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REVIEW 3 major objections 5 minor 1 cited by

Observational Multiplicity

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A closed-form estimate shows which probability predictions are arbitrary, because the observed labels themselves are a random draw.

desk verdict Introduces a useful regret measure for probabilistic classification with a rigorous logistic-regression variance bound, but the estimator answers model-implied uncertainty rather than the true-label regret of Definition 2.1 unless the model is well-specified. read the letter →

arxiv 2507.23136 v1 pith:7OQ7GVAN submitted 2025-07-30 cs.LG

Erin George , Deanna Needell , Berk Ustun This is my paper
classification cs.LG
keywords observationalmultiplicitypredictiveregretprobabilisticclassificationmodeluncertaintyquantificationlogisticregressionsafemachinelearning
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

This paper argues that a familiar source of arbitrariness in probabilistic classification is the randomness of the observed labels: a dataset is one draw of the labels, and a different draw would produce an equally valid but different model. The authors call this observational multiplicity and propose to measure it per individual as regret, defined as the variance of a point's predicted probability when the training labels are resampled from the true conditional probabilities. For logistic regression they prove that this variance is, up to a relative error, the closed-form quantity $p_i^2(1-p_i)^2 x_i^\top H^{-1} x_i$, where $H$ is the Hessian of the logistic loss. If the claim holds, a practitioner can estimate which individuals receive arbitrary predictions without collecting new labels, and can abstain on those points or collect their labels first. The stakes are safety and fairness: the burden of arbitrariness is not shared equally, and it concentrates on identifiable subgroups.

What carries the argument

The load-bearing object is the regret identity $Q_i = p_i^2(1-p_i)^2 x_i^\top H^{-1} x_i$, with $H = \sum_j p_j(1-p_j)x_j x_j^\top$ the Hessian of the logistic loss at the fitted parameters. It expresses per-point instability as the product of label noise, $p_i^2(1-p_i)^2$, and leverage in the inverse-Hessian metric, $x_i^\top H^{-1} x_i$, so regret is large near $p_i=1/2$ and in directions where the data barely constrain the model. The proof machinery is a quantitative self-concordance bound on one Newton step for logistic regression, which controls the difference between the retrained parameters and a single linearized step, plus Bernstein concentration on the resampled-label gradient. Algorithm 1 supplies the estimator: resample labels from the fitted model, retrain, record the point's predicted probability, repeat, and take the variance.

What would settle it

Take a dataset whose labels are generated by a known nonlinear probability model, fit a logistic regression to it, and compute both true regret (resampling labels from the known generator, retraining, and taking variances) and estimated regret from Algorithm 1. If the two diverge systematically, and in particular if estimated regret underestimates true regret for points whose fitted probabilities are extreme, then the well-specification premise fails and Algorithm 1 measures the model's internal plausibility, not the label randomness the definition targets.

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Extended reading notes

Core claim

The central discovery is that arbitrariness in probabilistic classification can be defined as a per-point variance and, for logistic regression, estimated without retraining on fresh data. Regret is the variance of $f_{A(D')}(x_i)$ when $D'$ is formed by resampling each label from the true conditional probability $p_i^*$; since those probabilities are unknown, Algorithm 1 resamples from the fitted model and averages the resulting predictions. Theorem 3.1 gives that when the resampling model is the initial logistic model, the resulting variance obeys $|\operatorname{Var}(\hat p_i) - Q_i| \le \epsilon Q_i$ with $Q_i = p_i^2(1-p_i)^2 x_i^\top H^{-1} x_i$ and $\epsilon$ a small constant depending on dimension, feature norms, and the Hessian's smallest eigenvalue, provided $\epsilon<1$. The paper shows this matches true regret in semi-synthetic experiments and uses it to flag high-regret points for abstention and targeted data collection, including on a large mortgage-loan dataset where estimated regret follows a heavy-tailed distribution.

Load-bearing premise

The load-bearing premise is that the fitted model's probabilities are the true conditional label probabilities, so that resampling labels from the model mimics resampling labels from nature; when the model is misspecified, the estimated regret describes arbitrariness implied by the model, not by the data-generating process, and the paper reports exactly this kind of underestimation for high-regret points in the bank dataset.

Editorial extensions

If this is right

  • For logistic regression, per-point regret can be computed or bounded from a single dataset using $Q_i$, so no fresh label collection is needed to know which predictions are arbitrary.
  • Regret concentrates: points with predicted probability near $1/2$ and points lying in low-curvature directions of the Hessian carry most of the instability.
  • Estimated regret supports selective abstention: withholding predictions for high-regret points reduces average KL divergence to the true probabilities, close to the best possible in the bank dataset.
  • Regret-guided data collection outperforms uniform sampling in two of the three semi-synthetic datasets, reaching full-dataset accuracy after a few additional samples.
  • On a real mortgage-loan dataset, 98% of points have negligible regret while a small tail follows an approximate power law, so arbitrariness is a minority phenomenon that is identifiable.

Reading between the lines

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

  • Inference: the same $Q_i$ formula suggests a cheap diagnostic for feature engineering: adding features that raise the minimum eigenvalue of $H$ in a cluster's subspace should shrink regret there, which is testable.
  • Inference: regret as defined is a property of the pairing of a model class and a dataset, so comparing regret across model classes on the same data would separate label noise from model flexibility; the paper's gradient-boosting experiment on loan data is a step in that direction.
  • Inference: because the estimator resamples from the deployed model, a model that is confidently wrong will report low regret; calibration audits should run alongside regret estimates before using abstention in high-stakes decisions.
  • Inference: the power-law tail found on the loan dataset, if it persists, implies that fairness audits should report regret distributions per group rather than aggregate accuracy, since group-level means can hide a small set of arbitrarily predicted individuals.
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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 / 5 minor

Summary. Across probabilistic classification, the paper argues that a single observed label vector is one draw from a stochastic process, so different draws produce different plausible training sets and different models. It defines observational multiplicity as the resulting arbitrariness and formalizes a per-example regret (Definition 2.1) as the variance of the prediction f_A(D')(x_i) when D' is created by resampling labels from the true conditional probabilities. Algorithm 1 estimates this variance by resampling labels from a fitted model, and Theorem 3.1 gives a finite-sample bound for logistic regression: if epsilon < 1, the variance of the resampled prediction is within a factor (1 +/- epsilon) of Q_i = p_i^2(1-p_i)^2 x_i^T H^{-1} x_i. The paper validates the estimator on three semi-synthetic datasets, applies it to 13 million Fannie Mae loans with a GBDT, and demonstrates abstention and active-learning uses.

Significance. The paper tackles a timely and important question: when multiple equally performant models disagree on an individual, how much of that disagreement is irreducible randomness in the observation process? The proof of Theorem 3.1 is a genuine technical contribution, applying Bach's self-concordance bounds in a nontrivial way and producing an explicit, actionable formula. The semi-synthetic study is careful, and the Fannie Mae experiment shows the framework scales. The significance is conditional, however, on the identification of the model-resampled variance with the true-probability variance of Definition 2.1; the paper itself acknowledges the proxy, and the experimental evidence does not close the gap under misspecification. With a redefinition or a transfer theorem, the work would be a solid contribution to model multiplicity and uncertainty quantification.

major comments (3)
  1. [2.3, Algorithm 1, Definition 2.1] Definition 2.1 defines regret as variance under resampling from the true conditional probabilities of the labels, but Algorithm 1 resamples from the fitted model's own probabilities and Theorem 3.1 analyzes exactly that model-resampled process (labels ~ Bern(p_i) from an initial logistic fit). Section 2.3 concedes that the proxy is 'exactly the model we were attempting to learn in the first place.' These are different stochastic mechanisms unless the fitted model coincides with the true conditional probability function; under misspecification, Algorithm 1 estimates a property of the deployed model rather than the quantity in Definition 2.1. The underestimation of high-regret points in the bank data is consistent with this divergence. The paper should either prove a bound that transfers from model-resampled variance to true regret under well-specification, or reframe regret and the safety applications as explicitly model-implied rather than as estimates of the true observational multiplicity.
  2. [Theorem 3.1 and Section 4] The central theorem applies only when epsilon < 1, but the paper never reports whether this condition holds in any of its experiments, and the condition depends on d, Xmax/Xmin, ||theta*||_2, and the smallest Hessian eigenvalue lambda_min, so it is most likely to fail in exactly the ill-conditioned or high-dimensional settings where regret matters. In the loan experiment the model is a GBDT, which is outside the theorem's scope entirely. Please state how a practitioner can verify epsilon < 1 and provide at least one experiment, synthetic or real, where the condition and the bound are actually checked; otherwise the empirical support for the theorem's regime is missing.
  3. [4.1 and 4.2] The semi-synthetic validation in Section 4.1 generates ground-truth labels from a logistic model whose parameters are themselves fit to the original data, so the experiments are confined to the well-specified logistic family. The Fannie Mae experiment is fully real and therefore lacks any ground-truth regret to compare against, and the GBDT estimator has no supporting theory. There is no experiment that quantifies the gap between true regret (Definition 2.1) and Algorithm 1's output under a deliberately misspecified model (e.g., nonlinear label-generating function with a logistic estimator). Such an experiment would directly address the main weakness identified in Major Comment 1 and would determine whether the reported bank underestimation is an isolated finite-sample artifact or a systematic feature.
minor comments (5)
  1. [Abstract] The abstract contains several grammatical errors, including 'can can undermine', 'as a result of different training labels change', and 'estimating regret promote safety'; these should be corrected.
  2. [5.3] Lemma 5.5 states Pr(B_c) <= 2d exp(...), but the proof bounds the complement B_c^c; the event/complement notation is also inconsistent in the lower-bound part of Section 5.4, which uses 'B_c' for both the event and its complement.
  3. [Algorithm 1] Algorithm 1 does not list k (the number of resamples) as an input; the experiments use k = 300, but the pseudocode should state k explicitly.
  4. [4.2] The claim that the tail of the regret distribution 'appears to follow a power law' is supported only by an eyeballed line y = 3.5e-10 x^{-3}; a fitted exponent and goodness-of-fit measure would make the claim precise.
  5. [Figure 1] The caption 'The five points at the top and five points at the bottom all have an 80% probability of lying in the positive and negative class' is ambiguous about which points are positive and which are negative; please clarify.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the finite-sample bound is a genuine derivation, and the model-based resampling is an acknowledged approximation, not a hidden identity.

full rationale

The paper's derivation chain is not circular. Definition 2.1 defines regret as prediction variance under labels resampled from the true conditional probabilities. Algorithm 1 instead resamples labels from the fitted model's probabilities, and Section 2.3 explicitly flags this: "This approximation, of course, is exactly the model we were attempting to learn in the first place." The paper then reframes the output as uncertainty "directly implied by the model we are seeking to deploy." This is an honest, disclosed assumption rather than a disguised reduction. Theorem 3.1 is a substantive finite-sample bound for the model-resampled variance, proved from Bach's Proposition 2 and concentration arguments, not from the conclusion; it is not the variance formula written as an input. The semi-synthetic experiments validate the estimator against known ground-truth labels, providing independent evidence, and the bank dataset's reported underestimation of high-regret points is a misspecification symptom consistent with the acknowledged proxy gap. The only self-citations (e.g., references [10], [13], and [19]) appear in related-work or motivation and are not load-bearing. Therefore no claim reduces by construction to its own inputs; the central result has independent mathematical content.

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

The central quantity rests on an estimated model standing in for the unknown data-generating process: theta* and the resampling design are inputs, not derived. Validation further fits the true logistic parameters to the same data, so no independent ground truth is available for the real-world GBDT case.

free parameters (4)
  • initial logistic regression parameters theta* = not reported
    Q_i and epsilon in Theorem 3.1 depend on theta* fitted from the observed labels, and Algorithm 1 resamples labels from sigma(x_i^T theta*).
  • number of label resamples k = 300
    Algorithm 1 variance estimates use 300 resamples; no sensitivity analysis is reported.
  • ground-truth logistic parameters for semi-synthetic data = fit by regularized logistic regression
    Validation computes 'true' regret from parameters fit to the original labels, so the ground truth is not independent of the data.
  • GBDT hyperparameters for Fannie Mae experiment = not specified
    The real-world regret estimates use a gradient boosting decision tree classifier, but hyperparameters and preprocessing details are omitted.
assumptions (5)
  • domain assumption Labels are independent Bernoulli draws from true conditional probabilities p_i^*
    Section 2 assumes inherently stochastic events and defines regret by resampling labels as Bern(g(x_i)).
  • ad hoc to paper Fitted logistic model is well specified, so resampling from f_theta*(x_i) approximates resampling from true probabilities
    Algorithm 1 and Theorem 3.1 resample from the initial fitted model; without well-specification, estimated regret is not the Definition 2.1 quantity.
  • standard math Bach's Proposition 2 (self-concordance bound) is correct and applicable
    Section 5.3 restates and relies on Theorem 5.3 from Bach [1] for the Newton step bounds.
  • domain assumption Feature matrix is full rank and not linearly separable, so the Hessian is positive definite
    Section 5.1 imposes this condition so ERM has a unique minimum and H is invertible.
  • domain assumption Compact support and label probabilities bounded away from 0 and 1
    Corollary 3.2 requires compact support K subset (0,1) for its high-probability asymptotic guarantee.

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Cite this review

Pith. "Pith review of Observational Multiplicity." pith.science (2026). https://pith.science/paper/7OQ7GVAN

@misc{pith2026250723136,
  author       = {Pith},
  title        = {Pith review of: Observational Multiplicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7OQ7GVAN}},
  note         = {Machine review of arXiv:2507.23136}
}
abstract

Many prediction tasks can admit multiple models that can perform almost equally well. This phenomenon can can undermine interpretability and safety when competing models assign conflicting predictions to individuals. In this work, we study how arbitrariness can arise in probabilistic classification tasks as a result of an effect that we call \emph{observational multiplicity}. We discuss how this effect arises in a broad class of practical applications where we learn a classifier to predict probabilities $p_i \in [0,1]$ but are given a dataset of observations $y_i \in \{0,1\}$. We propose to evaluate the arbitrariness of individual probability predictions through the lens of \emph{regret}. We introduce a measure of regret for probabilistic classification tasks, which measures how the predictions of a model could change as a result of different training labels change. We present a general-purpose method to estimate the regret in a probabilistic classification task. We use our measure to show that regret is higher for certain groups in the dataset and discuss potential applications of regret. We demonstrate how estimating regret promote safety in real-world applications by abstention and data collection.

Figures

Figures reproduced from arXiv: 2507.23136 by the authors.

Figure 1
Figure 1. Three different linear classifiers are obtained from different draws of the labels in a same dataset. The points are plotted in the two-dimensional feature space. The five points at the top and five points at the bottom all have an 80% probability of lying in the positive and negative class, respectively. There are four points that each lie on different sides of the decision boundary across the different draws. The … view at source ↗
Figure 2
Figure 2. Theorized versus actual regret in semi-synthetic datasets (from left to right: bc, bank, apnea). Each blue dot represents one point in the dataset and the black dashed line is the line y = x, representing perfect correspondence. A full proof of this corollary is in Section 5.6. Theorem 3.1 and Corollary 3.2 apply both to estimated regret determined according to Algorithm 1 and the true regret we would obtain in a we… view at source ↗
Figure 3
Figure 3. Predicted regret versus actual regret in semi-synthetic datasets (from left to right: bc, bank, apnea). Each blue dot is the median of predicted regret for a point in the dataset over 100 trials, while the dark blue and light blue lines represent the interquartile and full ranges of regrets, respectively. The black dashed line is the line y = x, representing perfect prediction. 0.000 0.005 0.010 0.015 Regret (true) … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Theorized versus actual regret in the semi-synthetic dataset bank. The data is split into four groups. Group 1, 2, and 3 are determined by the intersection of specific categorical features. Group 0 is the reset of the dataset. estimates are good on average over the 100…
Figure 5
Figure 5. Figure 5: Mean error versus coverage in selective prediction on semi-synthetic datasets (from left to right: bc, bank, apnea). The median and interquartile range is shown and the selective prediction set is determined using true regret, estimated regret, and the lowest error. Th…
Figure 6
Figure 6. Figure 6: Excess error in active learning learning on semi-syntehtic datasets (from left to right: bc, bank, apnea). The additional collections are shown when selected according to true regret, estimated regret, and uniformly at randomly. The median and interquartile range of ex…
Figure 7
Figure 7. Figure 7: The distribution of the estimated regret of each point in loan plotted on a log scale (left) and a log-log scale (right). except for date and location derived features. In total, there are 13,026,081 points in the dataset and 22 features. We take a random subset of 75%…
Figure 8
Figure 8. Figure 8: The distribution of the estimated regret for each point in loan on a log scale, separated by observed training label. 5.1 Preliminaries We consider a logistic regression task in which feature matrix is full rank and not linearly separable. Here “not linear separable” m…

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Cited by 1 Pith paper

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  1. Using predictive multiplicity to measure individual performance within the AI Act

    cs.LG 2026-02 conditional novelty 5.0 of 10

    Providers of high-risk AI should report how often equally accurate models disagree on a person's case, using the proposed conflict ratio and δ-ambiguity metrics.

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