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Measuring consistency via ensemble margin and local prediction variability: Auditing decision systems in the presence of predictive multiplicity

Sinjini Banerjee, Tim Marrinan, Anand D. Sarwate

arXiv:2609.01397v1stat.MLcs.AIcs.LG

TL;DR

Predictive multiplicity complicates auditing because different near-optimal models can produce unstable ensemble predictions and existing consistency measures can miss inconsistent or jointly wrong predictions. This paper combines ensemble margin with local prediction variability, proves convergence from finite ensembles to the full Rashomon set under mild assumptions, and finds that modest ensembles reduce unchecked errors while moderately increasing diversions.

  • Problem

    Existing work insufficiently characterizes how predictive multiplicity affects auditing, while existing consistency measures can miss inconsistent instances and cannot ensure correctness.

  • Method

    The paper combines ensemble margin with local prediction variability and analyzes finite ensembles sampled from the Rashomon set.

  • Results

    Finite ensembles reduce erroneous predictions escaping human review, increasingly approximate full-Rashomon-set audit decisions, and the proposed measures correlate more strongly with multiplicity metrics than existing measures.

  • Takeaways & Limitations

    Modest ensembles can provide more stable auditing decisions and better capture predictive multiplicity than auditing a single model.

  • Takeaways & Limitations

    The theoretical and experimental framework restricts ensembles to uniformly averaged model predictions; alternative weighted or input-dependent ensembles require separate analysis.

Abstract

from arXiv · show

The Rashomon effect is a machine learning phenomenon where equally accurate models produce different predictions for the same inputs (predictive multiplicity). Existing work primarily focuses on multiplicity within individual models, but in more complex decision systems, the impact of the Rashomon effect is less well understood. In this work, we study multiplicity from the perspective of auditing incorrect ensemble predictions, where the decision to divert an instance for human review is based on a consistency criterion that combines the ensemble margin with a measure of local prediction variability for each constituent model. With mild assumptions about stability and smoothness, we show that the consistency scores of finite ensembles converge to the corresponding consistency score of the expected model from the Rashomon set as the ensemble size and the number of samples used to measure local prediction variability increase. To demonstrate the efficacy of the proposed criterion, we evaluate the framework with respect to transformer models applied to natural language understanding tasks and parameter-efficient fine-tuning of large language models used for tabular data classification tasks. Our experiments show that ensembling models from the Rashomon set substantially reduces the risk of incorrect predictions going unchecked compared with auditing a single model, while incurring only a moderate increase in the number of diversions. Moreover, the auditing behavior of the full Rashomon set can be closely approximated by finite ensembles of relatively modest size, with the risk approaching zero for some datasets. We further demonstrate that the proposed measure exhibits stronger agreement with established predictive multiplicity metrics than existing consistency measures, providing a more reliable way to capture multiplicity in the Rashomon set.

1 Introduction

Modern model marketplaces make deployment easy, but the Rashomon effect means equally performing models can disagree, creating predictive multiplicity and auditing challenges. This paper proposes consistency-based diversion using finite ensembles, combining ensemble margins with local prediction stability and providing guarantees under mild assumptions.

  • Motivation: Model marketplaces expose many high-performing pretrained models, making “best” model selection appear straightforward while overlooking predictive multiplicity.The Rashomon effect describes equally well-performing models producing different predictions on held-out instances.
  • Motivation: Predictive multiplicity can produce inconsistent diagnoses, credit approvals, and risk assessments, undermining fairness and accountability in high-stakes domains.The paper links disagreement among competing models to deployment concerns in healthcare, finance, and criminal justice.
  • Auditing and diversion: Auditing systems must balance missed diversions of incorrect predictions against the expense and feasibility constraints of sending cases to human experts.Ground-truth labels are typically unavailable in practice, so auditing relies on proxy measures derived from model behavior.
  • Proposed framework: The proposed (β, σ)-consistency measure combines ensemble margin information with local perturbation stability in an input’s embedding space.Unlike single-model multiplicity measures, sampling models helps characterize how a finite retail ensemble approximates the expected warehouse model.
  • Theoretical guarantees: Under stability and local Lipschitz assumptions, finite retail ensembles assign high consistency with probability increasing exponentially in ensemble size M and perturbation samples S.The result connects consistency under the expected model to consistency estimates from finite ensembles.
  • Limitations of prior measures: Existing consistency measures can mistake inconsistent instances for consistent ones, and agreement among models does not imply correctness, causing missed diversions.The paper therefore distinguishes prediction stability from correctness when designing auditing criteria.

2 Problem Setup

The setup models binary classifiers, defines an empirical Rashomon set of similarly performing models, and treats finite ensembles sampled from it as random predictors whose instance-level variability must be audited.

  • Statistical setting: Binary classification uses feature vectors in R^d, labels in {-1,+1}, i.i.d. training and test data, and probabilistic model outputs with associated hard decisions.
  • Rashomon set: Randomized training can generate different model functions, motivating analysis of a Rashomon set rather than a unique trained predictor.
  • Rashomon set: The empirical Rashomon set contains models whose empirical error is within tolerance ϵ of a reference model’s error, although their individual predictions may differ.
  • Finite ensembles: Repeated training and rejection sampling produce conditionally i.i.d. model samples from a distribution τ over the Rashomon set, called a retail set.
  • Finite ensembles: A retail ensemble averages constituent model outputs, but different finite samples can yield different predictions at the same test point.

3 Proposed consistency measure

The proposed consistency measure combines ensemble margin with local prediction variability to identify instances whose predictions may change across Rashomon-set ensembles. Under stability and local smoothness assumptions, finite ensembles increasingly recover the expected model’s consistent instances as ensemble and perturbation sample sizes grow.

  • Local variability: Local perturbations of x probe ensemble behavior near the test instance, using input noise or embedding perturbations depending on the model setting.
  • Measure definition: The (β, σ)-consistency measure targets stability across independently sampled retail sets by combining distance from the decision boundary with local prediction variability.
  • Assumptions: The guarantees assume Rashomon-set stability at x, local Lipschitz continuity, and consistency of the expected model under perturbations.
  • Guarantee: Theorem 1 gives a probability increasing exponentially in ensemble size M and perturbation count S that a consistent expected-model instance remains (β − ϵ1, σ)-consistent for a retail ensemble.
  • Guarantee: For instances outside Dcons_test, the theorem is inconclusive because local variability may be large, the margin may be near the decision boundary, or both.
  • Measure definition: Unlike hard-label agreement or single-model confidence, the margin-based formulation preserves soft information and evaluates whether predictions remain separated from the decision boundary.
  • Empirical interpretation: Combining the measure with ensembling reduces predictive multiplicity, audit-decision variability, and the risk that incorrect predictions go unchecked.
  • Empirical interpretation: Across 500 sampled retail ensembles, recovery of consistent instances is near one even for small M and becomes more concentrated near one as M increases.

4 Experimental Setup

The experiments test whether the proposed consistency measure supports auditing, whether finite retail ensembles approximate Rashomon-warehouse decisions, and whether the measure captures predictive multiplicity. Results show that increasing ensemble size reduces audit risk, improves agreement with warehouse outcomes, and strengthens correspondence with multiplicity metrics.

  • Experimental questions: The experiments evaluate the proposed consistency measure as an auditing criterion, emphasizing trade-offs between audit risk and unnecessary diversions.The goal is not perfect identification of incorrect predictions, because consistency alone does not imply correctness.
  • Threshold analysis: Increasing the threshold β substantially reduces audit risk before sharply increasing the diversion fraction; from −0.5 to 0.25, audit risk falls from approximately 0.174 to almost 0.0.The threshold controls how conservative the auditing system becomes.
  • Finite-ensemble approximation: For retail ensembles, audit risk decreases toward warehouse values as M increases, while audit cost increases toward the warehouse audit cost ratio.At M = 1 and M = 3, mean audit risk ratios are approximately 0.4 and 0.25, compared with around 0.15 for the warehouse ensemble.
  • Finite-ensemble approximation: Audit risk fraction decreases rapidly with ensemble size, with most improvement between M = 3 and M = 10, after which the curves largely stabilize.For M = 1, the audit risk fraction is close to 0.07, whereas the warehouse value is close to 0.
  • Multiplicity analysis: The proposed consistency measure and lmcσ achieve the highest correlations with multiplicity metrics across ensemble sizes, while combining margin and local variability outperforms either component alone.The proposed measure also shows stronger and more stable correlations than hcσ, whose performance deteriorates substantially on SST2.

5 Conclusion

The paper presents a consistency-based auditing framework for predictive multiplicity and supports it with theoretical convergence guarantees and empirical findings. It also identifies extensions to weighted ensembles, mixture-of-experts, regression, and generative foundation models.

  • 5 Conclusion: The proposed consistency measure combines ensemble margin with local prediction variability to identify predictions stable across competing models.Finite-ensemble estimates converge to the full Rashomon-set consistency score as ensemble size and perturbation samples increase, under mild assumptions.
  • 5 Conclusion: Finite ensembles trade audit cost and audit risk: larger ensembles stabilize decisions and increasingly approximate the full Rashomon set.Single models are cheaper but show greater auditing-decision variability, while individual models allow more erroneous predictions to escape review and produce fewer unnecessary diversions than full-Rashomon-set ensembles.
  • 5 Conclusion: Even relatively small ensembles correlate more strongly with established predictive multiplicity metrics than single models.The experiments therefore find that aggregating independently trained models provides a more faithful characterization of multiplicity.
  • 5 Conclusion: A margin-based consistency measure penalized by local prediction variability outperforms confidence-based approaches across most experimental settings.
  • 5 Conclusion: The theoretical analysis is limited to uniformly weighted ensembles, while the empirical and framework scope focuses on probabilistic classification.Future work includes extending guarantees to weighted ensembles and mixture-of-experts architectures, and extending the framework to regression and generative foundation models.

A.1 Notation Summary

This section recaps the notation used in the paper’s proofs in a table.

  • A.1 Notation Summary: Table 3 summarizes the notation used throughout the proofs.

A.2 Measure concentration tools

This section states McDiarmid’s and Hoeffding’s inequalities in the forms used to support the paper’s concentration arguments.

  • A.2 Measure concentration tools: The paper cites Ledoux (2001) for the proof of Hoeffding’s inequality while restating both tools for self-containment.
  • A.2 Measure concentration tools: McDiarmid’s inequality bounds deviations of a function of independent random variables when changing one input has a bounded effect.The stated form bounds the probability that the function differs from its expectation by at least t.
  • A.2 Measure concentration tools: Hoeffding’s inequality applies to independent random variables bounded almost surely within an interval [a, b].It is stated for any ε > 0 and provides a concentration bound for their aggregate behavior.

A.3 Probabilistic guarantee on our proposed consistency measure

The appendix analyzes concentration of the proposed consistency quantities for randomly sampled ensembles and perturbations under stability, consistency, and local smoothness assumptions.

  • A.3 Probabilistic guarantee on our proposed consistency measure: The Rashomon consistency assumptions require i.i.d. ensemble sampling, η-local Lipschitz models, and test instances satisfying (β, σ)-consistency and ε2-stability.These assumptions define the subset Dcons of test instances analyzed by the guarantees.
  • A.3 Probabilistic guarantee on our proposed consistency measure: The ensemble model is defined for a sampled Rashomon-set ensemble, and its concentration is analyzed using McDiarmid’s inequality.The proof verifies bounded differences by replacing one sampled model and using ε2-stability at the instance.
  • A.3 Probabilistic guarantee on our proposed consistency measure: The analysis studies concentration over both randomly sampled ensemble models and uniformly sampled perturbations.The ensemble models are drawn i.i.d. from the Rashomon set, while perturbations are sampled uniformly from the local perturbation region.
  • A.3 Probabilistic guarantee on our proposed consistency measure: Hoeffding’s inequality bounds empirical variability terms because the model differences over perturbations are bounded by ησ.The deviation analysis decomposes the empirical variability gap and separately controls its component terms.

A.3.1 Proof of main results

Theorem 1 establishes a probabilistic consistency guarantee: under the stated assumptions, a finite ensemble’s consistency score does not fall far below the Rashomon-set score. The proof decomposes this deviation into ensemble-margin and local-variability terms and bounds them using concentration arguments.

  • Theorem 1 bounds the probability that a finite ensemble’s consistency score falls significantly below the corresponding Rashomon-set consistency score.
  • Because the Rashomon-set score is at least β, controlling the deviation by ϵ1 yields a finite-ensemble score of at least β − ϵ1.
  • The guarantee analyzes random perturbations V sampled around x and an ensemble prediction formed from the sampled models.
  • The proof measures the score gap as the absolute difference between finite-ensemble and Rashomon-set consistency scores.
  • The deviation is decomposed into a margin difference and a term capturing prediction variation under perturbations.

A.4 Local consistency

This section defines local consistency by combining expected perturbed predictions with a penalty for local prediction variability. It then states assumptions and a concentration guarantee showing that finite ensembles approximate the empirical Rashomon-set measure.

  • Definitions: Local consistency combines the expected prediction under perturbations with a penalty for expected prediction variability around x.
  • Definitions: A finite retail set receives a local-consistency score based on its ensemble prediction over perturbed samples.
  • Assumptions: The guarantee assumes i.i.d. model sampling, local Lipschitz continuity, local consistency, and local stability at the test point.
  • Guarantee: The concentration result is stated for a fixed test point in the locally consistent subset.
  • Guarantee: Theorem 2 bounds the probability that a finite retail ensemble’s local-consistency score deviates substantially below the empirical Rashomon-set score.

A.4.1 Probabilistic guarantee on local ensemble consistency

The probabilistic analysis separates variability caused by perturbation sampling from variability caused by sampling models into the retail ensemble. It then uses bounded-difference concentration arguments to control the resulting deviations.

  • The proof targets concentration of the local margin term before establishing the local-consistency guarantee.
  • The deviation probability is bounded by requiring at least one component gap to exceed half the target threshold.
  • Two randomness sources are analyzed separately: perturbation samples V and the sampled retail set R.
  • The gap is decomposed into components associated with perturbation variability and model-sampling variability.
  • McDiarmid’s inequality is applied to ensemble predictions and expected local predictions after verifying bounded differences.
  • The bounded-difference arguments use local Lipschitzness and local stability when individual perturbations or models are replaced.

A.4.2 Proof of Theorem 2

The proof of Theorem 2 bounds the deviation between finite-ensemble and empirical-Rashomon local consistency. It uses the fact that the empirical score is at least β and controls the deviation through separate margin and variability terms.

  • Theorem 2 concerns the deviation between finite-ensemble and empirical-Rashomon local-consistency scores.
  • For locally consistent test points, the empirical-Rashomon score satisfies lmcσ(R, x) ≥ β.
  • Bounding the deviation by ϵ1 yields the finite-ensemble guarantee cσ(R, V, x) ≥ β − ϵ1.
  • The proof rearranges the deviation into a perturbed-sample margin term and remaining variability terms.
  • The argument uses a triangle-inequality decomposition in which at least one of the margin or variability deviations must exceed ϵ1 when the total deviation does.

A.5 Multiplicity metrics

This section distinguishes predictive multiplicity metrics that measure disagreement at an instance from those that examine disagreement across an input neighborhood. It also introduces local discrepancy as a neighborhood-based metric.

  • Predictive multiplicity captures disagreement among models in the Rashomon set on an individual test instance.
  • Pairwise disagreement measures average disagreement between pairs of models in the empirical Rashomon set.
  • Discrepancy measures the fraction of model predictions conflicting with a reference model at a test instance.
  • Local discrepancy averages conflicting predictions across perturbed samples in the neighborhood of an input.
  • Prediction variance and prediction range quantify multiplicity through predicted probabilities rather than hard predicted labels.

A.6 Related measures

The paper defines consistency measures that combine ensemble margins with local behavior, then evaluates how they support auditing and capture Rashomon-set multiplicity. Larger ensembles increasingly reproduce warehouse auditing behavior, while the proposed measures correlate most strongly with multiplicity metrics.

  • A.6 Related measures: The proposed consistency measure combines ensemble margin information with local perturbation stability in the input embedding space.
  • A.7.3 Understanding retail and warehouse audit disagreements: Increasing the audit threshold β makes the system more conservative and substantially reduces audit risk across datasets and model architectures.
  • A.7.3 Understanding retail and warehouse audit disagreements: Finite retail ensembles show convergence toward warehouse audit metrics as ensemble size M increases across experimental settings.
  • A.7.3 Understanding retail and warehouse audit disagreements: Aggregate audit metrics quantify overall performance but do not reveal why retail and warehouse audit decisions differ.
  • A.7.3 Understanding retail and warehouse audit disagreements: Without ensembling, more than 16.6% of instances are in Cost decreases versus 4.2% in Cost increases, while close to 5% are in Risk increases.
  • A.7.3 Understanding retail and warehouse audit disagreements: 90.9% of test instances fall in the Baseline category at M = 30, compared with 73.1% at M = 1, as retail audit outcomes increasingly match warehouse outcomes.
  • A.7.5 Additional correlation tables from Section 4.3: The proposed consistency measures achieve the strongest agreement with warehouse multiplicity metrics, with correlations becoming stronger as ensemble size increases.
  • A.7.5 Additional correlation tables from Section 4.3: Combining ensemble margin with local prediction variability outperforms using either margin or local variability alone for characterizing predictive multiplicity.
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