Rethinking Pareto Frontier: On the Optimal Trade-offs in Fair Classification
Authors: Junyi Chai
Shenyu Lu
Xiaoqian Wang
Venue: ICLR 2026 Poster (2026)
Keywords: fairness-accuracy tradeoff
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Fairness has become an arising concern in machine learning with its prevalence in decision-making processes, and the trade-offs between various fairness notions and between fairness and accuracy has been empirically observed. However, the inheritance of such trade-offs, as well as the quantification of the best achievable trade-offs, i.e., the Pareto optimal trade-offs, under varied constraints on fairness notions has been rarely and improperly discussed. Owing to the sub-optimality of fairness interventions, existing work fails to provide informative characterization regarding these trade-offs. In light of existing limitations, in this work, we propose a reformulation of the model-specific (MS) Pareto optimal trade-off, where we frame it as convex optimization problems involving fairness notions and accuracy w.r.t. the confusion vector. Our formulation provides an efficient approximation of the best achievable accuracy under dynamic fairness constraints, and yields systematical analysis regarding the fairness-accuracy trade-off. Going beyond the discussion on fairness-accuracy trade-offs, we extend the discussion to the trade-off between fairness notions, which characterizes the impact of accuracy on the compatibility between fairness notions. Inspired by our reformulation, we propose a last-layer retraining framework with group-dependent bias, and we prove theoretically the superiority of our method over existing baselines. Experimental results demonstrate the effectiveness of our method in achieving better fairness-accuracy trade-off, and that our MS Pareto frontiers sufficiently quantify the two trade-offs.

Published as a conference paper at ICLR 2026
RETHINKING PARETO FRONTIER: ON THE OPTIMAL TRADE-OFFS IN FAIR CLASSIFICATION
Junyi Chai, Shenyu Lu & Xiaooian Wang"
Elmore Family School of Electrical and Computer Engineering Purdue University
West Lafayette, IN 47907, USA
{chai28, 1u Joywang) @purdue. edu
ABSTRACT
Fairness has become an arising concern in machine learning with its prevalence in decision-making processes, and the trade-offs between various fairesss notions and between fairness and accuracy has been empirically observed. However, the inheritance of such trade-off's, as well as the quantification of the best achievable trade-offs, i.c., the Pareto optimal trade-offs, under varied constraints on fairness notions has been rarely and improperly discussed. Owing to the sub-optimality of fairness interventions, existing work fails to provide informative characterization regarding these trade-offs. In light of existing limitations, in this work, we propose a reformulation of the model-specific (MS) Pareto optimal trade-off, where we frame it convex optimization problems involving faimesss notions and accuracy w.r.t. the confusion vector. Our formulation provides an efficient approximation of the best achievable accuracy under dynamic fairness constraints, and yields systematical analysis regarding the fairness-accurace trade-off. Going beyond the discussion on aacurrcy trade-offs, we extend the discussion the trade-off between fairness notions, which characterizes the impact of accuracy on the compatibility between fairness notions. Inspired by our reformulation, we propose last-layer retraining framework with group-dependent bias, and we prove theoretically the superiority of our method over existing basclines. Experimental results demonstrate the effectiveness of our method in achieving better fairness- accuracy trade-off, and that our MS Pareto frontiers sufficiently quantify the two trade-offs.
INTRODUCTION
As machine learning becomes increasingly integrated into various socictal domains, growing concerns have emerged regarding fairness issues (De-Arteaga et al.] 2022] Barocas al.) [023) Chen et al.) 2023a), where models can potentially reflect or perpetuate real- woold discrimination without proper regularization. Work has shown that fairness cannot achieved simply through unawareness of sensitive information (Pessach & Shmuelij 2022 Mehrabi et al., (021). owing to the implicit correlation between input features and sensitive information. Consequently, multiple fairness criteria and fairness interventions (Hort et al. 2022 Li al. 2023a Pessach & Shmueli, (023) have been proposed to quantitatively measure and rectify the disparities between sensitive groups.
Empirical observations indicate that improvements in fairness may be accompanied by deterioration in utility, leading to the accuracy trade-off (Zliobaite) 2015)) Menon & Williamson, 2018))
Zafar al. 2019). It has been shown that there exists inherent trade-off between demographic parity (DP) (Dwork et al. 2012) and accuracy, under the variations in base rates of different groups (Zhao & Gordon 2022). Recent work demonstrates the intrinsic trade-off in fair representation
learning, where the removal of sensitive information can be detrimental owing to its correlation with downstream tasks (Li al. 2023b). Despite the insight of such theoretical analysis, fails to quantify such phenomena under dynamical setting, i.e., how the accuracy changes as the constraint on fairness changes. In addition, existing work has not yet theoretically characterized the trade-offs
Corresponding author.

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between fairness and accuracy for error- based fairness notions, including equal opportunity (EOp) and equalized odds (EOd) (Hardt et al. 2016). Dutta et al. (2020) suggests that there is no necessary trade-off between error- brsor-baeed fairness notions and accuracy; however, the analysis requires strong assumptions regarding data and the classifier, hindering the generalizability to real-world scenarios.
Furthermore, existing work has pointed to the tension in enforcing different fairness notions simul- taneously, resulting in the trade-off between fairness notions. Different notions of fairness have been shown conflict with each other, given the variations in base rates (Kleinberg et al. [0166] Chouldechova 2017). Such conflict is also known as the 'impossibility results'. Going beyond
the impossibility, more recent work (Reich & Vijaykumar, 2202 Gultchin al.] 2022) states the
possibility of achieving various fairness notions concurrently, albeit with certain levels of faimess violations. Nonetheless, existing discussion has yet to quantitatively characterize the inherency of the trade-off between fairness notions. Moreover, while the correlation between such trade-offs and accuracy has been recognized in existing literature, remains unclear how the trade-off changes as the accuracy values differ, impeding the discussion on balancing between fairness notions.
Following existing literature (Martinez et al. 2020 Wei & Niethammer, 2022), for a specific network structure, we seek to find the best achievable accuracy under different fairness notions and under varied fairness constraints, also referred to as the model-specific (MS) trade-off. For instance, when choosing ResNet-50 as baseline, the 'model-specific" refers to all fairess interventions applied to ResNet-50. We refer to Section for a detailed discussion. This concept is also known as the Pareto optimality (Hochman & Rodgers 1969), where action or allocation is available that makes one individual better off without making another worse off. And the corresponding fairness-accurace curve by varying the fairness constraints is referred to as the Pareto frontier. Work including (Kim) et al. 2020; Jang et al. (202)) first addresses such topics by solving the MS optimization problem based on a post-processing framework (Hardt et al., (016) to approximate the MS Pareto frontier. Curves of methods that are located in the upper right region under the frontier correspond to higher values accuracy and lower values in fairness discrepancy, i.e., a better aairuac trade-off,
as shown in Fig. 2
Although it provides realistic quantification regarding the fairnes-- accuracc trade-off, there are several issues with the analysis. First, due to the potential suboptimality of post-processing, the obtained frontier can be suboptimal, with accuracy values along the curve falling below the best achievable. This leads to troublesome usage of the MS frontier, the curves of different methods can go beyond the frontier. We refer to Fig. 2 for the demonstration. Second, since the post-processe processsin framework is constructed by random flipping, the feasible regions are empirically determined by the confusion matrix of each sensitive group. This makes post-processing on testing data problematic (Kim et al. 2020 Jang et al. 2022): owing to the discrepancy between training and testing data, the performance on training data and testing data may exhibit variations, leading to infeasible solutions for post-processing on training data, as shown in Fig. | Consequently, the obtained frontier does not lead to meaningful upper bound regarding either the prot-procsssing method or alternative faimess interventions. Recent work characterize the MS Pareto frontier via an in in-processing intervention (Dehdashtian et al. 2024). However, as pointed out by the author, the corresponding MS frontier is suboptimal, leading to improper assessment the Pareto optimal trade-off. Wang et al. (2024) proposes approximate the model-agnostic (MA) Pareto frontier by directly optimizing the confusion matrix, corresponding to the Bayesian optimal classifier. However, obtaining the Baycsian optimal classifier can be very challenging, and the dependence the estimation of joint posterior probability can lead to large variations owing to the uncertainty involved. Furthermore, existing work on the Pareto optimality has been limited to the fairness-accuracy trade-off, whereas the Pareto optimal trade-off between fairness notions has not been properly addressed.
In this work, we propose unified framework obtain the MS Pareto optimal frontiers regarding both the fairness-accuracy trade-off and the trade-off between fairness notions. We theoretically discuss the suboptimality in (Kim et al. 2020 Jang et al. (022),, where we show the existence of better MS frontier under mild assumptions. Following the discussions, we reformulate the Pareto optimal trade-offs as constrained optimization problems w.r.t. the confusion vector without reliance on specified faimesss interventions or exterior estimation tools, bypassing the problem of algorithmic optimality. Based on our reformulation, we propose fair retraining for the last layer by adjusting the last linear layer using group-dependent bias. We summarize our contribution as follows:
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We theoretically discuss the suboptimality of the existing MS frontier and propose refor- mulation of the MS Pareto frontier.
We extend our formulation to the trade-off between fairness notions, where we show the dependence of the compatibility of faimess notions accuracy.
We propose novel framework for retraining the last layer, and we prove theoretically the superiority of our method over post post-processing baselines. Moreover, we show theoretically how our method deviates from the Pareto frontier, validated on real-worle datasets.
We validate from experiments the optimality of our MS Pareto frontier, as well as the superiority of our fair retraining framework in terms of haimess-accuragy trade-off on four benchmark datasets for both binary and multi-class classification tasks.
2 RELATED WORK
Fairness Notions and Trade-offs in Fair Clas- sification. There are three widely adopted fair- ness notions in the community: individual fair-
ness (Kusner al. 2017 [[vento 2019 Mukher- jee et al. 2020). group fairness (Chouldechova,
2017; Bellamy et al. 2018 Narayanan 2018
Goad0 tatigg Coan lnfing
and mini-max fairness (Martinez et al., 2020
Diana et al. 2021 Abermethy et al. [2022] Yang
al. (2023). In this study, explore group fair- ness criteria, specifically focused on DP (Dwork
al. 2012) and EOd (Hardt al. 2016). DDP is
defined as ý II o, where i is the prediction of classifier and denotes the sensitive attribute, and EOd is defined by ū IT a where is the ground truth label.
Figure Demonstration of the infeasibility of post-processing testing data. The feasible re- gions for post-processing on training and testing data specified by the triangles of different col- ors. When the solutions on testing space fall into the non-overlapping parts between training regions and testing regions (the gray regions the fig- ure), they cannot be achieved by the same post- processing techniques on training data.
Existing work empirically observes that achiev- ing a certain level of faimess typically comes at the cost of accuracy degradation (Calders et al. Zafar al. 2017 Menon Williamson, 2018 Barlas et al., 2021p. Ib, and achieving cer-
2009
tain fairness notion comes at the cost another fairness notion (Kleinberg al. 2016 Choulde- chova 2017). ). In response, Kim et al. (2020);
Jang et al. (2022); Dehdashtian al. (2024); Wang al. (2024)
the Pareto frontier to characterize this trade-off. Xian et (2023) states the inherent trade-off in achieving DP in fair classification.
Contrasting with these findings, some studies argue that enforcing fairness in machine learning
models does not necessarily reduce utility (Wick et al. 2019 Dutta et al. (020). and the possibility
of improving several fairness notions simultaneously (Reich Vijaykumar 2021 Gultchin al.) 2022).
Posc-Processing and Last-Layer Retraining in Fair Classification. Fairness through post- processing is prevalent approach in addressing bias in machine learning models. This method involves adjusting the predictions of pre- pre-trained network meet specified fairness constraints, typically through random flipping (Hardt et al. 2016) and adjusting thresholds (Corbett-Davies et al.
2017
Menon & Williamson 2018| Jang al. 2022 Xian al. 2023).
2022 observes that re-training the last linear layer could efficiently correct bias learned by an ERM model. Building on this, LaBonte et al. (2023) suggests that retraining the last linear layer with misclassified data could correct biases inherent in the network. Following this line of research, Chen et al. (2023b) proposes a strategy that leverages machine unlearning. specifically applied the last linear layer, effectively eliminate the learned biases, Mao et al. (2023) proposes a last-layer retraining framework incorporating fairness regularizations and reweighting.
Recent work (Menon et al. 2020
Kirichenko et al.
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Oaa
Base rate of group a.
B
Proportion of samples from sensitive group a = 1.
Z
Confusion vector.
A:
Matrix for calculating accuracy from the confusion vector.
App;
AP
Matrices for calculating DP from the confusion vector.
ABod
Matrix for calculating EOd from the confusion vector.
Zb
Confusion vector obtained by vanilla training.
2
Confusion vector obtained by vanilla training group
Table 1: Summarization of notations.
PROBLEM FORMULATION
Let (T,, yr, a,), 1 be the training set where I; E RRd is the input feature, y4 E {0, 1} the label and 0; 10, the sensitive information, let fo Rd - [0, 1] be the function of classifier with 9 the parameters, a fair classification problem can be formulated as
arg min Lete, s.1. faar & c,
where Eds = + Ca is the average classification loss and Lyair N Di=1 (Fai Uc, 0) is the fairness constraint. Let {(x,,, yl, a.), < n) be the test- ing set, when Lets is chosen as the classification error and Lfair is chosen as a fairness notion, given a fixed classifier structure f and its associated feasible hypothesis space H'' H specified by the training data, the following reformulation of equation] corresponds to the MS Pareto optimal fairness-accures accuracy trade-off:
arg min Ciser 8.1. Ct fair & BEN
where Ciie- - (as ut: 0) and E. fir + Cfair (2 af: We defer the exten- sion to multi-class classification to Appendix 12 The MS trade-off between fairness notions can be similarly formulated based on equation 2 Given the feasible hypothesis space H' and two fairness notions Laer and farr we aim to find the best achievable measure w.r.t. one faimess notion under the designated sacrifice the other fairness notion and in accuracy on the testing set:
arg min flairr s.1. Cjjir & e, Lise & 4 DE7
(3)
CHARACTERIZING TRADE-OFFS IN FAIR CLASSIFICATION
4.1 PRELIMINARIES
Inspired by (Kim et al. 2020), observe that accuracy, well various notions of fairness, be represented as linear transformations of the confusion matrix, allowing us to rewrite the intractable problem in equation 2 into simple convex optimization problems. Let M,, be the confusion matrix of group and z TPRR, TNR,, TPR;, TNR, J' be the corresponding confusion vector by vectorizing Mg, Let a, := Pr(Y 1
be the base rate of group a and == Pr[A- 1] be the marginal distribution of sensitive information, we show below reformulations of standard concepts of group fairness and accuracy:
Accuracy can be written as Acc A.z, where A. = [[ao(1 B)) (1) (3 013, (1 -a1) encodes the marginal distribution of data.
DP can be expressed as DP = |Arzz + Abp - z)), where App [0, (1 ao); 0, -(1 - an)]
Jau, 0, -2,, 0] and Abp =
0 -1
EOd can be formulated as EOd |ABou where A
0
Owing to the intractability of 0 loss, it is hard to directly obtain the Pareto optimal trade-off in
equation 2 Alternativlly, Kim al. (2020) reduces the Pareto optimal problem in equation 2 to
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post-processing pre-trained classifier testing data:
arg min(1 - A. + A ||||
where KC corresponds to the group-dependent parallelograms on the FPR-TPR plane constructed by random flipping (Hardt al. 2016) and À is the tunable hyperparameter. While such relaxation provides tractable solutions for quantifying the EOd-accuracy trade-off, it fails to quantify the Pareto optimal trade-off between fairness and accuracy, as discussed in the following lemma:
Assumption 1.
The group-dependent ROC curves are concave.
Lemma Under assumption [ let & be the feasible region constructed by combining random Slipping and threshold adjustment, let &, (F)..(T) (PR,(T) be the function of ROC curve for group a where T E [0, 1] is the decision threshold, for any & obtained by equation there always exists a strictly better solution 2 within K:
A < A.2,
| = | ||, & = 2 + + (0,6,0,8),
where 6 = min 0 (11,0,0,0Ez) (1 - 1,,1,0000 0/2)
| 0, ([0, -1
[0, 0,0, 1]2)[}.
We defer full proof to Appendix 14 Regarding Assumption 1 we make this assumption out of the property that the optimal ROC curves of a classifier are expected be concave (Bradley 1997). In practice, when the ROC curves are not strictly concave, we still observe a better trade-off under such construction. We refer to Fig. for the empirical validation. Consequently, the MS frontier by equation 4 does not provide accurate approximation of the MS Pareto optimal fairness- accuracy trade-off, where curves of different fairness interventions can possibly go beyond the frontier, and therefore shall not be thought of as proper baseline for evaluating the fairness-accuracy trade-offs.
4.2 PARETO OPTIMALITY IN FAIRNESS-ACCURAL TRADE-OFF
The MS fairness-accuracy
trade-off in equation arg max A.z, s.t. |.t.||Anp ||aoo/l|1 < C, arg maX A.z, s.t. Appz + App -]]] < c, where Ag. Agod. App and App are determined by the marginal distributions on the testing set, rather than the training set. Owing to the non-linearity between z and 0, we instead seek to find the supremum of the MS Pareto optimal trade-off. We first state observations that help tighten the upper-bound:
can be rewritten as
Optimality of accuracy in vanilla training. Regarding fairness interventions, imposing non- con-trivial constraints shall result in no better classification loss compared with vanilla training. Consequently, let 24 be the the confusion vector obtained from the baseline model, have the following upper- bound regarding z: A.z & A. (0), i arg ming 1/ ΣN Cete (24, 9:).
Optimality of group-dependent accuracy. The classification accuracy within each sensitive group shall be upper-bounded by the accuracy under vanilla training, and imposing fairness interventions on sensitive groups shall not lead to improved in-group accuracy. Owing to the potential overlap of information between different sensitive groups, we consider two different strategies for upper- bounding group- denenden accuracy, where the supremum is taken over the resulting accuracies: vanilla training on the corresponding group alone; 2. vanilla training on the whole training data. Let 25 [TPR,, TNR., O, 0] be the confusion vector of group and A [a,, (1 aa), O, O), we have the following upper- upper-bound bound regarding group-wisc accuracy: AAz & A. Açz
max ( (0), A. (0*) where 6 = arg min, [(H);=a]] E(ro,=u) Lets (I4,
Based on the constraints on group- wise accuracy and overall accuracy, we reformulate the optimiza- tion problem w.r.t. 0 into the convex optimization problem w.r.t z as follows:
arg max A.z, s.t. ||ABoa|]1 <.c.
zEK arg max A. s.l. ApPZ + Abr(1 - 2) E zEK
where : {z|0 < z < 1; 1, A.z > A.z] specifies the feasible region of z. Since we do not impose any specific fairness relaxations on f, formulation naturally corresponds to the
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supremum of the Pareto optimal fairness- accuracy trade-off. The values of and z determined under multiple randomized initializations of f to approximate the best achievable testing accuracy within H' specified by training, rather than testing data.
The reformulation in equation 5 provides us with a quick estimation regarding z due to the low- dimensionality.. Specifically, under each inequality constraint 6; E E = [+10 & T) uniformly sampled within the interval [o, 1] where 1 is the sampling interval, we solve the convex optimization problem in equation 5 to approximate the best-achievable accuracy A.z, under such faireess violation. The corresponding (G;; Azz,'s s then form the Pareto optimal frontier on the aacuueccy plane.
4.3 PARETO OPTIMALITY IN THE TRADE-OFF OFF BETWEEN FAIRNESS NOTIONS
Following our discussion the fairness-accuus accuraac trade-off, reformulate the optimization problem in equation w.r.t. z. Specifically, regarding the trade-off between DP and EOd, we characterize the problem follows:
arg min ADpZ Ap z) s.t. || <, Acz n. zeK
By varying e and 7, we are able to approximate the best-achievable trade-off between fairness notions set of contours the BOd-DP plane. Owing
under different accuracy values, corresponding the dependence on A.z, one natural question is, under what accuracy does the trade-off between DP and EOd vanish? The following lemma states the incompatibility of achieving both fairness notions simultaneously:
Lemma 2. Ifoo j on, the optimal classifier that achieves both DP and EOd will always have the same accuracy as constant predictor.
We defer full proof to Appendix 15 Lemma points to the inherence the EOd-DP trade-off: consequently, when evaluating fairness interventions, instead of expecting small violations in both DP and EOd, a more reasonable evaluation can be based on the Pareto optimality in the EOd-DP trade-off, where curves of methods that lie closer to their Pareto optimums indicate better improvement in fairness. We refer to Section 6.3 for the detailed discussion.
FAIR RETRAINING
5.1 OPTIMIZATION PROBLEM
Inspired by our formulation of the MS Pareto frontier, seek to directly optimize over the fairness notions so as to achieve better trade-offs. Specifically, the Lagrangian function of equation 5 can be written
LEOu A) = A. = (A (A Ac), (Dr(2, A) A. - AA Ap 2) Ac).
For chosen classifier structure f, we may think of it as the composition of two separate models: the encoder R - Th, which maps the input feature to the latent space, and linear classification head 8 + [o, 1]. which maps the latent representation to the soft prediction. For a pre-trained fixed encoder g. the optimization objective becomes much simpler owing to the linearity of the last layer. Nonetheless, the density estimations in high-dimensional latent space could still be challenging. We therefore consider the following two-layer reformulation of h: linear projection head h Rm - dim which projects the latent representation to a low-dimensional space, and linear classification head ha IR classification head /22 while the projection head remains frozen.
[0, 1]. During retraining, only update the linear
Inspired by work threshold adjustment (Jang et al. 2022), seek to further increase the degree of freedom when updating the classification head. Let [ = h g(x;) be the output of h and & be the input feature of h22, We consider the alternative modelling of the projected data:
[+0 #!, [a; = 0J, 1 [a; = 1]].
Such formulation allows us to apply group-dependee bias terms, which correlates with tuning group- dependent thresholds. Correspondingly. let [wo, bn, by] be the parameters of he where b, represents the

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group-dependent bias, let Pya be the estimated distribution of & in the subgroup {ilys y, a}. we can approximate z as the integral form: TPR, = - 2+0. Plas TNR, = Sws+t. <0 Poa- And the optimization problem regarding f22 can be formulated as arg max Ace(Sya: w; bo, b1) XEOd pSaa: w; bu, b1);
(6)
arg max Acc (f w, by: b1) ADP yaa w, by b1).
bybu,b2
Let {w", bn} be the parameters of classification head by the baseline model, when the normal vector of h2 is chosen as w", our formulation is equivalent as thresholding (Jang et al., 2022):
h2x3);) a(w" x; bg) = (h5h(( + Ca). where ea by b" is the group-dependent threshold
and or (z) is the sigmoid function.
5.2 THEORETICAL ANALYSTS
While our retraining framework naturally includes thresholding, its connection with post-processing under random flipping (Kim et al. 2020) remains unclear. We 'll focus on the EOd optimum, which quantifies the best achievable accuracy under zero EOd:
Definition The EOd optimum r of a fairness intervention corresponds to the between-gre group intersectional point in FPR TPR plane with the optimal accuracy (e.g., the red star in Fig.
max Acc, s.t. EOd 0,
rRER
where R2 is the feasible region of the fairness intervention in FPR TPR plane determined by the training data.
And the following theorem states the superiority of our EOd optimum over alternative basclines: Theorem Under assumption ] the EOd optimum z by our method achieves strictly better accuracy compared with the EOd optimum z' by random flipping.
We defer full proof to Appendix 16 Theorem esttalishhe that, under mild assumptions, our method
achieves superior performance compared to the average performance of flipping, i.c., random flipping. Specifically, our method attains not only better near-optimality, as proved in (Jang et al. 2022). but also strictly better accuracy in terms of EOd optimum compared with random flipping.
Comparison with thresholding. Owing to the deupddeenndent bias, when choosing w w", our method recovers the same ROC curves as the threshold adjustment (Jang et al.) 2022), indicating better or comparable EOd optimum.
We move discuss the optimality of retraining framework compared with the Pareto frontier. Following conventional modelling (Sagawa et al. 20200 Yao et al.) [2022] Wang & Wang 2024), we make the following assumption regarding logits of each subgroup:
4 NM 2²), i {ily = y, a, =a}
where the logits different subgroups share same variance s" and differ only in group mean Mye-
While the assumption equal variance seems bit strong, recent work has pointed to the convergence of group variance under vanilla training (Lu et al. 2024). The following theorem quantifies the accuracy drop of our method where the EOd optimum is attained:
Lemma 3. Let ry == ag + a1 - an33, under fixed pya and 8, the deuu-dee threshold ca of EOd optimum can be written as
52
(1-7)
1
+ 2p10 M111
PoI
5
1 (1
111 + 1411
log
G
log
+
Mnn - Pol
2
P11 - Po1
2
Theorem 2. Let ls = arg min, 11 erf(x)) tanh(kr)]l and ply : G- let Zs be the v2s
confusion vector under vanilla training and x be the confusion vector of EOd optimum under our method, we have the following estimation of accuracy drop when attaining EOd optimum:
(vbb+))) 1) + (1-27) ( (e21ó c214
2A.25 -1
A A,z"
2
(62t) + 1) (c2ui + +1)
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ABE
082
Qun POU Fordlee
POU Forticy
0.02
Fordee
Bar
0:2
Dur 20U
0.00
0.1
SOLF
SELT
6:4-
5141
036
16'
(a) Adult
(b) COMPAS
(c) CelebA
Figure 2: EOd-accuracy trade-off on three datasets. 'Ours-EOd-frontte corresponds to the MS EOd-accuracy Pareto frontier by equation and 'Ours-EOd' corresponds to our framework in equation 6
0.61
012
0.64
1.10
0.38
3.22
6:8
Cui-O7
Das
6476
(a) Adult
(b) COMPAS
(c) CelebA
Figure 3: accuurcyy trade-off on three datasets. Ours--D-froon corresponds to the MS DP- accuracy Pareto frontier by equation 5 and 'Ours-DP' corresponds to our framework in equation
CEVLA
Acc 0.55
0.2
Aoc 0.555
365
F00
8:0
RORLOP AE
0.50
0.55
03.04
0.855
Enseiin
Base ine
0.02
030
2hi
10
204
Adult
(b) COMPAS
(c) CelebA
Figure 4: EOd-DP trade-off on three datasets under different accuracy values. The Pareto optimums form a decaying curve on the EOd-DP plane, and the triangles correspond to the Pareto optimums under various accuracy values (tan for the baseline, and the others for our method).
We defer full proof to Appendix 17
Theorem 2 enables us to empirically estimate the decrease in accuracy when achieving the BOd optimum. Specifically, since our method achieves EOd optimum no worse than thresholding, by estimating Hya and using the latent representation and the readily apply Theorem ( to our retraining framework to obtain the
classification head w", supermum of the accuracy drop on different datasets, without calculating h2. We refer to Appendix 11 for detailed discussion.
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6 EXPERIMENTS
6.1 EXPERIMENTAL SETUP
We validate our method on four benchmark datasets: COMPAS Larson et al. 2016). Adult (Dua & Grafft 2017). CelebA (Liu et al. al.,2011) and Drug (Dua Graff 2017 The Drug dataset is for multi-class classification. We use accuracy as utility measure, DP and EOd as fairness measures. We
defer details of experimental setup (8) detailed results (9). ablation study (10). empirical validation
of theoretical results (II) and the extension to multi-class classification (2) to the Appendix.
We compare our method with the following related methods on post-processing and last-layer retraining: Baseline: Neural network without faimesss regularization. FACT: Post-processing by random flipping (Kim et al. 2020). This method focuses on the EOd-accuracy trade-off. Eq. Odds: Post processocs by finding the EOd optimum (Hardt et al. 2016). This method focuses the EOd-accuracy trade-off. DFR: Deep feature reweighting by last-layer retraining (Kirichenko et al.) 2022). This method focuses on the EOd-accuracy trade-off and is applied for both binary and multi-class classification. SELF: Last-layer retraining by selective fine-tuning (LaBonte et al. 2023). This method focuses on the EOd-accuracy trade-off and is applied for both binary and multi-class classification. G-STAR: posst-prcessing by thresholding (Jang et al. (022). This method focuses on both the EOd-accuracy trade-off and the DP-accuracy trade-off. FOC: Poot-progs based on the score functions (Xian et al. 2023). This method focuses on the DP-accuracy trade-off.
We consider two different approaches in comparison with our MS Pareto frontier for EOd-accuracy trade-off: FACT: The objective is optimized over training, rather than testing data. FACT + G-STAR: The boundaries of feasible regions are determined by the group-depended ROC curves, as discussed in Lemma I We do not include other frontiers for DP-accuracy trade-off or EOd-DP Trade-off as neither FACT nor G-STAR includes such discussion.
6.2 FATRNESS-ACCURAN TRADE-OFF
Results on EOd-accuracy trade-off are shown in Fig. 2 The objective of our method is chosen as equation[ojnn this part. Compared with the frontiers by FACT and by FACT G-STAR, our MS Pareto frontier shows an almost flat curve, without dramatic decrease in accuracy. Such counter-intuitive results point the non-inherency of EOd-accuracy trade-off:: since the disparities are measured by error rates, instead of the predictions alone, eliminating between-group disparities does not necessarily result in decrease in accuracy. On the contrary, neither FACT nor FACT G-STAR provides proper approximation of the Pareto optimal EOd-accuracy Tradc-off, as curves of several methods go beyond their frontiers. Compared with alternative methods, the curve of our retraining framework lies closer to the MS Pareto frontier, indicating a better EOd-accuracy trade-off. This validates the effectiveness of our group-dependent formulation for bias: compared with last-layer retraining and post-pre post-processing. our method obtains better flexibility during the fine-tuning, thereby leading to better or comparable performance in both fairness and accuracy.
Results on DP- Occcurayyy trade-off shown in Fig. The objective of method is chosen as equation 7 in this part. Compared with the EOd-acyyracy frontier, the DP-accuracy frontier shows a DP become more stringent, in line with previous
noticeable decrease in accuracy constraints work on the inherent trade-off between DP and accuracy (Zhao & Gordon 2022). In comparison, our formulation ensures to quantify the trade-off under dynamic scenarios, rather than under perfect DP only. Compared with other methods on DP, our method shows better or comparable trade-off in terms of both fairness improvement and accuracy, which validates the effectiveness of our method.
6.3 EOD-DP TRADE-OFF
Results on EOd-DP trade-off are shown in Fig. 4 When the accuracy greater than that of constant predictor, there is a clear trade-off between the two fairness notions, and as the accuracy decreases, the EOd-DP frontiers gradually approach the EOd-axis, indicating that the fairness measures become aligned and no longer present trade-off when the constraints on accuracy are sufficiently relaxed. Moreover, under each accuracy value, we observe an Pareto optimum (i.e., the inflection point along the curve), where no other points achieves both better DP and better EOd under the same accuracy.
Code available https:/ggitaaaccoomoly///feefffe

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Our observation of such Pareto optimums also indicates novel quantification for faimess inter- ventions taking into account both fairness notions. Specifically, regarding the EOd-DP curve of certain fairness intervention, we identify its Pareto optimums based the corresponding accuracy values, and the curve lying closer to its Pareto optimums implies better improvement faimess. We accordingly modify the objective of our fair retraining framework take into account both DP and EOd, and compared with the baseline, the curve of our method lies closer to its Pareto optimums, indicating improvement in fairness.
CONCLUSION
The trade-off problem in fair classification is important yet less studied. In this paper, we prove theoretically the non-optimality of existing work on the Pareto optimal trade-off, and we propose novel formulation of MS Pareto optimal trade-off regarding both fairness- aacressy and between fairness notions, which sufficiently approximates the upper-bound of the best achievable trade-off. Based on our formulation, bias, and we show theoretically the superiority of our retraining framework over post-processing baselines, as well as the deviation of our retraining framework from the Pareto frontier. We validate from experiments that our formulation leads to meaningful and informative quantification of the potential trade-offss for given classifier, and we show from experiments that our proposed retraining framework achieves better trade-offs, compared with state-of of-teoffthe-ar art alternatives. Future direction includes model-agnostic Pareto optimal trade-off and extension to the aacuuss- trade-off' under distribution shift.
propose last-layer retraining framework under group-dependent
ACKNOWLEDGEMENTS
This work was partially supported by the EMBRIO Institute, contract #2120200, a National Science Foundation (NSF) Biology Integration Institute, NSF IIS #2146091, IIS #2345235, and USDA #2023-67021-41368.
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EXPERIMENTAL SUPPLEMENT
8.1
DATASETS
The COMPAS dataset (Larson et al. 2016) contains 7,215 samples with 11 attributes. Following previous works on fairness (Chouldechova (017), we only select black and white defen- dants in COMPAS dataset, and the modified dataset contains 6, 150 samples. The goal is to predict whether a defendant reoffends within two years, and we choose race as sensitive attributes.
COMPAS:
Adult: The Adult dataset (Dua & Grafi 2017) contains 65,123 samples with 14 attributes. The goal to predict whether an individual's 's income exceeds 50K, and we choose sex as sensitive attributes.
CelebA: The CelebA dataset (Liu et al.) 2015) contains 202,599 face images, each of resolution 178 x 218, with 40 binary attributes. We choose attractiveness as labels and sex as sensitive attributes.
Drug: The Drug Consumption dataset (Dua & Grafi 2017) contains 1,885 samples with 12 attributes. The goal is to predict whether an individual has consumed Meth within one year, over one year ago or has never consumed Meth, and choose race sensitive attributes. This dataset is for multi-class classification.
8.2 IMPLEMENTATION
The network structure is chosen as MLP for Adult, COMPAS and Drug datasets and ResNet-50 for CelebA dataset with 80%-20% training-tsstingg partition of data. We repeat experiments dataset three times and report the average results. The input dimension dim of h2 set as 2 during retraining, and the hyperparameters of compared methods tuned suggested by the authors.
each
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We implement method in PyTorch 2.0.0 RTX-3090 GPU. The optimizer for ResNet-50 is chosen as SGD with learning rate 10-3 and weight decay 10 10 For MLP, the optimizer is chosen as SGD with learning rate 10" and momentum 0.8. The hyperparameter range for method is set as (0, 10).
AUXILIARY EXPERIMENTAL RESULTS
We include the detailed results in terms of EOd, DP and accuracy Tab.
Method
Accuracy
EOd
Baseline
63.581
38.8912.72%
FACT Kim et al.
2020
62.35
7.34%%.15%
Eq. Odds (Hardt et al. 2016
62.34+0.56%
9.92++1.14%
DFR (Kirichenko al. 20222
63.73+0.39%
3.34+1.52%
SELF LaBontte et al. 2023
0..68%
4.95++1.29%
G-STAR Dang et al. Ours
2022)
63.62+00.6%
2.18++0.55%
63.91
1.04:10.52%
Table 2: Experimental results of EOd-accuracy trade-off on COMPAS dataset.
Method
Accuracy
FOd
Baseline
83.16+00.30%
20.72++1.811 1.81%
FACT Kim et al. 2020
82.87=00.66%
10.29+
Eq. Odds (Hardt et al. 2016
82.12:10.77%
10.75+1.569%
DFR (Kirichenko a.
2022
83.18
3.44.1.288%
SELF LaBonte et al.
2023
83.51+1.51%
7.14
G-STAR (Jng et al. Ours
2022)
2.95 % 1..13+1.25%
83.47 -.533% %
3.1 3.11+-0.54%
Table 3: Experimental results of 6Od-accuracy trade-off on Adult dataset.
Method
Accuracy
EOd
Baseline
82.6710.63%
26.42+11.61%
FACT Kim et al.
2020
80.87+1.18%
12.29#1.12%%
Eq. Odds (Hardt et 2016)
79.56+
11.76+t
DFR (Kirichenko et al.
2022
81.24 %
4.79+11.679
SELF LaBonte cl al.
2023
81.68
4.14:10.599
G-STAR (Jng et al. 2022)
80.84+1.2794
5.45
Ours
81.53+0.77%
3.04
Table Experimental results EOd-accuracy trade-off on CelebA dataset.
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Method
Accuracy
DP
Baseline
65.58=1.21%
22.89+11.54%
FOC (Xian et al. 2023)
63.96=0.75% 6..6%% 0.644.0.64%
1.06=:0555%
G-STÄR (Jang al. 2022)
2.311:0.2%%
Ours
1.31+:0466%
Table 5: Experimental results of Daaaccuracy trade-off on COMPAS dataset.
Method
Accuracy
DP
Baseline
8..3%%
17.67+1.444 1.44%
FOC (Xian et al. 2023)
83.25+00.69
3.17+1.17%
G-STAR (Jang al. 2022)
82.20=10.44%
3.1610.93%
Ours
83.6410.72%
3.03 %
Table 6: Experimental results of DP-accuracy trade-off on Adult dataset.
Method
Accuracy
DP
Baseline
82.67+0.63%
18.62:t1.34%
FOC (Xian et al. 2023)
81.27
3.14:0065%
G-STAK (Jag al. 2022)
81.13:0.39%
5.1610.84%
Ours
81.17=0.55%
2.93:1
Table 7: Experimental results of DP-accuracy trade-off on CelebA dataset.
10 ABLATION STUDY
Regarding our retraining framework, the input dimension of h2 can vary based on the linear projection head h1. Results of varying the input dimension are shown in Tab. 8 Compared with other choices
(dim 4, dim 6), dim 2 in general provides better comparable performance in terms of fairness and accuracy, which validates the feasibility of our framework.
Method
Accuracy
DP
EOd
Ours (dim=2)
64.67:0.77%
3.39%
1.04.0..22%
Ours (dim=4)
64.24:10.40%
1.59:10.49% 4.45+t0.67%
3.79-10.22%
Ours (dim=6)
64.72+t0.33%
1.33:t0.66%
Table 8: Experimental results COMPAS dataset. The distribution of each subgroup is approximated by multivariate normal distribution.
Dataset
Method
Accuracy
DP
EOd
Adult
Baseline
85.16=0030%
17.67
20.72+1.8155
Adult
Ours (dim=2)
83.36:20.51%
1.43+
3.54
Adult
Ours (dim=4)
83.40-0.311%
5.28+1.788%
4.41
Adult
Ours (dim=6)
83.16=20.8%%
5.32+
6.23+1.86%
Table 9: Experimental results Adult dataset. The distribution of each subgroup is approximated by multivariate normal distribution.
Moreover, it is possible to multiple classification heads h2 for different groups. The optimization problem then becomes
arg max Ace(fya; wo, W1, bo. b.) AEOd(fyce Wo, W1, by b1):
Results of using multiple classification heads shown in Tab. 10 Compared with using single classification head, using group-dependent classification head achieves marginal improvement in fairness.
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Dataset
Method
Accuracy
DP
EOd
COMPAS
Baseline
63.58 % 64.67:10.77%
22.89:11.54%
38.89=2.72%%
COMPAS
Ours (w)
3.31=0.59%
1.04:10522%
COMPAS
Ours (w, 's)
64.74.1..64%
3.24:10.32%
1.02.L0.57%
Adult
Bascline
85.16:20.30%
17.67
20.72+1.819%
Adult
Ours (wr)
83.36++0.519%
1.28%
6.54+110669
Adult
Ours (wa, ()
83.47++0.33%
4.12+1..66%
3.11+0.54%
Table 10: Experimental results on COMPAS and Adult dataset. The distribution of each subgroup is approximated by 2D Gaussian. Experiments are repeated three times.
EMPIRICAL VERIFICATION OF THEORETICAL RESULTS
We estimate the accuracy drop relative vanilla training when attaining the EOd optimum using our method. Following the discussion in Theorem 2 we leverage the pre-trained encoder and linear classifier w" to estimate the values of j Results are shown in Tab. Compared to the results on the COMPAS and CelebA datasets, our method on the Adult dataset exhibits larger accuracy drop when attaining the EOd optimum under retraining, consistent with our empirical observation that fair retraining Adult dataset leads to larger deviation from the Pareto frontier compared with the other two datasets.
Dataset A,z"
A,Zz
A,
COMPAS 64.5%
1.3%
CelebA
79.73%
2.94%
Adult
80.87%
4.36%
Table 11: Estimations of accuracy drop when achieving EOd optimums on three datasets. Azz" is estimated by w(th p4) in Theorem 2
12 EXTENSION TO MULTI-CLASS CLASSIFICATION
We move on discuss the extension non- non-binary classification. Given y E k] the class label, let Accga == Pr(f(x) y. a. be the classification accuracy in the subgroup {ilm = y.a, = a} and z : AAc ..., Accian Acc.1..... Accaı] the corresponding confusion vector, let {a)a..... axa} be the marginal distribution of label in group a, the DP and EOd constraints under multi-class classification (Denis al. 202T) can be formulated as
EOd max k'E[k]
Agodæ z| c,
(8)
DP
max k k'E[k]
A
A (1 z)] &
where ABodR 10. 1y K.a 0 0..... 1 K. 11, JJ.......
0), App
[0.... ow k. OJ...., ow 1/y K'a 1]. and A = 1 +
k'[[ao..... Crgo a11...., a.1]. Regarding multi-class classification. the disparities are quanti- fied by the largest discrepancy amongst all the subgroups. Accordingly, the MS Pareto optimal fairness-accuracy trade-off can be formulated as
arg max A.Z, 8.l. |A < Vk E [k). ZER
arg max A, 8.l. ADP.N + A D 2) e, Ak' E [k), zEK
where A. = (1-3(111))00) (1-3)a00: (--3)aao.B011...... 3e11,...,,ean and the fairness constraints are taken over all the subgroups to accommodate with the maximum notions in equation g Let b = [bi....., bxal be the bias vector for group a, we similarly formulate our fair retraining framework as follows:
arg max Acc w. by. b1)
XEOd(faa: w. by. b1),
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Drug
0.72
0.705
0.70
0.700
0.60
Accuracy
Rccuracy
0.095
0.66
0.690
Ours frontier Beseline
Ours-DP-frontier
0.685
0.62
DFR
Ours-DP
0.680
10
10
EDd
(a) EOd-Acc
(b) DP-Acc
Figure 5: EOd-accuracy trade-off and DP-accuracy trade-off Drug dataset. We extend our discussion to multi-class setting compared with existing works on fairness-accuracy trade-off.
arg max Acc(S w. bo. b,) ADP(faa w. bo, b1):
Regarding multi-class classification, we do not consider the post-processing baselines for comparison as neither includes a direct extension to such tasks. We consider two benchmarks:
DFR: Deep feature reweighting by last-layer retraining (Kirichenko et al.
2022
SELF: Last-layer retraining by selective fine-tuning (LaBonte et al. 2023).
Results on EOd-accuracy trade-off and DP-accuracy trade-off for multi-class classification are shown in Fig. 5 Compared with alternative methods for last-layer retraining, method achieves better EOd-accuracy trade-off. Moreover, compared with the baseline, our method shows improvement in DP with relatively small degradation in accuracy. This validates the effectiveness of our method
under multi-class classification. Detailed results are in Tab. 12
Method
Accuracy
EOd
Baseline
70.63 67.64+0.55%
22.57111.69
DFR KKircheenoo al. 2022
6.17=1111%
SELF LaBonte et al. 2023
68. 13.55888
6.34+0.87%
Ours
5.04 1.04 %
Table 12: Experimental results of EOd-accuracy trade-off on Drug dataset.
Method Accuracy
DP
Baseline 70.63 79%
26.3442.25% 5.67+1078%
Ours 66.48:10.67%
Table 13: Experimental results of Dcaaccryy trade-off on Drug dataset.
13 EXTENSION TO ALTERNATIVE NOTIONS
Our formulation can readily generalized to alternative group fairness notions. For example, calibration withinn within-grrups notion (Kim et al.) 2020) can be formulated
CG
| A1 2 21))[l,,
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where
1
U1
0
Aco
1 V0
20
0
0
1
U1
V0
-V0
Too
(1
3)
0
0
0
0
(1
(1)
(3)
Ao
0
0
0
0
0
0
0
(1
a1)
0
000 (1
()
0
0
(1
Crg (1 - 8)
0
0
0
A
0
a1
0
(1
a1)
0
Specifically, 0a Pr[Y 1/A and Pr(A 1]. and vo. vı score representatives such
that O & wo, 01 < 1.
14 PROOF FOR LEMMA 1
Proof. Let ú be the original prediction and ÿ be the flipped prediction, let mea, be the measures under flipping and mea, be the measures without flipping, have
P = 1/y, a] = Plù = 1/0 1, P = lly. + Plù = 1/ú = 0, a) Plù 0ly, a].
Therefore, for y = 1, we have
TPR, = P y = 1yy = 1,a|= Pl = 11 = TPR. + Plū = 119 = 0, aNJR..
And y = 0, we have
FPR., Plù = 0,a] Plū = 10 FPR. + Plÿ = 1 = 0, a]TNR
Consequently, the feasible region of (Kim et al.) 2020) is determined by the group-dependent parallelograms in the FPR TPR plane with the vertices as
{(0,00 0), (FPRo, TPRo), (TNRo, FNR;), (1, [1, for group 0, {(0, 0), (FPR,, TPR;), (TNR), FNR1), (1, 1)]; for group
where (FPR,, TPR,, TNR., FNR,) are obtained from the baseline model. We first make few simplifications regarding the feasible regions the parallelograms.
Proposition 1. The (FPR,, TPR,) solution pair under random flipping always fall into the upper-half of the parallelograms.
For any {ro, ) = (FPRR.. TPR.), & & 1} pair in the lower-half of the parallelograms, since solutions lie below the line segment TPR = FPR, we have
FPR. TPR,, Va.
Consider the symmetric point rå of r. about the line segment TPR FPR. have r (TPR,, FPR.). Consequently, by equation 9 we have
EOd(r.) |TPR, TPR FPPR. FPR; FOO(ra);
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training
TPR
Group training
FPR
Figure 6: Demonstration of the (FPRg, TPR,) solution. The solution pair in the upper-hal half regions indicates better accuracy (higher TPR and lower FPR) and same EOd owing the symmetry.
Acc(ra) = (1 () + (1 - ao) TPRo)) + ((a + (1-ai) TPR;)) > (1 (oo + (1 (1) - FPR,)) + 3(01 TPR; + -1 (1-01)(1 -FPR;)) = Acc(ra)-
This shows the existence of strictly better solutions in the upper-half of the parallclograms under the symmetric construction, leading to the same EOd but better accuracy. as demonstrated Fig. 6
Proposition 2. At least one of the (FPRe, TPR,) solution fall on the boundary of the parallelograms.
Group 0, training
TPR
Group training
FPR
Figure 7: Demonstration of translation along the FxRs. The solution pair after translation obtains better accuracy (same TPR and lower FPR) and same EOd owing to the equidistant translations.
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For any (ro,ri) (FPRR,, TPR.), & 1) pair where both ro and r1 lie within the group- dependent parallelograms, we consider the following equidistant translation along the FPR-axis:
FPR. TPR,, FPR, - FPR (TPR, -PRR.) FPR, TPR,
r/ = (FPR, TPR.), 5 min FPR,
TPR.
Under such translation, least one of the solution ra touches the boundary of its parallelogram. This leads to the same EOd but better accuracy, owing to the decrease in FPR. Therefore, any solution pair that lie within the group-dependent parallelograms will be sub-optimal, as shown in Fig. 7
Group 0, training
TPR
Group 1, training
FPR
Figure 8: Considering the combination of random flipping and threshold adjustment, for any solution pair obtained by random flipping, there always exist a strictly better solution under equidistant translation along FPR-axis, where at least one of the new solution points touches the group-dependent ROC curve.
By Prop. [ andd we may simplify the discussion to the boundary of different fairness interventions. Since the ROC curve of each sensitive group is concave, let ROC, be the ROC curve group a, consider the combination of group-dependent thresholding and random flipping, the feasible region for each sensitive group now becomes a convex hull with its boundary the ROC curves, rather than line segments. Moreover, since both the ROC curves and the group-dependent parallelograms pass {(PPR, TPR,), (TNR,, FNR,)) } owing the concavity of ROC curves, for any solution pair (FPR,, TPR.) obtained from FACT Kim et al. 2020). we can always translate them along the direction of the negative half of the FPR axis up distance 8 such that at least one of the solution touches the ROC curve, as shown in Fig.
8 = min 05 (TPR)) FPR ' FPRR
Consequently, regarding EOd, we have
|TRR, TPRil + FPR, - FPR1 = TPR,, TPR. + (FPR, - 5) - -(FPR) - 8)}-
And regarding the accuracy, we have
(1 TPR (1 a1 FPRo)) 3(a, (1 a1) -FPR)))
< (1 - (aa + (1 a0) - FPR, + 8)) + ((1 TPR; + (1 - a1) -FR, + 8)).
Therefore, such translation preserves the EOd gap but improves accuracy, thereby indicating a strictly better EOd-accuracy trade-off than (Kim et al. 2020)
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15 PROOF FOR LEMMA 2
Proof. Consider the confusion vector 2 [TPRo: oNRo, 1PR1, TNR1]". Under zero violation in EOd, we have
TPRo 1PR TPR, TNR, TNR: TNR.
Correspondingly, let ag = Y 1|A = a| be the base rate of group a, under zero violation in DP,
have
Ouy TPRo + (1 - av) TNR,) = 01 1PR, + (1 - (1) -NR1).
(10)
Since ao # a1, we can further simplify equation 10
TPR
TNR 1.
Let a = Pr|Y = be the base rate of testing data, we have the accuracy under zero violation of DP and EOd as
Acc TPPR (1 a) TPR) (2a 1)TPR (1 a).
This shows that under zero violation of DP and EOd, the accuracy can be written as a linear function w.r.t TPR. When 0.5, the best accuracy will be a, which corresponds an all-positive predictor.
When a < 0.5, the best accuracy will be (1 a),, corresponding to an all all-neeative predictor. O
16 PROOF FOR THEOREM
l2
TPR
4
Group Group
Lo
FPR
Figure 9: Demonstration of the feasible regions by random flipping. The line segment Io passes through (0, 0) and (FPRg, TPRo), and line segment passes through (FPR,, TPR,) and (1, 1).
Proof. Let a := Pr(y be the base rate of training data, the EOd optimums under different accuracy values can be expressed line segments in the FPR TPR plane:
Acc (1 -a
TPR-
FPR
We denote such line segment passing through the EOd optimum by random flipping l2. When & 3, so long as the EOd optimum by random flipping achieves better accuracy than constant prediction, i.e., Acc > (1 a), the intercept of la will be strictly positive, and the slope of l2 is strictly smaller tham lg- When
2: when setting FPR 1, have TPR Ase 1, which indicates that the slope of l, shall be strictly greater than lo as shown in Fig. 99] Consequently, l2 always fall within the region between and I, where the slopes of line segments passing through the
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EOd optimum by random flipping smaller than lo but greater than 41- Regarding the EOd optimum by threshold adjustment, since it always fall within the upper-left region of the EOd optimum by random flipping, i.e., the shady region in Fig. [ proved in Theorem 5 of (Jang et al. 2022). it always lead to better accuracy compared with the EOd optimum by random flipping as a remains constant. Moreover, the EOd optimum achieves better accuracy than our method, we thereby conclude that the EOd optimum r" by our method achieves strictly better accuracy compared with r'a by random flipping.
17 PROOF FOR LEMMA 3 AND THEOREM 2
Proof. Given group mean Pya and variance s², under group-dependent thresholding, have the group-wise accuracy as
Ca Mya
erf
for y =
V2s
Accya
1
Ca
Mya
erf
for y =
2s
Since remains constant before and after thresholding, to achieve zero EOd, we have
Co M110 Ci Pil
Co = cı - P01,
which simplifies to 1410 - M111 = Moo Mul. Let aa == Ply
a] be the base rate within
group and let B : P(A = 1J. we have the accuracy under post-processing as
Acc =a1 BAccI1 11
(1
ai 3 + a0o11 Acc0 + (1 a0) -C)0o
=01
3
1
erf
C1 1-11-1111
1111 )) + (1-1a)) a1
1 1
2 jer
C1 M411
2
V2s
V2s
+ a0(1
3)
1 erf
(- C1 110
)) (1 -0) a1) B]
2 zerf
1
(-
C1 1400
2
V2s
V2s
Since Acc11 = Acc1n and Acco1
= Accao, we can further simplify Acc as
1
1
c
M111
) (1--a))
C1 Mol
Acc =a8
erf
erf
2
2
V2s
v2s
1
1
C1 1111
)) + (1 ao) (1 B)
1
1
( (1 P41
erf
+ erf
2
2
22s
2
2
v 22s
Taking derivative of Acc w.r.t. C1, we have
AAcc exppxp
(-(') (a--mi M11
2
(-( Pol 2
)
+ (1 -1-00 (1-a1) B1)
dc1
V22
V2s
222
a0(1 B) exp
(-(-)
M11
) + (1-a)) (1) exp
2)
exp
(-(2)
140
v 2s
22s
Setting
aAæ 8e,
O, have
(-($i-z)))
M11
2)
= = - ao + 3n - ai (113)exx exp (-(ct Hoo
ci
2
(oo - 3n + 01 () exp
v2s
2s
which simplifies to of
2 log
1-antag--op
Since oj = of + 1410 مul
an an-ay
log we have oj Moi-Moi threshold pair under post-procsssing.
tag
+
2 un tr and (cs, of) gives us the optimal
-0- -ajl-aa
2
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Accordingly, let zy be the confusion vector by vanilla training, have the change in accuracy under (c), ct) as
A.
())
- 1111
ci M41
=, [1
-11 3
V28
V28
(11)
+00 Br
c -
M10
0n) (1 3) erf
c Mo0
1 2
v2s
v2s
Owing to the intractability of enror function, we instead consider the following second-order Pade approximation (Baker Gammel 1961) error function for quantitative analysis, which yields
maximum absolute error of 0.013:
crf(x) = tanh(kx),
where k - arg min; -o erf(() - taxh(k:
Let 1y == k ا and y on + + - 0g B, let z be the confusion vector of EOd optimum under group-dependent thresholding,
22
rewrite Eq. equation II as
e-i-e-r 14 Ac Acti Ao%s - 7 eti + + (1 7) e// - 16
eH6 + e
which simplifies to
A Agt" 2A.26 - ((nbtan) - 11) + (17) (62t's e214
2
(c2to + +1) (eZMi +1)
23


Reviews:
[{"review_id": "q9FIaJcqSq", "reviewer": "Reviewer_ygdX", "summary": "This paper presents a theoretically grounded and empirically validated framework for characterizing the Pareto-optimal trade-offs in fair classification, specifically focusing on the fairness-accuracy and fairness-fairness (e.g., DP vs. EOd) trade-offs. The authors identify and rigorously address the suboptimality of existing post-processing methods (e.g., FACT, G-STAR) in approximating the true Model-Specific (MS) Pareto frontier. Their core contribution is a reformulation of the problem as a convex optimization over the confusion vector, which bypasses the limitations of specific intervention algorithms. Building on this, they propose a novel last-layer retraining method with group-dependent bias and provide strong theoretical guarantees for its superiority over existing baselines. The experimental results on multiple datasets are comprehensive and convincingly support the claims.", "questions": "Q1: In Lemma 1 and the corresponding proof, you assume concave ROC curves. Could you provide empirical evidence in the appendix (e.g., a plot) showing the ROC curves for your models on one of the datasets to validate that they are indeed approximately concave?\nQ2: The hyperparameter λ in your retraining framework (Eq. 6,7) is crucial. Could you provide more details on the sensitivity analysis? How robust is the performance to the choice of λ?\nQ3: In the multi-class experiments on the Drug dataset (Table 11, 12), why are there no baselines for the DP-accuracy trade-off other than your own method? Is it because the cited works (DFR, SELF) do not target DP?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper correctly identifies a key flaw in prior work: the use of sub-optimal fairness interventions (like post-processing with random flipping) leads to an inaccurate and pessimistic approximation of the MS Pareto frontier. The proposed reformulation using confusion vectors and convex optimization is elegant, efficient, and provides a more meaningful upper bound on achievable performance.\n2. The theoretical contributions are substantial, and the empirical Validation is rich.", "weaknesses": "1. The entire theoretical framework in Section 4.1 relies on the assumption that group-dependent ROC curves are concave. While the authors justify this by stating that optimal classifiers are expected to have concave ROC curves, this may not hold perfectly in practice for complex, deep neural networks, especially on real-world data with noise. The paper would be strengthened by a more detailed discussion of the implications when this assumption is violated and perhaps an empirical analysis of the concavity of the ROC curves in their experiments. \n2. The convex optimization for the frontier is performed over a 4-dimensional confusion vector for binary classification. The extension to multi-class (Section 13, Appendix) increases the dimensionality significantly (k classes × 2 groups). The computational cost and feasibility for large k should be briefly discussed. Is the optimization still tractable for, say, 100 classes?\n3. In the multi-class extension (Equation 8), the definitions of A_EOd,k' and A_DP,k' could be explained more clearly. The current notation is somewhat dense and could benefit from a sentence or two of intuitive explanation regarding how these matrices capture the worst-case disparity across all classes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yil6aBEgJG", "reviewer": "Reviewer_FUh2", "summary": "This paper addresses the fundamental challenge of quantifying and optimizing trade-offs in machine learning fairness. While prior studies have observed empirical trade-offs between fairness and accuracy—and among different fairness notions—they lack a principled characterization of the best achievable (Pareto optimal) trade-offs. To fill this gap, the authors reformulate the model-specific Pareto optimal trade-off as a convex optimization problem based on the confusion vector, enabling efficient approximation of optimal accuracy under dynamic fairness constraints. Their framework also extends analysis to the trade-off between fairness notions, revealing how accuracy influences their compatibility. Building on this formulation, they propose a last-layer retraining method with group-dependent bias and provide theoretical guarantees of its superiority. Experiments validate that the proposed approach achieves improved fairness-accuracy balance and effectively quantifies both types of trade-offs.", "questions": "1. How sensitive is the computed MS Pareto frontier to the estimation of $z_b$ and $z_{ab}$? Could suboptimal $z_b$ estimates (e.g., from local minima) substantially loosen the frontier? Please report the variance across multiple $z_b$ estimates.\n2. For datasets or models where ROC curves are not concave, how often does Lemma 1 fail to hold? Are there diagnostic tools to detect when post-processing frontiers may already be tight?\n3. The Gaussian equal-variance logit assumption in Theorem 2 is convenient but potentially unrealistic. Can you provide empirical evidence (on your datasets) showing that subgroup logit variances are comparable, or quantify how deviations affect the accuracy–drop bound?\n4. What is the computational cost of the proposed approach—particularly solving the convex program for multiple $\\epsilon$ samples and $z_b$ initializations? For example, how long does the process take in the ResNet-50 CelebA experiments?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed low-dimensional convex program over confusion vectors is both efficient and interpretable, providing a principled way to estimate model-specific Pareto frontiers.\n2. The last-layer retraining approach with group-dependent biases is easy to implement and consistently outperforms post-processing and other last-layer baselines in the experiments.\n3. The paper includes solid theoretical results—such as proofs of suboptimality for prior frontiers, superiority of the proposed retraining method over random flipping, and an analytic formula for accuracy–fairness trade-offs—that reinforce the empirical findings.", "weaknesses": "1. Strong modeling assumptions: The theoretical results rely on restrictive assumptions (e.g., concave ROC curves and equal-variance Gaussian logits), which may not always hold in practice. The robustness of the approach under violations of these assumptions could be further investigated.\n2. Dependence on $z_b$ / $z_{ab}$ estimation: The convex program depends on reliable upper-bound estimates obtained from multiple training restarts. The paper does not sufficiently analyze how sensitive the resulting frontier is to these estimates.\n3. Notation overload: The manuscript uses a large number of mathematical symbols, which can make it difficult to follow. Including a concise table summarizing all symbols and variables would significantly improve readability.\n4. Experimental detail and reproducibility: Some implementation details—such as the number of random initializations for $f$, the sampling strategy for the $\\epsilon$-grid, and runtime or computational cost—are missing or only briefly mentioned in the appendix. Providing these would enhance transparency and reproducibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JFFUdLERbl", "reviewer": "Reviewer_Nxm5", "summary": "The paper studies Pareto-optimal trade-offs in the fair classification task. It argues that prior model-specific (MS) Pareto frontiers can be suboptimal and may not reflect the true best-achievable trade-offs between accuracy and fairness, or between different fairness notions. To address this challenge, this paper reformulates MS Pareto frontiers as low-dimensional convex programs in terms of a confusion-vector parameterization, extends this to trade-offs between demographic parity and equalized odds under accuracy constraints, and proposes a last-layer retraining framework that introduces group-dependent bias terms. The theoretical comparisons and empirical results on multiple datasets validate the proposed method.", "questions": "1. For Assumption 1: it is likely that the ROC curves of a classifier is concave. But it might be common that all group-dependent ROC curves are all concave. Even though there is a empirical validation, I wonder the explicitly conditions behind Assumption 1. Is it possible that if the majority of the group-dependent ROC curves are concave, then Lemma 1 is correct?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The problem formulation is clear. The formation from accuracy/fairness metrics to a linear form of a confusion matrix is clean and leads to tractable convex optimization.\n2. The proposed research extends fairness-accuracy trade-off to a general fairness-fairness analysis with accuracy constraints.\n3. The theoretical results and experimental results validate the proposed methods.", "weaknesses": "1. It would be promising to discuss more fairness metrics beyond EOd and DP.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "m6q4qIDD4i", "reviewer": "Reviewer_JZwm", "summary": "The paper reformulates model-specific Pareto trade-offs for fairness–accuracy and DP–EOd via convex optimization over the confusion vector, and proposes a group-dependent last-layer retraining method with theoretical guarantees and empirical gains.", "questions": "see above", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Clear statement and intuitive novelty;\nMultiple datasets evaluated;", "weaknesses": "The problem doesn’t have a single fix. The right mix of fairness and accuracy changes by case. Even on the Pareto front, the final pick depends on the use case, so it isn’t a standalone answer.\n\nStandard fairness contraints already let you tune the balance between fairness and accuracy. So even if this study shows a better way to optimse them together, theoratically, I don‘t think it will change much in what we got after solving the problem.\n\nPeople have talked about accuracy vs fairness a lot in past years. I don’t think this is the main gap now; some newer work even tries to separate the two.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)