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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

Auditing ensemble predictions is challenging when similarly accurate models disagree, a setting less understood than multiplicity within individual models. The paper proposes an audit criterion combining ensemble margin with local prediction variability and finds that finite Rashomon ensembles stabilize auditing and reduce unchecked errors relative to single-model auditing.

  • Problem

    How predictive multiplicity affects auditing in complex decision systems is less understood, despite ensemble predictions varying across Rashomon-set samples.

    Source: §2.3 · p. 8

  • Method

    The paper combines ensemble margin with local prediction variability in a consistency measure and proves finite-ensemble estimates converge to full-Rashomon-set scores under mild assumptions.

    Source: §3.1 · p. 9, §5 · p. 21

  • Results

    Finite ensembles make auditing decisions increasingly stable and approximate full-Rashomon behavior, while trading higher audit cost for lower audit risk than single-model auditing.

    Source: §5 · p. 21

  • Takeaways & Limitations

    Finite ensembles offer a practical approximation to full-Rashomon auditing, whose set is infeasible to enumerate.

    Source: §4 · p. 12

    The study analyzes uniformly averaged model predictions; alternative ensemble strategies require separate theoretical analysis.

    Source: §4 · p. 12

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

The introduction frames predictive multiplicity as a reliability problem: similarly performing models can disagree on individual cases, and finite ensembles may not eliminate that instability. The paper proposes auditing ensemble predictions using consistency measures that account for both ensemble margin and local prediction variability.

1 Introduction

  • The Rashomon effect means equally well-performing models can predict differently on held-out cases, creating unstable decisions in high-stakes applications and motivating tools to flag uncertain predictions.The paper focuses on missed diversions of erroneous predictions and argues that ensembles can improve decision reliability.Source: §1 · pp. 1–2

1.1 Auditing and diversion to human experts

  • Auditing must trade unnecessary expert referrals (audit cost) against missed erroneous predictions (audit risk), while candidate-system decisions should approximate baseline diversions.The baseline is an idealized gold-standard model, although cost or resource constraints may require deploying a cheaper candidate.Source: §1.1 · pp. 2–3

1.2 Our contributions

  • Under stability and local Lipschitz assumptions, high baseline (β, σ)-consistency implies finite ensembles assign high scores with probability increasing exponentially in ensemble size M and perturbation samples S.The proposed measure combines ensemble margin information with local perturbation stability in the input embedding space.Source: §1.2 · pp. 3–4

1.3 Related work

  • Prior consistency analyses examine model agreement, prediction variance, perturbations, and local stability, but consistency alone does not ensure correctness and can leave errors undiverted.Models may agree and still be jointly wrong, a limitation with particular risk in high-stakes settings.Source: §1.3 · pp. 5–6
  • Small Rashomon-set ensembles retain Monte Carlo error, so independently sampled ensembles can produce unstable labels near decision boundaries despite asymptotic convergence.Practical cost, latency, or licensing constraints can limit ensemble size, motivating instance-specific measures of prediction stability.Source: §1.3 · p. 6

2 Problem Setup

The setup formalizes binary prediction, defines near-best models relative to a reference, and treats finite samples from that set as random ensembles whose differing predictions motivate consistency auditing.

2.1 Binary classification

  • The framework considers binary classification with i.i.d. training examples from an unknown distribution and randomized training that produces probabilistic models with associated hard decisions.The model output is interpreted as the predicted probability of the positive class.Source: §2.1 · p. 6

2.2 The empirical Rashomon set and predictive multiplicity

  • The empirical Rashomon set contains models whose test error is within ε of a reference model, yet they may disagree on individual instances, a phenomenon called predictive multiplicity.The error tolerance ε determines which models count as similarly performing and is application dependent.Source: §2.2 · p. 7

2.3 Ensembles using the Rashomon set

  • Finite ensembles average outputs of conditionally i.i.d. models sampled from the empirical Rashomon set; their predictions can vary across realizations at the same test point.The paper uses this randomness as the core difficulty its consistency measure addresses.Source: §2.3 · pp. 7–8

3 Proposed consistency measure

The proposed consistency measure combines an ensemble’s margin from the decision boundary with constituent models’ prediction variability near the test instance. Under stability and local smoothness assumptions, finite ensembles increasingly recover instances consistent under the expected Rashomon-set model as ensemble size and perturbation samples grow.

3.1 Measuring ensemble consistency

  • Local behavior is assessed by perturbing each test input and evaluating ensemble predictions, using raw-input noise or transformer-embedding perturbations as appropriate.The proposed (β, σ)-consistency characterizes whether the expected model’s margin remains large after accounting for local variability.Source: §3 · p. 8, §3.1 · pp. 8–9
  • Under stability, local Lipschitz continuity, and expected-model consistency, a sampled retail ensemble is consistent with high probability increasing exponentially in ensemble size M and perturbation count S.The resulting score is allowed a margin reduction of ϵ1; outside the assumed consistent-instance set, the theorem provides no guarantee.Source: §3.1 · p. 10

3.2 Interpretation of the consistency measure

  • The measure combines the ensemble prediction’s margin gap from the decision boundary with a penalty for average constituent-model prediction variability between x and nearby perturbed inputs.The margin captures distance from the decision boundary, while local variability tests whether agreement at x persists nearby.Source: §3.2 · p. 10
  • Across 500 sampled retail ensembles, the fraction of expected-model-consistent instances recovered concentrates near one as ensemble size increases.Most sampled ensembles recover nearly all such instances, even at small sizes.Source: Figure 1 · p. 11, §3.2 · p. 11

4 Experimental Setup

Across language and tabular classification, the experiments assess consistency-threshold auditing, finite ensembles as approximations to Rashomon-warehouse behavior, and consistency measures against warehouse multiplicity metrics. They also describe how model sets are constructed and sampled.

4.1.3 Computing retail set consistency

  • Retail ensembles use M = [1, 3, 5, 10, 30]; scores are estimated with cached libraries of 200 perturbations and 500 resamples, using uniform prediction averaging rather than weighted or input-dependent ensembles.The latter ensemble strategies change the estimator and require separate theoretical analysis.Source: §4 · p. 12, §4.1.3 · p. 14

4.1 Scenario Specification

  • The evaluation spans NLP sentiment (SST-2) and paraphrase detection (MRPC) with BERT and RoBERTa, plus Adult and Bank Marketing tabular classification with T-Few, T0, and IA3 adapters.T-Few serializes tabular features and values as natural language with task-specific prompts.Source: §4.1 · pp. 12–13

4.1.1 Empirical Rashomon set generation

  • For each dataset-model pair, rejection sampling retained 100 models within ε = 0.03 of the pool’s lowest empirical error to form a warehouse approximation, since enumerating the Rashomon set is infeasible.The authors note that the choice of 100 models was influenced by available GPU resources and per-job walltime limits.Source: §4.1.1 · p. 13, §4.1.2 · p. 13

4.1.2 Selecting the radius σ of local perturbations

  • Consistency is computed in representation space, with local perturbations scaled to nearest-neighbor distance; σ = 0.01 worked across settings and keeps perturbations within each instance’s local neighborhood.This setup targets local prediction stability rather than behavior under unrealistically large perturbations.Source: §4.1.2 · pp. 13–14

4.2 Operationalizing the consistency measure through auditing

  • The auditing rule diverts inputs whose consistency score meets the threshold; relative metrics track unnecessary audits and missed errors among model predictions, while absolute metrics report diversion and audit-risk fractions over the test set.Increasing the threshold makes auditing more conservative, reducing missed errors at the expense of reviewing more correct predictions.Source: §4.2 · pp. 14–15, §4.2.1 · pp. 15–16

4.2.2 High consistency thresholds substantially reduce audit risk for Rashomon warehouse

  • Audit risk fraction drops from approximately 0.174 at β = −0.5 to almost 0.0 at β = 0.25; across datasets and models, threshold selection targets risk near 0.05 without substantial added diversions.As β increases, audit risk falls faster than audit cost rises; the selection procedure caps diversions at prediction error plus a 0.15 tolerance.Source: §4.2.2 · p. 16, Figure 2 · p. 17

4.2.3 Retail set audit outcomes match warehouse audit outcomes as we increase M

  • ≈0.4 (M=1) and ≈0.25 (M=3) audit risk ratios decline toward the warehouse’s ≈0.15 as M grows; audit risk fraction falls from ≈0.07 (M=1) toward zero, mostly by M=10.Diversion fraction also moves toward the warehouse value as ensemble size increases, and the results indicate that relatively modest retail ensembles can approximate warehouse auditing behavior.Source: §4.2.3 · pp. 17–18, Figure 3 · p. 18

4.3 Comparison of proposed consistency measure against warehouse multiplicity metrics

  • The multiplicity comparison tests whether a single-retail-ensemble consistency score is a practical proxy for warehouse metrics, using Spearman rank correlations against related consistency measures and established multiplicity metrics.Comparators include local ensemble consistency, margin, local margin, local variability, and Hamman et al. (2025)’s single-model confidence-based measure.Source: §4.3 · pp. 18–19
  • Across all multiplicity metrics and ensemble sizes, cσ(x) and lmcσ(x) achieve the highest correlations among the consistency measures.Source: §4.3 · p. 19
  • Correlations with warehouse multiplicity metrics generally rise as retail ensemble size increases from M = 1 to M = 5 and M = 10, with larger gains from one to five models.The passage suggests relatively small retail ensembles capture much of the multiplicity information in the Rashomon warehouse.Source: §4.3 · p. 19
  • Measures combining margin and local variability outperform either alone: margin misses local instability, while variability misses proximity to the decision boundary.High-confidence predictions can still show substantial local multiplicity, and predictions with low local variability can remain vulnerable when their margins are small.Source: §4.3 · p. 20

4.3.1 A note on hcσ(x)

  • hcσ(x) has moderate correlations on MRPC but substantially lower ones on SST2; confidence-based scoring can mark low-multiplicity instances inconsistent, with scores near 0 or 1.Figure 4 shows this behavior for pairwise disagreement; hcσ(x) evaluates confidence in the predicted class and penalizes its local variability.Source: §4.3.1 · pp. 20–21, Figure 4 · p. 20
  • Instances with pairwise disagreement close to 0 receive proposed consistency scores above 0.4, reflecting large ensemble margins and low local prediction variability.Source: §4.3.1 · p. 21
  • Defining consistency for a fixed class avoids hcσ(x)'s changing target when sampled models predict different classes, enabling theoretical analysis of concentration toward the Rashomon-set value.Source: §4.3.1 · p. 21

5 Conclusion

The paper introduces an ensemble consistency framework for auditing predictive multiplicity and shows that finite ensembles can approximate the full Rashomon set. Experiments find trade-offs in audit risk and cost, stronger multiplicity measurement with ensembles, and better performance from margin-based consistency penalized by local variability than confidence-based approaches.

5 Conclusion

  • Under mild assumptions, finite-ensemble consistency estimates converge to those of the full Rashomon set as ensemble size and perturbation-sample count increase.The framework combines ensemble margin with local prediction variability to identify predictions stable across competing models.Source: §5 · p. 21
  • Even modest ensembles stabilize audit decisions toward the full Rashomon set, while single models let more erroneous predictions escape review but cause fewer unnecessary diversions.Single models also have considerably lower deployment cost than larger ensembles approximating the full Rashomon set.Source: §5 · p. 21
  • Across most experimental settings, margin-based consistency penalized by local prediction variability outperforms confidence-based approaches.Source: §5 · p. 21
  • Even relatively small ensembles achieve substantially stronger correlations with established predictive multiplicity metrics than a single model, making them a more faithful characterization of multiplicity.The results support aggregating independently trained models rather than relying on a single model.Source: §5 · p. 21
  • Future work includes extending concentration guarantees to weighted ensembles and mixture-of-experts, and extending consistency-based auditing to regression and generative foundation models.The theoretical analysis currently considers uniformly weighted ensembles and the framework focuses on probabilistic classification.Source: §5 · p. 21

A Appendix · A.1 Notation Summary

Appendix A.1 recaps the notation used in the proofs, referring readers to Table 3.

A.1 Notation Summary

  • The appendix summarizes proof notation in Table 3.Source: §A.1 · p. 27

A.2 Measure concentration tools

This section states concentration tools used in the manuscript: McDiarmid’s inequality for bounded-differences functions of independent variables and Hoeffding’s inequality for independent bounded variables.

A.2 Measure concentration tools

  • McDiarmid’s inequality bounds deviations of a bounded-differences function of independent variables from its expectation exponentially.The lemma gives a tail bound for the event that the absolute deviation is at least t.Source: §A.2 · p. 27
  • Hoeffding’s inequality gives a concentration bound for independent random variables that each lie almost surely in [a, b].The lemma states the bound for any ε > 0.Source: §A.2 · p. 27

A.3 Probabilistic guarantee on our proposed consistency measure

This section establishes probabilistic concentration results for the proposed consistency measure under assumptions on the Rashomon set and test instance. The proofs address randomness in both sampled ensemble models and local perturbations.

A.3 Probabilistic guarantee on our proposed consistency measure

  • The assumptions require i.i.d. Rashomon models that are η-locally Lipschitz, with the test instance in the set where the Rashomon set is (β, σ)-consistent and ϵ2-stable.This subset is denoted Dcons_test.Source: §A.3 · p. 27
  • Lemma 3 uses ϵ2-stability to verify bounded differences for the ensemble prediction, enabling application of McDiarmid’s inequality.The argument compares ensembles that differ by replacing one sampled model.Source: §A.3 · pp. 27–28
  • The main result involves showing measure concentration over perturbations and ensemble models, with models drawn i.i.d. from R and perturbations drawn uniformly from S(d), i.i.d. in the stated random-set setup.Source: §A.3 · p. 28
  • The empirical-to-population variability gap is bounded by splitting it into gA and gB and applying a probability bound that requires at least one term to exceed ϵ4/2.The proof applies Hoeffding’s inequality to the perturbation-sample term and uses total probability for a bound uniform over model-set realizations.Source: §A.3 · pp. 29–30

A.3.1 Proof of main results

The proof establishes a probabilistic guarantee for finite-ensemble consistency relative to the Rashomon-set score. It derives a lower bound from the consistency condition by controlling the deviation between the finite and population scores.

A.3.1 Proof of main results

  • Theorem 1 bounds the probability that a finite ensemble’s consistency score falls significantly below the Rashomon-set score; controlling the deviation by ε1 yields a score of at least β−ε1 for a (β,σ)-consistent instance.The proof states that it suffices to upper-bound gmeasure(R, V, x) by ε1.Source: §A.3.1 · p. 30
  • The argument assumes the Rashomon-set models are independently sampled, as are the input perturbations used to measure local prediction variability.The supplied proof passage specifies i.i.d. models and i.i.d. perturbations.Source: §A.3.1 · p. 30
  • The proof decomposes the score deviation into a difference in ensemble margins and a difference in average local stability between the finite ensemble and its population counterpart.It uses the triangle inequality to bound the deviation through these terms.Source: §A.3.1 · pp. 30–31

A.4 Local consistency

The section defines local consistency by combining an ensemble’s average local margin with a penalty for prediction variability. Under stability and smoothness assumptions, finite retail-set ensembles retain consistency with high probability as model and neighborhood-sample counts grow.

A.4 Local consistency

  • Local consistency combines the ensemble’s expected prediction around an instance with a penalty for the expected difference between predictions at the instance and its perturbations.Definition 9 applies this measure to the empirical Rashomon set; Definition 10 defines the corresponding measure for a finite retail set and perturbed samples.Source: §A.4 · pp. 31–32
  • The guarantee assumes models are locally Lipschitz and locally stable at the test instance, which must also meet the Rashomon set’s local-consistency threshold.The concentration results are stated for a fixed test instance satisfying these assumptions.Source: §A.4 · p. 32
  • Theorem 2 guarantees that finite-ensemble local consistency stays close to the empirical Rashomon-set measure with high probability, exponentially in the ensemble size and neighborhood sample count.The guarantee applies to test instances that satisfy the stated local-consistency and stability conditions.Source: §A.4 · pp. 31–32

A.4.1 Probabilistic guarantee on local ensemble consistency

This section establishes a probabilistic guarantee for local ensemble consistency by proving concentration results for perturbed ensemble predictions and the local-margin term. The analysis accounts for randomness in both the perturbation samples and the ensemble.

A.4.1 Probabilistic guarantee on local ensemble consistency

  • The main result is a concentration bound for the average of perturbed ensemble predictions, supporting a probabilistic guarantee for local ensemble consistency.The proof first establishes lemmas used to obtain the guarantee.Source: §A.4.1 · p. 32
  • The proof applies McDiarmid’s inequality after verifying bounded differences, using ensemble stability and local Lipschitzness assumptions.The bounded-differences arguments address variation in the ensemble sample and in the perturbation sample.Source: §A.4.1 · pp. 33–34
  • The local-margin concentration analysis treats perturbation-sample randomness and randomness in the sampled ensemble as two distinct sources.It decomposes the gap between the local margin and its target into contributions from these sources.Source: §A.4.1 · p. 34

A.4.2 Proof of Theorem 2

Theorem 2 bounds the probability that a finite ensemble’s local consistency score falls substantially below the corresponding score of the full Rashomon set.

A.4.2 Proof of Theorem 2

  • Theorem 2 bounds the probability that a finite ensemble’s local consistency score falls significantly below the full Rashomon set’s score.The supplied theorem statement does not include the bound’s explicit probability or parameter dependence.Source: §A.4.2 · p. 35
  • Under the stated (β, σ)-local consistency setup, the proof targets cσ(R, V, x) ≥ β − ε1, reducing it to upper-bounding gmeasure(R, V).Source: §A.4.2 · p. 35
  • The proof decomposes the deviation into a margin difference and terms capturing prediction variation under input perturbations.It then applies the triangle inequality and Lemma 5 to bound the terms.Source: §A.4.2 · p. 36

A.5 Multiplicity metrics

The section frames disagreement among well-performing Rashomon-set models as a sign that an instance may be difficult, and introduces local discrepancy to measure multiplicity around an instance. It also describes pairwise disagreement and discrepancy as existing measures.

A.5 Multiplicity metrics

  • Disagreement among well-performing Rashomon-set models on a test instance may indicate that the instance is fundamentally difficult.Source: §A.5 · p. 36
  • Pairwise disagreement averages disagreements between model pairs in the Rashomon set, while discrepancy measures conflicts with a reference model (Black et al., 2022a; Marx et al., 2020).Source: §A.5 · pp. 36–37
  • Local discrepancy measures average disagreement with a reference model over samples in an instance’s neighborhood.The section introduces it as a new metric for assessing multiplicity around x.Source: §A.5 · p. 37

A.6 Related measures … A.7.5 Additional correlation tables from Section 4.3

Across the additional experiments, larger retail ensembles increasingly reproduce warehouse audit behavior, while the proposed consistency measures track warehouse predictive multiplicity and outperform measures based only on margin or local variability. The appendix also defines related measures and reports how audit outcomes vary with the consistency threshold.

A.6 Related measures

  • The appendix defines margin, local margin, and local variability as quantities related to the proposed consistency and local-consistency measures.It also introduces HDMLD local stability as a related measure.Source: §A.6 · pp. 37–38

A.7.1 Additional plots showing high consistency thresholds substantially reduce audit risk for Rashomon warehouse

  • Across settings, increasing threshold β makes auditing more conservative, substantially reducing audit risk before audit cost and diversions rise sharply.The appendix reports this qualitative behavior across the remaining datasets and model architectures.Source: §A.7.1 · p. 38, Figure 5 · p. 39

A.7.3 Understanding retail and warehouse audit disagreements

  • For MRPC/BERT, over 16.6% of instances fall in Cost decreases versus 4.2% in Cost increases, while close to 5% fall in Risk increases; mixed and neutral outcomes quickly disappear as M grows.The lower audit cost of a single model therefore comes primarily at the expense of increased audit risk; mixed outcomes have incorrect audit decisions for both models, while neutral outcomes have correct decisions for both.Source: §A.7.3 · p. 39

A.7.2 Retail set audit outcomes match warehouse audit outcomes as M increases

  • As retail ensemble size M increases, audit cost, audit risk, diversion, and audit-risk-fraction outcomes show the same convergence toward warehouse behavior across experimental settings.The appendix reports these metrics as functions of M for the remaining datasets and model architectures.Source: §A.7.2 · p. 38, Figure 6 · p. 40

A.7.5 Additional correlation tables from Section 4.3

  • The additional correlation tables compare retail-ensemble measures at M = 1, 5, and 10 with warehouse multiplicity metrics across MRPC and SST2 RoBERTa tasks and Bank and Adult T0 tasks.The multiplicity metrics are pairwise disagreement, discrepancy, local discrepancy, prediction variance, and prediction range.Source: §A.7.5 · p. 41, Table 9 · p. 51, Table 10 · p. 52
  • Across these settings, the proposed consistency measure has strong negative correlations with warehouse multiplicity metrics, strongest for prediction variance and prediction range, and correlation magnitudes rise with ensemble size.Higher consistency corresponds to lower predictive multiplicity; lower consistency identifies instances with greater warehouse disagreement.Source: §A.7.5 · p. 41, Table 9 · p. 51, Table 10 · p. 52

A.7.3 Understanding retail and warehouse audit disagreements

  • For MRPC/BERT, Baseline outcomes rise from 73.1% at M = 1 to 90.9% at M = 30, while Risk increases falls from almost 5% to close to 1%.Cost decreases falls from 16.6% to 7.9%; similar trends are reported for other architectures and datasets.Source: §A.7.3 · p. 39, Table 4 · p. 44

A.7.4 Visualizing relation between proposed consistency measure and different multiplicity metrics

  • The proposed consistency measure assigns high scores to low-multiplicity instances, with stronger negative correlations as ensemble size grows; hcσ assigns some such instances low scores instead.The visual comparison is for MRPC paraphrase detection with independently fine-tuned BERT models.Source: §A.7.4 · p. 40, Figure 7 · p. 49, Figure 8 · p. 50

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