Fairness-Aware Meta-Learning via Nash Bargaining
Authors: Yi Zeng
Xuelin Yang
Li Chen
Cristian Canton Ferrer
Ming Jin
Michael Jordan
Ruoxi Jia
Venue: NeurIPS 2024 poster (2024)
Keywords: Fairness
Bargaining Game
Nash Bargaining Solution
Meta-Learning
Primary area: fairness
TL;DR: The paper introduces a two-stage fairness-aware meta-learning method based on Nash Bargaining that notably improves performance and fairness, supported by theoretical analysis and empirical results.

Abstract: To address issues of group-level fairness in machine learning, it is natural to adjust model parameters based on specific fairness objectives over a sensitive-attributed validation set. Such an adjustment procedure can be cast within a meta-learning framework. However, naive integration of fairness goals via meta-learning can cause hypergradient conflicts for subgroups, resulting in unstable convergence and compromising model performance and fairness. To navigate this issue, we frame the resolution of hypergradient conflicts as a multi-player cooperative bargaining game. We introduce a two-stage meta-learning framework in which the first stage involves the use of a Nash Bargaining Solution (NBS) to resolve hypergradient conflicts and steer the model toward the Pareto front, and the second stage optimizes with respect to specific fairness goals.
Our method is supported by theoretical results, notably a proof of the NBS for gradient aggregation free from linear independence assumptions, a proof of Pareto improvement, and a proof of monotonic improvement in validation loss. We also show empirical effects across various fairness objectives in six key fairness datasets and two image classification tasks.

Fairness-Aware Meta-Learning via Nash Bargaining
Yi Zeng". Xuelin Yang*,, 2 Li Chen², Cristian Canton Ferrer', Ming Jin', Michael I. Jordan', Ruoxi Jia!
Virginia Tech, Blacksburg, VA 24061, USA University of California, Berkeley, CA 94720, USA Meta AI, Menlo Park, CA 94025, USA
Abstract
To address issues of group-level fairness in machine learning, it is natural to adjust model parameters based on specific faimesss objectives over a sensitive- attributed validation set. Such an adjustment procedure can be cast within a meta-lcarning framework. However, naive integration of fairness goals via meta- learning can cause hypergradient conflicts for subgroups, resulting in unstable convergence and compromising model performance and fairness. To navigate this issue, we frame the resolution of hypergradient conflicts as a multi multti-player cooperative bargaining game. We introduce two-stage meta- leta-learning framework in which the first stage involves the use of Nash Bargaining Solution (NBS) to resolve hypergradient conflicts and steer the model toward the Pareto front, and the second stage optimizes with respect to specific fairness goals. Our method is supported by theoretical results, notably a proof the NBS for gradient aggregation free from linear independence assumptions, a proof of Pareto improvement, and a proof of monotonic improvement in validation loss. We also show empirical effects across various fairesss objectives in six key fairesss datasets and two image classification tasks.
Introduction
The traditional formulation of machine learning is in terms of system that improves its predictive and decision- decision-making performance by interacting with an environment. Such formulation is overly narrow in emerging applications- lumps the social context of learning system into the undifferentiated concept of an "environment" and provides no special consideration of the collective nature of learning. Such social context includes notions of scarcity and conflict, as well as goals such as social norms and collaborative work that are best formulated at the level of social collectives. The neglect of such considerations in traditional machine learning leads to undesirable outcomes real-world deployments of machine learning systems, including outcomes that favor particular groups of people over others [44, 7, 31, 10, 38, 51], the amplification of social biases and stereotypes [28, 14, 47], and an ongoing lack of clarity regarding issues of communication, trust, and faimess.
Our focus of the current paper is fairness, and we take perspective on fairness that blends learning methodology with economic mechanisms. The current favored methodology for addressing fairness recognizes that it is not a seeesiiz-iittaalll concept- different fairness notions are appropriate for different social settings [49, 32, 50]-5000 and treats fairness via meta-learning ideas. Meta-learning is implemented algorithnically with the tools of bi-level optimization. Specifically, fairness-aware meta-learning employs outer optimization to align with specific fairness goal over small, demo-
"uelin and Yi contributed equally. Yi's work is partially done at Responsible AI, Meta AL. Corresponding: Li Chen and Ruoxi Jia. Code; Nash-Meta- -etannn
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Aggeragated Vig
:0
Vanilla
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Stage
Stage
Stage Resalve confiicts bargaining
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Prioritizing the dominant group when (X) Minnimizing parity; (m)) An uncommon
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alignment rate vs. spocks
Hypergradient conflicts in standard one-stage fairness-awan meta-learning may lead to unstable performance
Identified a new Problem:
This Paper: Two-stage meta- learning with Nash bargaining can resolve conflicts for better faireess and performance at the same time
Figure I: Overview: We illustrate the problem of hypergradient conflicts in conventional one-stage fairess-awware meta-learning. which we find can lead to erratic performance and/or convergence at suboptimal, unfair local minima. Left: Graphical depiction of group-wise hypergradient conflicts (I. II) showing different scenarios where confficts arise in one-stage meta-learning affecting performance stability; (III) provides a depiction of the contrast case where the aggregated direction is not conflicting with any of the groups, which leads to more stable, fair, and performant model. Right: (a, b) Comparison of conventional one-stage meta- weta-lee (Vanilla showcased by FORMI, highlighted in gray) with our proposed two-stage meta-learning approach (Ours, highlighted in green ) which resolves hypergradient conflicts through bargaining process. In our evaluation, we show the efficacy of our method in enhancing fairness-aware meta-learning, with improvements in performance by up to 10% and fairness by up to 67%. by initially focusing on conflict resolution in Stage 1 to steer the model towards the Pareto front followed by focusing on fairness goals in Stage 2.
graphically balanced validation set to adjust a set of hyperparameters. while the inner optimization minimizes the hyperparameteeraddistee training loss [43, 52, 53]. This approach addresses two central challenges in group-level algorithmic fairness. First, it can integrate distinct fairness goals into the outer optimization. This flexibility allows customization of the focus of objectives, including enhancing the averaged loss across demographic- and label-balanced groups [43] and minimizing disparities [53]. This a conceptual improvement over methods confined a single fairness objective [18, 27]. Additionally, meta-learnin reduces the reliance on sensitive attribute labels in the training data. This circumvents the label dependence in conventional methods [30, 22, 21] and addresses ethical concerns over the acquisition of sensitive attributes [1].
Although these arguments suggest that meta-learning approach is promising for group-level faimess, it stops short of providing an economic mechanism which embodies fairness in terms of allocations and the management of conflict. Our work aims to bridge this gap by bringing a concept from economic mechanism design that of Nash bargaining into contact with meta-lcarning. Specifically, in our initial empirical explorations of gveta-lcarning algorithms, we found that performance and fairness can vary substantially according to the choice of fairness metric across different datasets, suggesting form of conflict that is not being resolved effectively via basic meta-learning procedures. Investigating further, we identified a phenomenon that we term hypergradient conftict which we believe is a pivotal factor in driving the contrast in effectiveness among different fairness goals when integrated with meta-learning. Briefly, the aggregated gradient of the outer optimization objective (the hypergradient) conflicts with the desired update associated with particular groups (Figure 1). To address this, we propose novel framework that resolves hypergradient conflicts as cooperative bargaining game. Specifically, we present a two-stage meta-learning framework for fairness: first incorporate the Nash Bargaining Solution (NBS) at an early training stage mitigate conflicts and steer the model toward the Pareto front, and then engage in the pursuit of specified fairness goals.
Our work also introduces a new derivation of the NBS for gradient aggregation, one that dispenses with gradient independence assumptions made in the past work so that it is applicable to broader contexts. This derivation may be of independent interest, and accordingly we present material on ghme-theo justification, Pareto improvement, and monotonic improvement in validation loss that help to relate the NBS to our bradeent-based learning setting. Our analysis sheds light on the convergence exhibited in empirical studies and provides an understanding of how our method improves meta-learnin for fairness.
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Pareta
Iniaal Points
Final Points
Buseline
LtR
FORME
Meto
Adult Income 13),
(easitive utribone: race)
Overall AUC (T)
0,668
(+20%%)
0,710 (-6.2%)
0,778 (-166%
2
Max (1)
Worst-gAUC (f)
0,544
0,225
0.0M0 9616)
(+3596
0,290 (+2916)
(0.7%))
0.694 0.694 (+286s)
0,163 (-28%)
13
15
Titanic Survival (12),
(sessiivve atribute: sex)
Overall AUC (T>
0,972
0.967 (-055%)
0,950 (-2.3%)
0:961
(-1.15)
10 20 30
Grop
10 20
30 20
30
Max-gAL CD (1)
0,056
0.065 (-21%)
(.015--415) (-41%)
0,033 (4115)
(l) Group-besed Averaging (LtR)
(II) Minimidng
(II) Max-Min (Meta-gDRO
Worst-gAUC (f) 0,944
0.964
0,933 1.2%)
0.944G
the Max Max-Gap (FORML)
(a) Comparison of the model performance of three conventional one-stage meta-learning methods (LtR,
(b) Trajectory of 1000-step optimizations. and y-axis: validation losses for groups and 2, rsp. Gray curves (lower left): the Pareto front. x=y: fairness (equal
FORML, and Meta-gDRO) across various fairness no- tions (results averaged from runs, % relative the baseline only using the training set w/o meta-learning)
validation loss). Ideal endpoint: their intersections.
Undesirable endpoints: ">>". 6 random initials.
Figure 2: The unreliable performance of conventional one-stage faire meta-learning.
Finally, we present a thoroughgoing set of empirical studies that evaluate our method using synthetic and real-world datasets, encompassing six key fairness datasets and two image classification tasks. As we'll show, our framework uniformly enhances the performance of one-stage meta-learning methods, yielding up to 10% overall performance improvement and 67% drop in disparity (Figure I).
The remainder of this paper structured as follows: Section 2 describes the problem. Section details our method, including the bargaining game formulation, solution, and theoretical analyses ($3.1-3.3), and our two-stage meta-learning framework, its theoretical foundations, and dynamical issues ($3.4-3 (53.4.3.6). Section 4 presents empirical results and analysis. Related work is deferred to Appendix A.1.
Problem Statement
Let 0 denote vector of model parameters, let (ttrain) denote the training data and let D(nal) validation set, with K sensitive groups D(val), for i E [K]. Let be vector-valued function where each entry corresponds to the per-example conventional loss (e.g., cross-entropy). Define group-wise validation loss [(val) vector of size K where !.val) | (()( (w))
(the averaged loss over samples in group i). Define L3 - T ((val) as the fairness loss with vector 8 of size K encoding the faimness objectives under consideration. Following established work [43, 53], we target group-level fairness objectives via meta- meta-learning follows:
w' arg min Ly (D((aa)) 10" (w)),
w20
("(w) arg min w L(D(train))
(2)
The vector w is set of hyperparameters that reweigh each training example in the current minibatch when updating 0, optimized on ((val) improve group-level fairness.
Existing fairness-an aware meta-learning work can be characterized as different protocols for 3. LtR [43] computes the average of all group-losses with demographic and label balanced (vaal) (i.e., BLR A 1). FORML [53] calculates the difference between the maximum and minimum group- loss (i.c., SPORML has 1 for max, -1 for min, and 0 otherwise), emphasizing parity. Meanwhile, group-level Max-Min fairness inspired from [41] focuses solely on the maximum group-loss [44], yielding a procedure referred to as Meta-gDRO, with MeewggRO set equal to one for the max and zero otherwise. Throughout the training process, the hypergradient of these methods, 10t (w)). is derived by applying the aggregation protocol 3 to the group-wise hyper-
gradients, V..L. (wal)) (D (+)))))) (w)). Their training process, wherein the aggregated hypergradient is iteratively utilized to update w, is referred to one-stage method. They differ from our two-stage method, which employs distinct hypergradient aggregation rules at two separate stages.
Unreliability of one-stage meta-learning for fairness. The traditional approach of plugging fairness objective into Lg seems natural. However, we find that the effect of this approach on performance and faireess can be unstable. We evaluate the above three one-stage fairness-aware
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methods based on targeted faimess metrics (detailed settings in 84.2): Overall Area Under the Curve (Overall AUC) for LtR which also measures prediction performance, maximum group AUC disparity (Max-gAUCD) for FORMI,, and worst group AUC (Worst-gAUC) for Meta-gDRO (Figure 2a).
Delving into the cause. After training on real data for several epochs, find that most subsequent epochs have fewer than 3% of the aggregated hypergradients aligned to the optimization objectives of each subgroup. That is, we see hypergradient conflicts:
3i, s.t. 9; Lg 0,
(3)
with the group-wise gradient 9i = VuLi (oal) (D(val) ((w)) and aggregated direction VL3
V (D(va(( [(w)). The prevalence of intrinsic hypergradient conflict in one-stage methods is unsurprising, because their aggregation methods are unable to prevent overlooking or incorrectly de- de-prioritizing certain groups. We study this phenomenon in synthetic settings to isolate structural issues from randomness in stochastic optimization (Figure 2b, settings Appendix A.6); We define the performance goal as convergence to the Pareto front, while the fairness goal corresponds to the line x = y. As observed, alignment issues synthetic experiments are prominent (Figure 2b): (I) Averaging (LtR) may induce oscillatory dominance among groups. (II) Parity-based method (FORML) produces conflicting hypergradients as it subtracts the loss of one group, necessitating performance trade-off's for fairness. (III) Minimizing the worst-group loss (Meta-gDRO) often exhibits toggling dominance as it solely prioritizes the current east-performing group, which may create conflicts and cannot land the Pareto front.
Hypergradient conflict resolution. Given the observation of hypergradient conflicts and convergence issues in one-stage meta-leaming, we turn to cooperative bargaining and propose two-stage method that seeks to attain more reliable improvements by resolving confiits at the early stage of training. Our methodology draws inspiration from the Nash Bargaining Solution (NBS), cornerstone of axiomatic bargaining in game theory, known for its general applicability and robustness [34]. Nash Bargaining is chosen for its desirable axiomatic properties, which prohibit unconsented unilateral gains by Pareto Optimality, and its principled approach of effectively balancing interests, making it appealing for practical deployment. We provide additional discussion of the game-theoretic perspective and additional empirical comparisons in Appendix A.2.
While the NBS has been studied in multi-task learning [37], it has yet to be explored in fairess- meta-learning. Unlike the settings in [37], applying the NBS our context challenges the assumption of linear independence among tasks, which is generally untenable for group-wise utility towards the same goal of performance gain (i.e., the settings of fairness). This drives our exploration into novel proofs and applications of the NBS in hypergradient aggregation, aiming to circumvent the need for linear independence and optimize shared outcomes through strategic negotiation and nested optimization.
3 Methodology
3.1 Nash Bargaining framework
We start with some preliminaries. Consider K players faced with set A of alternatives. If all players reach a consensus on a specific alternative, denoted as a in set A, then a will be the designated outcome. In the event of failure to agree an outcome, predetermined disagreement result, denoted as d, will be the final outcome. The individual utility payoff functions are denoted u, : A AUR [D) - R, which represent the players' preferences over A. Denote the set of feasible utility payoffs as ux (a)) : E A C K and the disagreement point as d = (w1 (D), (D)..... ug (D)). Nash proposed to study solutions to the bargaining problem through functions f (S, d) + R. The unique Nash bargaining solution (NBS), originally proposed for two players [34] and latter extended to multiple players [15], maximizes the function S(S, d) [I,(;; - d:). where .;; is the bargained payoff and d, is the disagreement payoff for player i. The NBS fulfills four axioms: Pareto Optimality, Symmetry, Independence of Irrelevant Alternatives, and Invariant to Affine Transformations. See Appendix A.3 for detailed definitions, and A.4 for additional assumptions.
S = (wy
In our problem, want to find á in Algorithm I, intermediate vector for the optimal w= reweigh each training sample. Let T. (val) be the validation loss and gi = V T, (val) be the hypergradient of group i. Let G be the matrix with columns g,. The central question is to find the protocol a as the
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weights applied to individual group i's loss. It associates an update step V La to i that improves the aggregated validation loss among all groups.
We frame this problem as a cooperative bargaining game between the groups Define the utility function for group as
U4 (L.)) 9i VLa.
The intuition is that the utility tells how much of proposed update is applied in the direction of hypergradient of group i (as it can be written as ||g: |l L cos 5 with angle 5 between 9i and V.,). This gives the projection of V La along Ji- or the "in effect" update for group If gi and VI., aligns well, the utility of group is large (or vice versa). Denote B, the ball of radius centered O. We are interested in the update VL in the agreement alternative set A {VL VL Be. VL, 9: D 9: > 0, Vi E [K]]. Assume A is feasible and the disagreement point is D (i.e. the update aa = O, staying at ti). The goal is to find a Va that maximizes the product of the deviations of each group's payoff from their disagreement point. Since u, forms (shifted) linear approximation at vi, we are essentially maximizing the utility of La locally. We provide further discussions the problem setup and assumptions in Appendix A.4.
3.2 Solving the problem
In this section, we will show the NBS is (up to scaling) achieved at VL. = EiE(K] a.go where a R solves G Ga by the following two theorems, with full proofs available in Appendix A.5: Theorem 3.1. Under O, arg maNv (u, d,) is achieved
Σ
gi gr
LV for some > 0.
(5)
VL
KE(K]
Proof Sketch. We employ the same techniques in Claim 3.1 of [37]. Theorem 3.2. The solution to Equation is (up to scaling) Vo = EiEK 0:Gi where
T Ga= a
(6)
with the element-wise reciprocal !.
However, whereas [37] relied on the linear independence of the gi 's to uniquely determine each coefficient (x 9;)-1, our technique makes no such assumption. Instead, we multiply both sides of Equation by
Proof sketch. Let x VL.. In line with [37] we observe that E
9:)
9j and obtain Ei<[K (x g:) g 9j) = ya gi [K]. Set a; (x gg) gg) This is equivalent to 03 ΣiErK) 9:0; 0,- which is the desired solution when written in the matrix form.
While our solution aligns with that of multi-task learning [37], our proof of Theorem 3.2 circumvents the necessity for linear independence among 9r. one of the core initial assumptions in the previous work. Linear independence does not hold in general as the goals for individual groups (or tasks) might overlap (such as sharing common underlying features) contradict each other (when there negative multiplicity). Our proof removes this assumption and sheds light on the effectiveness of updates based on the NBS in general cases. See Appendix A.5 for extended discussions on linear independence.
Furthermore, derive two useful properties of the NBS in additional to its four axioms, with full proofs available in Appendix A.5:
Corollary 3.3. (Norm of bargained update) The solution in Theorem 3.2 has &-norm VK. Corollary 3.4. If 9j a-bounded for j E [K]. ||a7 I|l
for [K].
Informally, the NBS has implicit e regularization (Corollary 3.3) which substantiates our empirical observation that separate f--oommalization on VL, for meta-learning rate adjustment yields better performance than e--oormaiization (the conventional setting [43]). Note that we do not impose any assumptions on the boundedness of the hypergradient 9i, ensuring the stability even when certain hypergradients are extreme. Furthermore, when 9s is bounded, Corollary 3.4 implies that ||a,]| is bounded below and no groups are left behind.
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Algorithm 1: Two-stage Nash-Meta-Learningg Training
Input:
0 D(train); D = [D (ual) ..... xxaa)
Parameters: 9o (fairness protocol); r7 (step size);
Thar (bargaining steps)
for step E (1..... T) do
Minibatch k sampled from ((trat))
/+ 1. Unrolling Inner Optimization */
( (w) + BackwardADo (w- L(D()))))))))))))
(w) t -00 -n (V())
3 Sp:
i Th har then
/+ 2. Validation Group-Utilities +/
for group & (l..... K) do 9h BackwardAD (L. (1
3. Bargaining Game +/
Set G with columns (
Solve for a: ( a 1 to obtain (1);
10
if a is not None then
11
// bargaining succeeded
3 a!);
< BackwardAD (L (0()
13
4. Weighted Update +/
Normalize( max(- 0)):
VO() BackwardAD (w L(D()))())
15
0(+1) < g(t) VD();
()))))
Output: ((T)
3.3
Game-theoretic underpinnings
The NBS provides incentives for each player to participate in the bargaining game by assuming the existence of at least one feasible solution that all players prefer over disagreement (Ez E S s.t. d). This aligns players' interests in reaching agreement. The constraint 9, VL, 0 resolves hypergradient conflict upon agreement.
Second, note that Equation 6 shows relationship between the individual and interactive components:
||a:9:ll
(a.)))) (a:g;) = ju
for E [K]. The relative weights (4 emerge from a player's own impact (loag: (12) and interactions with others ((a,g;) (a)g;)). This trade-off embodies individual versus collective rationality. Positive interactions (i.e., 9.1 >> 0) incentivize collective improvements by downweighting a4, Negative interactions (i.c., g; 9j < 0) increase a, to prioritize individual objectives. Furthermore, each player accounts for nontrivial contribution to the chosen alternative VL, under mild assumptions (Corol- lary 3.4). The negotiated solution balances individual and collective rationality through participation incentives and conflict resolution. This equilibrium encapsulates game-theoreen bargaining.
3.4 Two-stage Nash-Mettal.earning
We now present our two-stage method (Algorithm 1) that incorporates Nash bargaining into meta- learning training. Previous one-stage algorithms fix predetermined Lg; our two-stage method assigns 3 in the NBS in Stage and sets it back to the original 30 in Stage 2. Define ((() the
'To give theoretii justifications. build on similar expression (Equation 2, [37]).

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model parameter at step t, T as the number of total steps, and T (harr) S T as the number of bargaining steps in Stage 1. Let BackwardAD, (L) be the backward automatic differentiation of computational graph L w.r.t. 0, and Normalize( be 2 n--ormmllzation function. Each step is constituted by four parts: The first part is unrolling inner optimization, a common technique to approximate the solution to nested optimization problem [17]. We compute a temporary (unweighted) update ((t) on training data, which will be withdrawn after obtaining the udpated w. w is initialized to zero and included
(t) which ascertains the descent direction of each training data locally on the validation loss surface. The third part is to aggregate the hypergradients as the update direction for ū by a from the NBS. In Stage 1, successful bargaining grant the update of V by the NBS. If the bargaining game is infeasible or if we are in Stage 2 (not in the bargaining steps), we calculate t based on the fairness loss La of our choice.
into the computation graph. The second part calculates the hypergradient 9k
The last part is the update of parameter ((++1) using the clipped and normalized weights w for each training data in the minibatch.
3.5 Theoretical Properties
Theorem 3.5. (Update rule of 0) Denote L,
[train) = L(D! (0(0)
E R for the i-th sample in
training minibatch D at step I. 0 is updated as 0((+1) = O(1) 17(1)
DDD0 1=1
AP' with A0'
[
Viriirain) max ((Vol(BU)) T) ((U))))
(train) O)Voll (train))
Theorem 3.6. (Pareto improvement of w) Use at for the update. Assume (e is Lipschitz- (t) smooth with constant C and gi is a-bounded at step t. If the meta learning rate for is satisfies sor E [K], then I.(ual) (1)l+1)) S (.(a)) (16(0) for any group i [K] 110) CKa.
Theorem 3.7. Assume (eaa) is Lipschitz-smooth with constant C and VoL (train) is o-bounded.
If the learning rate for 0 satisfies r7()) S T Jaz, then Lyces (0(++1) S (((t)) for any fixed vector 3(0) with finite [1,35)] used to update ((t)
2
Informally, the closed-form update rule of indicates that the weight of a training sample is de- termined by its local similarity between the 3-reweighed validation loss surface and the training loss surface (Theorem 3.5). Under mild conditions, i yields Pareto improvement for all groups for the outer optimization using the NBS (Theorem 3.6). For the inner optimization, under mild conditions, the fairness loss T.g00 monotonically decreases w.r.t. ,300 regardless of the choice of protocol (Theorem 3.7). This generalizes [43] from 3 * any B with finite norm and provides a uniform property for the fairness-aware meta-learning methods. It entails the flexibility of our two-stage design that switches 3 between phases. Setting 3 a gives the desired property for the NBS. The validation loss surface reweighed by the NBS has the maximum joint utility, which empirically boosts the overall performance when used to update 6. Full proofs are given in Appendix A.5.
3.6 Dynamics of Two-stage Nash-Metta-Learning
Our method captures the interplay and synergy among different groups. Specifically, previous methods like linear scalarization (i.e., assigning fix weight to each group) are limited to identifying points the convex envelope of the Pareto front [5). Our method offers more adaptable navigation mechanism by dynamically adjusting the weight with the NBS, which accounts for the intricate interactions and negotiations among groups. Moreover, the optimal L. maximizes the utilization of information on validation loss landscape and leads to empirical faster and more robust convergence to the Pareto front even with distant initial points. Although first-order methods typically avoid saddle points [16, 39], if one is encountered, switching to the fairness goal upon unsuccessful bargaining offers a fresh starting point for subsequent bargaining iterations and helps to escape (Figure 6d, Appendix A.7). Our synthetic experiments show that Stage 2 training focused solely on the fairness goal does not deviate the model from the Pareto front (Figure 3). Specifically, the worst group utility y. La tends to concentrate around zero, ensuring the model stays in the neighborhood of the Pareto front and implying the robustness of our approach. The theoretical understanding of this interesting phenomenon is open problem for future study.
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Evaluation
We evaluate our method in three key areas: synthetic simulation ($4.1) for Pareto optimality vis-à-vis fairness objectives, real-world fairness datasets (f4.2), and two imbalanced image classification scenarios (Appendix A.7.3).
4.1 Simulation
Under the synthetic settings ($2 and Appendix A.6), we observe the convergence enhancements compared to Figure 2b and the effect of continuous bargaining throughout the entire training (another case of one-stage). Figure 3 demonstrates that Nash bargaining effectively resolves gradient conflicts and facilitates convergence the Pareto front in comparison to Figure 2b. This is evident from the reduced number non-Pareto-converged nodes in both continuous bargaining (Figure 3a) and early-stage bargaining (our final solution, Figure 3b). Notice that one-stage NBS doesn't 't always enhance fairness, evidenced by the observation that nodes at the Pareto front tend to stagnate. The NBS does leverage any information about specific fairness objectives. Our two-stage approach built on the bargaining steps can further push the model to the ideal endpoints.
Initial Points
Final
30
30
30
20
30
Goonn loos
Gooup Flod
10
10
10
0,
0r 0
10
20
30
10
20
10
30
30
20
30
10
20
10 20 30
Group
Group
Group
Group
Group
Group
Group based
Minimizing the Max Gap (FORML)
Max
Group based
Minimizing the Max Gap (FORML)
Averaging (LtR)
(Meta-gDRO)
Averaging (LtR)
(Mota- DRO)
(a) Continuous bargaining throughout the entire training
(b) Two-stage method (Stage 1: laargaining: Stage 2: falmess goals)
Figure 3: Synthetic illustration of the bargaining effects. "> final point not close to the fairness goal
(x=y). ">>" final point not at the Pareto front. (a) Bargaining across all 1000 steps; (b) Bargaining
only included in the first 100 steps (two-stage method).
4.2 Standard fairness benchmarks
We test our method on six standard faimess datasets across various sectors of fairness tasks: financial services (Adult Income [3], Credit Default (54)), marketing (Bank Telemarketing [33]), criminal justice (Communities and Crime [42]), education (Student Performance [11]), and disaster response (Titanic Survival [12]). Test sets comprise 3% of each dataset (10% for the student performance dataset with 649 samples) by randomly selecting demographically and label-balanced subset. See Table 2 in Appendix A.6 for data distribution specifics.
General settings and metries. We compare our two-stage Nash-Meta-Leariine with conventional one-stage fairness- aware meta-learning (i.e., LtR, FORML, Meta-gDRO), baseline training, and Distributional Robust Optimization (DRO) [18]. All methods share the same model architecture and training hyperparameters on each dataset. Our approach features a 15-epoch bargaining phase within the total 50 epochs. See Appendix A.6 for training details. Unlike synthetic experiments, the Pareto Front of real world datasets could be computationally intractable, so we cannot directly evaluate regarding this. Three metrics from 52 are used: Overall AUC (t). Max-gAUCD (L), and Worst-gAUC (t). corresponding to the goal LtR, FORML, and Meta-gDRO, respectively.
Results and analysis. Our NBS-enhanced two-stage meta-learning improves the overall performance, fairness objectives (in color), and stability, as in Table and with 95% CI in Table 4, Appendix
A.7. While the results without bargaining majorly agree with the prior work, bargaining increases FORML's Overall AUC by 10.34% (from 0.706 to 0.779) with a tighter 95% CI (from 0.202 to 0,054), and decreases Max-gAUCD by 26% (from 0.039 to 0.029) with tighter CI (from 0.046 to 0.018). However, our method faced challenges in two datasets: Credit Default, where performance and fairness occasionally declined, and Communities and Crime, where minimal improvement was observed (in particular, the Meta-gDRO). We diagnose that these two dataset's s validation set contains low feature-label mutual information, leading to noisy outcomes (Appendix A.7) and affecting
8

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Baseline
DRO
LUR
FORML
Meta-gDRO
(ours)
Inconse [3], (senative antribune: sex)
Overall (T)
0.778
0.761
0.830
DU
04
0810
0J837 (+-027)
MaxgAUCD (
0.016
0,029
0,050
0.047 (-003)
0.075
0,031 (.048)
0,052
0.046 (-006)
(orst-gADC(1) (1)
0,770
0,747
0,805
0,807
(+ 002)
0.763
0,795 (+032)
0.809
0J514 (+-006)
Adult Income [3], (seniiive atribute: race:
one group rminnngg data contains one positive (favomblle) lahel.
Overall AUC
0.668
0.652 0.256
0,803
0,805
0.710
0.775
0,793 (4.115)
Mak-gAUCC
0.225
0.098
(-156)
0.158
Worst-gAUCC(1)
0,544
(441))
0,903
III. Bank Telemarketing [13], (semstivve atiribute: age)
Overall (T) (T)
0.697
0.686
0.724
(+ 004)
0.706
0.779 (+ (+0075)
0,698
0.722 (+:0224)
Max-AUCC (1)
0.013
0,025
0,099
0.0K3 (- (066
0.009
0.029 (:010)
0.098
0.079 (.019)
Worst-gAUC
[2699]
0.691
0,675
0.686 (1.01))
01086
0.764 (o.078)
0.649
0.683 (4.0344
Crodit Defoult (54), (sensitive anvibute:
noisy others, analysis Table Appendis
(T)
0.624
(-04))
0.611
0.661 (:021)
Max-
B:024
B:022
0.037
0117 0.545
Worst-gAUC(11 (1)
0.622
0,613
0,612
0,803 (-03
0.595 (+:050)
0.674
0.650 (-024)
V. Communities and Crime [42]. (semativee atriibue: blackgtópct;
val data motay, meamwhil hile one group training datas ure all posilve labels
AUC Max-gAUCD
Overall AUC
0,525
0,568
0,8799
0.554 0.129 (+0558
(+:001+:21))
0.565 (+:014)
0.686
0.686 (+,000)
0,050
0.136
0,071
0.107
0.136 (+-029)
0.114
0.114000000
Worst-gAUC
0.500
0,500
0,643
0.636
0,500
0,500
0.629
Tifanic Survival (12), (sensitive attribute: sex)
Overall (T)
0,972
0.983
0.967
(+
0.950
0.972
0.972 (+:011)
Max-AAUCD (1)
0,056
0,033
0,044
0.044 (-0.0)
0,033
0 (.022)
0.033
0033 (-000)
Worsi-gADCC (T)
0.944
0.967
0.944
(+-06
0.933
0.967 (+-034)
0.944
0.966 (+:012)
VII Student Performance [11], (scnsitive attrilune: sex)
Overall ADC (1)
0.784
0,816
0,900
0.900 000)
0.828
0.822 (-006)
0.909
0.912 (+,003)
Max-gAUCD
0.119
0.106
0,013
0.057 024)
0,031
0.025 (-000)
Worst-gAUCC.11)
0.725
0,762
0,881
0.900
Table 1: Comparison on standard fairness datasets (averaged from 5 runs). Each of LRR FORMI. Meta-gDRO is paired with { Overall AUC,, Max-gAUCD, Worst-gAUC rsp. for aligning intended fair-
ness goals. Top results of each row in bold.
most tested methods (e.g., LIR, FORML). Additionally, for Communities and Crime, our method is influenced by the low bargaining success/feasible rate, possibly due to the lack of favorable (positive) training samples for the minority groups (Table 2, Appendix A.6). Conversely, our method still yields the anticipated bargaining results on the adult income dataset with only one positive Amer Seer-Indiann an sample. These insights emphasize the importance of validation set quality and representative samples in the training.
Effects of bargaining on hy hypergradient con- flicts. Bargaining enhances hypergradient align- ment by varying degrees among different one- stage algorithms (Figure 4, 6). For instance, LtR's alignment rate improves from 60% to 80%, and FORML jumps from 0 to 76.9% accompa- nied with more substantial performance and fair- ness gains on Bank Telemarketing (Figure 4), FORML. consistently benefits more from bargain- ing compared to the other two, likely due to its optimization goal that could intensify hy hypergradi- ent conflicts. Moreover, our approach uniformly promotes hypergradient alignment during Stage I (Figure 6). We show that early-stage bargaining, accompanied by its hypergradient conflict resolu- tion, is crucial for enhancing model performance and fairesss. Further illustrations and analyses in Appendix A.7.
gDBO
LRR w/
Nash
FORME w/ Noshs
MetegDDOO w/ Nash
Frequescy (tog)
103
A Augmmnt Tan (5
103
30
101
30t
Align 0.603
Rate:
Conti gre
Cono (thAlggee
0.000
07769
Gor kgn
0,210
0.847
Epoch
8
15
(a), Worst-group hypergradient alignment value
(b).. per epoch
Figure 4: Effects on hypergradient alignment (Bank Telemarketing). (a) Smallest g, Lg. Por- tion of positive values (Align. Rate) at the bottom. (b) Hypergradient alignment rate per epoch.
Discussions on scalability. To scale up models and datasets, one natural question is whether the bargaining game can still feasible. For larger number of groups, although the likelihood of getting unresolvable conflict may get higher due to the fact that more players are getting involved,

---

conjecture that the feasibility of A depends more on the group structure rather than the number of groups. For example, the interdependencies and shared factors between groups may cause dependency in hypergradients to enable the feasibility of A. When goals of groups rely on common resources or have shared objectives at a higher level, it is still likely to have nonempty A.
Discussion & Conclusion
In conclusion, our study offers several key insights: We identified hypergradient conflict as pivotal issue undermining stable performance in one-stage tuimess-aware meta-learning algorithms. To mitigate this, we proposed a two-stage framework that initially employs the NBS to resolve these conflicts, and then optimizes for fairness. Our assumption-frs proof of the NBS extended its applicability to a broader range of gradient aggregation problems. Empirical results demonstrated our method's superior performance across synthetic scenarios, seven real-world fairness settings on six key faimess datasets, and two image classification tasks.
Future directions. First, addressing the absence of specific label in the training subgroup and the low quality of the validation set that affected our method's effectiveness may be mitigated by fairness-aware synthetic data [48] or data-sifting methods [56]. Pairing our method with them and exploring more resilient solutions adapted to these extreme cases can be a promising direction. Second, Theorem 3.7 establishes the validity to switch the choice of 3 during training. Future work can focus on designing and flexibly choosing outer optimization goals to delicately improve performance, fairness, other metrics in interest. Third, we derive the NBS under = 0 for conflict resolution. Future study could investigate general D & O, which might not be useful for conflict resolution (the scope of our paper), but could be used in other cases as gradient aggregation method that gains advantages from axiomatic properties, as discussed in Appendix A.4.
Acknowledgments
We acknowledge Aram H. Markosyan and Vitor Albiero at Meta AI for their insightful discussions. Ruoxi Jia and the ReDS lab acknowledge the support from the National Science Foundation under Grant No. IIS-2312794. Ming Jin acknowledges the support from the National Science Foundation under Grants ECCS-233177 and IIS-2312794 Michael Jordan wishes acknowledge support from the European Union (#RC2202-YYC-OCEANN-000)
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A Appendix
In this section, we first provide background and related work (SA.1), along with context on the motivation of choosing the NBS ($A.2). Then, we give formal definitions of some preliminaries mentioned in the main text ($A.3). We supplement discussions on the gargaining game setup and assumptions (8A.4), followed by the full proofs of corollaries and theorems ($A.5). We then provide detailed experimental settings ($A.6) and the additional results and analysis on with 95% CI, test noise, gradient conflict resolution, and experiments on the imbalanced image classification task (5A.7). Finally, we discuss the broader impact and limitations (5A.8).
A.1 Background & Related Work
Bargaining game in ML. Bargaining has been broadly applied across various ML contexts such as multi-task learning [37], multi-agent reinforcement learning [46, 40, 9, 45], multi-armed bandits [2], feature selection [23], and Bayesian optimization [4]. For pargaining-relatee gradient aggregation, a key distinction between our work and prior research is our provision of proof for the NBS that does not rely on the linear independence assumption, and we further extend from the basic gradient descent setting to bi-level fairness- aware meta-learning [37]. In the context of meta-learning for fairness, we critically examine and build upon the work of [43], detailed in $2. To the best of our knowledge, we are the first work to incorporate Nash bargaining into fairness-aware meta-learning
Bias mitigation. Traditional bias mitigation methods such as relabeling [30], resampling [21], or reweighting [22] hinge upon the availability of sensitive attributes training data. However, such attributes can often be inaccurate, incomplete, or entirely unavailable due to privacy and ethical concerns in the collection process [1]. In response, researchers have developed two lines of proxy- based strategies over the past decades. The first line of work adopts indirect features associated with the sensitive feature of interest, such as zip codes for ethnicity [13] or sound pitch for gender [26]. Despite cireumventing the requirement of sensitive attributes in training data, the effectiveness of these proxies critically hinges on their correlation with the actual sensitive attributes. The second line of work aims at aligning with the Max-Min fairness principle and uses the worst-performing samples as the proxy of the most vantaaad groups [41, 18, 27, 8]. Challenges also arise in the potential bias toward mislabeled data when targeting porft-perfomming samples [51], Most importantly, with the proxy group only aligned with Max-Min fairness, these lines of mitigation cannot be generalized or adapted to help other fairness notions.
Scope and Notions of Fairness. Our exploration of fairness is specifically tailored to group- level notions, fairess--awe aware meta-learning inherently relies a representative set of samples organized by group information for model updates. While Max-Min fairness was chosen as a common evaluation metric, our paper also encompasses two additional mainstream faimess notions, including demographic parity [53] and average performance across equally represented groups [43], both used as one-stage baselines (Section 2). Our proposed two-stage method is also evaluated its effectiveness reducing disparity and improving the overall performance compared to these two baselines, respectively, as shown in Table 1.
Max-Min Fairness and Rawlsian Justice. Group-level Max-Min fairness a concept originated from Rawl's 's definition of fairness or, equivalently, justice [41]. Rawls defines the least advantaged
group by primary goods with objective characteristics, which are independent of specific predictors. The Difference Principle in Rawlsian Justice requires that the existing mechanism always contributes to the least advantaged group. While Rawlsian Justice has been extended to specific utilities in the context of ML group-level fairness [18. 27, 51], the worst-performing group in ML fairness may vary over epochs and depend on the optimization status, unlike the least advantaged group in the original Rawlsian context. Consequently, the group-level Max-Min fairness approach in ML may not necessarily create safety net for the least advantaged group as the original Rawlsian Justice intends. Rather, the ML gevup-level Max-Min fairness provides a dynamic optimization strategy that maximizes the minimum performance across all groups at each epoch. This approach ensures that the worst-performing group, which may change over time, is prioritized during the optimization process. While this interpretation of fairnsss differs from the original Rawlsian context, it remains a valuable technique for promoting equity in MI. systems by preventing any group from being consistently disadvantaged.
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A.2 Additional Discussions
Why NBS for hypergradient conflict resolution? - A game theory perspective. Game theory offers two main categories: cooperative and noncooperative games. Cooperative games involve players forming alliances achieve common objectives, while noncooperative games (e.g.. Nash equilibrium (36]) focus on players acting independently without contracts. Our problem aligns better with the cooperative category, where players collaborate to maximize their collective gains from the proposed hypergradient update. Cooperative bargaining, a subset of cooperative games, studies how players with distinct interests negotiate to reach mutually beneficial agreements, directly corresponding to our goal of conflict resolution. Among cooperative bargaining solutions, the NBS stands out for its general applicability, robustness [34], and unique solution satisfying desirable axioms including Pareto Optimality, Symmetry, Invariance to Affine Transformations, and Independence of Irrelevant Alternatives [35]. These properties make the NBS an attractive choice for resolving hypergradient conflicts in principled and fair manner.
Initial Paints
Final Points
30
30
30
30
Group 1osss
Group 2 losss
Group losss
Group lous
20
20
20
20
20
Group
10
10
10
10
10
30
10
20 Group
10
20 Group
10
Group
Group loss
Group loss
loss
(a) NBS (ours)
(b.1) GM (p=1)
(b.2) GM (p=-1)
(c) PCGrad
(d) CAGrad
Figure 5: Synthetic illustration of each gradient aggregation method in resolving gradient conflicts and their implication in converging to Pareto front. Red circles imply the nodes that cannot converge the Pareto front after 1000 steps of updates.
Why NBS for hypergradient conffict resolution? An empirical comparison. As far as we understand, there is no simple way to project or rescale a proposed update (such LtR, FORML, or Meta-gDRO) to guarantee conflict resolution. Moreover, the NBS finds a balanced solution on the Pareto front where no participant can unilaterally improve their position without others' agreement, which cannot be achieved by standard way of linear scalarization (e.g., Example 2.27 in [5], and [20]) or other gradient aggregation methods like Generalized Mean (GM) [6]. In synthetic experiments, learning with the NBS effectively steers the model toward the Pareto frontier during early training stages, which is the foundation of our proposed second stage (in our "two-stage" faimess-aware meta-learning) to achieve fairness goals without compromising model performance (the efficacy is illustrated in Figure 3). Empirically comparing optimization trajectories of the NBS and other baselines of gradient aggregation using the same settings in Section 2: GM, PCGrad [55], and CAGrad [29] (Figure 5), we find that these methods, except for the NBS, often favor groups with larger loss magnitudes, resulting in inefficient convergence towards the Pareto frontier. In contrast, the NBS, with its Pareto Optimality axiom, guides all nodes towards efficient convergence by considering the relative importance of different objectives and maximizing joint gains while avoiding favoring large loss value groups at the expense of others' utility.
A.3 Preliminaries
We give formal definition of Pareto Optimal and related terms, followed by the details of the four axioms of the NBS.
We write r > as x, 2 y, for all entry for vector x, and x > y if 2 and x # y. We use || denote noomm.
Pareto optimality. Consider a set of function fa...., fx we want to minimize using some parameter 0 O. Let vector valued function f(0) (0) ... fx (f)]. # is Pareto optimal if for all other g' E θ satisfies f(0') > f0). That is, no other 6' E θ satisfies f(0') < f(), meaning that objectives cannot be jointly improved without sacrificing any of them.
Pareto front. The set of Pareto optimal models forms the Pareto front. There is no preference the models in the Pareto front, unless with extra customized criteria (such fairness) introduced.
15

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Pareto improvement. 6' is an Pareto improvement to 0 if f(0)) x f(0).
The four axioms of the NBS:
1. Pareto Optimality: The solution is Pareto optimal in S. For solution x E S, it does no exist y > x: for y E S, y # x.
2. Symmetry: The solution is invariant to players' order.
3. Independence of Irrelevant Alternatives: The solution retains upon superset expansion if it remains in the original set. That is, if we expand the feasible set S and we know the solution stays in S, then the solution stays the same.
4. Invariant to A ffine Transformations: If we apply affine transformations to the utilities, the new solution is the original solution with transformed utilities accordingly. Consider affine transformations .I..... .K.. The original solution has utility payoffs [xI,...,II]] If we 9K transform the utilities from u, to g a ug the new solution has payoffs [g] (xx) (xx))
A.4 Problem Setup & Assumptions
Additional assumptions. Assume that S is convex and compact, and that there exists an x S such that 2; > de for all players. We also assume that all players have complete information over the game parameterized (S, d). Assume without loss of generality 9i # for i e [K].
Definition of A. Recall that in Section 3 we define A (VL : VL E Br, VL - DJ g, > 0, Vi E } with D = 0. As a result, A C B. and A's outer boundary lies on the boundary of B., For example. in 2D case, A is a set of circular sectors. A natural question is what happens when considering solutions in Be. First note that cannot directly take logarithm to obtain Equation 9 as some VL 9: - D g. are non- non-positive. Yet, an alternative way is to adjust assumptions on D such that VL D 9; > 0, i E [K]. For example, a way to guarantee the feasibility of A is set D = arg min B, x gi for E [K] such that (VL, d) TT gi for all i [K]. Here, the feasible region can always be B., Assumptions on D will be discussed later in the section. Under current assumption D 0, we can use the axiom of Independence of Irrelevant Alternatives deduce a result on extending A to B,: if the new NBS stays in A, then it remains unchanged. This would serve as a theoretical guarantee if one wants to make assumptions on the solution under the constraint set B, instead.
Feasibility of A. Another assumption is that A is feasible. If A is empty, then we cannot construct the Bargaining game because it doesn't 't satisfy Nash's 's assumption that there always exists one payoff a in the feasible payoff' set such that æ is better than d for each player. In this case, we use the default fairness protocol 3o instead of the improved bargained outcome a as in Algorithm 1. Experiment results show that our mechanism can handle this situation adaptively. In fact, switching to the fairness protocol upon unsuccessful construction of bargaining offers a fresh starting point for subsequent bargaining iterations and helps to escape the saddle points (Figure 6d, also discussed in Section 3.6).
Assumption D = O. We assume disagreement payoff D = O. This is to address the hypergradient conflict such that the NBS satisfies VL, g; 0, E [K]. We now discuss why we don't 't manually adjust D to make the bargaining problem feasible: Relaxing the constraint set will sacrifice the gradient alignment guarantee and deviate from our focus of conflict resolution. For D o O, we are unsure about whether it would help resolve hypergradient conflict. This is because it may flip the sign of some terms, which might result in some VL, 9i < 0 and instead cause hypergradient conflict.
General D. Though general D may not be suitable for conflict resolution (the problem identified in our paper), it may be useful for other scenarios as a gradient aggregation method (which is out of the scope of this work, but can be future direction). Here, we include a discussion for completeness: If A is feasible under general D. then we are able to get Σ, log (V - D) because each term is positive. The derivative w.r.t. VL remains E: DDL 9. Di- Though closed-form solution may not be guaranteed because of the applicability of the tangent slope argument of Equation 5 (that depends on the shape of A), but can be solved as an optimization problem. Note that this may give very different set coefficient because it is measured in terms of the improvement on D.
16

---

A.5 Proofs
Proof of Theorem 3.1: Theorem. Under D = 0, arg maxv [ (VLa) - di) is achieved at
9: LVVLas for some y > 0.
(8)
VLa
9s
Proof. We follow the same steps in Claim 3.1 of [37], along with additional characterization on the shape of under D Note that this theorem may not work under general D.
By the positivity of each term in A, this is equivalent to maximizing the summation of logarithms:
arg max L..BB iE[K]
Σ
log(V V La 9:)-
Taking the derivative of the objective w.r.t VL gives ΣIEK VL, Tg
9:. For any i E [K], we know x T 9i > if and only if (cx) g > with any given e > This means that a point is A, then all points in its radial direction in B, in A (i.e. the boundary of A is subset of the boundary of B,). By the Pareto Optimality, the optimal solution must lie on the boundary of B, as the utility is monotonically increasing in L,, for V A. The optimal points the boundary of B, have tangent slope 0 (i.e. gradient having the same direction as its normal). Hence, we know the normal is in parallel VL and the desired V T.a satisfies
9. VLs for some y 0.
(10)
KE[K]
Lea
91
Proof of Theorem 3.2:
Theorem. The solution to Equation is (up to scaling)
VLa = = ag where
G Ga 1
(11)
a
with the element-wiss reciprocal
Proof. Multiplying both sides with 9j. this is equivalent solving for in ΣjE[K] 9.
= gj
a.
for j E [K]. Observe that x a linear combination of gi (i.e. x = EiE(K] (x 91) -t))) It suffices solve for coefficients 9r by the linear system
LOT iE
91) (x) -1) gj
(12)
for E [K]. Without loss of generality, set 7 1 to ascertain the direction of x. Let [a ... axl
with a₁ = (x: (eTg:))- Equation 12 becomes
9:04 025"
(13)
*E[K]
for j [K]. equivalently,
Ga T 1 a
(14)
with the element- wise reciprocal a concluding the proof. Note that -a is also a solution when a is one, yet preserve RK (i.c. positive contribution of each g.) in implementation.
17

---

Additional Note on Linear Independence Condition for Theorem 3.2:
In Section 3.2, highlight that linear independence does not hold in general. We also wanted to note that fine-grained subgroup definitions can make linear independence assumptions more realistic as they break down larger groups into smaller, distinct units. If subgroups are well-defined and distinct in their goals, characteristics, or attributes, the vectors representing their goals are more likely to be linearly independent. However, real-world scenarios may still exhibit interdependencies or correlations between subgroup goals, especially in the context of fairness, even if subgroups are finely defined. For example, goals may still rely on common resources or have shared objectives at a higher level. The interdependencies and shared factors may cause dependency in hypergradients, Specifically, if one goal can be expressed as combination of the goals of other groups, then the corresponding vectors are linearly dependent. Additionally. if the number of subgroups exceeds the dimension of the hypergradient (i.e. definition too fine- grainedindd, it's impossible to have linear independence. Therefore, while fine-gres ahe-ggrieedd subgroup definitions can make linear independence assumptions more realistic, it's essential to carefully analyze the specific context and relationships between subgroups to determine the extent to which linear independence holds.
Proof of Corollary 3.3:
Corollary. (Norm of bargained update) The solution in Theorem 3.2 has 22-oorm VK.
Proof. It follows that
Σ K
a:gill = ΣΣ
agagg, 9j = K.
(15)
jEK
Proof of Corollary 3.4: Corollary. If gj is o-bounded for j E [K], ||a,5|| is (VKo)-bounded for j E [K].
Proof. By inequality, we have
|a7 = 99 Σ
oggi
< |10 |1
Σ aggi (E(K]
S VKo.
(16)
[E(
Proof of Theorem 3.5: Theorem. (Update rule of 6) Denote L (train) = L(D("10(()) E R for the i-th sample in train-
ing minibatch ( at step t. is updated as g(++1) = g(r) n00)
D ElDi0 Agi with Agi =
max ((Vo(((())TT ( VoL (train)))) V) (train)
(train)
Proof. Since we use the meta-learning framework in [43] with customized (30), we first evaluate the effect of g(t) in the computation graph, following similar steps in Equation 12 and Appendix A of [43]. We initialize i to 0. For data sample in training batch D(t) at step t, we perform a single gradient update in Algorithm 1:
a !
w
(B(0) T (waa)) (Dival) ((0)
(17)
=0
8 09
(3(0) T I, [((aa)) (D)c)))Dvva) 0
a
(
(w)
(1)
(18)
Ow (e)
(=p())
16 10
9 (3(0)) (all) (D
i (=gte)
x 00
((D(010)
10)
(19)
20
(=p(t)
18

---

Line 18 comes from chain rule; Line 19 comes from
@vi (1) (ūx!
①
g(0)
a
(0()
17()
Veco L(D(0)(0())
(20)
avi (1)
=-
8 00
(
(21)
(0=(())
Informally, it means that ú; or w, equivalently, is jointly determined by the (80) -wighhe gradient on the meta set and the gradient on the training batch. Analogous to Equation 30 of [43], the update rule of 0 is
D
0(++1) =0(1) - -ny(1)
1
Σ
max (-w -000)
(22)
00
D
r)(()
((())
A0i
(23)
D
with A max ((Vo((B()) ( (vat)))
VoL.
0) VoLltrain)
(train)
L
(train)
(24)
Note that losses and gradients are taken w.r.t. o and without loss of generality we can disregard 17(*) it absorbs the normalization constant.
Proof of Theorem 3.6:
Theorem. (Pareto improvement of ti) Use a* for the update. Assume L (val) is Lipschitz-smooth with constant C and g; is o-bounded at step t. If the meta learning rate for i satisfies r7()) s CKa5 (0)
for j [K] then ( (m((+)) I. (val) (x()) for any group E [K].
Proof. We follow similar (and standard) steps in Theorem 5.4 of [37], yet retrieve a slightly different (and group-wise tighter) upperbound for learning rate. In Theorem 5.4 of [37], the upperbound for learning rate was minke(K] CKa Our bound is better with multiplicative constant 2 than that of
[37].
Write Au ((++1) (0 and use a well-known property of Lipschinz-smoothness:
I (val) (1)(+))) - I, (val) (w)(t))
I + I2
(25)
with 1 = 10 ( v L. (wal)
(1) T A
(26)
K
= (0 T Σ
(t)
9)
(27)
j=l
(28)
(t)
a;
C
I3
|17 (2) Awll2
(29)
By Corollary 3.3,
I - 1 (1) Aujl ((n(t)))
C
(9)
Aw
(30)
2
Observe that - + I2 < equivalent
17(1)
(31)
2
(1)
0,
2
1 <
(32)
CKa
(t)
19

---

We also have 2
CKa S 2vKo 20 by Corollary 3.4. Note that (val) (w(t+1) S L
((ta))( (z)(t)
(cal)
CVR is strict when Inequality 32 is strict, which yields Pareto Improvement. We just perform one step of gradient update of w in Algorithm 1, yet this result is applicable to single level optimization that directly optimize the parameter in interest.
Proof of Theorem 3.7: We will use the following corollary in the proof:
Corollary A.1. Assume f Rd - RK Lipschitz-smooth with constant C. Fix 8 E RRK finite || For g: RK + R. g(x) B f(x). Then g is Lipschitz-smooth with constant C||,3||.
Proof. This is a classical result in Real Analysis. By chain rule, we have Vg(x) = [V(x)=(VV())))) Then for any x, E IR., by
inequality,
[V)(()) Vg(y>وl = II - (V) [lv ((())) = \// = C|
vll-
(33)
Thus g is Lipschitz-smooth with constant C||/1|.
Theorem. (Monotonic improvement of validation loss w.r.t. 0) Assume L (eal) is Lipschitz-smooth with constant C and VoL, (train) is o-bounded. If the learning rate for 0 satisfies 17( - 11
2|
then gus (f(++1) < ggt (0(+) for any fixed vector g(t) with finite || used to update g(t),
Proof. We incorporate ((() in the computation when following the same (and standard) steps in Lemma 1 of [43]. We drop the superscript of 3(0) for simplicity. By Corollary A.I, we know Lg Lipschitz-smooth with constant C/|/81|. Similar to Theorem 3.6, we use the gradients derived in Theorem 3.5 and have
((( s (0(())) - 11 + I2
(34)
with 17 = (VLo) 20
(35)
(1))
[VL3)]
Σ
max (V V V.ltraii) 00 VL (rain))
(train)
(train)
(36)
|DO)
iED
1/())
max ((VL))))))))))))))(())))))))))) TV (train)
(tranin 0),
(37)
|D
(ED())
I2 = C|I
||0
(38)
2
CII
e)
max ((
1 ( (train) 10 I. (train)
(39)
2
DD
DD
C
())
Σ
(V
v L (train)
0) L (trii)) 2
(40)
2
DOO TED!
CIIB1||
(1)
Σ max(((VLLg)
T V (traén) ( 220)
V VL (tras)) 2
(41)
2
DOP
(ED
(1)311
Σ
max T V (train) /00).
(42)
2 D1012
tDDI)
Line 40 and 41 come from the triangle inequality and Cauchy-Schwand inequality. Note that here the gradients are taken w.r.t. 0. We know -1, + 12 S is equivalent to
2))())
() < C C1
(43)
concluding the proof. Here, we generalize Lemma of [43] from a fixed protocol to any protocol 3 with finite norm. Setting the bargained to 3 gives a new desirable property of the NBS in
20

---

mete-learning which establishes its validity as meta-learning protocol. As such, it is a novel result of the NBS on the monotonicity of validation loss, which has not been presented before as [37] did not focus on meta-learning.
A.6 Detailed Experimental Settings
Synthetic settings (used in 82, 84). We provide here the details for the illustrative example of Figure 2h and 3. We use a modified version of the illustrative example in [37]. We first present our modified learning problem: Let 0 = (01.02) E R2, and consider the following objectives:
(0) c1
and (0) cı (0) 02 (0))2())), where
fi( = log [og(max(00.5(-6) - 7)
5e 6)) + 6,
f2( log log(max((0.5(-))) + 3) tanh(-02) + 2), 6)) + 6,
91 (0) ((-01 + 7) 0.1 (-02 8)3)/10 20,
92 (2) = ((-0) - 7 (-02 8)3)/10 20,
C1 (0) =
0) and c0) = max(tanb(-00556).
We now set L1 é and L = l2 as our objectives to simulate the case where two participated groups have the same scale of loss, as a closer reflection of the loss scope may encountered during inter-group bargaining. In particular, with the loss scale being set to the same, the fairness goal of the evaluated notions, i.c., LtR, FORML, and Meta- Meta-gDRO, becomes the same as x=y. We use six different initialization points {{ 8.5, 7.5), (0.0,0.0), (9.0, 9.0), (-7.5, -0.5), (9, - -10), (9. 220)]. We use the SGD optimizer and train each method for 1000 iterations with a learning rate of 0.1.
Standard fairness benchmark settings (used 84). The detailed fairness dataset settings are provided in Table 2. We access these fairness datasets via Library for Cemi-Auttmated Data Science: https: //gi//gttttt /gggttu.bb... com htt//IMM/aale The library is under Apache 2.0 License.
Dataset
A samples
features
Favor label
Attribute
Trainning Set
Test set (31s of whole data)
Validation set
M 23365.
9555))
((D: 366, 36633
Mc.00:
14060,
1450])
366))
Dlock ((0: 3972.
428))
Black
Black ((0:
Adult
106
White ([0: 31005,
Asian (f0: 969,
282))
10458)
( 146 [
White 311
(J0 353
Other 2353)
Other
Other ((0
Amna-nniia ((0: 289,
Amiz--ndian ((0: 146. 146))
Ames-haiia
3))
Bank
Telemorketing
45211
51
Ago
age ((1: 38563, 4632])
age 25 (f1: 715,0: 108))
age
age ((1: 3391)
((1: 339))
Age 25
Age (5: 3, 3/3
Credit
M 14121, 3563))
Mil1: 215, 2253)
McC;::
Default
S0X
2656))
F([1: 225))
F((1:
Communittes
Crime
1929
blackgtōpet
Truc (11: 841, O: 1015)
(f 1800, 0:
((1: 14/3
14))
False [[1: 1. 1))
Thue ((1: 13)
Sursival
Titanic
1526
M: 672) (1: 229,229,0 0: 1185)
9,0: 9): F(
C 1]
(( L.0:
Student
200 333
Mig
3,0 33
Performunce
seX
F(11:315,00 02))
PofE: 16. 165)
F((1: 2])
Table 2: We detail the number of samples per group, categorized by protected attributes and labels. The student performance dataset allocates 10% for testing due to its smaller size, while others reserve 3% Notably, in the adult income training dataset with race as a protected attribute, only one Amer-Indian sample with the positive (favorable) label. Additionally, after balancing test and validation sets, the community and crime dataset has samples in the "False" group labeled "0". These training data distribution-wise problems are marked in yellow and were maintained to examine how extreme imbalances in training data impact various algorithms.
Training specifies for standard fairness benchmarks (used in $4). For the models applied to these faimess datasets, we consistently employ a 3-layer neural network architecture comprised of an input layer, one hidden layer, and an output layer. The input layer takes features and transforms them to a dimension of size 128. A ReLU activation function and dropout layer for regularization follow this. The hidden layer further processes the data, again followed by a ReLU and dropout layer. Finally, the output layer maps the representation from the hidden layer to the number of classes
21

---

specified (2 for the considered fairness datasets). This architecture utilizes dropout after each ReLU activation to reduce overfitting, making it suitable for classification tasks.
We fine-tuned the training hyperparameters based on the baseline model. Common hyperparameters across all algorithms include total of 50 training epochs, SGD optimizer momentum of 0.9, and a weight decay of 5e-4, with the bargaining phase limited to 15 epochs for the three settings incorporating proposed Wash-Meta-Learinng Hyperparameters that varied are detailed in Table
Switching phases and selection of Tar- In practice, we determine Terr by monitoring the bargain- ing success rate. We observed from real-data experiments that model's performance would stabilize when this rate stabilizes. This may serve as a sign of switching to Stage because no significant improvements could be brought by bargaining. In our real-data experiments, we set T to 15 epochs, as this allowed all evaluated settings to reach a stable bargaining success rate.
Computing. All experiments were conducted on an internal cluster using one chip of H-100.
Learning rate
Drupout Prohabiitty
Adult
1e-3
0.2
512
Income
Se-4
0.4
512
Telemarketing
ago
le-3
0.3
Credit
Default
lo-3
0.5
512
Communities
blockgrfquct
le-4
0.2
Titanic
Survival
le-3
0.1
Performance
Student
sex
le-3
0.06
32
Table 3: Hyperparameter settings for standard fairness datasets.
A.7
Additional Results
A.7.1 Results with 95% Confidence Intervals (CIs).
The main empirical results are averaged from five independent runs using different random seeds, the 95% CI marked in Table 4. In addition to the key findings shown in Table 1, we find the results from the two-stage method are often more stable than the one-stage baselines, reflected by tighter Cls.
A.7.2 Additional Analysis
Noise analysis. To analyze the noise presented in validation sets, as referenced in Table 5, we employ normalized mutual information [24]. This metric offers a measure of the shared information between features and labels, normalized by the sum of their individual entropies. The normalized mutual information norm between feature set and labels is computed as:
I(X;Y)
Toomm (X:Y) H(X) H(Y)
where I(X;)) is the mutual information between X and Y, representing the amount of shared infor- mation, and H (X) and (Y) are the entropies of the feature set and the labels, rsp. A higher Inorm. indicates less noise, signifying a stronger relationship between the features and labels. We referred to https://github com hom/mutee httm/mutallnnn o/mutual.la...ammmmmmaa info (Apache-2.0 license) for implementation.
Referring to Table 5, it is evident that the validation set used from Credit Default dataset and the Communities and Crime dataset exhibits a notably lower mutual information score, compared to the other datasets. This suggests a significantly weaker correlation between the features and labels in these particular validation sets and higher level of noise. Such disparity in mutual information underscores the challenge achieving high model performance and could explain the varying degrees of success observed across different faimess interventions within this dataset.
22

---

Baseline
DRO
LIR
FORML
Meta-gDRO
1w0-stige (ours)
cap-stage
1w0-tage
ono-slige
rwo-dlago (ours)
Incomee [3], Esensitive attribure: se8)
Overall (t)
0,778 1 0.004
0.761
0.010
0,330 1 0,002
0,830 a 0.004
0,801
0.009
0,100 1. 0.001
0810 1 0004
0.837 -: 0.004
Max-gAlCDD (
0.016 + 0.012
D/029
0.016
0.050
+ D.0S5
0.047
+ 0.003
0,075
0.042
0,011
0.000
0,052 + 0.077
0.046 + 0.044
(orst-gAUC(1) 0,770 1 0.011
0,747
0.018
0.505
1 0,004
0.807
1: 0.004
0.763
1 0.022
0.795 1 0.007
0.809 1 0.006
0.814 1 0:006
AdAdu Income [3), (sensitive attribuse: race:
group training data contains only positive (favorable) label.
Overall (t)
0,608 0.004
D.652
0.005
0.003 # D00s
0,805
0:710 0,025
0.008
0,775 0.033
0.793 + 0.004
Mux-gAUCD
0.225 0.027
0.236
0001
D.05
0.090
0.290 0.200::00006
0:0400
4:0.44
0.156 :: 0.00
Worst-gAUIC(1) 0,544 8.019
0.538 0.002
0.555 + 0.04
0.760 + 0.018
0.540 0.019
0.688 0.036
0.694 + 0.047
0,703 0.028
III. Bank Telemarkedngg [35], (sensttive attribune: age)
Overall (t)
0.697 1 0.004
0.686
0.015
0.221 1 0,009
0,728 1: 0,305
0,706 1: 0.101
0.779 1: 0.027
0.698 1: 0.006
0,722 1: 0.009
Max Max-gAliCD (J) 0.013 + 0.010
0.025
0.01D
0.099 + D412
0.083 + 0.300
0.099 0.025
0.029 + 0.009
0.09% + 0.010
0.079
+ 0.008
Worst-gAUC()) 0.691 0.008
0.691
0.008
0.675 1 0,013
0,686
0,005
0.686
0,090
0.764 1 0.025
0.649 0.000
0,683 0.010
Defoult [54), (senslive attribute: sex:
data more noisy than others, see Table 5, Appendix A.7.)
Overall (t)
0A544 6:004
10624
+
0.616
0.554 0:039
0.611 0.018
= 0.017
MaxgAUCD (1)
0,024 1: 0.011
0.022 0.010
0.087 4: 0,022
0.025 =: 0.012
0.017 +: 0.012
0.033 4: 0.018
0.016 4 0007
0.022 -: 0.016
Worst-gAUIC (1)
0.622 + 0.005
0.615 0.012
0.612 + 0.013
0.603 + 0.082
0.545 + 0.083
0.595 + 0.010
0.674 + 0.044
0.650 + 0.025
V. Commmnitiis and Crime [47). (senciiive stmbbute: 6pcckt
val data noiny, meinwhile one group in training data all positive lsbel
Overall (UC (T) Ma
0,525 1 0,008
0.568
0.006
0.679 1 0.022
0,700 a 0,032
0.554 I 0,010
0,568 1 0.018
0.686 1 0.035
0.686 1 0.025
0,050
0.015
0.136 0.012
0.045
0.129
0.107 0.020
0.136 0.057
0.073
0.114 =
Worst-gAUCCth
0.000
0,500
0.000
0.643 + 0,000
0.636
0.046
0.080
0,500 #: 0,000
0.629 # 0.042
0.629
0.022
VL Titanic Servival [12). (sensitive altribule: sex)
Overall (1)
0.972 + 0.000
0.983
0.022
0997
+ 0.010
0.978 + 0.010
0,950 + 0.018
0.972 + 0.022
0.961 +-0012
0.972 : 0006
Max MAKCAUUCD( (1)
0,056 1 0.000
0.033
0.024
0.04 1 0.019
0.044 1: 0.019
0.003 a 0.024
0.011 a 0.019
0.033 1 0.024
0.033
1: 0.024
Werst-gAUCC()) 0,944 + 0.000
0.967 0.024
0.944 + D.000
0.956 0.019
0.933 0.019
0.967 + 0.024
0.944 + 0.000
0,956 + 0.019
VII Student Performance [11] (sensitive amaibne:
(TC Max-gAUCD (
0:011
0,816
1
0.900
0.912
6:022
0:106
0.037
+
0.027
0.082
0.025
0.031
0.018
0.025
Worst-gAUC
0.725 =: 0.020
0,762 = 0.032
0.944 :: 0.013
0.881 a: 0.020
0,300 = 0.022
0.806 :: 0.020
0.893 1: 0.028
0,900 :: 0.020
Table 4: Performance comparison standard fairness datasets (averaged from 5 runs), with 95% CL.
Additional illustration of resolving hypergradient conffict and our dynamics. Figure 6 provides additional visualizations of the impact of our bargaining strategy on gradient conflict resolution, utilizing the adult income dataset with sex as the sensitive attribute. The data indicates that FORML and Meta-gDRO exhibit greater initial conflicts compared to LtR, but also receive more pronounced benefits from our bargaining steps. In Figure 6d, we observe instances where bargaining was unsuccessful, yet an external force, aligned with specific faimess goals (shown with Meta-gDRO as an example), aids in overcoming local sticking points. These observations not only substantiate the utility of bargaining steps in mitigating conflicts but also empirically demonstrate the dynamic process of opting for updates over stagnation, which aligns with the analysis of our method discussed in 83.6.
Atteibute
features
Normalized Mutual Information (Validation
Adult Income
105
31.007
105
43.337 e-5
Bank
Telemarketing
ago
679.576
Credit
Default
26.023 e-5
Communities Crime
blackgttipact
1929
9.962
Titanie
Survival
sex
1526
63.911
Student
Performance
253.199
Table 5: Mutual information of each dataset's validation set normalized by the summation of features' and labels' entropy.
Sensitivity the initial weighting protocol 3n- Figure demonstrates that the improvement from bargaining correlates with the initial hypergradient alignment rate (the portion of aligned batches).
23

---

LtR
FORML
Mctu-gDRRO
Hypergradients Aligned A Non-Empty
759 (59.5%)
0 (0%)
365 (28.692 (2.6%%
1046 (82.0%)
974 (76.4%)
1072 (84.1%)
Table 6: Ote-stage methods in the first 15 epochs on US Crime dataset. "Hypergradients aligned" means the proposed update of the batch lies in A (VI. 9: > 0 for all i). Percentage parethesis is the portion out of total number of batches (1275 batches). We observed that approximately 80% of time the A is nonempty, which gives room for improvement with bargaining.
W/O Nash
W/ Nash
Vl-=0
WD Nash
W/ Nath
9. V VL
124
Dorporyy (tay Soain
107
107
10
150
10
1:0
105
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0.007
0.355
10000
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0.000
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0.51
40524
4.095
2014
00:00
0.122
0:581
6111
m)ninss Vlgife (I.2.....)
2 (lii), madss Why (1.2.....)
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V (L....., (i). it 2 VL, (ii)). mades (Va (1.2.....)
(a) How bargaining helps resolve conflicts (LtR)
(b) How bargaining helps to resolve conflicts (FORML)
WIO Nash
Nash
V
10'
10'
107
100
prequeency
50
Stacking: Worsi-group AUC
Roding: Worst-group AUC
Propueey
Stokkig: Overall AUC
Beding: Overall
127
11F:2
10F
3.003
0.016
0/002
0,322
0.300)
0.12
0.548
0.621
0.694
0.767
0.840
O-nists V4y:id (L.2....)) & EA. VL, (iil). marts of (1. K}
AUC
(c) How bargaining helps to resolve confficts (Meta- gDRO)
(d) Bargaining failed (local sticking points) and how external force helps (Meta-gDRO with Nash)
Figure 6: Visualizations of how Nash bargaining improves gradient conflicts across different methods (adult income with sex as the sensitive attribute). With (i), (ii, (iii) in (a), (b), (c) depicting the least value of group-wise hypergradient alignment (minfg. (g. VLg : é E [K), the averaged group- wise hypergradient alignment value & Σie[K] 9. VLg), and the maximum value of group-wise hypergradient alignment (max(gg VL : E [K]}) respectively.
Baseline
12R w/ Nash
Mnta gnRO
DRD
PORML
Meta Nash
LIR
FORML w/ Nash
0.9
1.0
0.0
0.75
0.7
0.6
0.8
0.8
0.6
0.70
0.7
0.6
0.65
0.6
04
0.4
03
0.6
0.4
0.2
0.60
0.1
0.2
0.2
02
0.5
0.0
0.0
0.0
160
100
160
100
Epach
Epoch
Epoch
Epoch
Epoch
Epoch
(0. Overall ACC:
ACC Disparity)
gorst-group ACC1
()) Overall ACC:
(i). Disparityy (0i). Worst-group ACC
(a) CIFAR-10 imbalanced training results.
(b) CIFAR-100 imbalanced training results.
Figure 7: Comparative analysis of accuracy and fairness during CIFAR-10 and CIFAR- 100 imbal- anced training (results averaged from 5 seeds). Subfigures (a) depict the CIFAR-10 results and (b) show CIFAR- Each plot tracks the evolution of overall ACC, ACC disparity, and worst-group ACC over training epochs, highlighting performance dynamics under 99%- 1% class imbalance for CIFAR-IO and 95%-5% for CIFAR- 100. The shaded regions represent the 95% CI, offering insight into the consistency of each method. The efficacy of Nash bargaining in the two-stage approach is evident, particularly in enhancing the worst-group ACC, vital for fairness in imbalanced settings.
When this initial rate is low, the bargaining process yields significant improvements (for example, FORML). Conversely, when the initial alignment rate is high, the gains from bargaining are more
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modest. This relationship may provide insight into the varying effectiveness of our approach across different scenarios.
Computational costs. Our analysis shows that the entire training process takes approximately 1.2-1.4 times as long as training with bargaining does, with the bargaining steps takes 2-10 times as regular training takes per epoch. We view this as worthwhile trade-off given the enhanced performance and fairesss achieved.
A.7.3 Imbalanced Image Classification
To further validate our framework, we extend our evaluation to imbalanced image classification on subsets of the CIFAR- 10 and CIFAR- 100 datasets [25]. For CIFAR-IO, we construct a training set of 5000 samples with an imbalanced class distribution of 99% to 1%. Due to the smaller per-class sample size in CIFAR-100, we create a 500-sample training set with a 95%-5% split. Test sets are balanced, containing equal number of samples from each class (1000 for CIFAR-10, 100 for CIFAR-100), and validation sets are composed of five samples from each group. We utilize ResNet-18 [19] for modeling and employ SGD with learning rate of 5e-4, momentum of 0.9, and a weight decay of 5e-4. All methods are trained for 100 epochs with a batch size of 128, and for two-stage algorithms, the initial 30 epochs incorporate bargaining.
Figure 7a delineates the trajectory of accuracy (ACC) for different methods during training epochs under a 99%- 1% class imbalance on CIFAR-10. One-stage meta-learning methods encounter difficul- ties in converging to a performing and fair model, often underperforming terms of both ACC and fairness in this context. In contrast, our two-stage method not only achieves significant improvements in ACC and fairness but also enhances algorithmic stability, as indicated by the tighter CIs. The early application of bargaining secures a higher initial ACC, and this advantage is sustained even as the model later transitions to focus on specific fairness objectives. The consistently improved trajectory, echoing synthetic example insights (Figure 3), suggests that reaching the Pareto front and subsequently focusing on fairness does not degrade the model's performance. The above observations underscore the efficacy of the two-stage bargaining approach in managing the trade-offs inherent in fairness-focuss the granular nature of CIFAR-100 (more classes and fewer samples per class) (Figure 7b). Despite this, the two-stage Nash bargaining approach demonstrates substantial improvement in aligning model performance with fairness goals. In contrast, the one-stage baseline methods show limitations in addressing the imbalance, as evidenced by their fluctuating and often lower ACC trajectories. Nash bargaining yields a marked improvement in the worst-group ACC, which is critical in such imbalanced scenarios. This indicates that the bargaining phase helps the model to better navigate the complex decision space of CIFAR-100 100, providing more equitable distribution of predictive accuracy among the classes. Notably, the early-stage bargaining not only propels the model towards higher initial accuracy but also prepares to maintain performance when the fairness objectives become the primary focus in the later stages of training. This strategic approach ensures that the pursuit of fairness does not come at the expense of overall accuracy, a balance that is particularly challenging in the diverse and imbalanced CIFAR-10W 100 environment. The above results from CIFAR- reinforce the generalizable effectiveness of Nash bargaining in our proposed two-stage meta- meta-learning.
tasks. The challenge of achieving fairness across groups is amplified by
A.8 Broader Impact
In this study, we present advancements in ML. fairness through the development and analysis of new fairness-aware meta-learning methods. Our research strictly utilizes synthetic or publicly accessible open-source datasets, avoiding the use of human subjects. While our method demonstrates notable improvements over existing approaches in various experimental setups, recognize that it is not infallible. As such, any application of our method in real-world decision-makinn tasks must be approached with meticulous consideration of the specific context and potential repercussions. It's crucial to understand that achieving ML faimess is an ongoing, collective endeavor within the research community.
Moreover, we also acknowledge that the sainesss-wwre meta-learning methods developed this study are not panaceas for all fairnss--related issues in machine learning. There are still several challenges and open research directions in this area, such as addressing fairness in multimodal learning, developing better explainability and interpretability techniques, and ensuring faireess
25

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in model adaptation and transfer learning. Furthermore, there is need for more diverse and representative datasets evaluate the effectiveness and generalizability of fairness-awar methods. To address these challenges, we call for further research and collaboration among researchers, practitioners, and policymakers to advance the state-of-the-ar MI fairness and to ensure that Al systems are fair, transparent, and accountable for all individuals and groups.
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Reviews:
[{"review_id": "4eveMHns2C", "reviewer": "Reviewer_YN1k", "summary": "The paper addresses group-level fairness in ML with two-stage meta learning. The modeler is given access to a sensitive-group-labeled *validation* dataset and must simultaneously design how to (linearly) weight each group loss in validation via bargaining to resolve gradient conflicts, and how to translate those group weights into weights for each unlabeled sample in the training set (minibatch).\n\nThe paper is supported by a proof of their Nash bargaining solution that does not rely on linear independence and empirical results for their implementation.", "questions": "The experimental section enhances several approaches (LtR, FORML, Meta-gDRO) with the proposed bargaining stage. To my understanding, these all differ only in their original choice of per-group weights \\beta and do not incorporate any additional steps in the bargaining phase to ensure the original objective is being preserved. \n\nThe results of the enhanced algorithm for all of these experiments are quite different so this would suggest quite a strong sensibility of the overall bargaining procedure to the initial weight \\beta^0. I would like the authors to elaborate on this further and provide an intuitive or formal explanation on why the bargaining solutions exhibit such strong sensitivity to this initial parameter.", "limitations": "None", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The NBS solution proposed here is very well motivated and, to my knowledge, novel in this particular sub-field. The overall goal of the NBS is to find optimal weights \\beta for each group loss. The utility per group is computed as the inner product of the gradient wrt the group loss and the grad wrt the weighted group loss function; the disagreement payoff for each player (group) is to simply stop optimization and keep the model as-is. Essentially, group weights are negotiated such that each group's loss gradient has sufficient (positive) alignment, and axiomatically disallows any one group to be unilaterally overruled (the 'do nothing' disagreement outcome has a higher overall payoff than any solution where any player sees their utility alignment become negative).\n\nThe resulting algorithm is relatively straightforward, though compared to simpler two-stage approaches, the bargaining stage requires K second order gradients per minibatch, as well as solving for a non-linear, non-negative equation to devise the optimal weights alpha per bargaining round", "weaknesses": "The computational costs of the bargaining phase could be considerable (see strengths). To this extent, the authors actually only run the bargaining stage in some predefined set of bargaining rounds Tbar and then continue with the weights fixed. \n\nI did not see anything to suggest these weights would remain constant once a bargaining solution is achieved, or any particular guidance or intuition on how this Tbar parameter is selected.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their thoughtful comments and valuable feedback. We appreciate the opportunity to address these points and clarify our approach.\n\n\n**W1: Computational cost.**\n\nWe appreciate this insightful question. We agree the bargaining phase introduces additional complexity to the training process, primarily due to solving the optimization problem in each step. Our analysis shows that the entire training process takes approximately 1.2-1.4 times as long as training with bargaining does, with the bargaining steps (first 15 epochs) themselves taking 2-10 times as much regular training takes per epoch. We view this as a worthwhile trade-off given the enhanced performance and fairness achieved. We've updated our manuscript accordingly.\n\n\n\n**W2: Switching phases and selection of $T_{bar}$.**\n\nThank you for your question! We refer to our observations from synthetic experiments (Figure 3) to justify the reason to remain a fixed weighting protocol after bargaining: After using bargaining to reach the Pareto Front, switching objective does not diverge the model from the Pareto Front in simulations. This implies that achieving downstream fairness goals may not be at the cost of compromising overall model performance if the model has been converged to the Pareto Front.\n\nIn practice, we determine $T_{bar}$ by monitoring the bargaining success rate. We observed from real-data experiments that model's performance would stabilize when this rate stabilizes. This may serve as a sign of swtiching to Stage 2 because no significant improvements could be brought by bargaining. In our real-data experiments, we set $T_{bar}$ to 15 epochs, as this allowed all evaluated settings to reach a stable bargaining success rate. We've added clarifications in our manuscript.\n\n**Q1: Sensitivity to the initial weighting protocol  $\\beta_0$.**\n\nWe agree and appreciate your acute observation! First, we want to clarify that $\\beta_0$ is also used in Stage 1 when bargaining fails, which would preserve the fairness objectives meanwhile providing a fresh start for subsequent bargaining. Second, for the exhibited sensitivity, our analysis to Figure 4 demonstrates that the improvement from bargaining correlates with the initial hypergradient alignment rate (the portion of aligned batches). When this initial rate is low, the bargaining process yields significant improvements (for example, FORML). Conversely, when the initial alignment rate is high, the gains from bargaining are more modest. This relationship may provide insight into the varying effectiveness of our approach across different scenarios.", "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": "i0usaKVyan", "reviewer": "Reviewer_gYXi", "summary": "This paper proposes a novel method to solve the unfairness issue in machine learning. In particular, the authors observed that, existing methods may cause \"hypergradient conflict\" during the optimization process. To resolve the hypergradient conflict, this paper applies the Nash bargaining solution (NBS), and operates in two stages: The first stage resolves the hypergradient conflict and obtains a solution near the Pareto front, and the next stage applies the fairness constraints.", "questions": "In equation (4), the utility is defined as $u_i (\\nabla L_{\\alpha}) = g_i^{\\top} \\nabla L_{\\alpha}$. It is easy to see that if the value is non-positive, then there is a misalignment. However, if it is positive, does the magnitude tell us some information, e.g. if the value is large?", "limitations": "Yes, future directions are discussed at the last section.", "rating": 6, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. This paper presents a subtle yet important observation on hypergradient conflict, that can cause inefficiency while learning fair representations.\n\n2. The application of the NBS is novel and seems appropriate for this scenario.\n\n3. The algorithm is provided with extensive theoretical justifications (section 3.5).", "weaknesses": "Although the approach is novel, more technical explanations on the game theoretical model may be needed. In particular, as far as I understand the work, incentives play a critical role and they provide justifications of many important processes, e.g. the bargaining. However, the origins of payoffs, the set of feasible utility payoffs, and disagreement point (on Page 4) were not emphasized, so I am interested in how the payoffs are determined? Also, were they pre-determined and fixed, or do they change during the algorithm?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful feedback and constructive comments. We are grateful for the chance to discuss these questions.\n\n**W1: Technical explanations on the game theoretical model.**  \n\nThank you for your comment! The concepts you mentioned lacking emphasis on Page 4 have their detailed definition subsequently on Page 5 (Line 151-160). Additional discussion on how incentives play roles is in Section 3.3. We also supplemented further explanations on the problem setup in Appendix A.4, the technical preliminaries in Appendix A.3, and the game theory context in Appendix A.2. Please let us know if you have further questions.\n\n**W2: How payoffs are determined?**\n\nThank you for your question! Equation 4 gives the definition of payoffs (i.e. utilities). Given a proposed update, the payoff of one group is determined by the dot product between the proposed update and the hypergradient of this group. Though the way to compute payoff is pre-determined, the values of payoffs change during the algorithm and serve as a criterion to compute the bargained update.\n\n**Q1: Magnitude of payoff.**\n\nThank you for your question! The value of utility tells us how much of the proposed update is applied in the direction of hypergradient of a certain group. Informally, if it is positive and the magnitude is large, it means the \"in-effect\" update for such group is large and this group is likely to be improved much by the proposed update (vice versa). We refer to Line 151-160 for detailed explanations.", "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": "33Qko9LSRe", "reviewer": "Reviewer_hQxX", "summary": "This paper studies fair prediction tasks where fairness is defined on some partition of the data points into groups (by gender or race, for examples). The paper studies a meta-learning framework that only needs access to group labels for the validation set rather than the entire training dataset. An outer hypergradient optimization of minibatch-level example weights is used to optimize for fairness (on the validation loss), interwoven with standard gradient optimization of the model parameters for the typical minibatch loss (given the weights).\n\nPrevious work has deployed this approach with various predetermined weighting schemes for the outer optimization. The current work identifies a challenge with the stability of such approaches — in particular, often during an outer optimization there may exist a group whose validation loss hypergradient is not aligned with the overall hypergradient, leading to potentially unstable learning and failure to converge to the Pareto front (that is, the undominated groupies loss frontier). To address this challenge, the current work proposes a first stage (conducted for several epochs) wherein the outer optimization is conducted by Nash bargaining between the groups: Roughly speaking, a hypergradient is selected which maximizes the product of alignment between the groups, relative to a default alternative of no change to the re-weighing hyperparameters. This is done for some time, ideally until the agents are able to converge to the Pareto front, at which time a secondary optimization for a particular fairness objective similar to prior work is pursued as a fine-tuning.\n\nThe method is validated in theory by demonstrating Pareto improvement under smoothness and boundedness conditions. Also, experiments are conducted on synthetic and real-world data. The benefit of the synthetic experiments is that the Pareto front can be explicitly computed, and that the method succeeds in converging to the front. The real world experiments show some improvements, though not universal or always dramatic, in fairness for predictive tasks including with unbalanced data, compared to previous meta-learning approaches that do not utilize Nash Bargaining.", "questions": "Q1. Referring to the discussion around lines 151-160. What is the relative significance for the empirical optimization results for simply constraining the hypergradient optimization to the alternative set A [requiring improvement for all groups] versus the selection of the Nash objective in particular to optimize within A? \n\nQ2. Also, how common is it for A to be empty, and would you expect this to become increasingly the case when looking at larger numbers of groups? It seems that in the experiments (referring to the Appendix A.6 details) there are only ever <= 5 groups, is it possible that this approach could stall or encounter more challenges for larger numbers of groups?", "limitations": "No concern", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper does a good job of identifying a serious challenge to the learning stability of prior meta learning techniques for fairness. The challenges of hypergradient conflict is intuitive, coherent, and well described. The methods proposed, inspired by Nash Bargaining, are reasonable and original for resolving these conflicts.\n\nThe theoretical work is a nice contribution, providing both closed form solutions for the bargaining step and a well-grounded argument for Pareto improvement throughout the process.\n\nThe paper does a good job of presenting a lot of results in a clear way with attention placed on visualizations and comprehensive tables for the benefit of the reader.\n\nSeveral different datasets are used and presented to do a good job of not overstating performance based on a single dataset.\n\nIn terms of impact and significance, group fairness-aware predictive models are of clear importance to the community, and the paper addresses one promising technique for dealing with the difficult case where group labels are not generally available for the entire training dataset. The technique proposed seems promising for continued development in contexts of fairness as well as simple problems of class imbalance.", "weaknesses": "Lots of empirical results and observations on synthetic data and for a particular model architecture, but any specifics about the synthetic data and model architecture are hidden in the appendix. Of particular concern is the lack of discussion around limitations of the scale of the experiments, given that the proposed method introduces additional complexity into the training process designed to address instability of existing meta learning approaches. The real-world dataset examples use very small MLPs with a single 128 neuron hidden layer, and the image examples from CIFAR only use 5,000 or 500 training examples. Similarly, the real-world experiments use (I believe) at most 5 groups.\n\nDiscussion around line 275 “Our experiments show…does not deviate the model from the Pareto front…” I think it should be noted explicitly and emphasized that these experiments are on synthetic examples constructed for purpose of simplicity of analysis, so this trend should not be taken as a given in real-world data where one cannot necessarily even calculate the Pareto front to be able to verify the property.\n\nIt would be nice to see some evaluation of the frequency with which bargaining fails in the real-world data, as well as the relative importance for resolving hypergradient conflicts between (1) constraining optimization to the agreement set A, and (2) optimizing within A particularly along the Nash objective.\n\nTypos/grammar:\n1. “hypergradeint” instead of “hypergradient” on line 91.\n2. Line 144-145, found the phrasing a little confusing\n3. Line 271 “if encountered one”", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate the reviewer’s thoughtful feedback and recognition of our contributions. Thank you for giving us the opportunity to clarify our approach based on your insightful comments.\n\n---\n**W1: Discussion on the scale of experiments.**\n\nThank you for your comment! We've updated Section 4.2 to include the potential limitations of the scale of experiments (in particular, on the effect of model size, training size, and number of groups, as suggested). Though we do not extensively explore very large-scale problems due to the scope of this paper, we encourage future studies in this direction. \n\n\n---\n**W2: Emphasis on synthetic experiments.**\n\nThank you for pointing it out! We agree and have changed the phrasing to ``experiments on synthetic data\" in Line 275 and all other places (Line 108, 574, etc) to avoid overstating the observed properties from simulations. We've also added a clarification on real-world data as suggested.\n\n\n---\n**W3 / Q1: Constraining optimization to set $A$.**\n\nThank you for your question! Appendix A.2 gives the comparison with conflict resolution methods, which attempt to give more aligned updates but do not gaurentee to be within $A$. To strictly constrain a proposed update to a non-empty $A$, there are two potential ways as follows:\n\n   1. Choose one update from $A$. This is essentially a bargaining process (where players choose feasible points from $A$). First, a random choice from $A$ would lose the axiomatic property of Pareto Optimality and may result in performance degradation. Second, for axiomatic choices (i.e. solutions characterized by their axioms), we prefer the Nash solution among other prominent ones (including the Kalai-Smorodinsky solution and the Egalitarian solution). Specifically, the Nash objective aligns with our objective to maximize the joint in-effect update. It stands out for its general applicability, robustness [34], and as the unique solution satisfying desirable axioms. \n\n   2. Transform the update from previous weighting protocols (LtR, FORML, Meta-gDRO) to $A$. As far as we understand, there is no simple way to project or rescale the proposed update into $A$. \n\nWe've updated our manuscript to include this discussion accordingly.\n\n___\n**Q2.1: Feasibility of $A$.**\n\nThank you for your question! The following table indicates the number of non-empty $A$ in the first 15 epochs on the US Crime dataset as a showcase.\n\n|                        | LtR          | FORML       | Meta-gDRO    |\n|------------------------|--------------|-------------|--------------|\n| Hypergradients Aligned | 759 (59.5%)  | 0 (0%)      | 365 (28.6%)  |\n| $A$ Non-empty          | 1046 (82.0%) | 974 (76.4%) | 1072 (84.1%) |\n\nHere, \"hypergradients aligned\" means the proposed update of the mini-batch lies in $A$ ($\\nabla L_{\\beta}^\\top g_i > 0$ for all $i$). The percentage in parenthesis is the portion out of the total number of 1275 mini-batches. We observed that approximately 80% times the $A$ is nonempty, which gives room for improvement with bargaining. We've updated our manuscript accordingly.\n\n\n**Q2.2: Scalability with larger number of groups.**\n\nAlthough the likelihood of getting unresolvable conflict may get higher due to the fact that more players are getting involved, we think that the feasibility of $A$ depends more on the group structure rather than the number of groups. For example, the interdependencies and shared factors between groups may cause dependency in hypergradients to enable the feasibility of $A$. When goals of groups rely on common resources or have shared objectives at a higher level, it is still likely to have nonempty $A$. \n\nTheoretically, all presented results should still hold as the number of group increases, thanks to the novel techniques employed in Theorem 3.2 for making this possible. For exmaple, if the number of subgroups exceeds the dimension of the hypergradient, it’s impossible to have linear independence and the previous result [Theorem 5.4, 37] cannot apply. \n\nWe've updated our manuscript to include this discussion.\n\n___\n**W4: Typos.**\n\nThanks for pointing them out! We've fixed the typos and improved the phrasing on Line 144-145 as suggested.", "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": "n2URpyvaCN", "reviewer": "Reviewer_RvEe", "summary": "This paper tried to address the problem of hypergradient conflicts in fairness-aware meta-learning, where the overall validation loss gradient is not aligned with per-group validation loss. They do this by using Nash Bargaining to allow the different groups to achieve consensus on the gradient update direction. Their approach is to have a two-stage solution, where initial rounds of optimization drive the solution to a pareto front, following which they optimize for fairness. They show useful properties of this approach, including non-reliance on the linear independence assumption previously used in NBS-based gradient aggregation.  Finally, they show empirical results in multiple datasets by adding their two-stage approach to existing algorithms, showing improvement in multiple domains.", "questions": "1. Please address the questions from the weaknesses section.\n2. Since the closed-form update is identical to previous work, is the only novelty the removal of the linear independence assumption? Do previous methods' results also match what this paper's experimental section shows?", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is well-written. The visualizations for their experiments are able to convey a lot of information in a small figure, and decently complement the claims of the paper, allowing readers to intuitively understand what the approach is doing. They derive a closed-form update rule for the weights resulting from the Nash bargaining. The problem being tackled is also significant, adding weight to the contributions of the paper.", "weaknesses": "1. It is not very clear to me when the algorithm should move onto the second stage. How is the threshold number of steps determined before moving from the Nash bargaining to optimizing fairness? Or are both steps always performed? Some text makes it seem like they happen independently, but this is not clear from the description of the method, or through the algorithm.\n2. The histograms (e.g. Figure 1, right) are unclear. Are the two colors ever overlapping? If they are disjoint, why is that the case? If they do overlap, there should be some indication of the hidden bars.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the reviewer’s constructive feedback and acknowledgment of our contribution. We appreciate the opportunity to clarify our approach and address your concerns.\n\n____\n**W1: Transitioning to Stage 2 and selection of $T_{bar}$.** \n\nThank you for your question! In practice, we determine $T_{bar}$ by monitoring the bargaining success rate. We observed from real-data experiments that the model's performance would stabilize when this rate stabilizes. This may serve as a sign of switching to Stage 2 because no significant improvements could be brought by bargaining. In our real-data experiments, we set $T_{bar}$ to 15 epochs, as this allowed all evaluated settings to reach a stable bargaining success rate. We also want to clarify that $\\beta_0$ is also used in Stage 1 when bargaining fails, which would preserve the fairness objectives meanwhile providing a fresh start for subsequent bargaining. We've updated the manuscript to improve clarity and consistency.\n\n___\n**W2: Figure 1.b clarification.**\n\nThe two colors are overlapping. We've updated the figure in our manuscript.\n\n___\n**Q2.1: Theoretical novelty.**\n\nThank you for your question! The removal of linear independence in Thm 3.2 is *not* the only novelty. Compared to previous work [37], our additional theoretical contribution is as follows:\n\n - For Pareto improvement of $\\tilde{w}$, Thm 3.6 gives a tighter upper bound for the learning rate than [37]. We require $\\eta^{(t)} \\le \\frac{2}{C K \\alpha_j^{(t)}}$ for all $j \\in [K]$, compared to $min_{j \\in [K]} \\frac{1}{C K \\alpha_j^{(t)}}$ in [Theorem 5.4, 37]. Our bound is better with multiplicative constant 2. See Line 710-712 for details.\n - For meta-learning, Thm 3.7 is a novel result of the NBS on the monotonicity of validation loss, which has not been presented before as [37] did not focus on meta-learning. In fact, we generalized [Lemma 1, 43] from a fixed protocol to any protocol $\\beta$ with the finite norm. Setting the bargained $\\alpha$ to $\\beta$ gives a new desirable property of the NBS in mete-learning, which establishes its validity as a meta-learning protocol.\n - Compared to the setup in [37], we also provided a careful characterization of the feasible set $A$ and extended the discussion from a single disagreement point $D=0$ to general $D$ in Appendix A.4 (Line 611-643). This may provide directions for future gradient aggregation work (Line 350-353).\nWe updated our manuscript with the aforementioned comparison.\n\n**Q2.2: Experimental results matching previous works.**\n\nOur experimental results on real data majorly agree with the previous results. We've carefully characterized the pros and cons of previous methods and the effect of bargaining on different previous methods, along with their comparisons in Sections 2 and 4.2. We've updated our manuscript to include this point accordingly.", "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:

Meta-review: The paper studies imposing group fairness via two-stage meta-learning. The authors observe hypergradient conflicts (i.e., a group's hypergradient is in conflict with the overall hypergradient) as a problem during the optimization for group fairnes via meta-learning. To resolve this problem, the authors propose a two stage process: In the first stage, a hypergradient is selected using ideas from Nash Bargarining Solution with the goal of converging to a Pareto frontier. In the second stage, a particular fairness objective is persued (which is similar to prior work).

The reviewer liked this work and in particular, both the observation about hypergradient conflict and using Nash Bargaining Solution were well received. The reviewers, however, had some concerns. The first one is the scalability of the bargaining step to larger number of groups, bigger datasets, and bigger models. The second concern is the selection of $T_{\text{bar}}$ and switching to the second stage. The reviewers found the authors' response to both these questions satisfying. So I encourage the authors to incorporate the responses into the final version even if the response indicates limitations of the work (e.g., scaling).
Decision: Accept (poster)