Title: Kernel-based Equalized Odds: A Quantification of Accuracy-Fairness Trade-off in Fair Representation Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Fair Representation Learning

Kernel Methods

Equalized Odds

Demographic Parity

Fairness-Accuracy Trade-off
Abstract: This paper introduces a novel kernel-based formulation of the Equalized Odds (EO) criterion, denoted as $\operatorname{EO}_k$, for fair representation learning (FRL) in supervised settings.
The central goal of FRL is to mitigate discrimination regarding a sensitive attribute $S$ while preserving prediction accuracy for the target variable $Y$. 
Our proposed criterion enables a rigorous and interpretable quantification of three core fairness objectives: independence ($\widehat{Y} \perp S$), 
separation—also known as equalized odds ($\widehat{Y} \perp S \mid Y$), and calibration ($Y \perp S \mid \widehat{Y}$). 
Under both unbiased ($Y \perp S$) and biased ($Y \not \perp S$) conditions, we show that $\operatorname{EO}_k$ satisfies both independence and separation in the former, and uniquely preserves predictive accuracy while lower bounding independence and calibration in the latter, thereby offering a unified analytical characterization of the tradeoffs among these fairness criteria.
We further define the empirical counterpart, $\widehat{\operatorname{EO}}_k$, a kernel-based statistic that can be computed in quadratic time, with linear-time approximations also available.
A concentration inequality for $\widehat{\operatorname{EO}}_k$ is derived, providing performance guarantees and error bounds, which serve as practical certificates of fairness compliance. 
While our focus is on theoretical development, the results lay essential groundwork for principled and provably fair algorithmic design in future empirical studies.


Peer reviews:
comments: SUMMARY:
This paper proposes a kernel-based Equalized Odds (EQ) criterion for characterization fairness in machine learning. An empirical estimator is provided to estimate the kernel EQ, and the author establishes theoretical guarantees for the estimator as well as fairness-accuracy tradeoffs.

STRENGTHS AND WEAKNESSES:
## Strengths
  - The proposed kernel-based EQ could be a useful extension of EQ with robust theoretical guarantees.
  - An empirical estimator is provided for calculating the kernel EQ which is practical and scalable.
  - The authors show desirable theoretical guarantees for fairness-accuracy tradeoffs as well as convergence of the empirical estimator. 

## Weaknesses

  - The proposed fairness criterion and the associated results are dependent on the given kernel. If the kernel is misspecified, the criterion may yield false fairness claims. Kernel misspecification could be less relevant for prediction tasks, but a concern for a criterion. 
  - Kernel-based equalized odds and fairness-accuracy tradeoffs have already been studied in literature. Potential missing related work: 
Tan et al.,  Learning Fair Representations for Kernel Models. AISTATS, 2020

QUESTIONS:
Does kernel misspecification lead to false fairness claims using the proposed criterion?

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 2
DATE: 2025-06-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a fairness measure for fair representation learning, which is a kernel-based formulation of EO ($EO_k$) and provides theoretical properties of this new proposed formulation. This new formulation uses MMD to measure the differences in representations between two demographic groups, enabling rigorous and interpretable quantification of independence, separation (via EO), and calibration within a unified framework. The main contribution of the paper is to quantify the trade-offs between these mutually incompatible fairness constraints and to analyze their implications.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The new formulation is mathematically well-defined in a principled manner (Section 2.1), and the derivation of the generalization bounds and the closed-form empirical estimator is technically correct and solid.\n\nWeaknesses:\n- The main concern with this paper is the lack of empirical support (the submission contains no experiments, toy examples or any empirical validation). While the authors state that the work is theoretical, but provides a connection between theory and practice is still important. Without at least a simulated example demonstrating how $EO_k$ captures trade-offs in realistic scenarios, it's difficult to assess its practical significance. \n\n- Section 2 presents the main theoretical contribution, and the section reads like a disconnected sequence of tehnical points (defs, lemmas, bounds), each introduced in isolation without sufficient transitions or motivation. For example. section 2.1 introduces the formulation of $EO_k$ and after that immediately moves to its RHKS form with no clear intuition or example to help the reader undersatnd the object being defined. Lemma 2.2 and 2.4 define technical function classes and bounds, but the role in the overall framework is not explained until much later.  Also, the notations are compact and mathematical shorthand that obscure the main ideas and reduce the readability. After reading Section 2, it is still unclear what the main theoretical message is beyond a set of bounds and definitions. The paper would benefit greatly from: (1) adding a motivating paragraph at the beginning of Section 2; (2) add some clear transitions between subsections to explain why each result is introduced and (3) maybe provide a summary paragraph at the end that synthesizes the theoretical results and explains their implications for fair representation learning.\n\n- The necessity of using kernel-based MMD to formulate fairness constraints is not well justified. This design choice introduces additional complexity. In practice, simpler and more model-agnostic alternatives are often more interpretable and widely adopted.\n\n- Similar to the above one, while the theoretical unification of multiple fairness definitions is interesting from a statistical perspective, in practice, practitioners typically select one or two fairness notions based on task requirements. The need to balance all three (independence, separation, and calibration) is not sufficiently motivated, and the paper lacks examples where such trade-offs arise naturally or meaningfully. In other words, the paper introduces $EO_k$ without sufficiently motivating the limitations of existing metrics or demonstrating that this new formulation enables better fairness–accuracy trade-offs in practical tasks. It remains unclear what real-world problems $EO_k$ solves that cannot already be addressed with existing fairness constraints.\n\nQUESTIONS:\n- While the paper emphasizes that $EO_k$ is not designed to optimize all fairness metrics simultaneously but rather to quantify the trade-offs between them, this conceptual distinction remains abstract without concrete justification. Can the authors provide a realistic case or application where such trade-off quantification is actually useful? For example, how would a practitioner interpret or act on the value of $EO_k$ in a model selection or deployment scenario? More generally, how does $EO_k$ support fairness-aware decision-making beyond what simpler metrics (e.g., separate TPR/FPR gaps or post-hoc fairness curves) already provide? What specific limitations of existing fairness criteria does $EO_k$ address that cannot be handled using DP or EO? i.e., why is a new kernel-based fairness metric necessary?\n\n- In practical settings, fairness constraints are typically selected based on domain or specific requirements (e.g., some cases decision makers prefer equalized odds, some cases prefer accuracy parity), why is a unified definition preferable or more actionable?\n\n- Using kernel-based definition will increase computational cost compared with DP and EO, so what practical gains justify this additional complexity?\n\nLIMITATIONS:\nThe authors did not discuss the limitations of their approach. The main limitations are outlined in Weakness Point 1 and further elaborated in the questions I have raised.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a kernel-based reformulation of the equality of odds fairness metric, which permits a quantifiable characterization of the fairness-accuracy trade-off in fair representations learning. This new definition, dubbed $EO_K$, allows characterization of sufficiency, independence, and calibration fairness desiderata, which authors carry out under two settings of ground truth label and sensitive label independence and lack thereof. Various theoretical results are provided, including a generalization guarantee, a bound on balanced accuracy based on a maximum mean discrepancy (MMD) formulation of $EO_k$, and quantified comparisons between the fairness desiderata.\n\nSTRENGTHS AND WEAKNESSES:\nOverall, I think this is a valuable contribution. The writing has issues that would limit its impact, however. Also, quantitative comparisons with prior theoretical works are missing. I will enumerate **S**trengths and **W**eaknesses. \n\n- **S1. The kernel-based approach is novel.** The formulation as an affine map of the unit ball in the RKKHS allows expression of the equalized odds violation as a total probability of disparities over classes in the proposed metric $EO_k$. This is interesting and scalable to more classes and (likely) multiple sensitive attributes.\n\n- **S2. Quantitative trade-off between various fairness definitions is interesting.** While prior work shows how satisfying different fairness metrics is impossible, those results mostly consider exact fairness (which in and of itself is often hard to enforce in reality, even for a single metric). Having a quantified trade-off using the presented method is a more granular extension to these results that I find useful.\n\n- **W1. The paper is disorganized and lacks flow.** The paper has three sections, with the third section being the conclusion, and the first section being the introduction. The introduction runs 5 pages and includes related work in two separate subsections. The discussion mixes in prior work with contributed work constantly, making assessment of the novelty and contribution difficult. Most sections and subsections miss proper transitions, which makes reading the paper quite difficult. More detailed feedback on this front:\n\t\t- I question the necessity of Section 1.3 (the second related work section discussing a plethora of prior work without clear motivation). All and all, regardless of the results, the paper is not in a publishable state.\n\t\t- Section 1.1 is unnecessarily dense and inapproachable. I understand the authors sought to summarize their results, but this is largely unhelpful.\n\n- **W2. Quantitative comparison to prior work is missing.** While there is plenty of qualitative comparison to prior work, I think the paper is missing quantitative comparisons with any meaningful baseline methods. In the absence of empirical results (which on its own is fine), I expected a rate comparison at the very least. This leaves important questions unanswered: for example, since the paper claims to achieve good fairness-accuracy trade-offs, does it Pareto dominate prior work in any meaningful capacity?\n\t    \n- **W3. $EO_k$ cannot resolve the tensions in choosing a proper fairness metric.** A non-technical weakness is that the paper suggests that the use of $EO_k$ resolves the issue of selecting a fairness criterion compared to prior work. In particular, the paragraph starting on Line 150 reads: \"To address the ambiguity in selecting a fairness criterion, recent works have proposed metrics that aim to approximate multiple fairness notions simultaneously... In contrast, our method acknowledges this conflict and formalizes the trade-offs using a kernel-based statistic, EOk.\"   \n\t    \n   However, choosing a proper fairness definition should be a function of the application and ideally based on a legal and philosophical discussion of the fairness required. Prior work [1] has shown that the impossibility of satisfying multiple fairness definitions is due to their opposing philosophies. Therefore, I think it is wrong to suggest a single technical fairness definition is appropriate for all applications, regardless of the social context it is being used. I'll flag this for an ethics review and invite the authors to reflect on this question from beyond a technical standpoint.  \n\t    \n     I am afraid making comparisons between fairness definitions easier, beyond its value as a research endeavor, is going to invariably make \"shopping around\" for a fairness definition worse. This is a classical reification that is prevalent in algorithmic fairness literature that we should be very consciously avoiding.  \n\t    \n    [1] H. Heidari, M. Loi, K. P. Gummadi, and A. Krause, “A Moral Framework for Understanding of Fair ML through Economic Models of Equality of Opportunity,” *arXiv:1809.03400 [cs, econ, stat]*, Sep. 2018, Accessed: Jun. 25, 2019. [Online]. Available: [http://arxiv.org/abs/1809.03400](http://arxiv.org/abs/1809.03400)\n\nQUESTIONS:\n- Q1. Can you summarize the theoretical novelties that you have contributed? At times, I had difficulty pinpointing your original contributions from prior work.\n- Q2. Can you add quantitative comparisons to prior work in fairness-utility trade-offs? For instance, a Lagrangian approach such as FERMI [1] or an encoder-based fair representation learning approach such as FARE [2]? \n  \n  Note that FERMI is not a representation-based method, but the topic of fairness-utility tradeoff is well explored. It bears comparison across methodologies.  \n  \n  [1] A. Lowy, S. Baharlouei, R. Pavan, M. Razaviyayn, and A. Beirami, “A Stochastic Optimization Framework for Fair Risk Minimization,” Jan. 11, 2023, *arXiv*: arXiv:2102.12586. doi: [10.48550/arXiv.2102.12586](https://doi.org/10.48550/arXiv.2102.12586).  \n  [2] N. Jovanović, M. Balunović, D. I. Dimitrov, and M. Vechev, “FARE: Provably fair representation learning with practical certificates,” in *Proceedings of the 40th international conference on machine learning*, PMLR, 2023.  \n- Q3. Please comment on the W3 and consider adding an impact statement.\n\nLIMITATIONS:\nThe paper would have benefited from empirical evaluations (even small-scale experiments on tabular data).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a kernel-based formulation of Equalized Odds (EO_k) for fair representation learning in a supervised setting (the authors focus on binary classification). The EO_k criterion provides a unified characterization of three core fairness objectives used in the literature: independence, separation (also called EO), and calibration. When the data is unbiased -- i.e., the target variable Y is independent of the protected attribute S -- EO_k satisfies both independence and separation. When the data is biased (Y is not independent of S), EO_k provides a controllable level of predictive accuracy while lower-bounding independence and calibration. The latter is the primary contribution of this work, as it formalizes and quantifies the trade-off between different fairness objectives, providing a parameter/knob to trade-off fairness enforcement with accuracy degradation.\n\nThe authors define the EO_k statistic and analyze its theoretical properties, including connecting it to Maximum Mean Discrepancy (MMD) of reweighted group distributions. The connection to MMD enables a closed-form approximation of the heuristic that runs in quadratic (or linear) time; this can be added as a penalty term to your fair representation learning objective. The authors also provide a concentration inequality with error bounds that can serve as a certificate of fairness compliance. The paper does not include empirical studies, which are planned for future work.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths:\n\nThe motivation behind EO_k and the novel characterization it enables is clearly expressed. I really like how this statistic paints a clearer picture of the trade-off between the three fairness criteria (which are not achievable all at once except in degenerate cases). Your observation that independence and calibration are at odds with predictive accuracy makes a strong case for EO_k strong given its ability to accommodate predictive accuracy in a controllable way.\n\nI found Eq. 6 to be quite instructive, as it shows how satisfying EO perfectly necessarily leads to independence and calibration being violated. Therefore, to ensure some predictive accuracy, you set your \\Beta parameter to > 0. It would be helpful if you could explain in more detail how independence and calibration are violated when EO_k = 0.\n\nEO_k has a practical, closed-form approximation that can be applied as a penalty term to your learning objective. This approximation is compatible with SGD and comes with hyperparameter-free convergence guarantees. \n\n### Weaknesses:\n\nThe main weakness of this paper is that it has no case study or empirical validation of EO_k. There are many properties of EO_k that make it suitable for practical use: efficient closed-form approximation, hyperparameter-free convergence, controllable fairness/prediction accuracy trade-off, etc. But without any experiments, it is difficult to gauge the utility of these properties. I also think you will learn new things about EO_k when you try to apply it to a fair representation learning task. Although you make it clear that your focus is on theoretical development, even the main theory papers you build on (e.g., Dwork et al., Hardt et al.) include some case study or experiment. I'm also curious to see how the logarithmic relationship between deviation bound and input dimention plays out; you noted this as a computational advantage of EO_k over trasitional IPMs. \n\nIt would be helpful to recount the thought process that led to the design of EO_k. You partially explain this in Section 2.1, but then you immediatly jump to the \"weighted summation conditioned on Y\" without explaining how you arrived at that choice. \n\nThe \"universal approximation\" property of the RKHS (reproducing kernel Hilbert space) is quite strong, saying it can approximate any neural network predictor to arbitrary precision. Can you comment on the practical realizability of this approximation?\n\nWhich parts of your theoretical analysis would change if you went beyond binary classification to more general (multi-class) classification?\n\n### Additional comments:\n\n* It would be better to present Equations 1-3 in a more consistent way rather than attempting to follow the original formulations (e.g., the Z parameter in Demographic Parity is unused). \n* You define \"perfect accuracy\" for a predictor/classifier in two different ways mathematically (as a probability of a conditional expectation and as a probability); this adds unnecessary confusion.\n\nQUESTIONS:\n1. Is there a small case study or experiment you can run (even repeating one from prior work) that demonstrates the effectiveness of EQ_k in practice? As I mentioned above, there are many practical properties of EQ_k that would be useful to validate. \n2. Can you walk through an example of how setting \\Beta > 0 leads to independence and calibration being violated? \n3. Can you summarize how your theoretical analysis would change if you generalized it to multi-class classification?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-23"}