CIFAR10/100-LT, ImageNet-LT, iNaturalist, and DermaMNIST report gains in False Head Rate, accuracy, and calibration, with ablations over utility values, f(ny), particle number, and repulsive force.

Significance. If the theoretical claims are established, the framework is a valuable contribution: it offers a principled way to inject asymmetric error costs into long-tailed training and provides an uncertainty estimate through the particle posterior. The paper is strong empirically in breadth: it includes multiple benchmarks, a new cost-oriented metric, an ensemble comparison in Table 9, a robustness study of utility values in Table 15, and comparisons with cost-sensitive baselines in Table 14. The two main concerns below concern the optimality claim of the test-time rule and the relation between the derived and implemented objectives; both are fixable in revision.

major comments (2)

[Section 4.4, Eq. (16)] Equation (16) is presented as maximizing log G(d|x*,D), but it computes sum_j sum_{y'} U_{y',d} log p(y'|x*,theta_j), which is E_{q(theta)} log g(d|x*,theta). With the particle approximation of Eq. (13), log G(d|x*,D) is approximately log((1/M) sum_j exp(sum_{y'} U_{y',d} log p(y'|x*,theta_j))). Jensen's inequality gives log E_q g >= E_q log g, with strict inequality whenever the particles disagree, and the two objectives can have different argmax decisions. This is not a posterior-approximation artifact: it persists even if q equals the true posterior supported on the M particles. The paper's central assertion that RF-DLC makes optimal decisions according to Bayesian Decision Theory is therefore not established by the derivation. Please either add experiments using the correct log-sum-exp rule, or explicitly reframe Eq. (16) as an approximate decision rule and measure the decision gap relative to the exact rule.
[Appendix E.2, Eq. (24)] The implemented training objective in Eq. (24) is not the derived objective in Eq. (10). Eq. (10) weights the entire per-sample term by 1/f(n_{y_i}) and has no alpha; Eq. (24) displays neither the 1/f(n_y) weighting nor the KL term, and introduces a tuned rescaling 1/alpha on the utility term, with alpha set differently per dataset in Table 10. If the class-count weighting is applied in the actual code around Eq. (24), the manuscript needs to show that explicitly; as written, the empirical method is not the objective whose derivation forms the paper's theoretical contribution. Since the paper claims that the utility matrix and variational strategy are directly derived from Bayesian Decision Theory (Section 1), this discrepancy must be resolved, and the sensitivity of results to alpha should be reported.

minor comments (5)

[Section 5.3, Table 3] Because RF-DLC is an ensemble of M particles while most rows are single models, please state this in the main text and reference Table 9 at the first presentation of Table 3; as presented, the comparison is not apples-to-apples.
[Section 5.6, Table 7] The growth-rate column in Table 7 appears to be the ratio of the last-class weight to the first-class weight, but the text calls it growth rate without defining it; please clarify the definition to avoid confusion.
[Abstract and Section 1] The abstract and Section 1 use 'quantity' where 'quantify' is intended ('to quantity the mispredictions'); please fix the typo.
[Section 4.2, Assumption 1] The limitations section mentions non-uniform testing distributions but not the failure of Assumption 1 (p_train(x|y)=p_test(x|y)); since the importance ratio in Eq. (9) depends on that assumption, please add a sentence noting this limitation and its potential effect on the objective.
[Appendix E.2] The statement that code will be released after acceptance conflicts with the reproducibility emphasis of the empirical section; providing the code with the revision would substantially strengthen the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the training objective is a genuine ELBO-style derivation and the utility matrix is a user input, not a fitted target; the only notable issue is a test-time approximation error, which is a correctness concern rather than a circular one.

full rationale

The derivation chain is self-contained. Eq. (4) defines the integrated gain from Bayesian decision theory; Eq. (10) is obtained by Jensen's inequality (Appendix C), importance sampling under Assumption 1, and the 1/f(n_y) ratio; the utility matrix U is a pre-defined user input, not a parameter fitted to the evaluation metrics. The tail-sensitive utility is aligned with the FHR metric, but that is cost-sensitive objective alignment, not a fitted parameter renamed as a prediction; Appendix F.4 shows the FHR improvement is robust across utility values, and the method is also evaluated on external accuracy and calibration benchmarks (Tables 3-5). The self-citation to Li et al. (2022) is used as a baseline (TLC) and as an uncertainty competitor, not as load-bearing support for the derivation, so it does not raise the circularity score. One non-circular concern is Eq. (16): the paper writes argmax_d log G(d|x*,D) ≈ argmax_d Σ_j Σ_{y'} U_{y',d} log p(y'|x*,θ_j), which replaces log E_{θ~q} g by E_{θ~q} log g; this is a correctness or approximation issue (the two objectives can disagree), not a circular reduction, and it does not affect the training derivation in Eq. (10). The limitations section also explicitly acknowledges the uniform-test and dataset-shift assumptions, consistent with a non-circular derivation from stated assumptions.

Assumptions & free parameters 6 free parameters · 6 assumptions · 1 invented entities

The central method relies on standard Bayesian decision theory plus three hand-chosen ingredients: the product-form gain, the class-count reweighting f(n_y), and the particle ensemble with approximate entropy regularization. The utility rescaling factor alpha is the most consequential free parameter because it is needed to reconcile the derived objective with the implemented one.

free parameters (6)

Importance weighting form f(n_y) = linear f(n_y)=n_y (selected after ablation vs sqrt, log, effective number, constant)
Eq. 9 leaves f arbitrary; Section 5.6 selects linear because it gives best held-out accuracy on CIFAR100-LT, so the weighting in Eq. 10 is a tuned modeling choice.
Utility rescaling factor alpha = 0.002 (CIFAR10-LT), 0.3 (CIFAR100-LT), 50 (ImageNet-LT), 100 (iNaturalist)
Introduced only in Appendix E.2 Eq. 24 to control the utility term's effect on accuracy; grid-searched, not present in the theoretical objective Eq. 10.
Repulsive force weight lambda = 5e-4 (CIFAR), 2e-4 (ImageNet-LT and iNaturalist), swept in Table 8
Controls the entropy and diversity regularization; chosen by grid search.
Annealing time constant tau for repulsive force = 40 (CIFAR), 20 (ImageNet-LT and iNaturalist)
Controls decay of the repulsive force during training; chosen by grid search.
Number of particles M = 3 for most experiments; up to 6 recommended in Fig. 2b
Ensemble size is an architecture choice with a clear accuracy-cost trade-off; main gains relative to single-model baselines depend on it.
Utility values in tail-sensitive matrix = -1 for discouraged mispredictions, 1 for correct decisions
Hand-assigned; Appendix F.4 shows performance is robust to values, but the sign and magnitude are user inputs.

assumptions (6)

domain assumption Assumption 1: intra-class consistency p_train(x|y)=p_test(x|y)
Section 4.2 Eq. 8. Required to reduce the importance ratio to p_test(y)/p_train(y); if covariate shift exists, the weighting in Eq. 10 is invalid.
domain assumption Testing distribution is uniform over classes: p_test(y)=constant
Section 4.2 Eq. 9 and Background. Turns the discrepancy ratio into 1/f(n_y); the authors acknowledge in Limitations that non-uniform test distributions break this.
ad hoc to paper The decision gain has the product form g(d|x,theta)=prod_{y'} p(y'|x,theta)^{u(y',d)}
Section 4.1 Eq. 5. This is a definition chosen for stability and flexibility, not forced by decision theory; the standard alternative is the weighted sum in Cobb et al. (2018).
domain assumption The true label is treated as the optimal decision at training (d_i=y_i)
Section 4.3, set d=Y in Eq. 10. Plausible for classification, but it means the framework does not optimize decisions against a different expert policy during training.
standard math Jensen's inequality and Bayes' rule used to derive the lower bound
Appendix C Eqs. 18-20; standard and unproblematic.
ad hoc to paper Particle variational distribution with diagonal-covariance entropy approximation approximates the posterior
Section 4.3 Eqs. 13-15. The entropy is approximated by 0.5 log|Sigma_hat| with SWAG-diagonal covariance; this is a heuristic regularization, not an exact posterior approximation.

invented entities (1)

False Head Rate (FHR) metric independent evidence
purpose: Quantifies tail-to-head mispredictions as |P_head intersection G_tail| / |G_tail|, used to evaluate tail-sensitive risk.
A well-defined metric computable from any classifier's predictions; not a hidden physical entity, but a new evaluation construct introduced by this paper.

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Cite this review

Pith. "Pith review of Making Reliable and Flexible Decisions in Long-tailed Classification." pith.science (2026). https://pith.science/paper/H2TXSNH3

@misc{pith2026250114090,
  author       = {Pith},
  title        = {Pith review of: Making Reliable and Flexible Decisions in Long-tailed Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2TXSNH3}},
  note         = {Machine review of arXiv:2501.14090}
}

read the original abstract

Long-tailed classification is challenging due to its heavy imbalance in class probabilities. While existing methods often focus on overall accuracy or accuracy for tail classes, they overlook a critical aspect: certain types of errors can carry greater risks than others in real-world long-tailed problems. For example, misclassifying patients (a tail class) as healthy individuals (a head class) entails far more serious consequences than the reverse scenario. To address this critical issue, we introduce Making Reliable and Flexible Decisions in Long-tailed Classification (RF-DLC), a novel framework aimed at reliable predictions in long-tailed problems. Leveraging Bayesian Decision Theory, we introduce an integrated gain to seamlessly combine long-tailed data distributions and the decision-making procedure. We further propose an efficient variational optimization strategy for the decision risk objective. Our method adapts readily to diverse utility matrices, which can be designed for specific tasks, ensuring its flexibility for different problem settings. In empirical evaluation, we design a new metric, False Head Rate, to quantify tail-sensitivity risk, along with comprehensive experiments on multiple real-world tasks, including large-scale image classification and uncertainty quantification, to demonstrate the reliability and flexibility of our method.

Figures

Figures reproduced from arXiv: 2501.14090 by the authors.

**Figure 2.** Ablation studies on CIFAR100-LT. Comparing (a) different forms of class probabilities and (b) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Full utility matrix configurations on CIFAR10-LT. The two matrices are designed to particularly [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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45.27 1,408,784 RF-DLC (ours) 44.68 1,408,784 F Additional Experimental Results F.1 Full Experimental Results on Classification We list the full experimental results of top-1 accuracy in Table 12, including the results on iNaturalist (Van Horn et al., 2018). Classes are equally split into three class regions (head, med and tail). For example, there are 33...

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Pith tools

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

[1] [8]

Detection of flying birds in airport monitoring based on improved yolov5

Xiaohang Shi, Jun Hu, Xueyue Lei, and Shiyou Xu. Detection of flying birds in airport monitoring based on improved yolov5. In 2021 6th International Conference on Intelligent Computing and Signal Processing (ICSP), pp. 1446–

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We will release the code after acceptance

The optimal values of those hyper-parameters are determined by grid search. We will release the code after acceptance. Table 10: Hyper-parameter configurations. Dataset Base Model Optimizer Batch Size Learning Rate T raining Epochs Discrepancy Ratio λ τ α CIFAR10-LT ResNet32 SGD 128 0.1 200 linear 5e-4 40 0.002 CIFAR100-LT ResNet32 SGD 128 0.1 200 linear ...

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Learning to model the tail.Advances in Neural Information Processing Systems, 30,

15 Published in Transactions on Machine Learning Research (01/2025) Yu-Xiong Wang, Deva Ramanan, and Martial Hebert. Learning to model the tail.Advances in Neural Information Processing Systems, 30,

work page 2025

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multi-expert

and then explored by Liu et al. (2019a); Korba et al. (2020); D’Angelo & Fortuin (2020). Instead of directly modeling the gradient flow, our framework optimizes the particles through stochastic gradient descent (SGD), with repulsive force induced by the integrated gain objective. Compared to existing particle optimization, our method is easy and cheap to ...

work page 2019

[5] [15]

Therefore, the computational cost of our method is comparable to existing ensemble models

and TLC (Li et al., 2022), in which the first few layers in neural networks are shared among all particles. Therefore, the computational cost of our method is comparable to existing ensemble models. Besides, compared with gradient-flow-based BNN like D’Angelo & Fortuin (2021), which typically uses 20 particles, our model is far more efficient with no more...

work page 2021

[6] [16]

Classes are equally split into three class regions (head, med and tail)

45.27 1,408,784 RF-DLC (ours) 44.68 1,408,784 F Additional Experimental Results F.1 Full Experimental Results on Classification We list the full experimental results of top-1 accuracy in Table 12, including the results on iNaturalist (Van Horn et al., 2018). Classes are equally split into three class regions (head, med and tail). For example, there are 33...

work page 2018

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For example, the method in Section 2 of Elkan (2001) does not consider specific error types during the training phase, and fails to incorporate the utility matrix during testing

78.16 84.51 79.62 67.63 21.88 10.27 22.01 34.53 Bayesian (RF-DLC) 80.62 88.47 81.82 63.05 10.87 6.37 11.09 16.71 decision-making processes. For example, the method in Section 2 of Elkan (2001) does not consider specific error types during the training phase, and fails to incorporate the utility matrix during testing. The Bayesian method in Section 4 of El...

work page 2001

[8] [19]

It specifically avoids mispredicting some classes as other classes

is one of the key contributions of this paper. It specifically avoids mispredicting some classes as other classes. To understand the effect of utility functions, we add an ablation study in Table 16, where Eq. 16 with tail-sensitive utility is added to baselines at inference time. Adding Eq. 16 improves the FHR performance and maintains the ACC, which dem...

work page 2025