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REVIEW 2 major objections 5 minor 1 cited by

Feasible Learning

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

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Pith's one-line read The paper argues that training models to cap each sample's loss, not minimize the average, produces fewer very-high-loss predictions at little average-cost penalty.

desk verdict A genuinely interesting learning framework with a clean theory, but its own CIFAR10 results contradict the abstract's unconditional tail-behavior claim. read the letter →

arxiv 2501.14912 v1 pith:RF4XXKDA submitted 2025-01-24 cs.LG cs.AI

Juan Ramirez , Ignacio Hounie , Juan Elenter , Jose Gallego-Posada , Meraj Hashemizadeh , Alejandro Ribeiro , Simon Lacoste-Julien This is my paper · ORCID
classification cs.LGcs.AI
keywords FeasibleLearningper-samplelossconstraintsprimal-dualoptimizationResilienttailbehaviorConditionalValueatRiskpreferenceconstrained
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

Feasible Learning (FL) replaces the usual objective of minimizing average loss with a feasibility problem: find a model whose loss on every training sample is below a threshold. Because any model meeting the constraints is acceptable, the optimization algorithm itself shapes the solution, and the paper studies a primal-dual method that re-weights samples through Lagrange multipliers. To handle thresholds that make the problem infeasible, the paper introduces Resilient Feasible Learning (RFL), which relaxes constraints with minimal-norm slack variables. Across image classification, age regression, and LLM preference optimization, the paper reports that FL- and RFL-trained models keep average performance close to ERM while producing a less heavy-tailed loss distribution. The central claim is that per-sample constraints, not average loss, are a viable organizing principle for training, with tail behavior as the payoff.

What carries the argument

The mechanism is the per-sample loss constraint: FL solves $\min_{\theta} 0$ subject to $g_i(\theta) \le \epsilon$ for every training sample, where $g_i$ is the loss on sample $i$. The paper solves this through a Lagrangian game with one multiplier $\lambda_i$ per sample; gradient descent on $\theta$ with ascent on $\lambda$ re-weights samples by their historical difficulty, so hard samples dominate updates and easy samples receive little pressure once their constraint is satisfied. RFL adds slack variables $u \ge 0$ with an $\alpha\|u\|^2$ penalty, which Proposition 1 turns into a quadratically regularized Lagrangian and Proposition 2 equates to an ERM objective with clamped-and-squared loss. The machinery's work is to make constraint satisfaction tractable at neural-network scale and to let the optimizer select a non-interpolating solution.

What would settle it

Run FL and ERM on the same benchmarks with matched hyperparameter budgets and compare test-set Conditional Value at Risk at high quantiles; if FL does not beat ERM on held-out data in a majority of tasks, the paper's central claim fails. The CIFAR10 experiment in Appendix C is already one such comparison that does not go FL's way.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that models trained by solving a feasibility problem with per-sample loss constraints display improved tail behavior compared to ERM, with only a marginal impact on average performance. In Section 5.3, the paper reports that FL consistently achieves lower Conditional Value at Risk (CVaR), the average loss among the highest-loss samples, than ERM on the DPO task, and the conclusion states that FL yields a less heavy-tailed loss distribution than ERM. The paper presents this as an empirical discovery supported across image classification, age regression, and language-model preference alignment, not as a universal theorem.

Load-bearing premise

The load-bearing assumption is that compressing the high-loss tail on the training set also compresses it on fresh test data drawn from the same distribution; the paper's own CIFAR10 appendix shows one case where ERM has lower test tail losses than FL, so this transfer is not guaranteed.

Editorial extensions

If this is right

  • If the central claim holds, practitioners can train high-capacity models with per-sample loss caps at roughly ERM's computational cost and obtain test solutions with fewer extreme losses.
  • RFL gives a concrete recipe for infeasible settings: relax the constraints with minimal-norm slacks, which is equivalent to minimizing a clamped-and-squared loss, so standard ERM machinery can be reused.
  • The per-sample Lagrange multipliers carry a difficulty signal, so the final training run can identify hard or mislabeled examples, and the many zero multipliers suggest data pruning is possible.
  • In preference alignment for LLMs, RFL reduced maximum DPO losses while raising average loss only slightly, making it a candidate for deployments where a few bad responses matter more than average reward.

Reading between the lines

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

  • Beyond the paper: if the tail-compression effect generalizes, final multiplier magnitudes could be used as a data-quality score to flag mislabeled or duplicated examples, extending the paper's UTKFace illustration.
  • Beyond the paper: the paper's own CIFAR10 appendix shows the claim is not universal, so the immediate next experiment is a systematic comparison of training-set tail compression against test-set tail compression across benchmarks.
  • Beyond the paper: because RFL is equivalent to a clamped-and-squared ERM objective, existing theory for thresholded losses likely applies to FL's generalization, a connection the paper does not develop.
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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

2 major / 5 minor

Summary. The paper introduces Feasible Learning (FL), a training paradigm in which a model is found by solving a feasibility problem that requires the per-sample loss to stay below a threshold epsilon, and Resilient Feasible Learning (RFL), a relaxation that adds minimal-norm slack variables to restore feasibility. The authors prove that RFL is equivalent to a quadratically regularized min-max problem (Proposition 1) and to a clamped-and-squared ERM objective (Proposition 2), and they propose alternating primal-dual updates with per-sample Lagrange multipliers. Experiments are reported on CIFAR10 classification, UTKFace age regression, and DPO fine-tuning of Llama-3.1-8B and Zephyr-3B. The central empirical claim is that FL and RFL achieve average performance comparable to ERM while improving tail behavior, i.e., reducing the number of very-high-loss samples and lowering CVaR.

Significance. The conceptual contribution is attractive and timely: framing learning as sample-level constraint satisfaction is a clean idea, and the primal-dual implementation is practical and has training cost comparable to ERM. The two propositions are simple but correctly proven, the slack-based treatment of infeasibility is principled, and the release of code supports reproducibility. The experiments span three qualitatively different tasks. However, the paper's most prominent advertised conclusion — that FL consistently yields less heavy-tailed loss distributions than ERM — is directly contradicted by the paper's own CIFAR10 results. Since this tail-behavior claim is the main reason a reader would prefer FL over ERM, the empirical core needs revision before the paper can be accepted in its current form.

major comments (2)
  1. [§5.3, Abstract, Conclusion vs. Appendix C.1] The statement in Section 5.3 that 'FL consistently achieves lower CVaR values compared to ERM' and the conclusion that 'FL yields a less heavy-tailed loss distribution than ERM' are not supported by the paper's own CIFAR10 results. Appendix C.1 explicitly states: 'In training, we observe slightly better CDF and CVaR curves for ERM compared to FL and RFL... A similar trend is observed in the test set.' Table 7 reports ERM test Max CE 12.830 ± 0.443 versus FL 15.533 ± 0.642 and RFL 16.235 ± 1.510, and Figure 5 shows ERM below FL and RFL in the CIFAR10 CVaR curves. Because the abstract and Section 5.3 present tail improvement as a general property, this internal contradiction is load-bearing. The authors should either scope the tail-behavior claim to the DPO and UTKFace settings, or provide a criterion for when the effect holds (noting, for example, that RFL with α=10^-3 in Table 8 does produce a lower CIFAR10 test Max CE than ERM, unlike the α=1 configuration emphasized in the main text).
  2. [§5.3 and Appendix C] The paper provides no distributional argument or stated assumption under which training-set loss concentration transfers to the test set. Because FL's constraints are enforced only on training samples, the test-side tail improvement is an empirical pattern rather than a consequence of the formulation. Given the CIFAR10 counterexample, the authors should either state a concrete hypothesis or condition for when tail improvement transfers, or explicitly limit the empirical contribution to the tasks and configurations where the effect is observed.
minor comments (5)
  1. [Appendix C.1] The text refers to 'a training dataset of 60000 samples,' but Table 3 lists CIFAR10 training size as 50000; please reconcile these numbers.
  2. [Table 13] The DPO results are reported without error bars or an explicit statement of the number of seeds, despite Section 5 saying that 'unless stated otherwise, all reported metrics are averaged over 5 seeds'; please clarify whether the DPO runs are single-seed.
  3. [Figure 4] The caption says the figure shows 'the loss and multiplier values' for UTKFace samples, but the figure appears to display only images; please describe how the loss and multiplier values are visualized.
  4. [Title page and acknowledgements] There are minor typos ('Correspondance', 'Pennislvania'); please proofread.
  5. [§2.1] The sentence 'Once a constraint is strictly satisfied, the dual updates reduce the corresponding multiplier' is slightly imprecise, since the reduction happens in the gradient-ascent update only if the constraint remains satisfied; consider rewording for precision.

Circularity Check

1 steps flagged · score 2.0 of 10

No fitted constant is disguised as a prediction; FL's training-loss tail is bounded by its constraint set by definition, while the headline test-set tail comparisons are independent empirical claims that are not circular (though internally contradicted on CIFAR10, a correctness issue).

  1. self definitional [Section 2, Eq. (FL(epsilon)); Section 5.3, Q3]
    "FL considers an optimization problem with a trivial, constant objective while enforcing a loss constraint for each training sample: min_{theta in Theta} 0 s.t g(theta) <= epsilon, (FL(epsilon)) ... We observe that FL produces a more concentrated loss distribution across (training and test) samples, resulting in fewer instances with excessively high losses (Section 5.3)."

    The FL(epsilon) problem is defined by per-sample constraints g(theta) <= epsilon, so any feasible solution has training losses individually bounded by epsilon. Reporting that FL produces a 'more concentrated' training-loss distribution, or fewer very high training losses, is therefore a restatement of the constraint set rather than an empirical consequence derived from independent principles. This is only a minor definitional element, however: the paper's substantive tail claims are the held-out test CDF/CVaR comparisons, which are not implied by the training constraint, and the nonconvex primal-dual solver does not guarantee feasibility, so the training-side observation is not the whole result.

full rationale

The paper's derivation chain is mostly self-contained: FL is a feasibility problem, its Lagrangian dual is written down, GDA updates are derived, and RFL is shown by Propositions 1-2 to be algebraically equivalent to a quadratically-regularized Lagrangian and to a clamped-and-squared ERM objective. These equivalences are direct manipulations, not circular reductions. No fitted parameter is renamed as a prediction: the Lagrange multipliers are optimized dual variables, and the tail-behavior claims are empirical comparisons on held-out test sets, which are not forced by the optimization problem's definition. The self-citations (Sohrabi et al. 2024, Hounie et al. 2024, Gallego-Posada et al. 2022) appear as background or as optional algorithmic suggestions and are not load-bearing for the central equivalence or the main experiments; no uniqueness theorem from the authors' prior work is imported to force a choice. The only definitional element is that feasible FL solutions have bounded per-sample training loss, which makes the training-side concentration claim partly by construction; this is minor because the central empirical content involves test-set generalization and cross-task comparisons. The paper's own CIFAR10 results contradict the universal tail claim (Appendix C.1 reports better CDF/CVaR curves for ERM in both train and test), but that is an internal-evidence and correctness problem, not a circularity problem.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The method introduces two main hand-chosen quantities (per-sample threshold epsilon and slack penalty alpha), plus the dual step size. The theory uses standard convex-analysis results (saddle-point theorem, first-order optimality). No new entities are postulated.

free parameters (3)
  • epsilon (per-sample loss threshold) = 0, 0.51, 0.02, 0.15, etc. per task
    Chosen by hand per task; controls the trade-off between feasibility and regularization; affects all results.
  • alpha (RFL slack penalty) = 1 to 1e-4 per task
    Tuned per task; large impact on dynamics and final performance, e.g., UTKFace best at 1e-3, 1e-4 degrades.
  • dual step size eta_lambda = 1e-4 to 1e-1 per task
    Additional hyperparameter for FL and RFL that can cause divergence if too large or slow convergence if too small.
assumptions (3)
  • standard math Saddle-point existence theorem (Hiriart-Urruty and Lemarechal, Theorem 4.3.1) for min-max interchange in Proposition 1
    Used to prove strong duality for the inner problem over u and lambda.
  • domain assumption Convexity in u and concavity in lambda of the RFL Lagrangian
    The Lagrangian L_RFL is convex in u and concave in lambda; this holds for any loss g(theta) and enables Proposition 1.
  • domain assumption Differentiability of surrogate loss g_i(theta)
    Needed for gradient-based primal-dual updates in Equation (2).

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Cite this review

Pith. "Pith review of Feasible Learning." pith.science (2026). https://pith.science/paper/RF4XXKDA

@misc{pith2026250114912,
  author       = {Pith},
  title        = {Pith review of: Feasible Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RF4XXKDA}},
  note         = {Machine review of arXiv:2501.14912}
}
read the original abstract

We introduce Feasible Learning (FL), a sample-centric learning paradigm where models are trained by solving a feasibility problem that bounds the loss for each training sample. In contrast to the ubiquitous Empirical Risk Minimization (ERM) framework, which optimizes for average performance, FL demands satisfactory performance on every individual data point. Since any model that meets the prescribed performance threshold is a valid FL solution, the choice of optimization algorithm and its dynamics play a crucial role in shaping the properties of the resulting solutions. In particular, we study a primal-dual approach which dynamically re-weights the importance of each sample during training. To address the challenge of setting a meaningful threshold in practice, we introduce a relaxation of FL that incorporates slack variables of minimal norm. Our empirical analysis, spanning image classification, age regression, and preference optimization in large language models, demonstrates that models trained via FL can learn from data while displaying improved tail behavior compared to ERM, with only a marginal impact on average performance.

Figures

Figures reproduced from arXiv: 2501.14912 by the authors.

Figure 1
Figure 1. Fitting polynomials of varying degrees: While ERM tends to overfit with high-degree polynomials, FL still recovers smoother solutions. This occurs even though non-smooth solutions are also part of the FL solution set, highlighting the influence of the optimization algorithm on the final solution. Due to ill-conditioning, we solve both problems using a standard convex optimization solver. The data is generated by add… view at source ↗
Figure 4
Figure 4. Training samples from UTKFace with large [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 3
Figure 3. illustrates the dual variable informativity in a Two-Moons classification task. Samples near the decision boundary, which are harder to classify, have larger multiplier values at the end of training. Hence, similar to support vectors in SVMs, these samples play a more significant role in shaping the classifier, as their higher multipliers give them greater influence in the primal updates. In contrast, points far fro… view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: Empirical distribution of per-sample cross-entropy (CE) losses on a CIFAR10 classification task. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Average training losses at the end of training on the CIFAR10 classification task. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Empirical distribution of per-sample squared error (SE) losses on a UTKFace age regression task. [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Average training losses at the end of training for UTKFace age regression. [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Llama 3.1-8B train and test empirical CDF and CVaR. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Zephyr 3B train and test empirical CDF and CVaR. The plots illustrate the differences in loss [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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