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Fairness and Sparsity within Rashomon sets: Enumeration-Free Exploration and Characterization

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper establishes that the extrema of a fairness metric over an epsilon-Rashomon set, under a sparsity cap, can be computed exactly by solving two mixed-integer programs, provided the hypothesis class admits a mathematical-programming…

desk verdict Genuinely new exact-MILP approach to fairness/sparsity tradeoffs over Rashomon sets, but the 'certifiably exact' claims need solver gap documentation. read the letter →

arxiv 2502.05286 v2 pith:3I6Y3XAO submitted 2025-02-07 cs.LG

Lucas Langlade , Julien Ferry , Gabriel Laberge , Thibaut Vidal This is my paper · ORCID
classification cs.LG MSC 90C1168T05
keywords fairnessRashomonsetmixed-integerlinearprogrammingsparsityscoringsystemsdecisiondiagramsstatisticalparitybusinessnecessity
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 tries to establish that the full range of fairness values attainable inside an $\epsilon$-Rashomon set --- the set of models whose true 0/1 loss is within $\epsilon$ of the best achievable loss --- can be computed exactly, without enumerating models, whenever the hypothesis class can be expressed as a mathematical program. The method first solves the unconstrained learning problem to fix the reference loss, then solves a second optimization that minimizes or maximizes a fairness metric such as statistical parity or equal opportunity subject to a sparsity cap and the Rashomon performance constraint. Applied to scoring systems and decision diagrams, the approach produces certified minimum and maximum fairness values for every combination of accuracy tolerance and sparsity budget. The empirical payoff is a precise map of the accuracy\textendash fairness\textendash sparsity trade-off: wide fairness ranges exist within 1% of optimal training accuracy, but tightening sparsity or accuracy shrinks those ranges and can make discrimination against a protected group unavoidable for all models in the set. A reader should care because this turns the legal \u201cbusiness necessity\u201d defense into a checkable optimization statement rather than a claim subject to cherry-picked models.

What carries the argument

The load-bearing object is the $\epsilon$-Rashomon set, the set of models $h \in H$ with $\hat{L}_S(h) \le \hat{L}_S(h_S) + \epsilon$, where $\hat{L}_S$ is the empirical 0/1 loss and $h_S$ is a loss-minimizing model. The framework converts exploration of this set into a two-stage mixed-integer program: first solve the learning problem with no regularization to obtain the reference loss $\hat{L}_S(h_S)$; then optimize the fairness metric, expressed as a linear function of prediction variables, subject to the sparsity constraint $\mathrm{Sparsity}(h) \le \alpha$ and the performance constraint $\hat{L}_S(h) \le \hat{L}_S(h_S) + \epsilon$. Flipping the sign of the objective yields the opposite extreme, so the two solves bracket the whole achievable fairness interval. The instantiations ride on two existing exact formulations: SLIM for scoring systems and a decision-diagram mixed-integer formulation, both of which express predictions with binary variables and linear constraints.

What would settle it

Rerun one experimental configuration from the paper, for example scoring systems on the Default of Credit Card Clients dataset with $\alpha=9$ and $\epsilon=10\%$, using a longer time limit or a different solver, and find a feasible model whose statistical parity lies outside the reported $[-0.124, -0.009]$ range for the same split; such a model would disprove the exactness claim. More fundamentally, exhibiting a scoring system in the hypothesis class that cannot be encoded by the allowed integer coefficient domains $\Omega_j$ would break the formulation's equivalence.

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Extended reading notes

Core claim

The central claim is that one can certify, for a given dataset, hypothesis class, fairness metric, sparsity budget $\alpha$, and accuracy tolerance $\epsilon$, the exact minimum and maximum fairness value over all models in the Rashomon set, together with the models attaining them. The key move is to replace convex loss surrogates with the true 0/1 loss inside a mixed-integer program: because the fairness objective is optimized directly rather than by enumerating models, the Rashomon set is not shrunk by approximation. For scoring systems, the predictions are integer-coefficient linear rules encoded by the SLIM formulation; for decision diagrams, predictions are flows through a fixed directed acyclic graph encoded by a recent decision-diagram mixed-integer formulation. The paper reports that the resulting intervals are wide and often cross zero, that sparsity restrictions narrow them asymmetrically, and that on the Default dataset, scoring systems with at most nine non-zero coefficients within 10% of optimal accuracy necessarily predict default more often for women than for men --- a systematic disparate impact certified by the optimization.

Load-bearing premise

The framework's results are certified as exact only if the mixed-integer programs exactly represent every model in the hypothesis class and the solver proves global optimality in every run; the paper states that convergence was reached but reports no optimality gaps.

Editorial extensions

If this is right

  • For any hypothesis class with a mathematical-programming formulation, the minimum and maximum of a fairness metric over the $\epsilon$-Rashomon set, under any sparsity cap, are computable and certified, along with models attaining them.
  • A certified interval that stays entirely on one side of zero means no model in the Rashomon set can avoid favoring one protected group, so a \u201cbusiness necessity\u201d defense is either established or refuted for that class, dataset, sparsity, and tolerance.
  • Sparsity constrains fairness: smaller sparsity budgets narrow the achievable fairness range, and tight enough budgets can exclude all models that favor the disadvantaged group, sometimes making bias systematic.
  • Accuracy and sparsity interact asymmetrically with different fairness metrics: statistical parity and equal opportunity can respond differently to the same sparsity tightening because equal opportunity is conditioned on true labels.
  • The hypothesis class is itself a decision variable: because decision diagrams with multivariate splits contain all scoring systems, they admit wider fairness trade-offs and make disparate impact harder to excuse by business necessity.

Reading between the lines

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

  • The paper certifies fairness ranges on the training set; an immediate extension is to certify ranges that hold out-of-sample, for example by solving the same two-stage mixed-integer program on bootstrap samples or deriving high-probability bounds on test-set fairness extrema.
  • Because the solver returns the models attaining each extreme, the machinery doubles as a generator of minimally discriminatory models, not just an auditor; one can extract a deployable model with a specified target fairness value and sparsity.
  • The COMPAS scalability result --- more training data narrows the certified fairness range and amplifies the disadvantage --- suggests a testable hypothesis for other datasets: increasing $N$ can harden, rather than soften, the bias that sparsity imposes, because the Rashomon set of near-optimal models shrinks.
  • The same two-stage pattern should apply to any property expressible as a linear function of predictions, such as calibration differences across groups or robustness to bounded input perturbations, even though the paper only instantiates fairness metrics.
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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 / 4 minor

Summary. The paper proposes an enumeration-free framework for characterizing the extrema of fairness metrics (statistical parity and equal opportunity) within epsilon-Rashomon sets of a hypothesis class, subject to a sparsity bound. The idea is to solve a MILP whose objective is the fairness metric, with constraints enforcing membership in the hypothesis class, the sparsity limit, and a loss bound relative to the optimal 0/1 loss. The framework is instantiated for scoring systems via a modified SLIM formulation and for decision diagrams via a modified Florio et al. formulation. Experiments on Adult, Default of Credit Card Clients, and COMPAS report min/max fairness values across epsilon and alpha values and interpret them for business-necessity arguments. The core claim is that the method exactly and certifiably characterizes achievable fairness under accuracy and sparsity constraints.

Significance. If the certification claim can be substantiated, this is a valuable contribution: it replaces convex-surrogate-based Rashomon-set bounds with exact 0/1-loss optimization, handles sparsity explicitly, and applies to two interpretable model families with released code. The two instantiations are nontrivial, and the experiments cover three benchmark datasets and two fairness metrics. The main caveat is that the reported 'convergence' does not yet demonstrate the global optimality certificates on which the exactness claim rests. Since this appears to be an empirical verification issue rather than a flaw in the modeling idea, the contribution is worth publishing after the evidence is added or the claims are qualified.

major comments (3)
  1. [Section 5.1, 'Exploration of the Rashomon set'; Section 4.1, Problems (6) and (17)] The central exactness claim requires every Gurobi solve to terminate with a proof of global optimality (or a correct infeasibility certificate). Section 5.1 states only that 'Convergence was reached in all runs after at most five such iterations'; no optimality gaps, final incumbent-versus-bound values, or solver statuses are reported. The hot-starting procedure produces feasible incumbents and can help pruning, but it does not by itself establish optimality. If any run ended with a nonzero MIP gap, the reported min/max fairness values are bounds rather than certified extrema, and statements such as Result 3 ('certifiably inevitable') or the N/A entries in Table 3a (which require certified infeasibility) are not justified. Please report per-instance termination status and MIP gap, or qualify the 'certifiable' claim accordingly.
  2. [Section 4.2, Constraints (13)-(14)] The big-M constants O'_i and O_i are not defined. The correctness of the encoding z_i = 1[y_i x_i^T lambda <= 0] relies on these constants being large enough to enforce the implications in both directions and on the choice of the margin gamma. Please specify how O'_i and O_i are computed from the coefficient domains Omega_j and the data, and state the value of gamma; otherwise the formulation is not reproducible and a too-small choice could invalidate the loss computation.
  3. [Section 4.3, Equations (20)-(21)] The integrality of the flow variables w_iv is not stated. If w_iv are continuous, Constraint (20) does not force z_i to be binary, and the objective (17) would not exactly represent the statistical parity of a 0/1 classifier. Please state the domains of w_iv, f_{iuv}^+ and f_{iuv}^-, and t_{uv}^+ and t_{uv}^-, and either prove that the formulation's extreme points yield binary w_iv or add explicit integrality constraints.
minor comments (4)
  1. [Abstract] The phrase 'within less than 1% of the best possible training accuracy' is imprecise; the experiments use an epsilon schedule between the optimal loss and the majority-class loss, not a direct threshold on accuracy. Please rephrase to match the actual definition in Section 5.1.
  2. [Table 3a] Several numeric entries contain stray spaces (for example, '0 .077' and '0 .120'); please fix the formatting.
  3. [Section 4.2, after Eq. (15)] The sentence 'ensuring that (y_i x_i^T lambda) for all examples i is lower-bounded' is incomplete; it should say that the margin gamma bounds the score away from zero when the model is confident. Please reword.
  4. [Section 4.1, Equations (8) and (17)] The objective minimizes the raw difference of positive rates, but the connection between the binary predictions ^y_i and the indicators 1[h(x_i)=1] in Equations (2) and (3) is not stated explicitly before the objective is used. A one-sentence clarification would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed fairness extrema are computed by solving explicitly stated optimization problems whose constraints define the Rashomon set, sparsity, and hypothesis class, with no fitted parameter or self-citation chain standing in for the result.

full rationale

The derivation chain is self-contained. The paper defines the epsilon-Rashomon set in Equation (4) using the optimal 0/1 loss obtained by solving Problem (5) with C = 0, and then solves Problems (6), (8), and (17), whose objective is exactly the fairness metric and whose constraints are exactly the hypothesis-class membership, sparsity, and performance conditions. No parameter is fitted to the reported fairness extrema, and no prior result by the same authors is invoked to replace an argument: the SLIM formulation is an external published encoding of scoring systems, and the decision-diagram MILP of Florio et al. (2023), although co-authored by Vidal, is a published, independently checkable mathematical programming formulation of a hypothesis class rather than a statement of the target fairness characterization. The only notable caveat is empirical verifiability rather than circularity: Section 5.1 states 'Convergence was reached in all runs' without reporting Gurobi optimality gaps or solver logs, so a nonzero MIP gap would reduce the reported extrema to bounds; that is a correctness and evidence-reporting concern, not a circularity of the derivation. Accordingly, the honest finding is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The framework itself postulates no new entities. It relies on exact encodings of hypothesis classes from prior work, on solver optimality, and on the chosen definitions of fairness and utility. The user-specified coefficient domains and diagram skeletons are the main hand-chosen elements that shape the results.

free parameters (3)
  • Scoring coefficient domain Ω = {0, ±1, ±2, ±5, ±10, ±20, ±30, ±50}
    User-defined set of allowed integer coefficients for scoring systems; defines the hypothesis class and therefore the Rashomon set. Chosen by hand, not derived.
  • Decision diagram skeleton = (1,2,3,3,3), 12 internal nodes, depth 5
    Pre-specified maximum structure for the decision diagram MILP, fixed based on preliminary experiments. Bounds the hypothesis class and affects the observed trade-offs.
  • Rashomon set schedule (p) = 1%, 5%, 10%, 20% of the loss gap to the majority classifier
    The epsilon tolerance is defined as p times the difference between optimal loss and majority-classifier loss; p is chosen by the experimenter, not fitted to data.
assumptions (4)
  • domain assumption SLIM MILP formulation exactly encodes the class of scoring systems with 0/1 loss, as per Ustun et al. (2014).
    The fairness optimization is built on this formulation without re-deriving or verifying it; if the formulation admits models outside H or excludes valid ones, the Rashomon set characterization is inexact. Section 4.2.
  • domain assumption The decision diagram MILP of Florio et al. (2023) exactly represents the allowed decision diagram space for the given skeleton.
    The framework inherits the expressive power and constraints of this formulation; errors or omissions there propagate to the fairness bounds. Section 4.3.
  • domain assumption Gurobi returns proven optimal solutions for all instances within the given time and memory limits.
    The claimed certification depends on global optimality; the paper asserts convergence without reporting optimality gaps or solver logs. Section 5.1.
  • domain assumption Statistical parity and equal opportunity are the fairness criteria of interest, and the 0/1 loss is the utility measure defining the Rashomon set.
    These choices frame the entire analysis; other fairness metrics or losses would yield different sets and different bounds. Sections 2 and 3.

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Cite this review

Pith. "Pith review of Fairness and Sparsity within Rashomon sets: Enumeration-Free Exploration and Characterization." pith.science (2026). https://pith.science/paper/3I6Y3XAO

@misc{pith2026250205286,
  author       = {Pith},
  title        = {Pith review of: Fairness and Sparsity within Rashomon sets: Enumeration-Free Exploration and Characterization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3I6Y3XAO}},
  note         = {Machine review of arXiv:2502.05286}
}
read the original abstract

We introduce an enumeration-free method based on mathematical programming to precisely characterize various properties such as fairness or sparsity within the set of "good models", known as Rashomon set. This approach is generically applicable to any hypothesis class, provided that a mathematical formulation of the model learning task exists. It offers a structured framework to define the notion of business necessity and evaluate how fairness can be improved or degraded towards a specific protected group, while remaining within the Rashomon set and maintaining any desired sparsity level. We apply our approach to two hypothesis classes: scoring systems and decision diagrams, leveraging recent mathematical programming formulations for training such models. As seen in our experiments, the method comprehensively and certifiably quantifies trade-offs between predictive performance, sparsity, and fairness. We observe that a wide range of fairness values are attainable, ranging from highly favorable to significantly unfavorable for a protected group, while staying within less than 1% of the best possible training accuracy for the hypothesis class. Additionally, we observe that sparsity constraints limit these trade-offs and may disproportionately harm specific subgroups. As we evidenced, thoroughly characterizing the tensions between these key aspects is critical for an informed and accountable selection of models.

Figures

Figures reproduced from arXiv: 2502.05286 by the authors.

Figure 1
Figure 1. Minimum statistical parity (SP) and equal opportunity (EO) achievable on the UCI Adult Income dataset, within a [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Minimum and maximum statistical parity (SP) and equal opportunity (EO) achievable on the COMPAS dataset, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Minimum and maximum equal opportunity (EO) achievable on the COMPAS dataset, within [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Minimum and maximum statistical parity (SP) and equal opportunity (EO) achievable on the three considered datasets, [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Minimum and maximum statistical parity (SP) and equal opportunity (EO) achievable on the three considered datasets, [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

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