Benchmarking Stochastic Approximation Algorithms for Fairness-Constrained Training of Deep Neural Networks
Authors: Andrii Kliachkin
Jana Lepšová
Gilles Bareilles
Jakub Marecek
Venue: ICLR 2026 Poster (2026)
Keywords: Fair Machine Learning
stochastic approximation
Augmented Lagrangian
Sequential Quadratic Programming
benchmarking
Primary area: datasets and benchmarks
TL;DR: We provide a benchmark for comparing stochastic approximation algorithms, based on real-world fairness-constrained learning problems.

Abstract: The ability to train Deep Neural Networks (DNNs) with constraints is instrumental in improving the fairness of modern machine-learning models. Many algorithms have been analysed in recent years, and yet there is no standard, widely accepted method for the constrained training of DNNs. In this paper, we provide a challenging benchmark of real-world large-scale fairness-constrained learning tasks, built on top of the US Census (Folktables,  Ding et al, 2021). We point out the theoretical challenges of such tasks and review the main approaches in stochastic approximation algorithms. Finally, we demonstrate the use of the benchmark by implementing and comparing three recently proposed, but as-of-yet unimplemented, algorithms both in terms of optimization performance, and fairness improvement. We release the code of the benchmark as a Python package at https://github.com/humancompatible/train.

BENCHMARKING STOCHASTIC APPROXIMATION ALGO- RITHMS FOR CURRNES-CCOSTRRANE TRAINING OF DEEP NEURAL NETWORKS
Andrii Kliachkin, Jana Lepšová", Gilles Bareilles" & Jakub Mareeek Artificial Intelligence Center, Czech Technical University in Prague firstname name fel. cvut.cz
ABSTRACT
The ability to train Deep Neural Networks (DNNs) with constraints is instrumental in improving the fairness of modern machine-leaming models. Many algorithms have been analysed in recent years, and yet there is no standard, widely accepted method for the constrained training of DNNs. In this paper, we provide a chal- lenging benchmark of real-world large-scale built on top of the US Census (Folktables, Ding et al. (2021)). We point out the theoretical challenges of such tasks and review the main approaches in stochastic approximation algorithms. Finally, we demonstrate the use of the benchmark by implementing and comparing three recently proposed, but as-of-yet unim- plemented, algorithms both in terms of optimization performance, and fairness improvement. We release the code of the benchmark as a Python package https:: /ttp::://gtthuut com/ http://uaanncomooo ible/trainn. train.
learning tasks,
INTRODUCTION
There has been considerable interest in detecting and mitigating bias in artificial intelligence (AI) systems, recently. Multiple legislative frameworks, including the AI Act in the European Union, require the bias to be removed, but there is agreement what the correct definition of bias is or how to remove it. A natural translation of the requirement of removing bias into the formulation of training of deep neural network (DNN) utilizes constraints bounding the difference in empirical risk across multiple subgroups (Chen et al., 2018; 8; Nandwani et al., 2019; Ravi et al., 2019). Over the past five years, there have been numerous algorithms proposed solve convex and non-convex empirical-risk minimization (ERM) problems subject to constraints bounding the absolute value of empirical risk (Fang et al., 2024; Berahas et al., 202 la; Curtis al., 2024a; Oztoprak al., 2023; Berahas et al., 202 b;; Na et al., 2023a;b; Bollapragada et al., 2023; Curtis et al., 2024b; Shi et al., 2022; Facchinei & Kungurtsev, 2023; Huang et al., 2025; Huang Lin, 2023). Numerous other algorithms of this kind could be construed, based on number of design choices, including:
sampling techniques for the ERM objective and the constraints, either the same or different; use of first-order or higher-order derivatives, possibly in quasi-Newton methods;
use of globalization strategies such as filters or line search;
use of "true" globalization strategies including random initial points and random restarts in order to reach global minimizers.
Nevertheless, there is no single toolkit implementing the algorithms, which would allow for their casy comparison, and there is no benchmark to test the combinations of design choices on.
In this paper, we consider the constrained ERM problem:
min (x {)] ZER
s.t. E[c(,, 4) < 0,
where and are random variables. Further, we provide automated way of constructing the ERM formulations out of a computation graph of a neural network defined by Pyrorch or TensorFlow, the
Equull contribution, more junior authors listed first.

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Table 1: Particular formulations of the constraint function c enforce fairness.
Model
Our formulation
Accuracy equality
R Digroop 4] [ (x)), Y
Brigroup Ri (o))) Y]]l &
Equail opportunity Hurdt et al. (2016)
[Beeerupp A.Y +] fo (X),
Erigroup B,Y [L(fo(X) YI1l S
Equalized odds Hardt et al. (2016)
Ees(t.-) group A.1' [ ((fo(X)) Y). ED group B.Y [[fo(X), ()]
choice of the constraints (sec Table 1), and definition of the protected subgroups to apply constraints to. Specifically, we provide means of utilizing the US Census data via the Python package Folktables, together with definitions of up to 5.7 billion protected subgroups. This presents a challenging benchmark in stochastic approximation for the constrained training of deep neural networks.
Our contributions. The contributions of this paper are:
literature review of algorithms subject to handling (1);
toolbox that (i) implements four algorithms applicable in real-world situations, and (ii) provides
an easy-to-use benchmark on real-world fairness problems;
numerical experiments that compare these algorithms on a real-world dataset, and a comparison with alternative approaches to fairness.
Paper structure. The rest of the paper organized as follows. Section 2 reviews related works and presents the relevant notions of fairness. Section 3 introduces the algorithms. Section 4 reports on our experiments. Section concludes.
RELATED WORK, AND BACKGROUND IN FAIRNESS
In the literature on fairness, one distinguishes among pre-processing, in-processing, and post- processing. Pre-proced pre-processing methods focus on modifying the training data to mitigate biases (Tawakuli Engel, 2024; Du et al., 2021). In-processing methods enforce fairness during the training process by modifying the learning algorithm itself (Wan et al., 2023). Post-processing methods adjust the model's 's predictions after training (Kim et al., 2019). The constrained ERM approach (1) belongs to the class of in proccesssiing methods.
In-processing methods include several approaches. One trend consists in jointly learning predictor function and an adversarial agent that aims to reconstitute the subgroups from the predictor (Adel et al., 2019; Louppe et al., 2017; Madras et al., 2018; Edwards Storkey, 2016). Another approach consists in adding "penalization" terms to the empirical risk term. These additional penalization terms, commonly referred to as regularizers, promote models that are a compromise between fitting the training data, and optimizing faimesss metric. Differentiable regularizers include, among others, HSIC (Li et al., 2022), Fairret (Buyl et al., 2024), or Prejudice Remover (Kamishima et al., 2012).
Closer to setting, Cotter et al. (2019) consider minimizing the empirical risk subject to the so- called rate constraints based on the model's prediction rates on different datasets. These rates, derived from dataset, give rise to non-convex, non-smooth, and large-scale inequality constraints akin to (I). Cotter et al. (2019) argue that hard constraints, although leading to more difficult optimization problem, offer advantages over using a weighted sum of multiple penalization terms. Indeed, while the choice of weights for the penalization terms may depend on the dataset, specifying one constraint for each goal is easier for practitioners. In addition, a penalizaaion- based model provides predictor that balances minimizing the data-fit fit term and penalties in an opaque way, whereas constraint-based model allows for clearer understanding of the model design: minimizing the data-fit term subject to "hard" fairess constraints. Rate constraints differ from those in (1) in that they piecewise-constane rendering first-order methods unsuitable for solving them. We refer to the recent work of Ramirez et al. (2025) for detailed argument on why constraining ERM problems is preferable to penalizing the ERM with multiple terms.
Major toolboxes for evaluating the faimess of models for training models with faimess guarantees include AIF360 (Bellamy et al., 2018) and FairLearn (Bird et al., 2020). Delaney et al. (2024) compute the Pareto front of accuracy and faimness metrics for high-capacity models, and Buyl et al.
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(2024) provides differentiable fairness-inducine penalization terms. We also note the recent Cooper toolbox, closest to our setting, with Lagrangian- baaee methods focus (Gallego-Posie Posada et al., 2025).
Le Quy et al. (2022) provides detailed survey of fairness-orienter datasets, and Ding et al. (2021) derives new datasets. The benchmark of Han al. (2023) reviews the existence of biases in prominent datasets, finding that "not all widely used fairness datasets stably exhibit fairesss issues", and assesses the performance of a range of in- in-processing methods in addressing biases, focusing on differentiable minimization only. Other benchmarks of fairness methods include Defrance al. (2024); Fabris et al. (2022); Pessach & Shmueli (2022); Chen et al. (2024). Statistical aspects of the faimnss--oosstrinned Empirical Risk Minimization have only been considered recentl;; see e.g. Chamon al. (2022).
The template problem (1) encompasses faimess fairness-enforeing approaches that find applications in high- risk domains, such as credit scoring, hiring processes, medicine and healthcare (Chen al., 2023), ranking and recommendation (Pitoura et al., 2022), but also in forecasting the observations of linear dynamical systems (Zhou et al., 2023b), or in two-sided economic markets (Zhou et al., 2023a). In addition, solving (1) is of interest in other fields, such compression of neural networks (Chen et al., 2018), improving statistical performance of neural networks (Nandwani et al., 2019; Ravi et al., 2019), or the training of neural networks with constraints on the Lipschitz bound (Pauli et al., 2021). We note that all the aforementioned methodologies feature large-scale constraints.
Deep neural networks (DNNs). Consider a dataset of N observations D {(x, Y), We seek some function fo such that fx(X,) 2 Y, A typical formulation of this task is the following regression problem:
E ((o(())))))) R(0).
min SER
Here, R:Rx x IR R is loss function, such the logistic loss E(y; log(1 + v=v*), the hinge loss E(y: 2) max
y2]. the absolute deviation loss {(y: z) = ly 2/. or the square loss
Ely: 2) = (y 2)². The term R is a regularizer, and fe a deep neural network (DNN) of depth L
with parameters 0. The DNN So is defined recursively, for some input X, as
a0 = X, a = p=(((V(())a,))))) for every i I,....L, L, fo(X) = aL.
(3)
where Vi(+) are linear maps into the space of matrices, and Ps are activation functions applied coordinate-wise such as ReLU max(0, 1), quadratics L2,, hinge losses max (), and SoftPlus log(1 e²). A dataset D is described by attributes (or features), such as age, income, gender, etc. The attribute which the DNN is trained to predict is called the class attribute. We denote the class attribute by Y. whereas the predicted value given by the DNN is denoted by Y. Both Y and Ý are binary and take values in {+,
Fairness-aware learning applied to DNNs. The goal of this approach is to reduce discriminatory behavior in the predictions of DNN across diffcrent demographic groups (e.g., male vs. female). The demographic groups also reffered to subgroups. The attributes such as race gender which must be handled cautiously are called protected. We denote by S the protected attribute or more generally the set of groups defined by multiple protected attributes, $1, ... Sm its possible values the indicator of membership in the groups, and D(s;] the observations in D such that S = 8;. A way to impose fairness constraint choices are shown in Table 1. Choosing loss difference bound as the constraint, denoting || Ex,reD(e) (X), Y) for i I...., m, and setting 8 yields formulation:
the learned predictor is to equip (2) with suitable constraints. Some possible
min OER*
Σ
((o(Xi) +R)
(4)
1
s.t. -858
es (0) < 6, i= 1,
Bounding the distance between subgroup losses yields constraints. Other formulations possible, such as bounding the distance between every pair of subgroup, providing simpler individual constraints, but in greater number (mm( + 1)/2). Formulation (4) extends several protected
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Table 2: Three elementary notions of fairness
Independence
Separation
Sufficiency
E, plad
-
S
Enett
psr pet 5
E, m
attributes by adding the corresponding of equations; we omit this direct generalisation for clarity, Note that we employ constraints based on the loss function, as it is continuous function, amenable nonsmooth nonconvex optimization. For constraints involving directly discontinuous quantities such accuracies and rates, see Cotter et al. (2019).
Fairness metries. There exist tens fairness metrics (Verma & Rubin, 2018). However, Barocas et al. (2023, Ch. 3) pointed out that most fairness metrics are combinations of three elementary fairness criteria: independence, separation, and sufficiency. These criteria cannot be minimized simultaneously, and there is trade-off between attaining the elementary fairmess metrics and the prediction accuracy, i.e., the probability that the predicted value is equal to the actual value. Thus, seek optimal trade-off between attaining the fairness metrics and minimizing the prediction inaccuracy. We next recall the definitions of these three relevent metrics, following Barocas et al. (2023), and provide the formula for computing them in Table 2.
Independence (Ind) This fairness criterion requires the prediction Y be statistically independent of the protected attribute S. Equivalent definitions of independence for a binary classifier Y are referred to statistical parity (SP), demographic parity, and group faimesss Independence is the simplest criterion to work with, both mathematically and algorithmically. In a binary classification
task, independence implies the equality of Pind = P( = + + = s;) for all i i.... m.
Separation (Sp) Unlike independence, the separation criterion requires the prediction Ý to be statistically independent of the protected attribute S, given the true label Y. The separation criterion also appears under the name Equalized odds (EO). In binary classification task, the separation criterion requires that all groups experience the same true negative rate and the same true positive rate. Formally, we require the equality of pSp P( = SS- s; Y v) for every 1...., , m, and 1,2
v E {+.-)
Sufficiency (Sf) The sufficiency criterion is satisfied if the true label Y is statistically independent of the protected attribute 8, given the prediction Y. In binary classification task, the sufficiency criterion requires a parity of positive and negative predictive values across the groups. Formally, we require the equality of pst
E,5 P(Y + = v. S = s). for every i = I,..., m, and v E {+,-}
ALGORITHMS
We recall that we consider the optimization problem
min F(x:)
C(x) 0
(5)
where the functions F: 1 R and C F" RT are defined as expectations of functions f and which depend on random variables and Ç. respectively. Solving (5) has the following challenges:
large-scale objective and constraint functions, which require sampling schemes, the necessity of incorporating inequality constraints, not merely equality constraints (see fairness formulations in Table 1).
the necessity to cope with the nonconvexity and nonsmoothness of F and C, due to the presence of neural networks.
In this section, we identify the algorithms that address these challenges most precisely. However, we note that there exists currently no algorithm with guarantees for such general setting.
Recalls and notation. We denote the projection of point onto set X' by (x) arg min ||x u|I2. We denote by N ~ ((p)) sampling a random variable from the geometric distribution with parameter po, i.e., the probability that N equals po)" po for 0. We
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Table 3: Assumptions on objective and constraint functions, F and C, which allow for theoretical convergence proofs.
Objective function f
Constraint function C
C. with Lipschitz VF
C(a) 0 and C(aa) <
C. with Lipechitz VC
tame loc. Lipschitz
tame Inc. Lipschiit
weakly convex
weakly convex
stnchastic
stochastic
((x)
Algurithm
SGD
(V) (V)
Berahas et al. (2021b) Fang et al. (2024) Curtis et al. (2024a)
/
/ ViC
Na at. (2025a)
ViC
SS3NN3S
Shi et (2022) Curtis et. (2024b)
S
Na al. (2025b)
/ vie))
vicH)
Bollapragada al. (2023)
/ (+v
Oztoprak et al. (2023)
SSL-ALM Huang al. (2025)
SSST
SSSI
Stoch. Ghost Facchinei & Kungurtsev (2023)
Stoch. Switch, Subg. Huang & Lin (2023)
distinguish between the random variable associated with the objective function and the random variable associated with the constraint function. Their probability distributions are denoted by Pg and Pc. For an integer J E N, a set {Ey}ý=1 of independent and identically distributed random
variables , E rdd Pg is called mini-batch. Inspired by Na al. (2023a), notation for the stochastic estimates computed from a mini-batch of size J:
the following
J
v = = j=1
(z)= ((),)))
v'c(x)=
Σ 1=1
Velx,
(6)
3.1 REVIEW OF METHODS FOR CONSTRAINED ERM
We compare recent constrained optimization algorithms considering a stochastic objective function in Table 3. We note that most of them do not consider the case of stochastic constraints. Among those which do consider stochastic constraints, only three admit inequality constraints. Moreover, with the exception of Huang & Lin (2023), all the algorithms Table assume to be at least C². which makes addressing the challenge of nonsmoothness of F infeasible. Davis et al. (2020) leads us to conclude that assuming the objective and constraint functions be tame and locally Lipschitz is suitable requirement for solving (5) with theoretical guarantees of convergence. At this point, however, no such algorithm exists, to the best of our knowledge.
Consequently, we consider the practical performance of the algorithms that address the challenges of solving (5) most closely: Stochastic Ghost, SSL-ALM, and Stochastic Switching Subgradient.
3.2 STOCHASTIC GHOST METHOD (STGH)
Facchinei & Kungurtsev (2023) propose the Stochastic Ghost method, that combines deterministic method for solving (1) (Facchinei et al., 2021) with a stochastic sampling approch for nonlinear maps (Blanchet al., 2019). The deterministic method of Facchinei al. (2021) consists in solving subproblem (7) to obtain direction d, and then to preform line search. Here, TR." is vector with all elements equal to one, and / > are user-prescribed constants, and Kj is defined as certain convex combination of optimization subproblems related to C' and VC. The definition of кą enables to expand the feasibility region so that (7) is always feasible. As the problem (I) is stochastic, the subproblem (7) is modified to stochastic version (8), using the notation in (6);
mind VF(ax) ]|
mina A f(xx) d 1ldd|? (xx) V ((c(x)) I K 'e'e, ||d] < B.
s.t. C(xx) + VC(xx) d S KAC. | & 8,
(8)
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In the stochastic setting (8), unbiased estimate d(zx)) of the line search direction d' is computed us- ing four particular mini-batches as follows. To facilitate comprehension, we denote X = {XE.l a mini-batch of size J with the j-th element Xx.5 = (vr, Ex.j), c(Tk, (k.)), Ve(r&, (t.z)). First, sample random variable N ((po) from the geometric distribution. Then we sample the mini- batches X and x²** and we partition the mini-batch x²* of size 2 into two mini-batches addd(X***) and even(XX of size 2 Finally, we solve (8) for each of the four mini-batches, denoting by d(ra; x the solution of (8) for the corresponding mini-batch XE- We obtain
d(mxi x odd(E
d(r even ())
d(xx)=
d(xxi
(9)
(o-
An update between the iterations IyK and Fx+1 1 then computed as Tk+1 = 2 + agd( where the deterministic stepsize aę should be square-summable E of co but not summable Σx= Or 00. For more details, see Algorithm 1 (Appendix C).
3.3 STOCHASTIC SMOOTHED AND LINEARIZED AL METHOD (SSL-ALM)
The Stochastic Smoothed and Lincarized AL Method (SSL-ALM) was described in Huang et al. (2025) for optimization problems with stochastic linear constraints. Although problem (1) has non- linear inequality constraints, we use the SSL-ALM due to the lack of algorithms in the literature dealing with stochastic non-linear constraints; see Table The transition between equality and inequality constraints is handled with slack variables. Following the structure of Huang et al. (2025), minimize over the set X R" I The method is based on the augmented Lagrangian
(AL) function L,(x, y) = F(x) + y C(x) + 2 ([( see e.g., (Bertsekas Rheinboldt, 2014). Adding smoothing term with an additional variable E IR yields the proximal AL function
(2, 2) Lp(2, v)
! 2
=1/
The SSL-ALM method was originally proposed in Huang al. (2025) where it is interpreted inexact gradient descent step on the Moreau envelope. An important property of the Moreau envelope that its stationary points coincide with those of the original function.
The strength of this method is that, as opposed to the Stochastic Ghost method, it does not use large did mini-batch sizes. In each iteration, we sample { ~ 'e to evaluate the objective and Ç1, S2 Pg to evaluate the constraint function and its Jacobian matrix, respectively. The function
sid
G( y. 2; G1, f
Fo(a,<1) + /el( G1) c( (2) p(x 2)
(10)
is defined that, in iteration k Eg,ci.5z [G yk+1, 24: (2) V VKp, Mk+1, 2x). Denot- ing 7, T, and positive parameters, the update is
yk+1 = 1/k: + no(x, Ct):
#k+1 (F GG(Ik, E. G2)),
(11)
Fk+1 = 24 + B(rx - zx).
For more details, see Algorithm 2 (Appendix C).
3.4 STOCHASTIC SWITCHING SUBGRADIENT METHOD (SSw)
The Stochastic Switching Subgradient method was described in Huang & Lin (2023) for optimization over closed convex set X' C. Ed which is casy to project on. It allows for weakly-convex, possibly nonsmooth, objective and constraint functions. They consider subgradients instead of gradients.
The algorithm relies on a prescribed sequence of infeasibility tolerances e and of stepsizes v/f and vgf At iteration k; we sample (1, ÇI,..., ,, sid Pe to compute e' (xx)) If o (ax) (xx)) is smaller than ék, we sample g sid Pe and update using a stochastic estimate S (x E) subgradient of F:
T*+1 projx s (x),

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not, we sample Ç sid Ps and update using a stochastic estimate (z () a subgradient of C::
2k+1 projx (x,. ()).
In either case, the updates only saved starting from prescribed index kn and the final output is sampled randomly from the saved updates. The algorithm presented here is slightly more general than the one presented in Huang & Lin (2023): allow for different stepsizes for the objective and the constraint update, while the original method employs equal stepsizes T/L Mh- For more details, see Algorithm 3 (Appendix C).
4 EXPERIMENTAL EVALUATION
In this section, we illustrate the presented algorithms on real-world instance of the ACS dataset, comparing how they fare with optimization and fairness metrics.
4.1 DATASET FOR FAIR MI.
Ding et al. (2021) proposed large-scale dataset for fair Machine Learning, based on the ACS PUMS data sample (American Community Survey Public Use Microdata Sample). The ACS survey is sent annually approximately 3.5 million US households in order gather information features such as ancestry, citizenship, education, employment, or income. Therefore, it has the potential to give rise large-scale learning and optimization problems.
In our experiments, we use the ACSIncome dataset, and choose the binary classification task of predicting whether an individual's income over $50,000.
4.2
EXPERIMENTS
Numerical setup. Experiments are conducted an Asus Zenbook UX535 laptop with AMD Ryzen 7 5800H CPU, and 16GB RAM, using Python with the PyTorch package (Paszke et al., 2019).
4.2.1 BINARY PROTECTED ATTRIBUTE
Dataset and problems We use the ACSIncome dataset over the state of Oklahoma The dataset contains 9 features and 17,917 data points, and may be accessed via the Python package Folktables. We choose race (RAC1P) as the protected attribute. In the original dataset, it a categorical variable with values. For the purposes of this experiment, we binarize it to obtain the non-protected group of "white" people and the protected group "non-white" people. The dataset is split randomly into train (80% 14,333 points) validation (10% 1,792), and test (10% 1.792 points) subsets and it is stratified with respect to the protected attribute, i.e., the proportion of "white" and "non-white" samples in the training. validation, and test sets is equivalent that in the full dataset (30.8% of positive labels in group "white", 20.7% in the group "non-white"). The protected attributes are then removed from the data so that the model cannot learn from them directly. The data normalized using Scikit-Learn StandardScaler.
Note that ACSIncome is real-world dataset for which ERM-based predictors without fairness safeguards are known to learn biases (Han et al., 2023). Accordingly, Table 4 (line 1) shows that an ERM predictor without fairness safeguards has poor fairness metrics; see also Figure
Problems. We consider the constrained ERM problem (4) without any regularization R 0. As our data is divided into just two groups, we constrain the difference between the loss values gs directly, instead of taking the average. In addition, consider baselines the ERM problem (2) without any regularization, R = O, and with fairness inducing regularizer R that promotes small difference in accuracy between groups, provided by the Fairret library (Buyl et al., 2024). In all problems, we take as loss function the Binary Cross Entropy with Logits Loss
((fe, = -Y, ; log o (X))) - (1 - Ya) log - (Xe,)))))
(12)
where (2) I the sigmoid function, and the prediction function fo is neural network with 2 interconnected hidden layers of sizes 64 and 32 and ReI.U activation, with total of 194 parameters.
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Figure 1: Train (blue) and test (orange) statistics over time (8) on the ACS Income dataset for each algorithm: SGD (column 1), fairret-regulrrized SGD) (column 2), SSL.ALLM (column 3), ALM (column 4) Switching Subgradient (column 5), and Stochastic Ghost (column 6). The plots depict the mean values for loss (first row) and the constraint at each timestamp, rounded to the nearest 0.5 seconds, over 10 runs. The shaded area depicts the region between the first and third quartiles.
Algorithms and parameters. We assess the performance of four algorithms for solving the constrained problem (4):
Stochastic Ghost (StGh) (Sec. 3.2 parameters on 0.05, p 0.8, 1, 3 20, À 0.5,
á = 0.05)
SSL-ALM (Sec. 3.3 parameters μ = 2.0, p 1.0, 0.01, v y 0.05, 3 0.5, M, 10),
plain Augmented Lagrangian Method ALM (Sec. 3.3, smoothing term removed 0, otherwise the same setting as SSL-ALM),
Stochastic Switching Subgradient (SSw) (Sec. 3.4 7 0.05, 7f = 0.04, = 0.01, vK+I
The hyperparameter values were chosen by grid-search validation set; report the performance of the algorithms under different hyperparameter choices in appendix B.
We also provide the behavior of SGD for solving the ERM problem. both with no fairness safeguards (SGD), and with fairness regularization on accuracy as provided by the Fairret library (Buyl et al., 2024) SSGD-Fairret These methods serve as baselines. When estimating the constraints, sample an equal number of data points for every subgroup.
Optimization performance. Figure present the evolution of loss and constraint values over the train and test datasets for the four algorithms addressing the constrained problem (columns 3-6), as well as for the two baselines: SGD without fairness (col. 1), and SGD with fairness regularization and quartiles values.
(col. 2). Each algorithm is 10 times, and the plots display the
To a certain extent, the four algorithms (col. 3-6) succeed in minimizing the loss and satisfying the constraints on the train set. The AL-based methods (col. 3 and 4) demonstrate better behavior compared to StGh (col. 5) and SSw (col. 6). Indeed, StGh exhibits higher variability in both loss and constraint values. SSw satisfies the constraint the best of all methods, but fails to minimize the objective function to the extent that other algorithms do. See appendix B for exploration of the algorithms' behaviour under different hyperparameter choices.
SGD (col. 1) exhibits the lowest variability in the trajectory and minimizes the loss in least time, but, as expected, does not satisfy the constraints. Fairree-regularized SGD (col. 2) minimizes the objective function slower than some constrained algorithmss, while maintaining small, but increasing, constraint violation. Note that the penalty parameter was optimized for constraint satisfaction: other penalty values would allow faster minimization of the objective at the cost of higher constraint violation; see appendix B more for details. This observation is consistent with Ramirez et al. (2025).
The ALM and SSL-ALM schemes satisfy the constraints the train set. On the test set, however, they are slightly biased towards negative values. Such bias is expected on unseen data and reflects the generalization behavior of faimess-constrained estimators. This is beyond the scope of the current work; see e.g. Chamon et al. (2022).
Fairness performance. Table displays the faimess metrics presented in Section 2: independence (Ind), separation (Sp), and sufficiency (Sf), along with inaccuracy (Ina). The mean value and standard
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Table 4: Fairness metries (independence, separation, sufficiency), inaccuracy, and Wasserstein distances between groups (Wd) for the four constrained estimators and the two baselines.
Algrame
Sp
SGD
0,065exxass
01066seuma
0,185esans
SIGh
0,067
0,101
ALM
0,085
0 0.210
0.057asun3
0.06Sesaen
0.058
0,185
0.304
0,221
0,158aruz
SSL-LLM
0.074:0.466
0.091.8.0
0 0.2005
000012000
0,10820.008
0,223
0,04618,912
0.370
NNW
U 0,00
0.1
anvroe
0.001.0.co1
0.10322.00 pm
0, 0,10823306
0,21 Bon
0,008gn 1o
0, 0,018.9.005
SGD Ftinet
1,001
0,125
0,00068.003
0,151
0,141aaz0s
0,21
0,006
00000
deviation over 10 runs are presented for the four fairness-constrained models and the two baselines, both on train and test sets. For all metrics, smaller better. We provide additional details in appendices A and B.I.
Among the four fairness-come -onntess--ootttaineedd models, StGh performs best in terms of sufficiency, but worst in terms of accuracy. Overall, the constrained optimization models improve on the faimness metrics of the unconstrained SGD and the regularized SGD-Fairret models. SSw has both fairness metries and inaccurary comparable to that of the unconstrained SGD model. This is consistent with the observation that the optimization method, with our choice parameters, favored minimizing the objective over satisfying the constraints. The ALM and SSL-ALM methods provide the best compromise: they improve independence, separation, and sufficiency relative to the SGD model, while moderately degrading accuracy. SGD-Fairre slightly improves sufficiency relative to the SGD model. Similar observations hold for metrics on the test set.
4.2.2 MULTI-VALUED PROTECTED ATTRIBUTE
Dataset and problems. We again consider the dataset ACSIncome, but over the state of Virginia, and choose Mari Mariaag age as the protected attribute. This attribute takes fives values, as opposed to the binary attribute setup of section 4.2.1.
We consider three optimization problems as approaches to tackle the learning task. First, we consider the constrained learning problem as deseribed in eq. (4), with 5. Second, consider the unconstrained, but penalized, problem
min X)(( go (0)
8 (0)
(13)
where À penalization weight. Third, we consider the unconstrained and unpenalized problem, as described in eq. (2), for comparison.
Algorithms and parameters. We solve the constrained learning problem (4) with Stochastic Ghost, Switching Subgradient, and SSL-ALM. We solve the baseline penalized problem (13) and the basic unconstrained unpenalized problem (2) using SGD.
The hyperparameters for each algorithm were tuned on the validation set; we picked the values resulting in lowest loss and constraint satisfaction after 60 seconds. Our hyperparameter choices are:
Regularized SGD: 0.4.
SSL-ALM:
0.01, 8 1 -,
Stochastic Ghost: 3 = 1.0, 70 - 0.005, ( p 2.05,p=0.1,7-0
0.5.
SSw: 7 0.05 constant, 7/f diminishing with nb = 0.25, n/f
2h-1
k > O; constraint tolerance
vk
d c diminishing with 6p = 0.01, < vi
Optimization performance. We report in Figure 2 the evolution of the mean and quartiles of the train and test values over 10 runs. SGD on the unconstrained and unpenalized problem (2) (first row) converges to a model such that the constraint are consistently above the constraint bound for three values of the protected attribute (Wid, Div, and Nev). SGD the penalized problem (second row) manages to meet all constraints for the training set, but constraint Div on the test set is eventually violated. The three constrained methods minimize function values while keeping with the constraint

---

03.47
Cakling Mag...o
Figure Train (blue) and test (orange) statistics over time on the ACS Income dataset for each algorithm: SGD (column 1), regularized SGD (column 2), SSL-ALM (column 3), Switching Subgradient (column 4), and Stochastic Ghost (column 5). The plots depict the mean values for loss (first row) and constraints (second to last row) at each timestamp, rounded to the nearest 0.5 seconds, over runs. The shaded area depicts the region between the first and third quartiles.
bounds. The performance of SGD on penalized problem and the three constrained algorithms is comparable.
Note that the penalized problem required consequent preliminary computations in order to tune the penalization parameter A. We found that the performance of the estimator sensitive to the value of À in two aspects: (i) on the difficulty of the optimization problem for SGD, and (ii) in the trade-off between minimization of the empirical risk and satisfaction of constraints; details. In contrast, the algorithms for constrained minimization showed (i) smaller sensitivity their hyperparameters, particularly so for SSL-ALM, and (ii) controllable trade-off' between minimization of the empirical risk, and constraint satisfaction; again, see appendix B.2 for details. This observation consistent with the argument of Ramirez et al. (2025).
appendix B.2 for
5 CONCLUSION
To the best of our knowledge, this paper provides the first benchmark for assessing the performance of optimization methods on real-world instances of faimess constrained training of models. We highlight the challenges of this approach, namely that objective and constraints are non-convex, non-smooth, and large-scale, and review the performance of four practical algorithms.
LIMITATIONS
Our work identifies that there is currently no algorithm with guarantees for solving the fairness constrained problem. Above all, hope that this work, along with the Python toolbox for easy
10

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benchmarking of new optimization methods, will stimulate further interest in this topic. Also, we caution readers that the method present here is not a silver-bullet that handles all biases and ethical issues of training ML models. In particular, care must be taken that fair ML is part of interdisciplinary pipeline that integrates the specifics of the use-case, and that it does not serve as an excuse for pursuing Business-As-Usun policies that fail tackle ethical issues (Balayn al., 2023; Wachter et al., 2021).
REPRODUCIBILITY
Code to reproduce the experiments is provided in the Supplementary Material. This includes a readme file with instructions to reproduce experiments. Details on the computing environment provided in Section 4.
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A ADDITIONAL DETAILS ON THE FAIRNESS OF THE BINARY EXPERIMENT.
Figure 3 presents the distribution of predictions over both groups The distribution of prediction without fairness guarantees (col. 1) clearly does not meet the group fairness standard. Indeed, the "non-white"" group has significantly higher likelihood than the "white" group of receiving small predicted values, and the converse holds for large predicted values. Among the fairness-com ciiness-constrained models, the ALM and SSL-ALM distributions are the closest to the distributions of SGD without fairness, which is consistent with retaining good prediction information. The Fairret penalized formulation (col. 2) and SSw (col. 5) have center-heavy distribution, which, in this case, is evidence of poor objective minimization.
The SSL-ALM, ALM, and Stochastic Ghost (col. 3, 4, and 6) have closer distributions across groups than the unconstrained formulation.
Numerically, this is expressed in Table 4 (col. Wd), which reports the value of the Wasserstein distance between group distributions for each model.
Table 4 displays the fairness metrics presented in Section 2: independence (Ind), separation (Sp), and sufficiency (Sf), along with inaccuracy (Ina). The mean value and standard deviation over 10 runs are presented for the four faimess-constrrinng models and the two baselines, both train and test sets. Figure 4 presents the mean values as spider plots. For all metrics, smaller is better.
Rrecition
Padcsoos
Prekitor
Pedidion
Prad
Iredicions
Figure 3: Distribution of predictions for each algorithm. Left to right: SGD, SGD-Fairret, SSL-ALM, ALM, SSw, StGh, Blue and red denote "white" and "non-white" groups.
SEOWET CheP
Corc Chee
2.5
(a) Train set
(b) Test set
Figure 4: Average value of the three faimess metrics (independence (Ind), separation (Sp), and sufficiency (Sf), along with mean inaccuracy (Ina), and difference in accuracy between the two groups (DifAcc). For all metrics, smaller values are better.
B HYPERPARAMETER SENSITIVITY ANALYSIS
In this section, we present the performance of algorithms on the validation set, justifying our hyperparameter choices in the experimental section. For each algorithm, we provide table of loss and constraint violation values under different hyperparameter choices after a training run.
For each algorithm, perform 5 runs with each hyperparameter combination. Every 100 iterations, we save the model state (except for Stochastic Ghost, where we save every 10 iterations due to higher iteration cost). We then pick hyperparameters based the average loss value and constraint
16

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violation over the last 20 states over runs. For algorithms that exhibit exceptionally noisy trajectories, we also provide convergence plots.
B. EXPERIMENT BINARY ATTRIBUTE
We recall that the constraint bound å was set to 0.005. With random parameter initialization, initial loss value was = 0.7.
B.1.1 STOCHASTIC SWITCHING SUBGRADIENT
Stochastic Switching Subgradient admits choice of constant, diminishing, or adaptive stepsizes for both objective and constraint updates (1y² and vf,, respectively). In this experiment, we were unable find a combination of the above that would enable the algorithm to both minimize the objective on with other methods and maintain the feasibility of the solution. The run results are presented in table 5.
As suggested in the original paper, the diminishing stepsize follows the rule
10
Vk + 1
and the adaptive stepsize follows the rule
c(zk, () T/e [ 1)1
We use diminishing infeasibility tolerance, with e = 0.01, E3 = Vk+1
60
Loss
Constraint violation
Hyperparameters
mean
std
mean
std
T adaptive, 1f adaptive
0.681
0.006
0.000
0.000
' 0.05, 17" adaptive
0.679
0.006
0.000
0.000
r = 0.05, 77 0.05
0.626
0.006
0.000
0.000
T = 0.05, 17" = 0.045
0.589
0.01
0.0012
0.002
f = 0.05, T7 0.04
0.567
0.014
0.003
0.003
r 0.05. 17" 0.025
0.469
0.012
0.018
0.004
17 0.05. 1f dimin with 76 0.05
0.409
0.001
0.025
0.004
7 0.05, 17 dimin with T6 = 0.25
0.409
0.001
0.025
0.004
T/ 0.05, rf dimin with To 0.5
0.410
0.002
0.027
0.005
T/ 0.05, r/f dimin with 1f = 3.0
0.476
0.016
0.018
0.004
T dimin with Tál = 0.05, 17 dimin with 7/ = 0.05
0.660
0.002
0.000
0.000
r dimin with 1/6 = 0.5, 17 dimin with no 0.5
0.633
0.004
0.000
0.000
Table Loss and constraint violation on the validation set after 30-second runs of the Stochastic Switching Subgradient in the setup of Exp. 1, rounded to 3 digits.
We pick n 0.05, x7 0.04, as it results the best objective value with relatively low infeasibility; we then conduct
B.1.2 SSL-ALM
SSL-ALM allows the user to tune the primal (r) and dual (7)) parameter update rate, the penalty multiplier p. as well the smoothing update rate 3 and multiplier μ. We consider only constant stepsizes and 7). and fix = 1.The run results are presented in table
The lowest loss value with the least constraint violation is offered by 0.01, 0.05, μ 0; the second best combination T = 0.01, = 0.05, μ 2. As the original paper assumes μ 2 2, we test both options "ALM"" and "SSL-ALM". respectively.
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Hyperparameters
Loss
Constraint violation
7
μ
mean
std
mean
std
0.05
0.05
2
0.4376
0.0283
0.0194
0.0294
0.05
0.01
2
0,4240
0.0096
0.0112
0.0097
0.01
0.05
0.4465
0.0122
0.0074
0.0085
0.01
0.01
2
0.4416
0.0138
0.0133
0.0145
0.05
0.05
0.4560
0.0962
0.0282
0.0620
0.05
0.01
0
0.4222
0.0099
0.0111
0.0100
0.01
0.05
0.4448
0.0114
0.0070
0,0078
0.01
0.01
0
0.4407
0.0155
0.0157
0,0154
0.05
0.05
0.4377
0.0242
0.0180
0.0271
0.05
0.01
4
0.4252
0.0098
0.0121
0.0095
0.01
0.05
0.4471
0.0112
0.0073
0.0077
0.01
0.01
0.4387
0.0132
0.0143
0.0099
Table 6: Loss and constraint violation on the validation set after 5 30-second runs of the SSL-ALM variants the setup of Exp. rounded 4 digits.
part PAtEO
Figure 5: Loss (top row) and constraint (bottom row) evolution of the Stochastic Ghost the validation dataset with different hyperparameter choices. The line corresponds to the mean value over 5 runs, the shaded region to the area between the 1st and 3rd quartiles over runs.
B.1.3 STOCHASTIC GHOST
Among the algorithms discussed, Stochastic Ghost features the largest number of hyperparameters, which makes the algorithm difficult to tune. We set po, the parameter of the geometric distribution that controls the number samples taken at each iteration, to 0.4, as, in expectation, this matches batch size used for the other algorithms. As in the original paper Facchinei & Kungurtsev (2023), use the OSQP solver to solve the subproblems.
We attempt to tune the stepsize ag. the stepsize decay rate å, and two subproblem parameters 8 and
The results presented in table 7. Due to the level of noise in the algorithm's trajectories, add the plots for some of the more stable hyperparameter configurations in fig.
We pick the values dg 0.05, & 0.05, p 0.1, 3 20.
B.1.4 FGD+FAIRRET
For regularized SGD, we tune the regularization penalty multiplier; once we find a value that leads to minimizing both loss and constraint violation, try different stepsize values.
The run results are presented table 8.
We pick multiplier equal 3 and LR of 0.05.
18

---

Hyperparameters
Loss
Constraint violation
8
mean
std
mean
std
0.0100
0.0500
0.1000
10
0.6144
0.0287
0.0135
0.0115
0.0100
0,0500
0.1000
20
0.6390
0.0461
0.0287
0,0297
0.0100
0.0500
0.1000
0.6007
0.0448
0.0256
0,0170
0.0250
0.0250
0.1000
10
0.4781
0.0245
0.0504
0,0183
0.0250
0.0250
0.1000
0.4674
0.0169
0.0381
0,0175
0.0250
0.0250
0.1000
5
0.4896
0.0371
0.0367
0.0238
0.0250
0.0500
0.1000
0.5041
0.0402
0.0318
0.0249
0,0250
0.0500
0.1000
20
0.5246
0.0566
0.0326
0,0236
0.0250
0,0500
0.1000
5
0.5218
0.0301
0.0343
0.0229
0.0500
0.0500
0.1000
10
0.5054
0.0295
0.0280
0.0196
0.0500
0.0500
0.1000
0.5020
0.0804
0.0545
0.0510
0.0500
0.0500
0.1000
20
0.4745
0.0266
0.0152
0,0154
0.0500
0.0500
0.1000
5
0.4800
0.0338
0.0442
0,0243
0,0500
0.0500
0.8000
0.5700
0.0432
0.0441
0.0375
0.0500
0.0500
0,8000
20
0,6124
0.1448
0.0851
0.0744
0.0500
0.0500
0,8000
0.6139
0.0783
0.0647
0.0618
0.0500
0.1000
0.1000
10
0.5136
0.0413
0.0486
0.0297
0.0500
0.1000
0.1000
0.5124
0.0620
0.0379
0.0384
0.0500
0.1000
0.1000
5
0.5131
0.0505
0.0356
0.0197
Table Loss and constraint violation on the validation set after 30-second runs of the Stochastic Ghost in the setup of Exp. rounded to 4 digits.
Hyperparameters
Loss
Constraint violation mean
Multiplier
LR
mean
std
std
0
0.05
0.406
0.003
0.020
0.004
0.05
0.406
0.004
0.010
0.007
2
0.05
0.421
0.008
0.012
0.006
0.05
0.539
0.026
0.004
0.005
0.07
0.523
0.029
0.006
0.008
0.1
0.523
0.033
0.007
0.008
4
0.05
0.676
0.009
0.001
0.002
Table 8: Loss and constraint violation on the validation set after 5 30-second runs of fairret-regularized SGD in the setup of Exp. 1, rounded digits.
B.2 EXPERIMENT MULTI- VALUED ATTRIBUTE
We recall that in this experiment, we the constraint bound be 0.05. We test Regularized SGD, SSL-ALM, Stochastic Ghost, and Switching Subgradient.
B.2.1 STOCHASTIC SWITCHING SUBGRADIENT
The run results presented in table 9. We pick r 0.05 constant, o/f diminishing with ng 0.25 as it offers the lowest loss value while being close to feasibility.
B.2.2 SSL-ALM
Unlike Exp. 1, we do not tune the f hyperparameter, setting μ 2. The run results are presented in
appendix B.2.2. We pick - = 0.05, 0.01.
B.2.3 STOCHASTIC GHOST
The results presented in appendix B.2.3. We pick og 0.01, 0.2, 0.1, 5.
19

---

Hyperparameters
Loss
Max constraint violation
%
sy rule
ry rule
mean
std
mean
std
0.05
const
0,05
const
0.598
0.002
0.000
0.000
0.5
const
0.05
const
0.537
0.007
0.000
0.000
0.05
const
0.5
const
0.631
0.004
0.000
0.000
0.5
const
0.5
const
0.590
0.006
0.000
0.000
0.05
const
0.05
dimin
0.543
0.018
0.031
0.001
0.005
const
0.05
dimin
0.551
0.002
0.000
0.000
0.05
const
0.05
dimin
0.499
0.003
0.023
0.003
0.05
const
0.1
dimin
0.506
0.006
0.011
0.003
0.05
const
0.25
dimin 0.519
0.004
0.001
0.002
Table 9: Loss and constraint violation on the validation set after 5 60-second runs of the Switching Subgradient in the setup of Exp. 2, rounded to digits.
Hyperparameters
Loss
Max constraint violation
7)
mean
std
mean
std
0.05
0.05
0.536
0.007
0.000
0,001
0.05
0.01
0,530
0.006
0.000
0.001
0.01
0.05
0,542
0.003
0.000
0.000
0.01
0.01
0.538
0.003
0.000
0.000
Table 10: Loss and constraint violation on the validation set after 5 60-second runs of the SSL-ALM in the setup of Exp. 2, rounded to 4 digits.
Hyperparameters
Loss
Constraint violation std
3
mean
std
mean
0.0050
0.0100
0.1000
5
0.5948
0.0218
0.0016
0.0031
0.0050
0.0100
0.1000
0.6149
0.0193
0.0005
0.0012
0.0050
0.0500
0.1000
5
0.6161
0.0142
0.0001
0.0006
0.0050
0.0500
0.1000
0.6187
0.0191
0.0016
0.0033
0,0100
0.0100
0.1000
5
0.5373
0.0243
0.0275
0.0296
0.0100
0.0100
0.1000
0.5321
0.0280
0.0290
0.0259
0.0100
0.0500
0.1000
5
0.5442
0.0222
0.0180
0.0196
0.0100
0.0500
0.1000
0.5583
0.0449
0.0099
0.0166
0.0100
0.0500
0.1000
20
0.5498
0.0311
0.0106
0.0176
0,0100
0.0500
0,8000
10
0,5716
0.0235
0.0096
0,0123
0.0100
0.0500
0.8000
20
0.5571
0.0409
0.0181
0.0184
0,0100
0.1000
0.1000
5
0.5448
0.0297
0.0197
0.0198
0.0100
0.1000
0.1000
0.5522
0.0213
0.0146
0.0108
0.0100
0.2000
0.1000
5
0.5868
0.0181
0.0070
0.0071
0.0100
0.2000
0.1000
0.5898
0.0198
0.0016
0.0035
0,0100
0.2000
0,8000
5
0.5616
0.0330
0.0121
0,0119
0.0500
0.2000
1.0000
0.5144
0.0296
0.0199
0.0255
Table 11: Loss and constraint violation the validation set after 5 60-second runs of the Stochastic Ghost in the setup of Exp. 2, rounded to 4 digits.
B.2.4 REGULARIZED SGD
The results presented in appendix B.2.4, We pick À 0.4.
20

---

Loss
Constraint violation
Penalty multiplier
mean
std
mean
std
0
0.432
0.001
0.075
0.004
0,3
0,475
0.002
0.030
0,003
0,4
0.512
0.002
0.001
0.002
0.5
0.550
0.002
0.000
0.000
Table 12:
Loss and constraint violation on the validation set after 60-second runs of the Regularized SGD in the setup of Exp. 2, rounded to digits.
C ALGORITHMS IN MORE DETAIL
In this section, we provide the pseudocodes of algorithms presented in Section as Algorithms Recall that we denote by X = ( mini-batch of size J with the j-th element
XR.j
(< Eky) caa, (k.)), Velax, Ck.)))-
(14)
21

---

Algorithm Stochastic Ghost algorithm
Require: Training dataset D, constraint dataset C, initial neural network weights 20
Require: Parameters Po (0, 1), og. P. 7,
1: for Iteration k = to K do sid 2: Sample { <~ Pg and Ç Ps
3: Sample . (po)
4: Set J = 2
5:
Sample mini-batch {Gr) so that (1, (i....,() SJ ~ Ps
Sample a mini-batch {fi})=1 so that EI,..., EJ sid
6:
Pe
Set X and x²" using (14)
7:
8:
Compute d(xx)) from (9)
9: Set aę ak-1(11 daa=1)
10: Update rk++ + ord((x))
end for
Algorithm 2 Stochastic Smoothed and Linearized AL Method for solving (1)
Require: Training dataset 'D, constraint dataset C, initial neural network weights z
Require: Parameters Jr, 7), M, > 0, r, 3, p 2 0
1: for Iteration k 0 to K
do
2: Sample { ~ Pg and Ç1. Ç2 itd Pc
Rid
3: yk+1 = yk + nc( {1)
4:
if [|y My then
5:
Mk++1 =
6:
end
7: Tk+1 projx (2t TG(#k Mh+1, Znik, S2),, where G is defined in (10)
8:
7k+11 = =x + 3(0x - 2x)
9: end for
Algorithm 3 Stochastic Switching Subgradient Method
Require: Training dataset D, constraint dataset C, initial neural network weights 2g E X Require: Total number of iterations K, sequence of tolerances of infeasibility 6 2 0, sequences of stepsizes vig and v mini-batch size J, starting index ko for recording outputs,
1: for Iteration k = 0 to K - do
2: Sample mini-batch {cr {(r)/=2 so that SI...., Ps
sid
3: Set (xx) Σ c(ax. C3)
4:
if (Ix) I then
sid Sample ę ~ Ps and generate S (II: E)
5:
Set I++1 proj (zx ní. (fx, and, if 2 Ko, I IU {k}
6:
7:
else
Sample ( itd ~ Pc and generate 5 C) Set x proj (x S (*x, ()) and, if k 2 ko. 1 T (k}
8:
9:
10:
end if
11:
end for
12:
Output: x, with randomly sampled from I using P(r = k)) E.er n
22


Reviews:
[{"review_id": "FJkAp0M492", "reviewer": "Reviewer_1Hbo", "summary": "The paper presents a benchmark for evaluating stochastic approximation algorithms in fairness-constrained training of deep neural networks. Built on the US Census dataset, the benchmark enables large-scale experiments on fairness objectives formulated as constrained ERM problems. The authors review existing algorithms and implement three recent ones, namely Stochastic Ghost, Stochastic Smoothed and Linearized Augmented Lagrangian Method, and Stochastic Switching Subgradient, and compare them to SGD baselines. The study highlights the lack of unified toolkits and provides a first step toward standardized evaluation for fairness-constrained optimization prooblems.", "questions": "1. See some questions in Weaknesses.\n2. Could the work extend to multi-group or intersectional attributes?\n3. Are there plans to include computational efficiency or memory usage comparisons?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work provides a reproducible and extensible benchmark framework for fairness-constrained deep learning, filling a gap in the literature where no unified platform existed. \n2. The writing is very clear, and the notations are consistent. I appreciate Table 3, where the authors review a wide range of stochastic constrained optimization algorithms with a structured taxonomy and theoretical assumptions.\n3. The work evaluates multiple fairness criteria, independence, separation, sufficiency, and Wasserstein distance, showing nuanced trade-offs among methods.", "weaknesses": "1. The paper primarily implements existing algorithms rather than introducing a new one. While benchmarking is valuable, this may limit perceived theoretical contribution.\n2. Only one dataset with a binary protected attribute is used. The scalability and generalization to multiple attributes have not been tested.\n3. The presentation of the experimental results can be improved. The current figures are difficult to read.\n4. There is no discussion of hyperparameter search across algorithms. Why are the parameters set as in lines 366-375? The results could reflect suboptimal settings rather than intrinsic algorithmic differences.", "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": "TfdcgdzTnE", "reviewer": "Reviewer_Rxvy", "summary": "This paper provides a benchmark of several stochastic optimization algorithms for training deep neural networks under different fairness constraints. They consider datasets built on top of the US Census (Folktables) dataset. They consider the popular group fairness measures independence (statistical parity), sufficiency, and separation. Several experiments have been included in the paper that compare the performance of these existing optimization techniques in minimizing the measures of fairness over the datasets.", "questions": "Q1. What would be the gap in existing survey papers or benchmarking papers on algorithmic fairness that this paper fills?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "-- The paper has included a lot of experiments on different algorithmic (stochastic) variants of implementing fairness as a constraint during training. Methods considered include: (i) Stochastic ghost method; (ii) Stochastic smoothed and linearized AL method; (iii) Stochastic switching subgradient method, etc.\n\n-- They consider the three popular fairness notions, and also multiple datasets. \n\n-- Presentation is generally good.", "weaknesses": "-- While the vast experiments are highly appreciated, I believe this paper is more suitable as a dataset/benchmark paper. The stochastic optimization algorithms already exist in the literature and have also been used for constraint optimization. The paper applies these constrained optimization variants for the specific constraint of group fairness and studies their performance. \n\n-- Indeed, the paper is quite comprehensive in their experimentation. But, still, the novelty would be limited for such a venue since it is more like a survey of applying different existing techniques to the fairness constraint and seeing the performance. The paper could also be better suited as a survey paper. Though there do exist several other survey papers on fairness in literature, and it would be important to highlight what is the technical gap in existing survey/benchmarking papers that this paper fills.\n\n-- The measures of fairness are mainly the three popular ones. \n\n-- Some works compare tradeoffs between different group fairness measures. E.g. The possibility of fairness: Revisiting the impossibility theorem in practice. \nIt would be good to compare with them.", "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": "C1USHlXDff", "reviewer": "Reviewer_S3CZ", "summary": "This paper presents a benchmarking study and toolbox for evaluating stochastic approximation algorithms applied to fairness-constrained training of deep neural networks. The authors consider the general constrained empirical risk minimization (ERM) problem, where fairness is imposed through hard constraints (e.g., demographic parity, equal opportunity, equalized odds). They provide an open-source implementation integrating four major stochastic optimization algorithms (Stochastic Ghost (StGh), SSL-ALM, Augmented Lagrangian Method (ALM), and Stochastic Switching Subgradient (SSw)) within a unified framework built on PyTorch and Folktables datasets. The benchmark allows users to automatically construct constrained training formulations and apply fairness constraints across up to 5.7 billion protected subgroups from census-based datasets. Extensive experiments on the ACSIncome dataset compare the convergence speed, fairness violation, and test performance of each algorithm, as well as their robustness under different fairness metrics and constraint formulations. The work aims to standardize empirical evaluation practices in fairness-constrained deep learning and offer a reproducible experimental testbed for future research", "questions": "1- How does the framework handle multiple simultaneous protected attributes, especially when fairness constraints interact (e.g., intersectional fairness)?\n\n2- Are the results robust to different data distributions or dataset shifts (e.g., subsampled or noisy features)?\n\n3- How computationally expensive are these fairness constraints for large-scale DNNs compared to regularization-based approaches?\n\n4- Could the benchmark include group fairness metrics beyond accuracy, such as calibration or counterfactual fairness?\n\n5- How do hyperparameter settings for fairness constraints (e.g., δ thresholds) influence the convergence behavior of the tested algorithms?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1- Evaluates four distinct fairness-constrained optimization algorithms under identical experimental setups, providing valuable comparative insights.\n\n2- Offers a transparent, well-engineered implementation with all datasets, hyperparameters, and metrics clearly documented.\n\n3- Bridges fairness theory with realistic deep learning setups, enabling reproducible fairness experiments on real data.\n\n4- Covers three key fairness notions (independence, separation, sufficiency) and links them to optimization constraints.\n\n5- The benchmark includes stochastic ghost gradient methods, augmented Lagrangian, and subgradient-based solvers, giving a broad coverage of optimization paradigms.", "weaknesses": "1- Experiments are mainly conducted on a single dataset (ACSIncome), which restricts the scope of empirical validation. Inclusion of more varied domains (e.g., image or language tasks) would strengthen the claim of generality.\n\n2- While results are reported, deeper analysis of when and why certain algorithms perform better (e.g., under which fairness metrics or subgroup imbalances) is missing.\n\n3- Although billions of potential subgroup combinations are mentioned, the experiments do not convincingly demonstrate performance at that scale.\n\n4- The paper does not clearly discuss how the proposed framework will be maintained or integrated with existing fairness toolkits, which may limit its long-term impact.", "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": "kuci4hadD3", "reviewer": "Reviewer_2stX", "summary": "The paper introduces a new benchmark, based on the US census, for testing fairness-constrained learning tasks. The dataset is tested on various methods from the literature, which are implemented in a toolbox.", "questions": "- What are the main contributions of the paper? Is the proposed dataset an instance of the ACS dataset? What does it make it particularly suitable for fairness testing compared to the original ACS dataset?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- A standard benchmark specific for testing the fairness of machine learning algorithms can surely be an interesting contribution, and the paper proposes to address this gap", "weaknesses": "- While the main claimed contribution of the paper is to provide a benchmark for fairness-constrained learning, I found the structure of the paper unclear, with most of the text reviewing existing literature and not many details on the introduced benchmark and dataset and on what the key novelties and results of the paper are.\n\n- The proposed dataset is an instance of the ACS dataset, already published in [Frances Ding, Moritz Hardt, John Miller, and Ludwig Schmidt. Retiring adult: New datasets for fair machine learning. Advances in Neural Information Processing Systems, 34, 2021], and already used in the context of fairness applications. Consequently, the main contributions of the paper are the implementation and existing comparison of existing algorithms. In my opinion, this contribution, while potentially useful, is too incremental for publication in ICLR", "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)