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What Constitutes a Less Discriminatory Algorithm?

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that no data-only measurement can define a less discriminatory algorithm, and that courts must instead judge whether a proposed model would reasonably generalize before accepting it as an alternative to the status quo.

desk verdict A genuinely useful formal analysis of LDA search, with a central 'reasonableness' predicate that remains a placeholder. read the letter →

arxiv 2412.18138 v2 pith:46A2IQJP submitted 2024-12-24 cs.CY cs.LGstat.ML

Benjamin Laufer , Manish Raghavan , Solon Barocas This is my paper · ORCID
classification cs.CYcs.LGstat.ML
keywords lessdiscriminatoryalternativedisparateimpactdoctrinealgorithmicfairnessmodelmultiplicityselectionratedisparityreasonablenessstandardNP-hardnessfairness-utilitytrade-off
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 asks what should count as a less discriminatory algorithm (LDA) in disparate-impact law, a question regulators now press on firms and plaintiffs. Its central claim is that no purely quantitative definition can work: because courts lack held-out data, a plaintiff can always construct a decision rule that is perfectly accurate on the firm's training data and has zero selection-rate disparity on post-deployment data, so measured accuracy and disparity alone cannot separate a genuine LDA from an overfit artifact. The paper therefore defines an LDA as a model that is reasonably projected to be at least as accurate as the baseline and to have significantly lower disparity, and it argues that courts must apply a reasonableness standard for generalization. It then shows the mathematical limits on achieving zero disparity are real but rarely binding, that finding the least discriminatory alternative is NP-hard yet a close approximation is efficiently findable, and that simple randomized searches can reduce disparity out-of-sample, sometimes at no accuracy cost. The payoff, if correct, is a framework in which firms can proactively search for fairer models and plaintiffs can challenge failures to do so without pretending that data alone can settle the question.

What carries the argument

The argument runs on two engines. The first is the feasible utility-disparity polygon: for a finite population with group sizes and base rates, every binary classifier corresponds to a point in the $(\Delta, U)$ plane with $U(h;\lambda) = \mathrm{TPR} - \lambda\mathrm{FPR}$, and the Pareto frontier is traced by two types of label swaps away from the perfect classifier; this pins down the threshold $U^*$ in Theorem 1. The second is the reasonableness standard in Definition 1, which replaces measurement with projection: because no held-out data exists at litigation time, a candidate LDA must be judged by whether a court would reasonably expect it to generalize, with model complexity relative to the baseline as the main heuristic. On top of these, the paper's NP-hardness result is driven by a reduction from subset sum, while the positive approximation guarantee comes from a polynomial-time approximation scheme for the same knapsack-like problem.

What would settle it

Try to construct the Proposition 1 rule inside a restricted hypothesis class, for instance shallow decision trees of a fixed depth, on the Adult and German Credit datasets; if no such rule can simultaneously achieve perfect training accuracy and zero selection-rate disparity across groups, the impossibility result depends on unbounded model complexity rather than on the absence of held-out data, and a quantitative standard might suffice.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the LDA concept must move from measured performance to projected performance. Proposition 1 constructs a rule that is perfectly accurate on one dataset and selects a constant fraction of each group on another, showing that any definition based solely on observed accuracy and disparity can be satisfied by a pathological model. The paper's formal definition therefore says $h'$ is an LDA relative to $h_0$ when $h'$ is reasonably projected to have at least $h_0$'s accuracy and significantly lower selection-rate disparity; what makes the projection reasonable is left to a case-by-case judgment, not to a formula. The accompanying results give the boundaries of what is achievable: a utility threshold $U^*$ below which a zero-disparity alternative always exists even when base rates differ, an NP-completeness result for the full-information LDA search, a polynomial-time $(1+\epsilon)$-approximation that guarantees finding a significantly better model whenever the baseline is not already near-optimal, and empirical evidence on common datasets that simple random-seed or resampling searches within one model class reduce out-of-sample disparity, sometimes with a utility gain.

Load-bearing premise

The load-bearing premise is that courts can apply a reasonableness standard well enough to judge whether a proposed model would generalize, even though the paper does not define that standard precisely and concedes that formalizing it in full generality is beyond its scope.

Editorial extensions

If this is right

  • Courts should evaluate candidate LDAs by whether they would be expected to generalize, not by how they score on observed data, which means post-deployment measurements alone cannot establish liability.
  • When group base rates differ, perfect accuracy necessarily has nonzero disparity, but a zero-disparity alternative exists whenever the baseline's utility is below $U^*$; the paper's reading of Theorem 1 is that this trade-off is vacuous except at unusually high accuracy, so it rarely shields a firm.
  • Since the least discriminatory alternative is NP-hard to find but a $(1+\epsilon)$-approximation is computable in polynomial time, a firm's claim that the search is computationally impossible is a weak defense as long as the legal standard permits de minimis slack.
  • Simple searches by random seed or resampling within the same model class can reduce disparity on unseen data, sometimes improving utility as well, so firms need not rely on specialized optimization to find reasonable LDAs.
  • Because a good-faith search often turns up and rejects models that later prove less discriminatory out-of-sample, treating considered-and-rejected models as evidence of liability would discourage proactive searching.

Reading between the lines

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

  • The paper leaves implicit that the same gameability argument would apply to other fairness metrics, not just selection-rate disparity, so any purely data-based fairness definition in litigation may need an accompanying reasonableness standard.
  • Outside the courtroom, the reasonableness standard could be made operational by benchmarking generalization through temporal or cross-domain shifts, turning an opaque legal judgment into a measurable inductive question.
  • The approximation guarantee is proved for the full-information setting, where the firm knows the true population distribution; converting it into practical training algorithms for finite samples and restricted model classes is a natural next step the paper leaves open.
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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 / 6 minor

Summary. This paper examines how to define a "less discriminatory algorithm" (LDA) in the legal disparate-impact sense. It argues that LDA definitions based on measured accuracy and disparity on observed datasets are untenable: Proposition 1 constructs a decision rule that achieves perfect accuracy on pre-deployment data and zero disparity on post-deployment data, and the authors contend that any purely quantitative measured-performance definition admits similar pathologies. The paper therefore proposes Definition 1, under which a model is an LDA if it is "reasonably projected" to have at least the accuracy of the baseline and "significantly lower" selection-rate disparity, while acknowledging that the projection standard is not fully formalized. It then analyzes the feasibility of finding LDAs: Theorem 1 characterizes the attainable utility-disparity frontier under full information and shows a zero-disparity alternative exists unless the baseline utility is above a threshold; Theorem 2 proves the full-information LDA existence problem NP-complete via a reduction from subset sum, with an accompanying (1+epsilon)-approximation claim; and Section 5 presents empirical simulations showing that simple multiplicity-based searches can reduce out-of-sample disparity on the Adult dataset, though not on German Credit.

Significance. If the results hold, the paper makes three useful contributions: it gives a crisp negative result showing why measured-performance LDA definitions are insufficient, it provides a formal characterization of when accuracy/disparity trade-offs do and do not bind, and it offers evidence that practical LDA searches are often computationally feasible despite worst-case hardness. The NP-completeness reduction and the empirical open-source evaluation are concrete and reproducible. The main caveat is that the paper's positive definition rests on an unspecified "reasonableness" predicate, so the conceptual core is not yet fully operational; the paper's significance is therefore stronger as a negative/conceptual result than as a complete formal framework.

major comments (3)
  1. [Section 2.1, Definition 1] The definition's central predicates, "reasonably projected" and "significantly lower," are never given a formal or operational meaning, and Section 2.2 explicitly defers their full formalization as beyond the scope of the work. This is load-bearing: Proposition 1 only rules out purely measured-performance definitions, and without an operational account of reasonable projection, a court (or a firm) has no criterion for distinguishing the pathological Proposition 1 rule from a legitimate multiplicity-based alternative. The paper may establish a negative thesis, but the positive account of what constitutes an LDA is not yet established. The authors should either provide a formal operationalization for at least a restricted but nontrivial setting, or explicitly reframe the contribution as a diagnosis plus a research agenda.
  2. [Section 4 / Appendix C.3, Claim 5] The claim that a (1+epsilon)-approximate full-information LDA can be identified in polynomial time O(n^3 epsilon^{-1}) is stated without proof: Appendix C.3 says "the proof is deferred" and points only to standard subset-sum approximation schemes. This claim is load-bearing for the paper's conclusion that the NP-hardness result is "weak" and unlikely to be prohibitive in practice. As written, the reader cannot verify the central "weak computational limits" argument. A complete proof or a clear designation of the claim as conjectural is needed.
  3. [Section 3.2, Theorem 1] The second displayed formula in Theorem 1 is garbled: "Delta(h') = 1 - min[ n1/n+, lambda n2/n- ] Delta(h*) - Delta(h0)" is missing parentheses and at least one factor, so the claimed characterization of the minimum-disparity alternative at a given utility level cannot be verified from the statement as printed. The appendix proof may be correct, but the theorem statement must be restated cleanly before the feasibility result can be used as the paper intends.
minor comments (6)
  1. [Section 2.2] There is a typo in the first paragraph: "representative dastaset" should be "representative dataset."
  2. [Appendix C.2] The arithmetic derivation of the condition on alpha contains a garbled line with missing parentheses and a dropped factor (the line beginning "-lambda 1/N ..."), which makes the reduction harder to follow; the derivation should be rewritten cleanly.
  3. [Table 1] The column "Freq. min-disp." is not defined in the caption or the surrounding text; the caption should explain what this frequency measures.
  4. [References] Reference [54] contains a typo: "uidance" should be "Guidance."
  5. [Section 5] The sentence "These results imply that it may not be a good idea to punish firms for having considered and rejected models that are ultimately less discriminatory" goes beyond what the experimental simulation can support; the claim should be softened to a hypothesis or explicitly labeled as a policy interpretation.
  6. [Definition 2 vs. Theorem 2] Definition 2 lists the LDA input as <X, sigma, rho_g, h0> but the utility function U(h; lambda) is also part of the problem instance; the tuple should include lambda or the text should state that lambda is fixed.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; the central 'reasonableness standard' is left undefined, but this is an incompleteness, not circularity.

full rationale

The paper's formal results are derived from stated assumptions rather than from their own conclusions. Proposition 1 is an impossibility argument against purely measured-performance definitions and is presented as a simple observation, not as a consequence of Definition 1. Theorem 1 is proven in Appendix B from the stated population, classifier, and randomized-decision formalism, with no fitted parameters. Theorem 2 is proven by a polynomial-time reduction from Subset Sum in Appendix C.2, which is an independent computational argument. The approximation claim (Claim 5) is indeed asserted with its proof deferred, but deferring a proof is an omission, not a circular step. The empirical section uses train/evaluation/test splits and reports out-of-sample results, so the reported disparity and utility improvements are genuine held-out evaluations rather than fitted inputs renamed as predictions. The main weakness is that Definition 1 relies on 'reasonably projected' and 'significantly lower', which are not formally operationalized; Section 2.2 concedes that defining reasonable projections 'in full generality is beyond the scope this work.' This makes the positive account incomplete, but the paper does not claim to derive the reasonableness standard from the LDA definition, nor does it reduce any theorem to an assumption. The self-citation to Black et al. [7] (which includes a co-author of this paper) is used for a policy recommendation about proactive 'reasonable search' and is not load-bearing for the formal results. Overall, the derivation chain is self-contained; the score reflects minor incompleteness rather than circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The theoretical framework is abstract but does not introduce any new physical or formal entities beyond the standard definitions of classifiers, utility, and disparity. The main free parameters are the utility weight lambda and the approximation slack epsilon, both of which are left unspecified. The core conceptual innovation, the 'reasonableness standard,' is explicitly not formalized.

free parameters (2)
  • lambda (utility weight)
    The utility function U(h; lambda) = TPR - lambda * FPR is used throughout the theoretical analysis, and its value is not fixed by theory. The hardness result is stated for any positive lambda, but Theorem 1's threshold depends on lambda.
  • epsilon (approximation factor)
    The approximation algorithm and the definition of approximate LDA depend on an arbitrarily chosen epsilon, which represents a de minimis threshold for disparity improvement. There is no principled way to choose it.
assumptions (5)
  • domain assumption Binary protected groups and binary outcomes.
    The formal setting in Section 3.1 assumes a binary group g in {1,2} and a binary label y in {0,1}. This is a standard simplification in the fairness literature, but it may not capture multi-group or multi-class settings.
  • domain assumption Demographic disparity is measured as difference in selection rates.
    The paper defines disparity as SR1(h) - SR2(h), a specific fairness metric. Other metrics (e.g., false positive rate differences) are not considered. This assumption is shared with much of the cited literature on LDAs.
  • domain assumption The firm's business goal is captured entirely by the utility function U(h; lambda).
    The paper equates business efficacy with predictive accuracy and a scalar utility. It explicitly sets aside challenges to the dataset and target variable, which are acknowledged in Section 2 as possible legal challenges.
  • ad hoc to paper An LDA must be in the same model class as the original model for reliable comparison.
    In Section 2.2, the paper argues that models in the same class can be compared by measured performance, but models in different classes require 'case-by-case' judgment. This assumption is necessary to make the empirical search methods (random seed/sampling) interpretable, and it is not formally justified.
  • ad hoc to paper A 'reasonableness standard' can be applied by courts to evaluate projections of model performance.
    The paper's central definition of an LDA (Definition 1) relies on this standard, but the paper explicitly states that defining it fully is 'beyond the scope' of the work. This is the paper's weakest assumption.

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Cite this review

Pith. "Pith review of What Constitutes a Less Discriminatory Algorithm?." pith.science (2026). https://pith.science/paper/46A2IQJP

@misc{pith2026241218138,
  author       = {Pith},
  title        = {Pith review of: What Constitutes a Less Discriminatory Algorithm?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46A2IQJP}},
  note         = {Machine review of arXiv:2412.18138}
}
read the original abstract

Disparate impact doctrine offers an important legal apparatus for targeting discriminatory data-driven algorithmic decisions. A recent body of work has focused on conceptualizing one particular construct from this doctrine: the less discriminatory alternative, an alternative policy that reduces disparities while meeting the same business needs of a status quo or baseline policy. However, attempts to operationalize this construct in the algorithmic setting must grapple with some thorny challenges and ambiguities. In this paper, we attempt to raise and resolve important questions about less discriminatory algorithms (LDAs). How should we formally define LDAs, and how does this interact with different societal goals they might serve? And how feasible is it for firms or plaintiffs to computationally search for candidate LDAs? We find that formal LDA definitions face fundamental challenges when they attempt to evaluate and compare predictive models in the absence of held-out data. As a result, we argue that LDA definitions cannot be purely quantitative, and must rely on standards of "reasonableness." We then identify both mathematical and computational constraints on firms' ability to efficiently conduct a proactive search for LDAs, but we provide evidence that these limits are "weak" in a formal sense. By defining LDAs formally, we put forward a framework in which both firms and plaintiffs can search for alternative models that comport with societal goals.

Figures

Figures reproduced from arXiv: 2412.18138 by the authors.

Figure 1
Figure 1. Consider a given, finite population broken down by group belonging and outcomes (a; left). If randomized decision rules are feasible, then a polygon depicts the convex set of feasible, in-sample utility and disparity values (b; center). If solutions are restricted to deterministic classifiers over the dataset (c; right), the polygon bounds the achievable values. This section’s approach is to consider the set of all … view at source ↗
Figure 2
Figure 2. Results from a simple randomized search for a less discriminatory algorithm on the Adult dataset. The search procedure randomly samples with replacement from the training dataset and retrains a Random Forest classifier n times, for n ∈ {2, ..., 100}. From the n candidate models, it selects the minimum-disparity classifier, as measured using separate data from an evaluation dataset. As n increases, we find disparity … view at source ↗
Figure 3
Figure 3. Empirically observed achievable polygon using the Adult and German Credit evaluation datasets. In the case of Adult (left), a Random Forest classifier exhibits wide selection rate disparities. Randomly sampled alternatives exhibit small variations in utility and disparity. Analysis of performance on held out test data suggests that selecting the best alternative can statistically improve utility and reduce disparity… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Runtime (left, center) and accuracy performance (right) of exact and approximate algorithms for computing the least discriminatory alternative classifier, given oracle access to information about the population distribution. The exact algorithm is compared to three dif…

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Forward citations

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.