# fairnessaware_metalearning_via_nash_bargaining__57d5d43c.pdf

Fairness-Aware Meta-Learning via Nash Bargaining

Yi Zeng 1, Xuelin Yang 2, Li Chen3, Cristian Canton Ferrer3, Ming Jin1, Michael I. Jordan2, Ruoxi Jia1

1Virginia Tech, Blacksburg, VA 24061, USA 2University of California, Berkeley, CA 94720, USA 3Meta AI, Menlo Park, CA 94025, USA

To address issues of group-level fairness in machine learning, it is natural to adjust model parameters based on specific fairness objectives over a sensitiveattributed validation set. Such an adjustment procedure can be cast within a meta-learning framework. However, naive integration of fairness goals via metalearning 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.

1 Introduction

The traditional formulation of machine learning is in terms of a system that improves its predictive and decision-making performance by interacting with an environment. Such a formulation is overly narrow in emerging applications it lumps the social context of a 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 in 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 fairness.

Our focus of the current paper is fairness, and we take a perspective on fairness that blends learning methodology with economic mechanisms. The current favored methodology for addressing fairness recognizes that it is not a one-size-fits-all concept different fairness notions are appropriate for different social settings [49, 32, 50] and treats fairness via meta-learning ideas. Meta-learning is implemented algorithmically with the tools of bi-level optimization. Specifically, fairness-aware meta-learning employs outer optimization to align with a specific fairness goal over a small, demo-

Xuelin and Yi contributed equally. Yi s work is partially done at Responsible AI, Meta AI. Corresponding: Li Chen and Ruoxi Jia. Code: Nash-Meta-Learning.

38th Conference on Neural Information Processing Systems (Neur IPS 2024).

(a) Worst-group alignment value

(b) Hypergradient alignment rate vs. epochs

Stage 1 Stage 2 Stage 1: Resolve conflicts

via bargaining

Stage 2: Focus on fairness goals

+10 -67 % % Overall Performance Disparity

Up to: Align Conflict

Boosting hypergradient

alignment in

(I) Prioritizing the dominant group when

averaging (Lt R) [Ren et al., 2018]

Conflicting Conflicting Non-conflicting

: : Aggregated

(II) Minimizing parity

(FORML) [Yan et al., 2022]

(III) An uncommon case where there is

no conflict

Identified a new Problem: Hypergradient conflicts in standard one-stage fairness-aware meta-learning may lead to unstable performance

This Paper: Two-stage meta-learning with Nash bargaining can resolve conflicts for better fairness and performance at the same time

Figure 1: Overview: We illustrate the problem of hypergradient conflicts in conventional one-stage fairness-aware 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 conflicts 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 a more stable, fair, and performant model. Right: (a, b) Comparison of conventional one-stage meta-learning (Vanilla showcased by FORML, highlighted in gray ) with our proposed two-stage meta-learning approach (Ours, highlighted in green ) which resolves hypergradient conflicts through a 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 hyperparameter-adjusted 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 demographicand label-balanced groups [43] and minimizing disparities [53]. This is a conceptual improvement over methods confined to a single fairness objective [18, 27]. Additionally, meta-learning 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 a meta-learning approach is promising for group-level fairness, 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-learning. Specifically, in our initial empirical explorations of meta-learning algorithms, we found that performance and fairness can vary substantially according to the choice of fairness metric across different datasets, suggesting a form of conflict that is not being resolved effectively via basic meta-learning procedures. Investigating further, we identified a phenomenon that we term hypergradient conflict 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 a novel framework that resolves hypergradient conflicts as a 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 to 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 game-theoretic justification, Pareto improvement, and monotonic improvement in validation loss that help to relate the NBS to our gradient-based learning setting. Our analysis sheds light on the convergence exhibited in empirical studies and provides an understanding of how our method improves meta-learning for fairness.

Baseline Lt R FORML Meta-g DRO

Adult Income [3], (sensitive attribute: race)

Overall AUC ( ) 0.668 0.803 (+20%) 0.710 (+6.2%) 0.775 (+16%) Max-g AUCD ( ) 0.225 0.090 (-96%) 0.290 (+29%) 0.163 (-28%) Worst-g AUC ( ) 0.544 0.755 (+38%) 0.540 (-0.7%) 0.694 (+28%)

Titanic Survival [12], (sensitive attribute: sex)

Overall AUC ( ) 0.972 0.967 (-0.5%) 0.950 (-2.3%) 0.961 (-1.1%) Max-g AUCD ( ) 0.056 0.044 (-21%) 0.033 (-41%) 0.033 (-41%) Worst-g AUC ( ) 0.944 0.944 (-) 0.933 (-1.2%) 0.944 (-)

(a) Comparison of the model performance of three conventional one-stage meta-learning methods (Lt R, FORML, and Meta-g DRO) across various fairness notions (results averaged from 5 runs, % relative to the baseline only using the training set w/o meta-learning)

(I) Group-based

Averaging (Lt R)

(II) Minimizing the Max-Gap (FORML)

(III) Max-Min (Meta-g DRO)

Group 1 loss

Group 2 loss

Group 1 loss Group 1 loss

Group 2 loss

Group 2 loss

(b) Trajectory of 1000-step optimizations. X and y-axis: validation losses for groups 1 and 2, rsp. Gray curves (lower left): the Pareto front. x=y: fairness (equal validation loss). Ideal endpoint: their intersections. Undesirable endpoints: . 6 random initials.

Figure 2: The unreliable performance of conventional one-stage fairness-aware 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 a 67% drop in disparity (Figure 1).

The remainder of this paper is structured as follows: Section 2 describes the problem. Section 3 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 ( 3.4-3.6). Section 4 presents empirical results and analysis. Related work is deferred to Appendix A.1.

2 Problem Statement

Let θ denote a vector of model parameters, let D(train) denote the training data and let D(val) a validation set, with K sensitive groups D(val) i , for i [K]. Let L be a vector-valued function where each entry corresponds to the per-example conventional loss (e.g., cross-entropy). Define group-wise validation loss L(val) as a vector of size K where L(val) i = 1 |D(val) i |L(D(val) i |θ (w)) 1

(the averaged loss over samples in group i). Define Lβ = β L(val) as the fairness loss with vector β of size K encoding the fairness objectives under consideration. Following established work [43, 53], we target group-level fairness objectives via meta-learning as follows:

w = arg min w 0 Lβ(D(val)|θ (w)), (1)

θ (w) = arg min θ w L(D(train)|θ). (2)

The vector w is a set of hyperparameters that reweigh each training example in the current minibatch when updating θ, optimized on D(val) to improve group-level fairness.

Existing fairness-aware meta-learning work can be characterized as different protocols for β. Lt R [43] computes the average of all group-losses with demographic and label balanced D(val) (i.e., βLt R = 1 K 1). FORML [53] calculates the difference between the maximum and minimum grouploss (i.e., βFORML 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-g DRO, with βMeta-g DRO set equal to one for the max and zero otherwise. Throughout the training process, the hypergradient of these methods, w Lβ(D(val)|θ (w)), is derived by applying the aggregation protocol β to the group-wise hypergradients, w L(val) i (D(val)|θ (w)). Their training process, wherein the aggregated hypergradient is iteratively utilized to update w, is referred to as a 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 a fairness objective into Lβ seems natural. However, we find that the effect of this approach on performance and fairness can be unstable. We evaluate the above three one-stage fairness-aware

methods based on targeted fairness metrics (detailed settings in 4.2): Overall Area Under the Curve (Overall AUC) for Lt R which also measures prediction performance, maximum group AUC disparity (Max-g AUCD) for FORML, and worst group AUC (Worst-g AUC) for Meta-g DRO (Figure 2a).

Delving into the cause. After training on real data for several epochs, we 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:

i, s.t. g i Lβ < 0, (3)

with the group-wise gradient gi = w L(val) i (D(val)|θ(w)) and aggregated direction Lβ = w Lβ(D(val)|θ(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-prioritizing certain groups. We study this phenomenon in synthetic settings to isolate structural issues from randomness in stochastic optimization (Figure 2b, settings in 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 in synthetic experiments are prominent (Figure 2b): (I) Averaging (Lt R) 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-offs for fairness. (III) Minimizing the worst-group loss (Meta-g DRO) often exhibits toggling dominance as it solely prioritizes the current least-performing group, which may create conflicts and cannot land on the Pareto front.

Hypergradient conflict resolution. Given the observation of hypergradient conflicts and convergence issues in one-stage meta-learning, we turn to cooperative bargaining and propose a two-stage method that seeks to attain more reliable improvements by resolving conflicts at the early stage of training. Our methodology draws inspiration from the Nash Bargaining Solution (NBS), a 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 fairness-aware meta-learning. Unlike the settings in [37], applying the NBS in 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 a 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 a failure to agree on an outcome, a predetermined disagreement result, denoted as d, will be the final outcome. The individual utility payoff functions are denoted ui : A {D} R, which represent the players preferences over A. Denote the set of feasible utility payoffs as S = (u1(a), ..., u K(a)) : a A RK and the disagreement point as d = (u1(D), ..., u K(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 f(S, d) = Q i(xi di), where xi is the bargained payoff and di 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.

In our problem, we want to find w in Algorithm 1, an intermediate vector for the optimal w to reweigh each training sample. Let L(val) i be the validation loss and gi = w L(val) i be the hypergradient of group i. Let G be the matrix with columns gi. The central question is to find the protocol α as the

weights applied to individual group i s loss. It associates an update step Lα to w that improves the aggregated validation loss among all groups.

We frame this problem as a cooperative bargaining game between the K groups. Define the utility function for group i as

ui( Lα) = g i Lα. (4) The intuition is that the utility tells us how much of proposed update is applied in the direction of hypergradient of group i (as it can be written as gi Lα cos δ with angle δ between gi and Lα). This gives the projection of Lα along gi, or the in effect update for group i. If gi and Lα aligns well, the utility of group i is large (or vice versa). Denote Bϵ the ball of radius ϵ centered at 0. We are interested in the update Lα in the agreement alternative set A = { L : L Bϵ, L α gi D gi > 0, i [K]}. Assume A is feasible and the disagreement point is D = 0 (i.e. the update Lα = 0, staying at w). The goal is to find a Lα that maximizes the product of the deviations of each group s payoff from their disagreement point. Since ui forms a (shifted) linear approximation at w, we are essentially maximizing the utility of Lα locally. We provide further discussions on 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 Lα = P

i [K] αigi, where α RK + solves G Gα = 1

α by the following two theorems, with full proofs available in Appendix A.5: Theorem 3.1. Under D = 0, arg max Lα A Q

i [K](ui( Lα) di) is achieved at X

1 Lα gi gi = γ Lα, for some γ > 0. (5)

Proof Sketch. We employ the same techniques as in Claim 3.1 of [37]. Theorem 3.2. The solution to Equation 5 is (up to scaling) Lα = P

i K αigi where

with the element-wise reciprocal 1

Proof sketch. Let x = Lα. In line with [37] we observe that x = 1

i K(x gi) 1gi. However, whereas [37] relied on the linear independence of the gi s to uniquely determine each coefficient (x gi) 1, our technique makes no such assumption. Instead, we multiply both sides of Equation 5 by gj and obtain P i [K](x gi) 1(g i gj) = γx gi, j [K]. Set αi = (x gi) 1. This is equivalent to g j P

i [K] giαi = α 1 j , 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 gi, 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) or contradict each other (when there is 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, we 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 ℓ2-norm

Corollary 3.4. If gj is σ-bounded for j [K], α 1 j is (

Kσ)-bounded for j [K].

Informally, the NBS has implicit ℓ2 regularization (Corollary 3.3) which substantiates our empirical observation that a separate ℓ2-normalization on Lα for meta-learning rate adjustment yields better performance than ℓ1-normalization (the conventional setting in [43]). Note that we do not impose any assumptions on the boundedness of the hypergradient gi, ensuring the stability even when certain hypergradients are extreme. Furthermore, when gi is bounded, Corollary 3.4 implies that αj is bounded below and no groups are left behind.

Algorithm 1: Two-stage Nash-Meta-Learning Training

Input: θ(0); D(train); D(val) = {D(val) 1 , . . . , D(val) K } Parameters: β0 (fairness protocol); η(t) (step size); Tbar (bargaining steps)

1 for step t {1, . . . , T} do

2 Minibatch D(t) sampled from D(train); /* 1. Unrolling Inner Optimization */

3 θ(t)( w) Backward ADθ w L(D(t)|θ(t)) ;

4 ˆθ(t)( w) θ(t) η(t) θ(t)( w);

6 if t < Tbar then

/* 2. Validation Group-Utilities */

7 for group k {1, . . . , K} do

8 g(t) k Backward AD w L(val) k (D(val)|ˆθ(t)( w)) ;

/* 3. Bargaining Game */

9 Set G(t) with columns g(t) k ;

10 Solve for α: (G(t))T G(t)α = 1

α to obtain α(t);

11 if α(t) is not None then

// bargaining succeeded

13 w Backward AD w Lβ(D(val)|ˆθ(t)( w)) ; /* 4. Weighted Update */

14 w Normalize max( w, 0) ;

15 θ(t) Backward ADθ w L(D(t)|θ(t)) ;

16 θ(t+1) θ(t) η(t) θ(t);

Output: θ(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 ( x S s.t. x d). This aligns players interests in reaching an agreement. The constraint g i Lα > 0 resolves hypergradient conflict upon agreement.

Second, note that Equation 6 shows relationship between the individual and interactive components:

αigi 2 2 + X

j =i (αigi) (αjgj) = 1, (7)

for i [K]. The relative weights αi emerge from a player s own impact ( αigi 2 2) and interactions with others ((αigi) (αjgj)). This trade-off embodies individual versus collective rationality. Positive interactions (i.e., g i gj > 0) incentivize collective improvements by downweighting αi. Negative interactions (i.e., g i gj < 0) increase αi to prioritize individual objectives. Furthermore, each player accounts for a nontrivial contribution to the chosen alternative Lα under mild assumptions (Corollary 3.4). The negotiated solution balances individual and collective rationality through participation incentives and conflict resolution. This equilibrium encapsulates game-theoretic bargaining.

3.4 Two-stage Nash-Meta-Learning

We now present our two-stage method (Algorithm 1) that incorporates Nash bargaining into metalearning training. Previous one-stage algorithms fix a predetermined Lβ; our two-stage method assigns β = α in the NBS in Stage 1 and sets it back to the original β0 in Stage 2. Define θ(t) as the

To give game-theoretic justifications, we build on a similar expression (Equation 2, [37]).

model parameter at step t, T as the number of total steps, and T(bar) T as the number of bargaining steps in Stage 1. Let Backward ADθ(L) be the backward automatic differentiation of computational graph L w.r.t. θ, and Normalize( ) be ℓ2-normalization function. Each step t is constituted by four parts: The first part is unrolling inner optimization, a common technique to approximate the solution to a nested optimization problem [17]. We compute a temporary (unweighted) update ˆθ(t) on training data, which will be withdrawn after obtaining the udpated w. w is initialized to zero and included into the computation graph. The second part calculates the hypergradient g(t) k 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 w by α from the NBS. In Stage 1, successful bargaining grant the update of w by the NBS. If the bargaining game is infeasible or if we are in Stage 2 (not in the bargaining steps), we calculate w based on the fairness loss Lβ of our choice. The last part is the update of parameter θ(t+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 θ) Denote L(train) i = L(D(t) i |θ(t)) R for the i-th sample in

training minibatch D(t) at step t. θ is updated as θ(t+1) = θ(t) η(t)

|D(t)| P|D(t)| i=1 θi with θi =

max (( θ((β(t)) L(val))) θL(train) i , 0) θL(train) i .

Theorem 3.6. (Pareto improvement of w) Use α(t) for the update. Assume L(val) i is Lipschitzsmooth with constant C and g(t) i is σ-bounded at step t. If the meta learning rate for w satisfies η(t) 2 CKα(t) j for j [K], then L(val) i ( w(t+1)) L(val) i ( w(t)) for any group i [K].

Theorem 3.7. Assume L(val) is Lipschitz-smooth with constant C and θL(train) i is σ-bounded.

If the learning rate for θ satisfies η(t) 2|D(t)| C β(t) σ2 , then Lβ(t)(θ(t+1)) Lβ(t)(θ(t)) for any fixed

vector β(t) with finite β(t) used to update θ(t).

Informally, the closed-form update rule of θ indicates that the weight of a training sample is determined by its local similarity between the β-reweighed validation loss surface and the training loss surface (Theorem 3.5). Under mild conditions, w 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 Lβ(t) monotonically decreases w.r.t. β(t) regardless of the choice of protocol (Theorem 3.7). This generalizes [43] from β = 1

K 1 to any β 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 β between phases. Setting β = α 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 θ. Full proofs are given in Appendix A.5.

3.6 Dynamics of Two-stage Nash-Meta-Learning

Our method captures the interplay and synergy among different groups. Specifically, previous methods like linear scalarization (i.e., assigning a fix weight to each group) are limited to identifying points on the convex envelope of the Pareto front [5]. Our method offers a 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 Lα 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 gi Lβ 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 an open problem for future study.

4 Evaluation

We evaluate our method in three key areas: synthetic simulation ( 4.1) for Pareto optimality visa-vis fairness objectives, real-world fairness datasets ( 4.2), 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 to the Pareto front in comparison to Figure 2b. This is evident from the reduced number of 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 always enhance fairness, evidenced by the observation that nodes at the Pareto front tend to stagnate. The NBS does not 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.

Group 1 loss

Group 2 loss

Group 1 loss Group 1 loss

Group-based Averaging (Lt R)

Minimizing the Max-Gap (FORML)

Max-Min (Meta-g DRO)

Group 1 loss

Group 2 loss

Group 1 loss Group 1 loss

(a) Continuous bargaining throughout the entire training (b) Two-stage method (Stage 1: bargaining; Stage 2: fairness goals)

Group-based Averaging (Lt R)

Minimizing the Max-Gap (FORML)

Max-Min (Meta-g DRO)

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 fairness 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 a demographically and label-balanced subset. See Table 2 in Appendix A.6 for data distribution specifics.

General settings and metrics. We compare our two-stage Nash-Meta-Learning with conventional one-stage fairness-aware meta-learning (i.e., Lt R, FORML, Meta-g DRO), 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 2 are used: Overall AUC ( ) , Max-g AUCD ( ) , and Worst-g AUC ( ) , corresponding to the goal of Lt R, FORML, and Meta-g DRO, respectively.

Results and analysis. Our NBS-enhanced two-stage meta-learning improves the overall performance, fairness objectives (in color), and stability, as in Table 1, 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-g AUCD by 26% (from 0.039 to 0.029) with a 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-g DRO). We diagnose that these two dataset s validation set contains low feature-label mutual information, leading to noisy outcomes (Appendix A.7) and affecting

Baseline DRO Lt R FORML Meta-g DRO

one-stage two-stage (ours) one-stage two-stage (ours) one-stage two-stage (ours)

I. Adult Income [3], (sensitive attribute: sex)

Overall AUC ( ) 0.778 0.761 0.830 0.830 (+.000) 0.801 0.810 (+.009) 0.810 0.837 (+.027) Max-g AUCD ( ) 0.016 0.029 0.050 0.047 (-.003) 0.075 0.031 (-.044) 0.052 0.046 (-.006) Worst-g AUC ( ) 0.770 0.747 0.805 0.807 (+.002) 0.763 0.795 (+.032) 0.809 0.814 (+.006)

II. Adult Income [3], (sensitive attribute: race; * one group in training data contains only one positive (favorable) label. )

Overall AUC ( ) 0.668 0.652 0.803 0.805 (+.002) 0.710 0.775 (+.065) 0.775 0.793 (+.018) Max-g AUCD ( ) 0.225 0.236 0.090 0.090 (-.000) 0.290 0.134 (-.156) 0.163 0.158 (-.005) Worst-g AUC ( ) 0.544 0.538 0.755 0.760 (+.005) 0.540 0.688 (+.148) 0.694 0.703 (+.009)

III. Bank Telemarketing [33], (sensitive attribute: age)

Overall AUC ( ) 0.697 0.686 0.724 0.728 (+.004) 0.706 0.779 (+.073) 0.698 0.722 (+.024) Max-g AUCD ( ) 0.013 0.025 0.099 0.083 (-.016) 0.039 0.029 (-.010) 0.098 0.079 (-.019) Worst-g AUC ( ) 0.691 0.691 0.675 0.686 (+.011) 0.686 0.764 (+.078) 0.649 0.683 (+.034)

IV. Credit Default [54], (sensitive attribute: sex; * val data is more noisy than others, see analysis in Table 5, Appendix A.7. )

Overall AUC ( ) 0.634 0.624 0.630 0.616 (-.014) 0.554 0.611 (+.057) 0.682 0.661 (-.021) Max-g AUCD ( ) 0.024 0.022 0.037 0.025 (-.012) 0.017 0.033 (+.016) 0.016 0.022 (+.006) Worst-g AUC ( ) 0.622 0.613 0.612 0.603 (-.009) 0.545 0.595 (+.050) 0.674 0.650 (-.024)

V. Communities and Crime [42], (sensitive attribute: blackgt6pct; * val data is noisy, meanwhile one group in training data are all positive labels )

Overall AUC ( ) 0.525 0.568 0.679 0.700 (+.021) 0.554 0.568 (+.014) 0.686 0.686 (+.000) Max-g AUCD ( ) 0.050 0.136 0.071 0.129 (+.058) 0.107 0.136 (+.029) 0.114 0.114 (-.000) Worst-g AUC ( ) 0.500 0.500 0.643 0.636 (-.007) 0.500 0.500 (+.000) 0.629 0.629 (+.000)

VI. Titanic Survival [12], (sensitive attribute: sex)

Overall AUC ( ) 0.972 0.983 0.967 0.978 (+.011) 0.950 0.972 (+.022) 0.961 0.972 (+.011) Max-g AUCD ( ) 0.056 0.033 0.044 0.044 (-0.00) 0.033 0.011 (-.022) 0.033 0.033 (-.000) Worst-g AUC ( ) 0.944 0.967 0.944 0.956 (+.012) 0.933 0.967 (+.034) 0.944 0.956 (+.012)

VII. Student Performance [11], (sensitive attribute: sex)

Overall AUC ( ) 0.784 0.816 0.900 0.900 (+.000) 0.828 0.822 (-.006) 0.909 0.912 (+.003) Max-g AUCD ( ) 0.119 0.106 0.013 0.037 (+.024) 0.056 0.031 (-.025) 0.031 0.025 (-.006) Worst-g AUC ( ) 0.725 0.762 0.894 0.881 (+.013) 0.800 0.806 (+.006) 0.894 0.900 (+.006)

Table 1: Comparison on standard fairness datasets (averaged from 5 runs). Each of { Lt R , FORML ,

Meta-g DRO } is paired with { Overall AUC , Max-g AUCD , Worst-g AUC } rsp. for aligning intended fairness goals. Top results of each row in bold.

most tested methods (e.g., Lt R, 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-Indian sample. These insights emphasize the importance of validation set quality and representative samples in the training.

Worst-group hypergradient alignment value

0.603 0.800 0.000 0.769 0.210 0.847

Epoch (a). (b). % per epoch

Align Conflict Align Conflict Align Conflict Align Rate:

Figure 4: Effects on hypergradient alignment (Bank Telemarketing). (a) Smallest g i Lβ. Portion of positive values (Align. Rate) at the bottom. (b) Hypergradient alignment rate per epoch.

Effects of bargaining on hypergradient conflicts. Bargaining enhances hypergradient alignment by varying degrees among different onestage algorithms (Figure 4, 6). For instance, Lt R s alignment rate improves from 60% to 80%, and FORML jumps from 0 to 76.9% accompanied with more substantial performance and fairness gains on Bank Telemarketing (Figure 4). FORML consistently benefits more from bargaining compared to the other two, likely due to its optimization goal that could intensify hypergradient conflicts. Moreover, our approach uniformly promotes hypergradient alignment during Stage 1 (Figure 6). We show that early-stage bargaining, accompanied by its hypergradient conflict resolution, is crucial for enhancing model performance and fairness. Further illustrations and analyses are in Appendix A.7.

Discussions on scalability. To scale up models and datasets, one natural question is whether the bargaining game can still be 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, we

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.

5 Discussion & Conclusion

In conclusion, our study offers several key insights: We identified hypergradient conflict as a pivotal issue undermining stable performance in one-stage fairness-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-free 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 fairness datasets, and two image classification tasks.

Future directions. First, addressing the absence of a 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 β during training. Future work can focus on designing and flexibly choosing outer optimization goals to delicately improve performance, fairness, or other metrics in interest. Third, we derive the NBS under D = 0 for conflict resolution. Future study could investigate general D = 0, which might not be useful for conflict resolution (the scope of our paper), but could be used in other cases as a 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 Re DS 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-2331775 and IIS-2312794. Michael Jordan wishes to acknowledge support from the European Union (ERC-2022-SYG-OCEAN-101071601).

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In this section, we first provide background and related work ( A.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 ( A.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 ( A.7). Finally, we discuss the broader impact and limitations ( A.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 bargaining-related gradient aggregation, a key distinction between our work and prior research is our provision of a 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], as 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 in 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 proxybased 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 circumventing 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 disadvantaged groups [41, 18, 27, 8]. Challenges also arise in the potential bias toward mislabeled data when targeting worst-performing 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 grouplevel notions, as fairness-aware meta-learning inherently relies on 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 fairness 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 on its effectiveness in 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 is a concept originated from Rawl 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 a safety net for the least advantaged group as the original Rawlsian Justice intends. Rather, the ML group-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 fairness differs from the original Rawlsian context, it remains a valuable technique for promoting equity in ML systems by preventing any group from being consistently disadvantaged.

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 to 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 a principled and fair manner.

Group 1 loss

Group 2 loss

Group 1 loss

Group 2 loss

Group 1 loss

Group 2 loss

(a) NBS (ours) (b.1) GM (p=1) (b.2) GM (p=-1)

Group 1 loss

Group 2 loss

Group 1 loss

Group 2 loss

(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 to the Pareto front after 1000 steps of updates.

Why NBS for hypergradient conflict resolution? An empirical comparison. As far as we understand, there is no simple way to project or rescale a proposed update (such as Lt R, FORML, or Meta-g DRO) 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 a 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 fairness-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 a formal definition of Pareto Optimal and related terms, followed by the details of the four axioms of the NBS.

We write x y as xi yi for all entry i for vector x, y, and x y if x y and x = y. We use to denote ℓ2-norm.

Pareto optimality. Consider a set of function f1, . . . , f K we want to minimize using some parameter θ Θ. Let vector valued function f(θ) = [f1(θ) . . . f K(θ)]. θ is Pareto optimal if for all other θ Θ satisfies f(θ ) f(θ). That is, no other θ Θ satisfies f(θ ) f(θ), 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 of the models in the Pareto front, unless with extra customized criteria (such as fairness) introduced.

Pareto improvement. θ is an Pareto improvement to θ if f(θ ) f(θ).

The four axioms of the NBS:

1. Pareto Optimality: The solution is Pareto optimal in S. For solution x S, it does no exist y x for y 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 Affine Transformations: If we apply affine transformations to the utilities, the new solution is the original solution with transformed utilities accordingly. Consider affine transformations g1, . . . , g K. The original solution has utility payoffs [x1, . . . , x K]. If we transform the utilities from ui to g ui, the new solution has payoffs [g1(x1), . . . , g K(x K)].

A.4 Problem Setup & Assumptions

Additional assumptions. Assume that S is convex and compact, and that there exists an x S such that xi > di for all players. We also assume that all players have complete information over the game parameterized as (S, d). Assume without loss of generality gi = 0 for i [K].

Definition of A. Recall that in Section 3 we define A = { L : L Bϵ, L α gi D gi > 0, i [K]} with D = 0. As a result, A Bϵ and A s outer boundary lies on the boundary of Bϵ. For example, in 2D case, A is a set of circular sectors. A natural question is what happens when considering solutions in Bϵ. First note that we cannot directly take logarithm to obtain Equation 9 as some L α gi D gi are non-positive. Yet, an alternative way is to adjust assumptions on D such that L α gi D gi > 0, i [K]. For example, a way to guarantee the feasibility of A is to set D = arg minx Bϵ x gi for i [K] such that ( Lα d) gi > 0 for all i [K]. Here, the feasible region can always be Bϵ. 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 to deduce a result on extending A to Bϵ: 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 Bϵ 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 satisfy Nash s assumption that there always exists one payoff x in the feasible payoff set such that x is better than d for each player. In this case, we use the default fairness protocol β0 instead of the improved bargained outcome α 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 = 0. We assume disagreement payoff D = 0. This is to address the hypergradient conflict such that the NBS satisfies L α gi > 0, i [K]. We now discuss why we don 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 = 0, 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 L α gi < 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 a future direction). Here, we include a discussion for completeness: If A is feasible under general D, then we are able to get P

i log ( Lα D) gi because each term is positive. The derivative w.r.t. Lα remains P

i 1 ( Lα D) gi gi. 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 it can be solved as an optimization problem. Note that this may give a very different set coefficient because it is measured in terms of the improvement on D.

Proof of Theorem 3.1: Theorem. Under D = 0, arg max Lα A Q

i [K](ui( Lα) di) is achieved at

1 Lα gi gi = γ Lα, for some γ > 0. (8)

Proof. We follow the same steps in Claim 3.1 of [37], along with additional characterization on the shape of A under D = 0. 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 Lα Bϵ

i [K] log( Lα gi). (9)

Taking the derivative of the objective w.r.t. Lα gives P

i K 1 Lα gi gi. For any i [K], we know x gi > 0 if and only if (cx) g > 0 with any given c > 0. This means that if a point is in A, then all points in its radial direction in Bϵ are in A (i.e. the boundary of A is a subset of the boundary of Bϵ). By the Pareto Optimality, the optimal solution must lie on the boundary of Bϵ as the utility is monotonically increasing in Lα for Lα A. The optimal points on the boundary of Bϵ have tangent slope 0 (i.e. gradient having the same direction as its normal). Hence, we know the normal is in parallel to Lα and the desired Lα satisfies X

1 Lα gi gi = γ Lα, for some γ > 0. (10)

Proof of Theorem 3.2: Theorem. The solution to Equation 5 is (up to scaling) Lα = P

i K αigi where

with the element-wise reciprocal 1

Proof. Multiplying both sides with gj, this is equivalent to solving for x in P

i [K] g i gj x gi = γx gj for j [K]. Observe that x is a linear combination of gi (i.e. x = 1

i [K](x gi) 1gi). It suffices to solve for coefficients x gi by the linear system X

i K (g i gj)(x gi) 1 = γx gj (12)

for j [K]. Without loss of generality, set γ = 1 to ascertain the direction of x. Let α = [α1 . . . αK] with αi = (x gi) 1. Equation 12 becomes

i [K] giαi = α 1 j (13)

for j [K], or, equivalently,

with the element-wise reciprocal 1

α, concluding the proof. Note that α is also a solution when α is one, yet we preserve α RK + (i.e. positive contribution of each gi) in implementation.

Additional Note on Linear Independence Condition for Theorem 3.2:

In Section 3.2, we 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 a 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-grained), it s impossible to have linear independence. Therefore, while fine-grained 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 ℓ2-norm

Proof. It follows that

i K αigi 2 = X

j K αiαjg i gj = K. (15)

Proof of Corollary 3.4: Corollary. If gj is σ-bounded for j [K], α 1 j is (

Kσ)-bounded for j [K].

Proof. By Cauchy-Schwarz inequality, we have

Proof of Theorem 3.5: Theorem. (Update rule of θ) Denote L(train) i = L(D(t) i |θ(t)) R for the i-th sample in train-

ing minibatch D(t) at step t. θ is updated as θ(t+1) = θ(t) η(t)

|D(t)| P|D(t)| i=1 θi with θi =

max (( θ((β(t)) L(val))) θL(train) i , 0) θL(train) i .

Proof. Since we use the meta-learning framework in [43] with customized β(t), we first evaluate the effect of β(t) in the computation graph, following similar steps in Equation 12 and Appendix A of [43]. We initialize w to 0. For data sample i in training batch D(t) at step t, we perform a single gradient update as in Algorithm 1:

w(t) i (β(t)) L(val)(D(val)|ˆθ(t))

w(t) i =0 (17)

θ(β(t)) L(val)(D(val)|θ)

w(t) i ˆθ(t)( w(t) i )

w(t) i =0 (18)

θ(β(t)) L(val)(D(val)|θ)

θ=θ(t) θL(D(t) i |θ) θ=θ(t) . (19)

Line 18 comes from chain rule; Line 19 comes from

w(t) i ˆθ(t)( w(t) i ) =

θ(t) η(t) θ(t) w(t) L(D(t)|θ(t)) (20)

θL(D(t) i |θ) θ=θ(t) . (21)

Informally, it means that w, or w, equivalently, is jointly determined by the β(t)-weighed gradient on the meta set and the gradient on the training batch. Analogous to Equation 30 of [43], the update rule of θ is

θ(t+1) = θ(t) η(t)

i=1 max ( w(t) i , 0)L(D(t) i |θ(t)) (22)

= θ(t) η(t)

i=1 θi (23)

with θi = max (( θ((β(t)) L(val))) θL(train) i , 0) θL(train) i . (24)

Note that losses and gradients are taken w.r.t. θ(t), and without loss of generality we can disregard η(t) as if it absorbs the normalization constant.

Proof of Theorem 3.6: Theorem. (Pareto improvement of w) Use α(t) for the update. Assume L(val) i is Lipschitz-smooth with constant C and g(t) i is σ-bounded at step t. If the meta learning rate for w satisfies η(t) 2 CKα(t) j for j [K], then L(val) i ( w(t+1)) L(val) i ( w(t)) for any group i [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 mini [K] 1 CKα(t) j . Our bound is better with multiplicative constant 2 than that of

Write w = w(t+1) w(t) and use a well-known property of Lipschitz-smoothness:

L(val) i ( w(t+1)) L(val) i ( w(t)) I1 + I2 (25)

with I1 = η(t) L(val) i ( w(t)) w (26)

= η(t)g(t) i

j=1 α(t) j g(t) j (27)

α(t) i , (28)

2 η(t) w 2. (29)

By Corollary 3.3,

2 η(t) w 2 = C(η(t))2

2 w 2 = CK(η(t))2

Observe that I1 + I2 0 is equivalent to

α(t) i (31)

CKα(t) i . (32)

We also have 2 CKα(t) i 2

Kσ CK = 2σ C

K by Corollary 3.4. Note that L(val) i ( w(t+1)) L(val) i ( w(t))

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 is Lipschitz-smooth with constant C. Fix β RK with finite β . For g : RK R, g(x) = β f(x). Then g is Lipschitz-smooth with constant C β .

Proof. This is a classical result in Real Analysis. By chain rule, we have g(x) = ( f(x))β. Then for any x, y Rd, by Cauchy-Schwarz inequality,

g(x) g(y) = ( f(x))β ( f(y))β = ( f(x) f(y))β f(x) f(y) β C β x y . (33)

Thus g is Lipschitz-smooth with constant C β .

Theorem. (Monotonic improvement of validation loss w.r.t. θ) Assume L(val) is Lipschitz-smooth with constant C and θL(train) i is σ-bounded. If the learning rate for θ satisfies η(t) 2|D(t)| C β(t) σ2 ,

then Lβ(t)(θ(t+1)) Lβ(t)(θ(t)) for any fixed vector β(t) with finite β(t) used to update θ(t).

Proof. We incorporate β(t) in the computation when following the same (and standard) steps in Lemma 1 of [43]. We drop the superscript of β(t) for simplicity. By Corollary A.1, we know Lβ is Lipschitz-smooth with constant C β . Similar to Theorem 3.6, we use the gradients derived in Theorem 3.5 and have

Lβ(θ(t+1)) Lβ(θ(t)) I1 + I2 (34)

with I1 = ( Lβ) θ (35)

= ( Lβ) η(t)

i D(t) max{( Lβ) L(train) i , 0} L(train) i (36)

i D(t) max{(( Lβ) L(train) i )2, 0}, (37)

i D(t) max{( Lβ) L(train) i , 0} L(train) i

max{( Lβ) L(train) i , 0} L(train) i 2 (40)

i D(t) max{(( Lβ) L(train) i )2, 0} L(train) i 2 (41)

C β (η(t))2σ2

i D(t) max{(( Lβ) L(train) i )2, 0}. (42)

Line 40 and 41 come from the triangle inequality and Cauchy-Schwarz inequality. Note that here the gradients are taken w.r.t. θ. We know I1 + I2 0 is equivalent to

η(t) 2|D(t)|

C β σ2 , (43)

concluding the proof. Here, we generalize Lemma 1 of [43] from a fixed protocol to any protocol β with finite norm. Setting the bargained α to β gives a new desirable property of the NBS in

mete-learning, which establishes its validity as a 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 2, 4). We provide here the details for the illustrative example of Figure 2b and 3. We use a modified version of the illustrative example in [37]. We first present our modified learning problem: Let θ = (θ1, θ2) R2, and consider the following objectives:

ℓ1(θ) = c1(θ)f1(θ) + c2(θ)g1(θ) and ℓ2(θ) = c1(θ)f2(θ) + c2(θ)g2(θ), where f1(θ) = log(max(|0.5( θ1 7) tanh( θ2)|, 5e 6)) + 6, f2(θ) = log(max(|0.5( θ1 + 3) tanh( θ2) + 2|, 5e 6)) + 6,

g1(θ) = (( θ1 + 7)2 + 0.1 ( θ2 8)2)/10 20,

g2(θ) = (( θ1 7)2 + 0.1 ( θ2 8)2)/10 20, c1(θ) = max(tanh(0.5θ2), 0) and c2(θ) = max(tanh( 0.5θ2), 0)

We now set L1 = ℓ1 and L2 = 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.e., Lt R, FORML, and Meta-g DRO, 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, 1.0), (9, 20)}. We use the SGD optimizer and train each method for 1000 iterations with a learning rate of 0.1.

Standard fairness benchmark settings (used in 4). The detailed fairness dataset settings are provided in Table 2. We access these fairness datasets via Library for Semi-Automated Data Science: https://github.com/IBM/lale. The library is under Apache 2.0 License.

Dataset # samples # features Favor label Attribute Trainning Set Test set (3% of whole data) Validation set

sex M ({0: 22365, 1: 9555}) F ({0: 14060, 1: 1430}) M ({0: 366, 1: 366}) F ({0: 366, 1: 366}) M ({0: 3, 1: 3}) F ({0: 3, 1: 3})

Adult Income 48842 105 1 race

Black ({0: 3972, 1: 428}) White ({0: 31006, 1: 10458}) Asian ({0: 969, 1: 282}) Other ({0: 233}) Amer-Indian ({0: 289, 1: 1 })

Black ({0: 146, 1: 146}) White ({0: 146, 1: 146}) Asian ({0: 146, 1: 146}) Other ({0: 146, 1: 146}) Amer-Indian ({0: 146, 1: 146})

Black ({0: 3, 1: 3}) White ({0: 3, 1: 3}) Asian ({0: 3, 1: 3}) Other ({0: 3, 1: 3}) Amer-Indian ({0: 3, 1: 3})

Bank Telemarketing 45211 51 1 age age > 25 ({1: 38564, 0: 4632})

age < 25 ({1: 715, 0: 108})

age > 25 ({1: 339, 0: 339}) age < 25 ({1: 339, 0: 339})

age > 25 ({1: 3, 0: 3}) age < 25 ({1: 3, 0: 3})

Credit Default 30000 24 0 sex M ({1: 14122, 0: 3542}) F ({1: 8787, 0: 2656}) M ({1: 225, 0: 225}) F ({1: 225, 0: 225}) M ({1: 4, 0: 4}) F ({1: 4, 0: 4})

Communities and Crime 1994 1929 0 blackgt6pct False ({1: 1002, 0: 0 })

True ({1: 841, 0: 101})

False ({1: 14, 0: 14})

True ({1: 14, 0: 14})

False ({1: 1, 0: 1})

True ({1: 1, 0: 1})

Titanic Survival 1309 1526 1 sex M ({1: 151, 0: 672}) F ({1: 329, 0: 118}) M ({1: 9, 0: 9}) F ({1: 9, 0: 9}) M ({1: 1, 0: 1}) F ({1: 1, 0: 1})

Student Performance 649 58 1 sex M ({1: 200, 0: 33})

F ({1: 315, 0: 32})

M ({1: 16, 0: 16})

F ({1: 16, 0: 16})

M ({1: 2, 0: 2})

F ({1: 2, 0: 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 no 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 specifics for standard fairness benchmarks (used in 4). For the models applied to these fairness 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 in features and transforms them to a dimension of size 128. A Re LU activation function and a dropout layer for regularization follow this. The hidden layer further processes the data, again followed by a Re LU and dropout layer. Finally, the output layer maps the representation from the hidden layer to the number of classes

specified (2 for the considered fairness datasets). This architecture utilizes dropout after each Re LU 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 a total of 50 training epochs, an 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 Nash-Meta-Learning. Hyperparameters that varied are detailed in Table 3.

Switching phases and selection of Tbar. In practice, we determine Tbar 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 switching to Stage 2 because no significant improvements could be brought by bargaining. In our real-data experiments, we set Tbar 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.

Dataset Attribute Learning rate Dropout Probability Batch Size

sex 1e-3 0.2 512 Adult Income race 5e-4 0.4 512

Bank Telemarketing age 1e-3 0.3 512

Credit Default sex 1e-3 0.5 512

Communities and Crime blackgt6pact 1e-4 0.2 32

Titanic Survival sex 1e-3 0.4 32

Student Performance sex 1e-3 0.05 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 CIs.

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 Inorm between a feature set X and labels Y is computed as:

Inorm(X; Y ) = I(X; Y ) H(X) + H(Y )

where I(X; Y ) is the mutual information between X and Y , representing the amount of shared information, and H(X) and H(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/mutualinfo/mutual_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 a higher level of noise. Such a disparity in mutual information underscores the challenge in achieving high model performance and could explain the varying degrees of success observed across different fairness interventions within this dataset.

Baseline DRO Lt R FORML Meta-g DRO

one-stage two-stage (ours) one-stage two-stage (ours one-stage two-stage (ours)

I. Adult Income [3], (sensitive attribute: sex)

Overall AUC ( ) 0.778 0.004 0.761 0.010 0.830 0.002 0.830 0.004 0.801 0.009 0.810 0.004 0.810 0.004 0.837 0.004 Max-g AUCD ( ) 0.016 0.012 0.029 0.016 0.050 0.005 0.047 0.003 0.075 0.042 0.031 0.009 0.052 0.007 0.046 0.004 Worst-g AUC ( ) 0.770 0.011 0.747 0.018 0.805 0.004 0.807 0.004 0.763 0.022 0.795 0.007 0.809 0.006 0.814 0.006

II. Adult Income [3], (sensitive attribute: race; * one group in training data contains only one positive (favorable) label. )

Overall AUC ( ) 0.668 0.004 0.652 0.005 0.803 0.005 0.805 0.007 0.710 0.023 0.775 0.008 0.775 0.013 0.793 0.004 Max-g AUCD ( ) 0.225 0.027 0.236 0.011 0.090 0.015 0.090 0.018 0.290 0.006 0.134 0.040 0.163 0.044 0.158 0.029 Worst-g AUC ( ) 0.544 0.019 0.538 0.002 0.755 0.004 0.760 0.018 0.540 0.019 0.688 0.036 0.694 0.047 0.703 0.028

III. Bank Telemarketing [33], (sensitive attribute: age)

Overall AUC ( ) 0.697 0.004 0.686 0.015 0.724 0.009 0.728 0.005 0.706 0.101 0.779 0.027 0.698 0.006 0.722 0.009 Max-g AUCD ( ) 0.013 0.010 0.025 0.010 0.099 0.012 0.083 0.009 0.039 0.023 0.029 0.009 0.098 0.010 0.079 0.008 Worst-g AUC ( ) 0.691 0.008 0.691 0.008 0.675 0.013 0.686 0.005 0.686 0.090 0.764 0.025 0.649 0.009 0.683 0.010

IV. Credit Default [54], (sensitive attribute: sex; * val data is more noisy than others, see Table 5, Appendix A.7. )

Overall AUC ( ) 0.634 0.004 0.624 0.008 0.630 0.019 0.616 0.035 0.554 0.039 0.611 0.018 0.682 0.003 0.661 0.017 Max-g AUCD ( ) 0.024 0.011 0.022 0.010 0.037 0.022 0.025 0.012 0.017 0.012 0.033 0.018 0.016 0.007 0.022 0.016 Worst-g AUC ( ) 0.622 0.005 0.613 0.012 0.612 0.013 0.603 0.032 0.545 0.033 0.595 0.010 0.674 0.004 0.650 0.025

V. Communities and Crime [42], (sensitive attribute: blackgt6pct; * val data is noisy, meanwhile one group in training data are all positive labels )

Overall AUC ( ) 0.525 0.008 0.568 0.006 0.679 0.022 0.700 0.032 0.554 0.010 0.568 0.018 0.686 0.035 0.686 0.035 Max-g AUCD ( ) 0.050 0.015 0.136 0.012 0.071 0.045 0.129 0.073 0.107 0.020 0.136 0.037 0.114 0.073 0.114 0.073 Worst-g AUC ( ) 0.500 0.000 0.500 0.000 0.643 0.000 0.636 0.046 0.500 0.000 0.500 0.000 0.629 0.042 0.629 0.042

VI. Titanic Survival [12], (sensitive attribute: sex)

Overall AUC ( ) 0.972 0.000 0.983 0.012 0.967 0.010 0.978 0.010 0.950 0.018 0.972 0.022 0.961 0.012 0.972 0.016 Max-g AUCD ( ) 0.056 0.000 0.033 0.024 0.044 0.019 0.044 0.019 0.033 0.024 0.011 0.019 0.033 0.024 0.033 0.024 Worst-g AUC ( ) 0.944 0.000 0.967 0.024 0.944 0.000 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 attribute: sex)

Overall AUC ( ) 0.784 0.011 0.816 0.020 0.900 0.007 0.900 0.011 0.828 0.026 0.822 0.032 0.909 0.020 0.912 0.017 Max-g AUCD ( ) 0.119 0.032 0.106 0.037 0.013 0.013 0.037 0.027 0.056 0.032 0.031 0.025 0.031 0.018 0.025 0.011 Worst-g AUC ( ) 0.725 0.020 0.762 0.032 0.894 0.013 0.881 0.020 0.800 0.022 0.806 0.020 0.894 0.028 0.900 0.020

Table 4: Performance comparison on standard fairness datasets (averaged from 5 runs), with 95% CI.

Additional illustration of resolving hypergradient conflict 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-g DRO exhibit greater initial conflicts compared to Lt R, 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 fairness goals (shown with Meta-g DRO 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 3.6.

Attribute # features Normalized Mutual Information (Validation Set)

sex 105 31.007 e-5 Adult Income race 105 43.337 e-5

Bank Telemarketing age 51 679.576 e-5

Credit Default sex 24 26.023 e-5

Communities and Crime blackgt6pact 1929 9.962 e-5

Titanic Survival sex 1526 63.911 e-5

Student Performance sex 58 253.198 e-5

Table 5: Mutual information of each dataset s validation set normalized by the summation of features and labels entropy.

Sensitivity to the initial weighting protocol β0. Figure 4 demonstrates that the improvement from bargaining correlates with the initial hypergradient alignment rate (the portion of aligned batches).

Lt R FORML Meta-g DRO

Hypergradients Aligned 759 (59.5%) 0 (0%) 365 (28.6%) A Non-Empty 1046 (82.0%) 974 (76.4%) 1072 (84.1%) Table 6: One-stage methods in the first 15 epochs on US Crime dataset. Hypergradients aligned means the proposed update of the batch lies in A ( L β gi > 0 for all i). Percentage in 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.

(i). (ii). (iii).

(a) How bargaining helps to resolve conflicts (Lt R)

(i). (ii). (iii).

(b) How bargaining helps to resolve conflicts (FORML)

(i). (ii). (iii).

(c) How bargaining helps to resolve conflicts (Metag DRO)

(d) Bargaining failed (local sticking points) and how external force helps (Meta-g DRO 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 (min{g i Lβ : i [K]), the averaged groupwise hypergradient alignment value ( 1

i [K] g i Lβ), and the maximum value of group-wise hypergradient alignment (max{g i Lβ : i [K]}) respectively.

(i). Overall ACC (ii). ACC Disparity (iii). Worst-group ACC

(a) CIFAR-10 imbalanced training results.

(i). Overall ACC (ii). ACC Disparity (iii). Worst-group ACC

(b) CIFAR-100 imbalanced training results.

Figure 7: Comparative analysis of accuracy and fairness during CIFAR-10 and CIFAR-100 imbalanced training (results averaged from 5 seeds). Subfigures (a) depict the CIFAR-10 results and (b) show CIFAR-100. Each plot tracks the evolution of overall ACC, ACC disparity, and worst-group ACC over training epochs, highlighting performance dynamics under a 99%-1% class imbalance for CIFAR-10 and a 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

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 a worthwhile trade-off given the enhanced performance and fairness 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-10, 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 an 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 Res Net-18 [19] for modeling and employ SGD with a 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 difficulties in converging to a performing and fair model, often underperforming in 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 our 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-focused tasks. The challenge of achieving fairness across groups is amplified by the granular nature of CIFAR-100 (more classes and fewer samples per class) (Figure 7b). Despite this, the two-stage Nash bargaining approach demonstrates a 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, providing a 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 it 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-100 environment. The above results from CIFAR-100 reinforce the generalizable effectiveness of Nash bargaining in our proposed two-stage meta-learning.

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, we recognize that it is not infallible. As such, any application of our method in real-world decision-making tasks must be approached with meticulous consideration of the specific context and potential repercussions. It s crucial to understand that achieving ML fairness is an ongoing, collective endeavor within the research community.

Moreover, we also acknowledge that the fairness-aware meta-learning methods developed in this study are not panaceas for all fairness-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 fairness

in model adaptation and transfer learning. Furthermore, there is a need for more diverse and representative datasets to evaluate the effectiveness and generalizability of fairness-aware methods. To address these challenges, we call for further research and collaboration among researchers, practitioners, and policymakers to advance the state-of-the-art in ML fairness and to ensure that AI systems are fair, transparent, and accountable for all individuals and groups.

Neur IPS Paper Checklist

The checklist is designed to encourage best practices for responsible machine learning research, addressing issues of reproducibility, transparency, research ethics, and societal impact. Do not remove the checklist: The papers not including the checklist will be desk rejected. The checklist should follow the references and follow the (optional) supplemental material. The checklist does NOT count towards the page limit.

Please read the checklist guidelines carefully for information on how to answer these questions. For each question in the checklist:

You should answer [Yes] , [No] , or [NA] .

[NA] means either that the question is Not Applicable for that particular paper or the relevant information is Not Available.

Please provide a short (1 2 sentence) justification right after your answer (even for NA).

The checklist answers are an integral part of your paper submission. They are visible to the reviewers, area chairs, senior area chairs, and ethics reviewers. You will be asked to also include it (after eventual revisions) with the final version of your paper, and its final version will be published with the paper.

The reviewers of your paper will be asked to use the checklist as one of the factors in their evaluation. While [Yes] is generally preferable to [No] , it is perfectly acceptable to answer [No] provided a proper justification is given (e.g., error bars are not reported because it would be too computationally expensive or we were unable to find the license for the dataset we used ). In general, answering [No] or [NA] is not grounds for rejection. While the questions are phrased in a binary way, we acknowledge that the true answer is often more nuanced, so please just use your best judgment and write a justification to elaborate. All supporting evidence can appear either in the main paper or the supplemental material, provided in appendix. If you answer [Yes] to a question, in the justification please point to the section(s) where related material for the question can be found.

IMPORTANT, please:

Delete this instruction block, but keep the section heading Neur IPS paper checklist ,

Keep the checklist subsection headings, questions/answers and guidelines below.

Do not modify the questions and only use the provided macros for your answers.

Question: Do the main claims made in the abstract and introduction accurately reflect the paper s contributions and scope?

Answer: [Yes]

Justification: We have detailed the specific contributions and the scope of the focus in both the abstract and the introduction.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper. The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: See Section 5 and Appendix A.8, and additional discussion in Appendix A.1 on the difference between the Max-Min fairness and Rawlsian Justice.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. The authors are encouraged to create a separate Limitations section in their paper. The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren t acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: See Section 3 and additional discussions in Appendix A.4.

Guidelines:

The answer NA means that the paper does not include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced. All assumptions should be clearly stated or referenced in the statement of any theorems. The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: See Appendix A.6 for all the experimental details.

Guidelines:

The answer NA means that the paper does not include experiments. If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed. While Neur IPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm. (b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We provide the code at https://github.com/reds-lab/ Nash-Meta-Learning with detailed instruction included.

Guidelines:

The answer NA means that paper does not include experiments requiring code. Please see the Neur IPS code and data submission guidelines (https://nips.cc/ public/guides/Code Submission Policy) for more details. While we encourage the release of code and data, we understand that this might not be possible, so No is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark). The instructions should contain the exact command and environment needed to run to reproduce the results. See the Neur IPS code and data submission guidelines (https: //nips.cc/public/guides/Code Submission Policy) for more details. The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc. The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable). Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: See Appendix A.6

Guidelines:

The answer NA means that the paper does not include experiments. The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: See Table 4 and Figure 7.

Guidelines:

The answer NA means that the paper does not include experiments. The authors should answer Yes if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) The assumptions made should be given (e.g., Normally distributed errors). It should be clear whether the error bar is the standard deviation or the standard error of the mean. It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: See Computing in Appendix A.6.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn t make it into the paper). 9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the Neur IPS Code of Ethics https://neurips.cc/public/Ethics Guidelines? Answer: [Yes] Justification: The authors have read and ensured to follow the ethics guidelines. Guidelines:

The answer NA means that the authors have not reviewed the Neur IPS Code of Ethics. If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: See Section 5, Appendix A.8. Guidelines:

The answer NA means that there is no societal impact of the work performed. If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks. Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: See Appendix A.6, A.7. Guidelines:

The answer NA means that the paper does not use existing assets. The authors should cite the original paper that produced the code package or dataset. The authors should state which version of the asset is used and, if possible, include a URL. The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset s creators. 13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [Yes] Justification: See the supplemental material, clean code.zip, and the included clean code/READEME.md file. Guidelines:

The answer NA means that the paper does not release new assets. Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. The paper should discuss whether and how consent was obtained from people whose asset is used. At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA] Justification: The paper does not involve crowdsourcing. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. According to the Neur IPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [NA] Justification: The paper does not involve crowdsourcing. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the Neur IPS Code of Ethics and the guidelines for their institution. For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.