Fair Decisions from Calibrated Scores: Achieving Optimal Classification While Satisfying Sufficiency

arXiv:2602.07285 · cs.LG · Submitted 2026-02-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: We're looking at the paper: "Fair Decisions from Calibrated Scores".

Jane: Here's the gist of the paper. The gist The work presents an exact solution for optimal binary classification under sufficiency, assuming finite sets of group-calibrated scores, providing a geometric characterization and a simple post-processing algorithm to attain the optimal classifier with respect to various objectives.

Tom: Feasibility in the Unconstrained Setting The paper begins by characterizing all attainable (PPV, FOR) pairs ignoring the role of A.

Jane: Geometry of the Feasible Region The joint distribution of R and Y is completely determined by p, q and the selection rate of R, µ:= P(R = 1).

Tom: Boundary of the Feasible Region The boundary of C can therefore be divided into two parts.

Paper summary: Tom: Looking at the whole paper, "Fair Decisions from Calibrated Scores: Achieving Optimal Classification While Satisfying Sufficiency," it’s clear the authors provided a very precise way to characterize the limits of what a sufficient classifier can do in this setting. They give us that geometric characterization of feasible pairs between PPV and FOR.

Jane: What does that mean for the real world, outside of just getting some mathematical proof? It sounds like they are trying to bridge the gap between perfect fairness ideals and what’s actually implementable with real scores.

Lu: The implication is that when you have multiple protected groups, achieving perfect sufficiency might be impossible if you try to enforce it across every single group at once, because the feasible regions for those groups might not even overlap.

Meng: So, instead of aiming for a single rule that works perfectly everywhere, the path forward they suggest is looking for rules that balance things differently across groups while still respecting sufficiency as much as possible.

Jane: That seems like a realistic direction. It suggests we should be less focused on finding one absolute perfect solution and more interested in how the constraints interact when we have these diverse groups in our data.

Tom: The paper gives us an algorithm, a post-processing method that uses only the group-calibrated scores to attain this optimal classifier. That’s something tangible we can use to test if our current decision rules are falling short of what's possible under sufficiency.

Lu: And they also showed that even when scores aren't perfectly calibrated, the resulting predictive parity violation is bounded linearly by the maximum calibration error, which is a very useful piece of information for real-world deployment where perfect calibration isn't guaranteed.

Jane: So, in short, this paper gives us a geometric understanding of the trade-offs and an algorithm to find a classifier that respects sufficiency under these constraints. It pushes us to consider more nuanced fairness objectives when dealing with multiple groups.

Conclusion: Tom: So we're wrapping up "Fair Decisions from Calibrated Scores: Achieving Optimal Classification While Satisfying Sufficiency." Basically, this paper gives us a way to find the absolute best binary classifier when you have multiple protected groups and you want to keep things fair in a specific way—called sufficiency.

Jane: Sufficiency, that’s the idea that if your scores already look right for each group individually, then combining those rules is actually better than just picking one rule for everyone. It cuts down on the unfairness we usually see when we try to force equality across all groups simultaneously.

Lu: The authors tackle this by looking at a geometric characterization of all possible outcomes. They map out the feasible pairs between predictive value and false omission rate, which is pretty dense mathematically, but it shows exactly what’s possible for any given set of scores.

Meng: What’s interesting for me is how they use that geometry to create a simple post-processing algorithm. It doesn't just give you a big equation; it gives you a concrete rule using only the group-calibrated scores and group labels. That sounds deployable, which is what I care about at all.

Lalam: From my perspective, the paper’s core contribution is showing that even when scores aren't perfectly calibrated—which they never are in the real world—you can bound how much unfairness you’ll get based on how far off your calibration is. It’s a practical safety net.

Tom: Exactly, because we ship models with scores that aren't perfect, this means we can quantify the error. If we know the maximum calibration error, we know the maximum predictive parity violation we can expect from our final decision rule. That’s a big deal for production monitoring.

Jane: So what does this mean for someone just listening? It means that when you're building an AI system with different user groups, you don't have to worry about finding one single perfect decision rule that works everywhere. Instead, you focus on making sure the rules work well *within* each group and then use a simple method to combine them fairly.

Lu: The main limitation they point out is that this exact sufficiency approach only works when every single group's feasible region overlaps with every other group's. If those regions don’t intersect, you can't achieve perfect sufficiency across the board with this method.

Meng: That’s a crucial caveat for me. It means if our data splits into groups that are too distinct, this specific sufficiency algorithm might not find a solution at all; we might need to relax the strict requirement of perfect sufficiency then.

Tom: Right, so it’s not about finding one magical universal rule, but about optimizing the rules locally where they make sense for each group while still keeping things honest overall. That sets us up well for looking at how these individual groups interact with other fairness constraints next.

Etam Benger, Katrina Ligett

The Hebrew University of Jerusalem

cs.LG

Submitted: 2026-02-07

Updated: 2026-05-28

Comments: Accepted to ICML 2026

Journal ref: Proceedings of the 43rd International Conference on Machine Learning (ICML 2026), PMLR 306:7516-7540, 2026

Code: https://github.com/etambenger/fair-decisions-from-calibrated-scores

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist The work presents an exact solution for optimal binary classification under sufficiency, assuming finite sets of group-calibrated scores, providing a geometric characterization and a simple

Key concepts

Sufficiency: This constraint means the classification decision for one group is entirely determined by the scores of that group, independent of other groups. In this context, it simplifies finding fair decisions because it allows for specific rules tailored to each protected group.
Predictive Parity Violation: Even with perfectly calibrated scores (true probabilities), applying a single classification threshold can violate predictive parity across different groups. The paper shows how this violation is bounded linearly by the maximum error in the calibration of those scores.
Geometric Characterization: The achievable performance outcomes (Positive Predictive Value and False Omission Rate) are mapped onto a geometric shape called a feasible region. Analyzing this region allows researchers to precisely locate the optimal classification point that balances fairness and accuracy.

Summary

The gist The work presents an exact solution for optimal binary classification under sufficiency, assuming finite sets of group-calibrated scores, providing a geometric characterization and a simple post-processing algorithm to attain the optimal classifier with respect to various objectives.

Feasibility in the Unconstrained Setting

The paper begins by characterizing all attainable (PPV, FOR) pairs ignoring the role of A.

Geometry of the Feasible Region

The joint distribution of R and Y is completely determined by p, q and the selection rate of R, µ:= P(R = 1).

Boundary of the Feasible Region

The boundary of C can therefore be divided into two parts. First, a trivial part consisting of two straight edges at p = π and q = π, implied by the constraint q ≤ π ≤ p. The boundary consists of two straight segments at q = smin and p = smax, and, if m > 2, a sequence of hyperbolic arcs on each Jk with 1 < k < m.

Fair Decisions from Calibrated Scores

In the multi-group setting, the intersection C0 ∩ C1 is the main object of interest. By Equation (8), if (p, q) ∈ C0 ∩ C1 then max〈π0, π1〉

0, for any fixed µ, the expected loss is minimized by maximizing p.

Robustness to Calibration Error

The predictive parity violation that may result from using our method with approximately calibrated scores is bounded linearly by the maximum calibration error.

Conclusion

The geometric view highlights a limitation of exact sufficiency for binary classification with multiple protected groups. A sufficient classifier exists only when all the groupspecific feasible regions have a nonempty intersection. This suggests that relaxing the strict sufficiency criterion and seeking group-specific rules that minimize the differences of PPV and FOR across more than two groups, even when those quantities cannot achieve equality, may be a promising direction for future work.

Algorithm 1 Boundary Trace

Algorithm 1 iterates over ∂(C0 ∩ C1), while keeping track of the active group and the indices k and l, and evaluating a given objective function F.

Active Boundary Check

The condition for the active boundary on Jk,l,i is given by Φk,l(p) = Ak,lp2 + Bk,lp + Ck,l ≥ 0. The optimum lies on ∂C1 but in the interior of C0; hence, the selection rule for the “black” group is a threshold, while the selection rule for the “white” group cannot be realized by thresholding alone.

COMPAS

Applying Algorithm 1 with the 0–1 loss objective yields (p, q) = (0.6446, 0.3346) and accuracy 0.6563. The optimum lies on a breakpoint of ∂C1, corresponding to the deterministic threshold that selects all individuals with decile score at least 5 in the “black” group.

ACS Income

The post-processed classifier attains average group-wise values PPV0 = 0.7758, PPV1 = 0.7768, FOR0 = 0.1545, and FOR1 = 0.1549. The mean accuracy is 0.8167 [0.8164, 0.8171].

Synthetic Experiment

The optimal fair classifier (maximizing accuracy) attains p = 0.80, q = 0.31, with accuracy 0.7239. For ten values of 0.01 ≤ ϵ ≤ 0.10, we add random uniform perturbations to the scores such that maxa∈[m] maxi∈[ma] s a i − s̃ a i = ϵ.

Improvements for AI systems

Improve classification decision making by using a post-processing algorithm that takes only a score and group membership as input to output a binary decision, which attains the optimal classifier with respect to various objectives like loss minimization or minimizing deviation from separation.
Develop systems that achieve optimal binary classification under sufficiency for finite group-calibrated scores, allowing them to satisfy predictive parity across groups by identifying the feasible region of positive predictive value and false omission rate pairs.
Implement a method to find the classifier that minimizes deviation from separation while satisfying sufficiency, which is useful in scenarios where independence and separation are generally incompatible as a single threshold cannot satisfy both.
Create decision systems that can be robust to calibration errors by bounding the predictive parity violation linearly by the maximum calibration error, ensuring fairness even when scores are only approximately group-calibrated.

Abstract

Binary classification based on predicted probabilities (scores) is a fundamental task in supervised machine learning. While thresholding scores is Bayes-optimal in the unconstrained setting, using a single threshold generally violates statistical group fairness constraints. Under independence (statistical parity) and separation (equalized odds), such thresholding suffices when the scores already satisfy the corresponding criterion. However, this does not extend to sufficiency: even perfectly group-calibrated scores -- including true class probabilities -- violate predictive parity after thresholding. In this work, we present an exact solution for optimal binary (randomized) classification under sufficiency, assuming finite sets of group-calibrated scores. We provide a geometric characterization of the feasible pairs of positive predictive value (PPV) and false omission rate (FOR) achievable by such classifiers, and use it to derive a simple post-processing algorithm that attains the optimal classifier using only group-calibrated scores and group membership. Finally, since sufficiency and separation are generally incompatible, we identify the classifier that minimizes deviation from separation subject to sufficiency, and show that it can also be obtained by our algorithm, often achieving performance comparable to the optimum.