Audits Under Resource, Data, and Access Constraints: Scaling Laws For Less Discriminatory Alternatives

Thank you for your very constructive feedback. We appreciate the time you took to provide interesting questions. Below we respond point‑by‑point and indicate the revisions we will make should the paper be accepted.

Clarification on contributions. Before we address your specific concerns, we noticed some themes in reviews, so we wanted to clarify our contribution as well as how it meaningfully adds to existing work.

1. Novel theoretical result that does not exist in extensive literature on Pareto frontiers. Despite the extensive interest on Pareto frontiers and recent work on performance-fairness Pareto frontiers [1-3, 11], no prior works have closed-form expressions for Pareto frontiers (in complex settings beyond, e.g., low-dimensional, binary features). The existing literature on performance-fairness Pareto frontiers falls into two categories: (i) clever methods to find the frontier empirically via various multi-objective optimization techniques [4-9]; and (ii) analytical approaches in simple settings that pre-date large models [10-12]. This is (i) not feasible in our setting, as it would still require that claimants are able to train models at the same scale as large companies and thus would place prohibitive costs on claimants with limited data and computational resources; and (ii) also not applicable in our setting, as it cannot be applied to large models with flexible, high-dimensional inputs. Thus, even absent our setting (LDAs), our derivation of a closed-form tight bound is a novel result in the study of Pareto frontiers, multi-objective optimization, and fairness that we provided by combining insights from information theory and results on implicit regularization via data augmentation/manipulation.
2. Addressing a persistent issue in AI accountability and audits that appears repeatedly. The authors engage with and belong to both the ML and legal communities, and have engaged with stakeholders who seek to audit and/or make legal claims about AI systems. Across these, there is a persistent issue (as in our introduction) of an asymmetry between the resources, information, data, and access that claimants possess vs. those that defendants possess. This asymmetry is especially pronounced in disparate impact procedures; namely, step 3 involving LDAs. At the moment, no method exists that is able to address this issue at scale (for large models). So this setting is a strong, motivating case study and we provide the only known framework that addresses the above asymmetry.
3. Future work. We note that while our main result on a closed-form scaling law holds for our stated assumptions and fairness as demographic parity, it is an important first step given the state of current work, as described above. We therefore hope our work provides a roadmap for extensions and hope to extend these results to other notions of fairness and performance. As our main contributions are our proposed reframing of LDAs for large models and the novel theoretical result, we leave extensive experiments to future work.

We hope that these points help to clarify our contributions, and we will do our best to make these contributions clear/better contextual our contributions in a revision if given the opportunity.

Next, we highlight and respond to specific concerns that you raised:

• Lack of specificity on the conditions under which the scaling laws hold. Thank you for this point. We agree that our results, as presented, apply only for our stated loss (BCE) and fairness definition (demographic parity). As for the hypothesis class, the only assumption we place on it are the symmetry assumptions stated in our work (for an explanation of this assumption, see our last bullet below). Although our contribution in this work does not expand to other losses/fairness definitions, we believe it is an important first step and we seek to be upfront with our setup (i.e., that our results do not extend beyond it for now, except for what we show in the simulations, where we stress test the assumptions). Even so, we agree with the reviewer that the fact that our result lacks full generality is a limitation (though we do not believe that all losses and fairness definitions will result the same closed-form, so one would need to derive independent results for each setting). We will make this limitation even clearer in a revision, as we plan to include an explicit discussion and limitations section.
• More extensive experiments. We agree with the reviewer that our experiments are limited. We hope that the points at the top of this rebuttal address this in part. To add to it, we make the following clarifications. (1) Our main contribution is the reframing of the LDA problem in low-resource/information settings and ensuing theoretical scaling law. We hope that the reviewer recognizes this as an important contribution, as there previous works entirely on the empirics of scaling laws and entirely on the theory of scaling laws that precede ours [13-16]. (2) We are already taking steps to engage with employment agencies to apply this framework to a real dataset. We believe this warrants a standalone follow-up work. (3) We want to note that, even so, we believe our synthetic experiments are important in that the main “gap” in our theoretical result is whether the assumptions hold. So we stress-test the assumptions in Section 5 via the synthetic experiments that test whether the theoretical result holds under a different data generating process than assumed, whether implicit regularization using training data manipulation is a good approximation of actual training, and if Assumption 4.2 holds in practice. Although we discuss this in the Appendix (E.2), we understand this should be clearer and will update Section 5 to reflect this.
• Training small neural nets is non-trivial. We agree that training even small neural nets is non-trivial, but we want to emphasize our contribution compared to the status quo. At the moment, we are in a more dire situation in which a plaintiff must provide enough evidence that an LDA exists but often the resources/data involved to train one are impossible for them to obtain. They must produce some results that are convincing enough (if it’s too hand-wavy, it’s often not accepted), and thus it is impossible to avoid placing any burden on them.
• Formatting concerns. We used H for the loss in the appendix proof but L in the Theorem 4.2 statement. Thank you very much for pointing this out—we will fix it in the revision.
• Non-trivial to bound constant B in Theorem 4.2 to explicate how $c_1(\mathcal{F}, \mathcal{D})$ depends on $| \mathcal{F}|$ and $| \mathcal{D}|$. We are not sure we fully understood the reviewer’s question, but under existing scaling laws (that only discuss loss), this term often follows the form $1/N^\alpha + 1/D^\alpha$, where $N = | \mathcal{F}|$ and $D = | \mathcal{D}|$.
• Reasonability of symmetry assumptions given incentive of defendant to only release skewed portions of their training data. This is an interesting point. We believe that this assumption is fairly mild as it implies that the model class $\mathcal{F}$ is not biased/favorable towards any particular distribution. Although this assumption may not hold exactly in practice, it is reasonable for large deep learning models, because they are intended to be universal function approximators. That is, as a model class, they are designed to be relatively setting agnostic and over-parameterized such that they could be symmetric wrt the DGP. As for defendants being incentivized to release skewed portions of their training data, this type of dishonest behavior is impossible to prevent but typically countered by the fact that failing to follow the court’s orders would be flagrantly breaking the law and thus we generally presume that the defendant will not do this or otherwise be severely penalized. Again, preventing dishonesty is impossible in practice, but we typically use a “stick” to disincentivize it (e.g., penalties for perjury).

[1] Bertsimas et al. (2011). The price of fairness. Oper. Res.

[2] Menon & Williamson (2018). The cost of fairness in binary classification. FAccT.

[3] Kim et al. (2020). FACT: A diagnostic for group fairness trade-offs. ICML.

[4] Navon et al. (2020). Learning the Pareto front with hypernetworks. arXiv.

[5] Ruchte & Grabocka (2021). Scalable Pareto front approximation for deep multi-objective learning. ICDM.

[6] Singh et al. (2021). A hybrid 2-stage neural optimization for Pareto front extraction. arXiv.

[7] Rothblum & Yona (2021). Consider the alternatives: Navigating fairness-accuracy tradeoffs via disqualification. arXiv.

[8] Kamani et al. (2021). Pareto efficient fairness in supervised learning: From extraction to tracing. arXiv.

[9] Liu & Vicente (2022). Accuracy and fairness trade-offs in machine learning: A stochastic multi-objective approach. CMS.

[10] Xu & Strohmer (2023). Fair Data Representation for Machine Learning at the Pareto Frontier. JMLR.

[11] Liang et al. (2021). Algorithm design: A fairness-accuracy frontier. arXiv.

[12] Gillis et al. (2024). Operationalizing the search for less discriminatory alternatives in fair lending. FAccT.

[13] Hestness et al. (2017). Deep learning scaling is predictable, empirically. arXiv.

[14] Kaplan et al. (2020). Scaling laws for neural language models. arXiv.

[15] Hoffmann et al. (2022). Training compute-optimal large language models. arXiv.

[16] Bahri et al. (2024). Explaining neural scaling laws. PNAS.

A sincere thanks for your response. We further respond below:

1. (First bullet) We appreciate your update. If we are interpreting your response correctly, the suggestion is to incorporate further discussion into Section 5. We agree with this and will do so in the revision if given the opportunity. Our plan is to (i) make Section 5's main points and contributions much clearer (e.g., by adding \paragraph environments) and (ii) add a step-by-step explanation of how to fit a scaling law, linking that explanation to our simulations.

2. (Second and third bullets) We're glad that we were able to clarify these points.

3. (Improving clarity) We understand the feedback to improve clarity and carry out our promised revisions in this and other responses to reviewers. We take these promises seriously, as our goal is for this framework to be usable and applicable.

4. (Step-by-step procedure) Yes, absolutely. As mentioned above, we plan to incorporate a walk through/step-by-step procedure of precisely how one would use Theorem 4.2's result. At the moment, this is implied/scattered, but our goal would be to write it clearly in one place (either at the end of the main results section or at the start of the simulations section, which we may rename "Demonstration of Scaling Law" --> we plan to play around with both to determine which is clearest).

[Apologies for the deleted response above. It was a response to a different reviewer that was mistakenly sent in this thread.]

We appreciate your critique of both our theoretical and empirical work and the time you took with our submission. Below we respond to all of your points and note any revisions we will make shall the paper be accepted.

Clarification on contributions. Before we address your specific concerns, we noticed some themes in reviews, so we wanted to clarify our contribution as well as how it meaningfully adds to existing work.

1. Novel theoretical result that does not exist in extensive literature on Pareto frontiers. Despite the extensive interest on Pareto frontiers and recent work on performance-fairness Pareto frontiers [1-3, 11], no prior works have closed-form expressions for Pareto frontiers (in complex settings beyond, e.g., low-dimensional, binary features). The existing literature on performance-fairness Pareto frontiers falls into two categories: (i) clever methods to find the frontier empirically via various multi-objective optimization techniques [4-9]; and (ii) analytical approaches in simple settings that pre-date large models [10-12]. This is (i) not feasible in our setting, as it would still require that claimants are able to train models at the same scale as large companies and thus would place prohibitive costs on claimants with limited data and computational resources; and (ii) also not applicable in our setting, as it cannot be applied to large models with flexible, high-dimensional inputs. Thus, even absent our setting (LDAs), our derivation of a closed-form tight bound is a novel result in the study of Pareto frontiers, multi-objective optimization, and fairness that we provided by combining insights from information theory and results on implicit regularization via data augmentation/manipulation.
2. Addressing a persistent issue in AI accountability and audits that appears repeatedly. The authors engage with and belong to both the ML and legal communities, and have engaged with stakeholders who seek to audit and/or make legal claims about AI systems. Across these, there is a persistent issue (as in our introduction) of an asymmetry between the resources, information, data, and access that claimants possess vs. those that defendants possess. This asymmetry is especially pronounced in disparate impact procedures; namely, step 3 involving LDAs. At the moment, no method exists that is able to address this issue at scale (for large models). So this setting is a strong, motivating case study and we provide the only known framework that addresses the above asymmetry.
3. Future work. We note that while our main result on a closed-form scaling law holds for our stated assumptions and fairness as demographic parity, it is an important first step given the state of current work, as described above. We therefore hope our work provides a roadmap for extensions and hope to extend these results to other notions of fairness and performance. As our main contributions are our proposed reframing of LDAs for large models and the novel theoretical result, we leave extensive experiments to future work.

We hope that these points help to clarify our contributions, and we will do our best to make these contributions clear/better contextual our contributions in a revision if given the opportunity.

Next, we highlight and respond to specific concerns that you raised:

• Motivating computational hardness of work. Theorem 4.2 is the main result of the paper. There is no easy way to obtain the scaling law, and this is precisely our contribution. To contextualize this contribution, there have been foundational papers that focus entirely on empirical scaling laws or entirely on the theory of scaling laws, as we discuss in Section 6. With our method, we show that one can obtain the scaling law specified in Theorem 4.2 and hence have an easier solution to the problem of finding an LDA. The proof of Theorem 4.2 was quite involved and, to the best of our knowledge, there exists no other closed-form characterizing the performance-loss Pareto frontier, and the framework of establishing LDAs under low resources and information using scalings is entirely new.
• Reasonability of Assumptions 4.1 and 4.2. Assumption 4.1 is reasonable when considering very large models. It means that the model class $\mathcal{F}$ is not biased/favorable towards any particular distribution. Although this assumption may not hold exactly in practice (as most assumptions do not), it is a reasonable condition for deep learning models, especially large ones, because (large) deep learning models are meant to be universal function approximators. That is, as a model class, they are designed to be relatively setting agnostic and over-parameterized such that they could be symmetric wrt the DGP. A similar thought process holds for Assumption 4.2.
• Assumption 4.1 holding outside of Bayes optimal classifiers. Assumption 4.1 uses \hat{f} to denote any classifier, not just a Bayes optimal classifier. Bayes optimality is only used as a proof technique to trace the Pareto frontier (it is not an assumption).
• Details of Thm 4.2, intuition and justification for constant terms. The setup for Theorem 4.2 is provided between Sections 3.1 and 4.1. Constants (that may depend on complicated quantities like $p$) are a standard tool used in, e.g., statistical learning theory (see, e.g., Prop 2.5 or Thm 2.6 of [17]). There are two main reasons for this. (1) the exact value of the constant is context-specific! Providing a specific value would not be meaningful. (2) One could potentially provide a specific form for each constant, but that would require significant assumptions on the specific context. With respect to the scaling law literature, constants appear naturally to show that the scaling law takes a particular form, where certain constants must be “fit” based on data (e.g., see Equation 4.1 of [14]). Lastly, we note that one can trace the derivation of our constants in the proof on page 28 of the supplementary material.
• Take-aways from Fig 3, specify N, what does varying N solve. N is the number of parameters of the model. We defined N at the start of the section on line 262 but will define it again near Figure 3. The magnitude of N relates to the computational difficulty of training such a model, so smaller N corresponds to limited computational resources. This general approach in finding a form of the scaling law as a function of N is common in the scaling law literature (e.g., Kaplan et al. 2020, Sharma and Kaplan 2022, or Clark et al. 2022).
• Correctness of formulation, responsibility for LDA. Establishing an LDA is the responsibility of the plaintiff in all disparate impact claims under Title VII of the Civil Rights Act (see Lines 49-50). We would like to emphasize that the authors have significant engagement with the the legal community, and failing to ground our work in real stakeholder issues would be antithetical to our goals. This work was produced by first identifying a concern that is repeatedly raised, then seeking to provide a framework that can genuinely help plaintiffs and other claimants. We further point the reviewer to Sections 1, 2, and 6 (in particular, the “three steps of disparate impact cases” from lines 104-112). Section 6 identifies several related works that write about the fact that the burden of establishing an LDA falls on plaintiffs.
• Reduced information on vendor’s model. To contextualize the situation, the current debate on burden of proof issues for claimants is whether or not vendors should release their entire trained models and training data and whether, in the absence of this, plaintiffs have any mechanisms for relief. Thus, asking vendors to release the approximate N and D is a much lower ask that we expect companies to welcome.
• Regulator or court requesting report from vendor. While this may be true in some cases, in court cases involving disparate impact claims under Title VII of the Civil Rights Act, the burden of LDA proof falls squarely on the claimant. This is explicitly detailed in lines 110-112 and is precisely what we address. There is some discussion about whether this burden of search for an LDA should be shifted onto model developers (see lines 348-353), but for now the burden is entirely on the claimant.

[1] Bertsimas et al. (2011). The price of fairness. Oper. Res.

[2] Menon & Williamson (2018). The cost of fairness in binary classification. FAccT.

[3] Kim et al. (2020). FACT: A diagnostic for group fairness trade-offs. ICML.

[4] Navon et al. (2020). Learning the Pareto front with hypernetworks. arXiv.

[5] Ruchte & Grabocka (2021). Scalable Pareto front approximation for deep multi-objective learning. ICDM.

[6] Singh et al. (2021). A hybrid 2-stage neural optimization for Pareto front extraction. arXiv.

[7] Rothblum & Yona (2021). Consider the alternatives: Navigating fairness-accuracy tradeoffs via disqualification. arXiv.

[8] Kamani et al. (2021). Pareto efficient fairness in supervised learning: From extraction to tracing. arXiv.

[9] Liu & Vicente (2022). Accuracy and fairness trade-offs in machine learning: A stochastic multi-objective approach. CMS.

[10] Xu & Strohmer (2023). Fair Data Representation for Machine Learning at the Pareto Frontier. JMLR.

[11] Liang et al. (2021). Algorithm design: A fairness-accuracy frontier. arXiv.

[12] Gillis et al. (2024). Operationalizing the search for less discriminatory alternatives in fair lending. FAccT.

[13] Hestness et al. (2017). Deep learning scaling is predictable, empirically. arXiv.

[14] Kaplan et al. (2020). Scaling laws for neural language models. arXiv.

[15] Hoffmann et al. (2022). Training compute-optimal large language models. arXiv.

[16] Bahri et al. (2024). Explaining neural scaling laws. PNAS.

[17] Wainwright MJ (2019). Basic tail and concentration bounds. In High-Dimensional Statistics: A Non-Asymptotic Viewpoint, pp. 21–57. Cambridge Univ. Press.

Thank you - we appreciate your time and thoughtful feedback, and we will do our best to improve the writing around the technical contributions.

We sincerely appreciate your careful reading and constructive feedback. Below we address each concern and outline the concrete revisions we will make if the paper is accepted.

Clarification on contributions. Before we address your specific concerns, we noticed some themes in reviews, so we wanted to clarify our contribution as well as how it meaningfully adds to existing work.

1. Novel theoretical result that does not exist in extensive literature on Pareto frontiers. Despite the extensive interest on Pareto frontiers and recent work on performance-fairness Pareto frontiers [1-3, 11], no prior works have closed-form expressions for Pareto frontiers (in complex settings beyond, e.g., low-dimensional, binary features). Our work thus takes a step to fill this gap. To explain further, the existing literature on performance-fairness Pareto frontiers falls into two categories: (i) clever methods to find the frontier empirically via various multi-objective optimization techniques [4-9]; and (ii) analytical approaches in simple settings that pre-date large models [10-12], e.g., binary feature vectors of a fixed, low dimension. As we explained in our work, this (i) is not feasible in our setting, as it would still require that claimants are able to train models at the same scale as large companies and thus would place prohibitive costs on claimants with limited data and computational resources; and (ii) is also not applicable in our setting, as it cannot be applied to large models with flexible, high-dimensional inputs. Thus, even absent our setting/motivation (LDAs), our derivation of a closed-form tight upper bound is a novel result in the study of Pareto frontiers, multi-objective optimization, and fairness that we were able to provide by combining insights from information theory and results on implicit regularization via data augmentation/manipulation.
2. Addressing a persistent issue in AI accountability and audits that appears repeatedly. The authors engage with and belong to both the ML and legal communities. The authors have also engaged with stakeholders who seek to audit and/or make legal claims about AI systems. Across these conversations, there is a persistent issue (as identified in our introduction) of an asymmetry between the resources, information, data, and access that claimants possess vs. those that defendants/auditees possess. This asymmetry is especially pronounced in disparate impact procedures; namely, step 3 involving LDAs. At the moment, no method exists that is able to address this issue at scale (for large models). So this setting is a strong, motivating case study and we provide the only known framework that addresses the above asymmetry.
3. Future work. We note that while our main result on a closed-form scaling law holds for our stated assumptions and fairness as demographic parity, it is an important first step given the state of current work, as described above. We therefore hope our work provides a roadmap for extensions and hope to extend these results to other notions of fairness and performance. As our main contributions are our proposed reframing of LDAs for large models and the novel theoretical result, we leave extensive experiments to future work.

We hope that these points help to clarify our contributions, and we will do our best to make these contributions clear/better contextual our contributions in a revision if given the opportunity.

We now respond to specific concerns that you raised:

• Real application for theoretical results. The setting of this paper is grounded in the specific Title VII procedure that arises in disparate impact doctrine (lines 104-112) and thus that procedure is a self-evident example.
• Experiments using real data. Thank you for raising this. We hope that the points at the top of this rebuttal address this in part. To add to it, we make the following clarifications. (1) Our main contribution is the reframing of the LDA problem in low-resource/information settings and ensuing theoretical scaling law. We hope that the reviewer recognizes this as an important contribution, as there previous works entirely on the empirics of scaling laws and entirely on the theory of scaling laws that precede ours [13-16]. (2) We agree with the reviewer that an extensive real-world study is important to viability, and we are already taking steps to engage with employment agencies to apply this framework to a real dataset. We believe this warrants a standalone follow-up work. (3) We want to note that, even so, we believe our synthetic experiments are important as the main “gap” in our theoretical result is whether the assumptions hold. So the synthetic experiments test the assumptions in Section 5 to see whether the theoretical result holds under a different data generating process than assumed, whether implicit regularization using training data manipulation is a good approximation of actual training, and if Assumption 4.2 holds in practice. Although we discuss this in the Appendix (E.2), we understand this should be clearer and will update Section 5 to reflect this.
• Choice of algorithmic parity. We agree with the reviewer that there is no universally “right” fairness definition. We want to assure the reviewer that the authors have engaged with the fairness community and with stakeholders on how the law, courts, and other policymakers have interpreted/applied fairness definitions in practice. Given this, we offer two points. (1) In practice, simple definitions like demographic parity and equalized odds are often more accepted by courts than sophisticated ones. (2) As discussed above, our work provides the first closed-form expression characterizing performance-fairness Pareto frontiers despite a long history of related work—we believe this is a significant first step that provides a roadmap for future work. However, we do acknowledge that our result is for a specific setting (choice of fairness metric, assumptions, model class) and we hope that it will be generalized in future work. We understand your example and will suggest that practitioners make sure the bias metric is appropriate for their use case.
• Bias from using a measure not as well calibrated as ground truth. This is indeed a persistent issue that spans many fields. We appreciate the pointer and will add it to a limitations section. However, we do not believe that this issue precludes our framework from being applied. The concern the reviewer raises is, in fact, a more fundamental issue that arises anytime an outcome-based fairness definition is used and thus is not an issue specific to our work, though we fully agree that it is an important issue and would be a compelling extension.
• Better model that relies on easily collected but different input data. This is certainly a possibility, but our motivating scenario is a company using a proprietary large model (so the plaintiff does not have access to any training material), and the task of training a competing large model is infeasible for the plaintiff given their limited resources and common appeals to trade secrecy used by companies. This is the motivation of our work, which we describe in depth in Sections 1, 2, and 6.
• Use cases in which our approach is superior to other approaches to this task. We point the reviewer to the discussion at the top of this rebuttal as well as to our discussion of related work in Sections 6 and 2. Mainly, all other approaches characterize Pareto frontiers empirically (not feasible for low-resourced plaintiffs) or in settings with simple input spaces (not applicable to the increasingly large modern models that are the target of our work).
• Limitations. We appreciate this feedback and are happy to discuss our limitations in depth in a revision by specifically addressing that (i) our method works conditional on the validity of the performance metric (e.g., demographic parity) supplied by the claimant; (ii) it also assumes that the claimant has at an ability to train small models (like an MLP) on small datasets; and (iii) our specific theoretical result contains several assumptions that should be further explained. We want to note that we strive in the existing manuscript to state all assumptions, definitions, and score clearly, as we agree that hiding these can only hurt practitioners (which is against the spirit of this work: to provide claimants with a useful framework).

[1] Bertsimas et al. (2011). The price of fairness. Oper. Res.

[2] Menon & Williamson (2018). The cost of fairness in binary classification. FAccT.

[3] Kim et al. (2020). FACT: A diagnostic for group fairness trade-offs. ICML.

[4] Navon et al. (2020). Learning the Pareto front with hypernetworks. arXiv.

[5] Ruchte & Grabocka (2021). Scalable Pareto front approximation for deep multi-objective learning. ICDM.

[6] Singh et al. (2021). A hybrid 2-stage neural optimization for Pareto front extraction. arXiv.

[7] Rothblum & Yona (2021). Consider the alternatives: Navigating fairness-accuracy tradeoffs via disqualification. arXiv.

[8] Kamani et al. (2021). Pareto efficient fairness in supervised learning: From extraction to tracing. arXiv.

[9] Liu & Vicente (2022). Accuracy and fairness trade-offs in machine learning: A stochastic multi-objective approach. CMS.

[10] Xu & Strohmer (2023). Fair Data Representation for Machine Learning at the Pareto Frontier. JMLR.

[11] Liang et al. (2021). Algorithm design: A fairness-accuracy frontier. arXiv.

[12] Gillis et al. (2024). Operationalizing the search for less discriminatory alternatives in fair lending. FAccT.

[13] Hestness et al. (2017). Deep learning scaling is predictable, empirically. arXiv.

[14] Kaplan et al. (2020). Scaling laws for neural language models. arXiv.

[15] Hoffmann et al. (2022). Training compute-optimal large language models. arXiv.

[16] Bahri et al. (2024). Explaining neural scaling laws. PNAS.

We appreciate your quick and thoughtful response.

Response to Point 1. Given the imperfect nature of communicating via OpenReview, we will do our best to address what we think the reviewer is saying here. If we understand correctly, we completely agree with the spirit of the reviewer's point. What is often frustratingly clear is that the path to addressing issues that arise in CS/law is incredibly fraught and requires the buy-in of multiple stakeholders whose reactions are varied. This is actually precisely the motivation of our work, so we completely understand (and share) the skepticism that the framework is immediately applicable. Our hope is that this (and future works) can contribute to a larger body work that is essentially adding "options" to the mix so that claimants/factfinders are no longer constrained by not having choices/tools to pick from. They can survey and debate a variety of options, then select the one that seems best for the situation.

So, to be fully clear, we do not feel that this would be immediately applicable, and it requires building a body of work around this issue, which always begins with initial steps (and we believe that this paper is helping take those steps). In fact, our goals are indeed much more modest (than hoping this would be immediately applicable): given the strong resistance to granting access, information, and data around AI systems, our hope is that, if our method yields results that support the plaintiff's claim, this would just help plaintiff justify the request for more access that allows them to prove (or disprove) that claim conclusively.*

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Response to Point 3 (or 2?). Unfortunately, this is true! Although highly debated and very much not universally agreed upon, there is a long-standing "rule" known as the Four-Fifths rule, which compares selection rates and has been used in Title VII disparate impact cases. Although this standard might change, that does not change the reality of the situation. If one wants more recent evidence that that selection rates are still widely used, one needs only look at NYC's Local Law 144, which came into effect in 2023. To make the connection clear, the absolute difference in selection rates = the definition of demographic parity.

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A realistic use case. We are still confused by this point. The broader issue that we seek to address is that a lack of access to models, training procedures, data, and resources has prevented plaintiffs from winning their cases (or, even earlier, from gathering enough evidence to bring a case). Our realistic use case is embedding this in Title VII, and even more specifically, scoping ourselves specifically to (i) disparate impact claims and (ii) the LDA requirement within disparate impact cases. We would argue that our entire paper is meant to tackle a specific case study.

From here, making it more realistic is indeed a goal of ours, but this step (to our understanding) would be to empirically verify our framework on real data. This is, as mentioned, a very good suggestion and something we absolutely want to do! However, that step not within scope of this paper, as doing so would require multiple pages to describe how we acquired the data, how we cleaned it, issues that we ran into (which is always the case with real data), in-depth methodology, in-depth results, and ablations. To be clear, we don't think that the reviewer's point is that we should do this; the paragraph above is probably more applicable to addressing the reviewer's concern, and we believe this concern arises from a misunderstanding.

Compared to the cited paper, ours is very different. The cited paper puts forth a model (a game) of the setting, then uses that example to demonstrate how the model captures appropriate settings that fall within scope of the motivation. This is fully natural and conventional for this type of paper, but ours is very different. Our motivation is a legal setting (in fact, a specific requirement within this setting), and we expend more than a figure (multiple sections!) motivating it.

Thanks for your response. We do greatly appreciate your feedback - as you can understand, OpenReview is wonderful for allowing interaction, but it still is an imperfect way to communicate, so we are doing our best to fully understand your concerns given our reading of it. We always aim to sincerely engage with your (and reviewers') concerns.

We acknowledge that you are unlikely to engage further given that you have submitted your mandatory acknowledgment, but we wanted to respond below to engage with your feedback.

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Efforts to further understand your call for a realistic use case.

Your clarification is greatly appreciated because we think we finally understand the point of confusion from both ends. We believe that it may stem from an misunderstanding of the LDA setting we are tackling and would greatly appreciate the opportunity to explain why. In the LDA setting, we are firmly embedded in the disparate impact doctrine. In that doctrine, nothing much changes from the classical classification/algorithmic scoring setting, beyond the fact that the models used are larger.

For example, consider an LLM used to process resumes. They are still used to output score(s) and/or a recommendation to hire/not hire. This would be the same type of signal considered in classical discrimination/fairness works. The difference is that these large models can now effectively process/interpret complex, semantic text, like resumes or application short answers.

The reviewer's clarification indicates that perhaps the reviewer is viewing legal definitions of LLM discrimination as definitions that interpret the text that is produced, perhaps whether the text contains undertones of discrimination. However, this is the portion that is due to a misunderstanding of our paper. Such analyses would not fall into disparate impact doctrine - it would be considered as disparate treatment. This would be completely out-of-scope for our paper, as our paper looks at Title VII --> Disparate Impact --> LDAs. The LDA requirement does not have any meaning beyond the disparate impact doctrine. (We make a short note of this in Footnote 3.)

We hope this also addresses the reviewer's repeated concern about demographic parity not being the right metric. As we wrote above, demographic parity is indeed still used for disparate impact. Although there are countless other definitions for fairness and discrimination that have emerged (from equalized odds to individual fairness to envy freeness in matching markets to multicalibration), the evidence, as we wrote above, points to the fact that the courts often still prefer demographic parity. They favor it over concepts like equalized odds often because one cannot access the "ground truth" necessary to determine false positive rates/false negative rates.

We would like to mention that, though we understand your clarification, we believe that we did our best to interpret your initial feedback about providing "a realistic use case". We hope you understand that, given that our work is precisely motivated by a very specific, real use case, we were confused. We did our best to understand and interpret it.

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Revisions to the work

Relatedly, we greatly appreciate the reviewer's time to give feedback, and it helps us understand where we need to clarify our work and setting. We plan to edit our work to reflect your feedback, including: (i) the scope of our setting, perhaps incorporating Footnote 3 into the main text, (ii) a figure or short example describing how this would apply to, say, the use of LLMs in employment decisions, (iii) a step-by-step procedure of how to fit and apply a scaling law.

Thank you for your thorough and constructive feedback. We appreciate the time you took to engage with our submission. Below we respond point‑by‑point and indicate the revisions we will make in the camera‑ready version should the paper be accepted.

Clarification on contributions. Before we address your specific concerns, we noticed some themes in reviews, so we wanted to clarify our contribution as well as how it meaningfully adds to existing work.

1. Novel theoretical result that does not exist in extensive literature on Pareto frontiers. Despite the extensive interest on Pareto frontiers and recent work on performance-fairness Pareto frontiers [1-3, 11], no prior works have closed-form expressions for Pareto frontiers (in complex settings beyond, e.g., low-dimensional, binary features). Our work thus takes a step to fill this gap. To explain further, the existing literature on performance-fairness Pareto frontiers falls into two categories: (i) clever methods to find the frontier empirically via various multi-objective optimization techniques [4-9]; and (ii) analytical approaches in simple settings that pre-date large models [10-12], e.g., binary feature vectors of a fixed, low dimension. As we explained in our work, this (i) is not feasible in our setting, as it would still require that claimants are able to train models at the same scale as large companies and thus would place prohibitive costs on claimants with limited data and computational resources; and (ii) is also not applicable in our setting, as it cannot be applied to large models with flexible, high-dimensional inputs. Thus, even absent our setting/motivation (LDAs), our derivation of a closed-form tight upper bound is a novel result in the study of Pareto frontiers, multi-objective optimization, and fairness that we were able to provide by combining insights from information theory and results on implicit regularization via data augmentation/manipulation.
2. Addressing a persistent issue in AI accountability and audits that appears repeatedly. The authors engage with and belong to both the ML and legal communities. The authors have also engaged with stakeholders who seek to audit and/or make legal claims about AI systems. Across these conversations, there is a persistent issue (as identified in our introduction) of an asymmetry between the resources, information, data, and access that claimants possess vs. those that defendants/auditees possess. This asymmetry is especially pronounced in disparate impact procedures; namely, step 3 involving LDAs. At the moment, no method exists that is able to address this issue at scale (for large models). So this setting is a strong, motivating case study and we provide the only known framework that addresses the above asymmetry.
3. Future work. We note that while our main result on a closed-form scaling law holds for our stated assumptions and fairness as demographic parity, it is an important first step given the state of current work, as described above. We therefore hope our work provides a roadmap for extensions and hope to extend these results to other notions of fairness and performance. As our main contributions are our proposed reframing of LDAs for large models and the novel theoretical result, we leave extensive experiments to future work.

We hope that these points help to clarify our contributions, and we will do our best to make these contributions clear/better contextual our contributions in a revision if given the opportunity.

Next, we highlight and respond to specific concerns that you raised:

• Re: Figure 1**,** we thank the reviewer for pointing this out and will replace it with a more professionally rendered schematic. Regarding the suggestion to use empirical data, our objective with Figure 1 is to provide a schematic illustrating why the legal problem finding LDAs can be cast/formalized in terms of Pareto frontiers and the intuition behind scaling laws for Pareto frontiers. This aims to provide the reader with a mental picture of what the paper proposes, instead of a real-life application..
• Re: related works. We initially placed the Background and Related Work section after the introduction, but moved it to the end of the work, as has become common in ML works. As the reviewer’s concern indicates, this change would have warranted adding more citations to the introduction, and we will do so (and move Section 6 forward) in a revision. Irrespective of the placement, we want to assure the reviewer that our work engages deeply with the literature. We closely surveyed several related areas, including algorithmic fairness, anti-discrimination law, multi-objective optimization, and machine learning. The gap we address results from this survey. We also do our best to highlight the most closely related work on LDAs and hope that our discussion of them in Sections 2 and 6 contextualize our contributions relative to them.
• Leaving Empirical Validation for Future Work. This is a valuable question. We hope that the points at the top of this rebuttal address this in part. To add to it, we make the following clarifications. (1) Our main contribution is the reframing of the LDA problem in low-resource/information settings and ensuing theoretical scaling law. We hope that the reviewer recognizes this as an important contribution, as there previous works entirely on the empirics of scaling laws and entirely on the theory of scaling laws that precede ours [13-16]. (2) We agree with the reviewer that an extensive real-world study is important to viability, and we are already taking steps to engage with employment agencies to apply this framework to a real dataset. We believe this warrants a standalone follow-up work. (3) We want to note that, even so, we believe our synthetic experiments are important in that the main “gap” in our theoretical result is whether the assumptions hold. So we stress-test the assumptions in Section 5 via the synthetic experiments that test whether the theoretical result holds under a different data generating process than assumed, whether implicit regularization using training data manipulation is a good approximation of actual training, and if Assumption 4.2 holds in practice. Although we discuss this in the Appendix (E.2), we understand this should be clearer and will update Section 5 to reflect this.
• Model class being symmetric w.r.t. DGP. It means that the model class $\mathcal{F}$ is not biased/favorable towards any particular distribution. Although this assumption may not hold exactly in practice, it is reasonable for large deep learning models, because they are meant to be universal function approximators. That is, as a model class, they are designed to be relatively setting agnostic and over-parameterized such that they could be symmetric wrt the DGP.
• Description of constants. Thank you for this. To potentially overexplain, constants (that may depend on complicated quantities like $p$) are a standard tool used across, e.g., statistical learning theory (see, e.g., Prop 2.5 or Thm 2.6 of [17]). There are two main reasons they come in handy. First, the exact value of the constant is context-specific! Providing a specific value would not be meaningful. Instead, saying that the some variable Y (in our case, loss) depends on an independent variable X (in our case, fairness gap) according to our closed-form equation, where there are constants that need to be fit based on data is a traditional approach. Second, one could potentially provide a specific form for each constant, but that would require significant assumptions on the specific context. With respect to the scaling law literature, constants appear naturally to show that the scaling law takes a particular form, where certain constants must be “fit” based on data (e.g., see Equation 4.1 of [14]). Lastly, we note that one can trace the derivation of our constants in the proof on page 28 of the supplementary material.
• Line 305 varies with N. Yes, the figure shown above Line 305 varies only N while keeping D constant.

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