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REVIEW 4 major objections 6 minor 58 references

Category-wise influence vectors reveal a classifier's Pareto ceiling and how to push past it.

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2026-08-04 11:32 UTC pith:XLG4XYSK

load-bearing objection A straightforward category-wise extension of influence functions wrapped around an under-supported Pareto-ceiling criterion; the reweighting machinery is new but the load-bearing geometry is asserted, not proven. the 4 major comments →

arxiv 2510.03950 v2 pith:XLG4XYSK submitted 2025-10-04 cs.LG

What Is The Performance Ceiling of My Classifier? Utilizing Category-Wise Influence Functions for Pareto Frontier Analysis

Shahriar Kabir Nahin , Wenxiao Xiao , Joshua Liu , Anshuman Chhabra , Hongfu Liu This is my paper
classification cs.LG
keywords category-wise influencePareto frontierperformance ceilingsample reweightinglinear programminginfluence functionsdata-centric learningmulti-class classification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether a classifier has reached its best possible per-class performance, and if not, how to improve it without sacrificing any class. It extends influence functions to measure each training sample's effect on each class separately, producing an influence vector. From these vectors it derives a geometric criterion: a model is at its ceiling only when every training sample is beneficial to one class and harmful to another, so no combination of samples can help all classes at once. It then uses linear programming with a genetic-algorithm search to reweight training samples and achieve Pareto improvements. If correct, practitioners could know when further data reweighting is futile and how to target weak classes.

Core claim

The central claim is that the per-class influence vector of every training sample carries enough information to characterize the classifier's Pareto frontier over categories. In particular, a classifier has not reached its performance ceiling if any sample is jointly beneficial or jointly detrimental to all classes, or if samples lying in tradeoff regions can be combined into a jointly beneficial direction; the frontier is reached exactly when all influence vectors collapse to the line y = -x in the two-class case, meaning any improvement to one class necessarily hurts another. The paper proposes PARETO-LP-GA, which solves a linear program over sample weights to maximize influence on target

What carries the argument

The influence vector P(z) ∈ R^K, whose k-th coordinate is the influence of sample z on the validation loss of class k, computed via the standard influence-function formula I(z, S_k) = Σ_{v∈S_k} ∇ℓ(v)^T H^{-1} ∇ℓ(z). This vector converts the question of performance ceiling into a geometric question about the distribution of influence vectors, and provides the objective and constraints for the LP reweighting.

Load-bearing premise

The influence scores computed at the current epoch correctly predict how per-class accuracies will actually change when the training weights are changed and the model is trained for one more epoch.

What would settle it

On a deep network, compute influence vectors at an epoch, solve the LP, train the weighted next epoch, and compare predicted per-class accuracy changes to actual ones; if the Spearman correlation between predicted and actual changes is near zero or negative on a dataset like CIFAR-10, the proposed ceiling criterion and reweighting collapse.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the criterion holds, a user can check the influence vectors at any epoch and know whether reweighting can still produce a Pareto gain.
  • PARETO-LP-GA gives a concrete recipe to improve specified classes: solve the LP, train one epoch with the weights, and the target classes should improve while others degrade only slightly.
  • The course-correction setting means a practitioner can detect a 'bad epoch' where some classes dropped and reweight the next epoch to reverse that drop.
  • The same machinery provides a data-centric explanation of class tradeoffs during training, connecting sample selection to fairness and domain adaptation objectives.
  • The appendix shows iterative noise removal guided by category-wise influence pushes a linear model to its ceiling, suggesting a data-cleaning schedule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A practical convergence check for many classes would be to test whether the first principal component of the influence vectors explains nearly all variance; if so, the influence vectors lie on a one-dimensional tradeoff line, analogous to y = -x in two classes.
  • One can use the influence vectors to compute, at each epoch, an upper bound on achievable per-class accuracy improvement, effectively giving a speed limit for training a fair or balanced classifier.
  • If the influence estimates are stable across epochs, the LP weights might be reusable or transferable to a similar model trained on the same data, reducing the cost of repeated reweighting.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes category-wise influence functions that assign each training sample an influence vector in R^K by evaluating the classical influence function on each class-specific validation subset. It argues that these vectors reveal whether a multi-class classifier has reached a Pareto frontier across class accuracies, and introduces PARETO-LP-GA, a linear-programming plus genetic-algorithm reweighting method that claims to improve target classes while limiting degradation of others. The experimental sections validate category-wise influence on synthetic binary problems and on CIFAR-10, STL-10, Emotion, and AG_News, and report Pareto-style improvements for a ResNet-9 model on CIFAR-10 in two settings, Direct Improvement and Course Correction. The central conceptual claim is that the performance ceiling is characterized geometrically: for two classes, the frontier is reached when all influence vectors are 'close to the line y=-x'; for more classes the paper applies a first-PC explained-variance heuristic in Section 5.2.

Significance. If the proposed ceiling criterion were rigorously established, the paper would offer a practically relevant data-centric diagnostic: practitioners could tell whether reweighting training data can still improve all classes simultaneously, and then use the LP to obtain such a reweighting. The category-wise influence formulation itself is a natural and potentially useful extension of influence functions, and the empirical evidence in Section 5.1 that influence vectors correlate with class-wise performance changes (Spearman correlation > 0.8) is a real strength. The PARETO-LP-GA framework is concrete and the reported CIFAR-10 results show meaningful target-class gains with small non-target degradation. However, the paper's main theoretical contribution, the geometric ceiling criterion, is asserted rather than derived, and the PCA-based validation in Section 5.2 does not test the criterion. The paper also does not provide code or proofs, and it relies on influence-function estimates in deep non-convex models whose fragility is acknowledged in the related work but not mitigated in the reweighting algorithm.

major comments (4)
  1. [Sec. 3.3, Fig. 1] The central criterion 'all samples are close to the line y=-x' is asserted, not proven. The two-arrow example only shows that absence of joint-positive/joint-negative samples is not sufficient for the frontier, because a pair of tradeoff samples can be combined into a joint-positive one. It does not establish necessity or sufficiency of the line condition, nor does it characterize the frontier in terms of influence vectors. For K>2, the 'line y=-x' is undefined, so the criterion cannot be applied as stated to CIFAR-10, STL-10, AG_News, or Emotion. This is the load-bearing claim of the paper and needs a formal derivation or a precise high-dimensional definition with conditions.
  2. [Sec. 5.2, PCA check] The PCA argument is logically inverted. The paper states that a first-PC explained-variance ratio exceeding 0.2 shows that influence vectors 'do not fit a hyperplane and suggest room for Pareto improvement.' But a set of points lying on a (K-1)-dimensional affine hyperplane can have a very large first-PC ratio depending on the distribution along that hyperplane, and a one-dimensional line—which is the paper's own ceiling criterion for K=2—would give a ratio near 1. Thus the reported first-PC values do not provide evidence against the ceiling hypothesis. The real-data ceiling check in Section 5.2 is therefore not a valid test of the paper's own criterion.
  3. [Algorithm 1, Lines 4-7] The LP solves for weights using influence vectors as predictions of class-wise accuracy changes, yet the only validation of this premise is removal of the top 10% beneficial/detrimental samples in Section 5.1. Reweighting all samples by an optimized vector is a different operation, and the paper cites work (Bae et al. 2022, Basu et al. 2020, Epifano et al. 2023) that influence functions are fragile in deep non-convex models, but no mitigation is provided. The GA fitness uses the one-epoch measured deltas Δ_k to select α and w, so the reported improvements are selected on the same metric used to evaluate them; this circularity should be addressed, for example by reporting held-out validation performance or repeated runs with different random seeds.
  4. [Sec. 4, Fig. 2] The synthetic 'verification' is a sanity check of influence functions, not of the ceiling criterion. In the noisy dataset, mislabeled points are expected to have negative influence on both classes, and the paper shows that they do. In the non-linearly separable dataset, the paper observes that influence vectors lie 'roughly on a straight line' and treats this as confirming the Pareto frontier, but there is no independent ground-truth frontier against which the criterion is tested. To validate the ceiling criterion, the authors should construct a setting with a provably known Pareto frontier and check whether the line/PCA condition is both necessary and sufficient.
minor comments (6)
  1. [Eq. (1)] The notation H^{-1}_{hatθ} is used without defining the Hessian inverse as a matrix square or discussing regularization; for deep models, Hessian inversion is typically approximated, and this should be stated explicitly.
  2. [Sec. 3.3] The phrase 'close to the line y=-x' is not quantified. Is closeness Euclidean distance in influence-vector space, angular distance, or a projection threshold? Without a metric the criterion is untestable even for K=2.
  3. [Algorithm 1, Line 7] The fitness function notation '1[Δ≤0](-∞)' is informal; a large finite negative constant would be clearer, as would a description of how ties and identical fitness values are handled by the GA.
  4. [Appendix B.1] The heading mentions 'dataset augmentation' but the procedure only removes points; this is a misnomer and should be corrected.
  5. [Fig. 2 and Fig. 5] The captions reference 'SubfiguresABC' and 'SubfiguresAandD' without spaces and use 'blue and orange' while the figures use colors that may not be distinguishable in grayscale or for color-blind readers; please use textures or labels for accessibility.
  6. [Sec. 5.2] The claim that STL-10 is 'cleaner' than CIFAR-10 and therefore has less room for improvement is presented without quantitative support (e.g., per-class accuracies or influence-vector statistics).

Circularity Check

0 steps flagged

No circular reductions; the Pareto-ceiling criterion is unsupported and the PCA check is invalid, but these are correctness issues, not circularity.

full rationale

The derivation chain from Eq. (1) to the category-wise influence vector P_k(z)=I(z,S_k) is a direct application of standard influence functions; it does not smuggle the target conclusion into the definition. The Section 5.1 validation removes top beneficial/detrimental samples, retrains, and compares against actual per-class accuracy changes (Spearman > 0.8), which is an external check rather than a restatement of the fitted signal. In PARETO-LP-GA, the reported results are measured epoch-to-epoch accuracy changes; although the GA's fitness uses the same measured Delta for selection, the paper does not relabel that optimization objective as an independent prediction. The self-citations (Chhabra et al. 2024/2025, Xiao et al. 2024) appear only as related work and motivation, not as load-bearing uniqueness theorems or ansatz justifications. The 'all samples close to the line y=-x' condition in Section 3.3 is asserted rather than derived, is undefined for K>2, and the Section 5.2 PCA interpretation (ratio > 0.2 => 'do not fit a hyperplane') is logically inverted; however, an unsupported or even incorrect criterion is a soundness/rigor problem, not a circular reduction from the paper's own equations. No step meets the evidentiary bar for circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper depends on the standard influence-function approximation, an unproven Pareto-frontier condition, and a linear-transport assumption between influence scores and next-epoch accuracy. The two fitted objects (alpha thresholds and sample weights) are the method's own optimization variables. No new physical or external entities are introduced.

free parameters (2)
  • alpha_k class-wise performance thresholds = not reported (GA-searched)
    In Algorithm 1, alpha_k are initialized randomly and evolved via a genetic algorithm; the returned alpha* directly controls which LP weight solutions are admissible. These thresholds are fit on the same validation performance signal used to report the final gains.
  • per-sample weights w_i = returned as w*
    LP decision variables selected to maximize estimated target-class influence under alpha constraints. They are the actual intervention and are fit to the influence estimates, not independently verified.
axioms (4)
  • domain assumption Influence function approximation in Eq. (1) is valid: twice-differentiable loss, invertible Hessian, and small perturbation assumptions hold, even for deep non-convex models.
    Section 3.1 uses the standard Koh-Liang Taylor expansion. The paper cites work showing fragility in deep models (Bae et al. 2022, Basu et al. 2020, Epifano et al. 2023) but proceeds without mitigation.
  • ad hoc to paper If all influence vectors are close to the line y=-x, the classifier has reached its Pareto frontier.
    Section 3.3 states this as a 'new condition' but gives no formal definition of 'close', no proof, and no generalization to K>2 classes.
  • domain assumption Weighted sums of current-epoch influence scores predict class-wise accuracy changes at the next epoch.
    Algorithm 1 lines 4-6 solve the LP using P_k(z) and then assume the optimized weights will produce the estimated per-class improvements in a single training epoch.
  • ad hoc to paper First principal component explained variance exceeding 0.2 implies influence vectors do not fit a hyperplane and therefore the model has room for Pareto improvement.
    Section 5.2 uses this PCA statistic as evidence of room for improvement, but the statistic does not establish the claimed geometric or Pareto property.

pith-pipeline@v1.3.0-alltime-deepseek · 12652 in / 10291 out tokens · 87604 ms · 2026-08-04T11:32:25.920751+00:00 · methodology

0 comments
read the original abstract

Data-centric learning seeks to improve model performance from the perspective of data quality, and has been drawing increasing attention in the machine learning community. Among its key tools, influence functions provide a powerful framework to quantify the impact of individual training samples on model predictions, enabling practitioners to identify detrimental samples and retrain models on a cleaner dataset for improved performance. However, most existing work focuses on the question: "what data benefits the learning model?" In this paper, we take a step further and investigate a more fundamental question: "what is the performance ceiling of the learning model?" Unlike prior studies that primarily measure improvement through overall accuracy, we emphasize category-wise accuracy and aim for Pareto improvements, ensuring that every class benefits, rather than allowing tradeoffs where some classes improve at the expense of others. To address this challenge, we propose category-wise influence functions and introduce an influence vector that quantifies the impact of each training sample across all categories. Leveraging these influence vectors, we develop a principled criterion to determine whether a model can still be improved, and further design a linear programming-based sample reweighting framework to achieve Pareto performance improvements. Through extensive experiments on synthetic datasets, vision, and text benchmarks, we demonstrate the effectiveness of our approach in estimating and achieving a model's performance improvement across multiple categories of interest.

Figures

Figures reproduced from arXiv: 2510.03950 by Anshuman Chhabra, Hongfu Liu, Joshua Liu, Shahriar Kabir Nahin, Wenxiao Xiao.

Figure 1
Figure 1. Figure 1: Influence space for 2 categories. The answer to the above question is no, indicating that the current model can still be improved. At the individual sample level, if all samples may be located within tradeoff regions, involving or removing any sample will lead the tradeoff effect. However, when samples are considered collectively as a set, combining certain samples can yield a new sample that falls into th… view at source ↗
Figure 2
Figure 2. Figure 2: Validation of our category-wise influence function methods for analyzing the Pareto frontier on two synthetic [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Real-world data experiments on CIFAR10 [Krizhevsky et al., 2009] image dataset. Emotion [Saravia et al., 2018] and AG_News [Zhang et al., 2015]). For calculating category-wise sample influence vectors, we employ EKFAC [Grosse et al., 2023] due to its fast implementation on deep models. First, we evaluate whether category-wise influence serves as an effective indicator for measuring performance changes. To … view at source ↗
Figure 4
Figure 4. Figure 4: Real-world data experiments on Emotion [Saravia et al., 2018] text dataset [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment demonstrating the use of the category-wise influence function for dataset augmentation. The [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Real-world data experiments on STL-10 [Coates et al., 2011] image dataset [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Real-world data experiments on AG_News [Zhang et al., 2015] text dataset. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The numbers of beneficial and detrimental samples for each classes on [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

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