arXiv CSopen access

End-to-End Fairness Optimization with Fair Decision-Focused Learning

End-to-End Fairness Optimization with Fair Decision-Focused Learning Yu Wang School of Business Stevens Institute of Technology [email protected]

Violet (Xinying) Chen School of Business Stevens Institute of Technology [email protected]

arXiv:2607.29441v1 [cs.LG] 31 Jul 2026

Abstract Many real-world systems rely on predictive models to inform decisions, and fairness concerns arise in both the prediction and decision stages. We introduce end-to-end fairness optimization (E2EFO) as a unifying framework that integrates fairness across the prediction-to-decision pipeline. We focus on resource allocation with group-based fairness: the prediction task estimates allocation impacts while limiting accuracy disparity across groups, and the decision task distributes those impacts equitably by optimizing a group-based alpha-fairness measure. Within this framework, we propose fair decision-focused learning (FDFL), a training paradigm that jointly accounts for prediction accuracy, prediction fairness, and decision regret – the loss in decision fairness due to imperfect predictions. FDFL trains the predictor by gradient descent, combining the objective gradients through multi-task learning techniques. The core computational challenge is the decision Jacobian with respect to the predictor parameters: we derive exact closed-form formulas for a tractable class of fair allocation and apply a differentiable optimization layer in the general case. We further establish a finite-sample generalization bound for the scalarized FDFL objective. Numerical experiments on a healthcare-based single resource allocation and a synthetic multiple resource allocation illustrate the value of jointly accounting for prediction fairness and decision fairness in prediction-informed decision-making.

1

Introduction

Data-driven predictions increasingly inform real-world decisions in critical domains. In healthcare, forecasts of medical needs guide the allocation of scarce care resources (Obermeyer et al. 2019); in social services, predicted risk informs screening for intervention (Chouldechova et al. 2018); in finance, loan processing decisions account for the expected creditworthiness of applicants. These processes typically consist of two stages: a prediction task estimates unknown parameters, and a subsequent decision task uses these estimates as input. Many such applications require fairness, and it is worth distinguishing between fairness in how predictions are obtained and fairness in how decisions are made (Paulus and Kent 2020, Scantamburlo et al. 2024). Prediction fairness seeks to eliminate undesirable disparities in predictions, reflecting concerns about statistical properties of the predictor. Decision fairness aims at equitable decision outcomes, reflecting moral concerns about the justice of decisions. Kuppler et al. (2022) argue that both aspects should be jointly addressed. A stylized example illustrates the potential insufficiency of pursuing either fairness alone. A decision maker allocates a fixed amount of resources to recipients in an advantaged group A and a disadvantaged group D; each recipient derives a benefit from the resource, and a predictive model estimates these benefits from historical data carrying systemic bias against D. A standard accuracy-maximizing predictor tends to be more accurate on A and underestimate benefits in D. Suppose the decision maker responds with a fair decision policy that equalizes total benefits across the two groups. Because the policy takes the imperfect predictions as input, it may compensate D by allocating disproportionately more resources than their true benefits justify, creating unfairness against A. Alternatively, suppose the decision maker responds with a fair predictor that reduces the underestimation at the cost of noisier estimates, and allocates to maximize the total predicted benefit. The allocation now favors A, which contains more high-benefit recipients, leaving D under-served. Neither intervention suffices for fairness throughout the prediction-to-decision pipeline.

1

We study this joint problem, which we call End-to-End Fairness Optimization (E2EFO). The prediction task pursues accuracy for reliability while accounting for prediction fairness to mitigate unfair bias; the decision task, formalized as an optimization model that maximizes a decision fairness objective subject to feasibility constraints, seeks equitable outcomes. We focus on resource allocation as the decision task. An allocation instance is a set of stakeholders, partitioned into groups, with observable features and unknown allocation impacts. A predictor estimates the impacts, and the decision model distributes limited resources among the stakeholders. We specify each stakeholder’s utility to be linear in the resources received, the feasible allocation set to be convex and compact, and the decision fairness objective to be a group-based α-fairness measure of the induced utilities. With these specifications, the decision Jacobian with respect to the input prediction is computable (Section 4.1). E2EFO can be solved under two data-driven paradigms. Predict-then-optimize (PTO) methods address the two tasks separately (Bertsimas and Kallus 2020, Kannan et al. 2024): predictions are made first to achieve desirable accuracy and fairness, then input into a fairness-embedded optimization model. Decision-focused learning (DFL) methods instead train the predictor to directly optimize downstream decision quality (Donti et al. 2017, Wilder et al. 2019, Elmachtoub and Grigas 2022). In both paradigms, decision quality is measured by regret, the loss in the decision objective from acting on predictions rather than the ground-truth parameters (formalized in Section 3). DFL improves regret over PTO when the two tasks are misaligned, that is, when good predictions do not guarantee good decisions (Elmachtoub and Grigas 2022, Mandi et al. 2024). We mainly explore the DFL perspective because adopting different fairness perspectives in prediction and decision can expose E2EFO to such misalignment; our experiments identify the conditions under which it does. E2EFO evaluates a predictor along three dimensions: prediction error, prediction disparity measuring the gap in prediction error across groups, and decision regret, all to be minimized. Existing DFL methods target decision regret alone to pursue decision quality. Optimizing all three jointly is a multiobjective problem, for which multi-task learning (MTL) (Sener and Koltun 2018, Chen et al. 2024) offers a principled paradigm. MTL trains one machine learning model against several objectives by combining their gradients, via static scalarization (Kendall et al. 2018) or dynamic conflict-avoidant combination (Désidéri 2012, Yu et al. 2020). We propose Fair Decision-Focused Learning (FDFL), a family of training algorithms for E2EFO that draws on DFL methods to make training decision-aware and on MTL methods to balance the three objectives. We summarize our main contributions as follows. 1. We propose E2EFO as a unifying framework for incorporating fairness into prediction-informed decision-making. Within this framework, we focus on group-based fairness: prediction fairness measures the disparity in prediction accuracy using the mean absolute deviation (MAD) of groupspecific prediction errors, and decision fairness measures the equity of the utility distribution induced by allocations with a two-level group-based α-fairness function. 2. We develop FDFL algorithms to train predictors with the three E2EFO objectives. Under gradientbased training, we use MTL techniques, including static scalarization and dynamic conflict-avoidant combinations, to combine the objective gradients. The decision-regret gradient requires differentiating through the decision model: we derive an exact closed-form decision Jacobian for the single-budget α-fair allocation, and apply differentiable convex optimization layers from Agrawal et al. (2019a) for general convex feasible sets where no closed form is available. 3. We establish a finite-sample generalization guarantee for scalarized FDFL with high probp training:
ability, the composite excess risk of the empirical minimizer is O EΘ q/N , under boundedness and Lipschitz regularity conditions. We verify the required conditions for our α-fairness optimizing allocation tasks. 4. We evaluate FDFL methods and fairness-embedded PTO baselines on a single-resource allocation built from a healthcare context, and on a synthetic multiple resource allocation with group imbalance varied by construction. Experimental results demonstrate the value of multi-objective training and identify the regimes in which each fairness objective delivers its gains. The remainder of the paper is organized as follows. Section 2 reviews related work. Section 3 formulates the E2EFO problem and specifies the fairness measures. Section 4 presents the FDFL training algorithms, the decision Jacobian computation, and a generalization guarantee. Section 5 reports experimental results. Section 6 concludes. 2

2

Related Work

We review three lines of literature: (1) prediction-informed decisions under the PTO or DFL paradigms; (2) multi-task learning for handling multiple training objectives; and (3) fairness in prediction and in optimization. The first two lines provide the methodological foundation for our framework, while the third motivates it.

2.1

Prediction-Informed Decisions

Prediction-informed decisions belong to the broad field of data-driven prescriptive decision-making, also known as data-driven optimization or contextual optimization. We adopt the term prediction-informed decisions to emphasize the combination of prediction techniques from machine learning with the optimization methodology. The survey by Sadana et al. (2025) organizes this literature under three paradigms: decision rule optimization, sequential learning and optimization, and integrated learning and optimization. The latter two share a structure in which a prediction model is learned prior to optimizing decisions, which aligns with our setting, so we review these two paradigms. Sequential learning and optimization, also referred to as predict-then-optimize (PTO), first trains a prediction model to estimate uncertain parameters and then plugs the estimates into a decision optimization model. The predictor may produce a point estimate, yielding a deterministic view of the downstream optimization, or a conditional distribution, yielding a stochastic view that accounts for prediction uncertainty. In this sequential paradigm, the stochastic view is commonly adopted (Bertsimas and Kallus 2020, Kannan et al. 2024). As Sadana et al. (2025) note, the two views coincide when the decision objective is linear in the predictions, since estimating the conditional distribution reduces to estimating its expectation. We adopt the deterministic view in this work. Integrated learning and optimization, also referred to as decision-focused learning (DFL), trains the predictor to directly optimize decision performance, enabling the prediction component to anticipate its impact on the decisions. DFL has been studied for a variety of decision tasks, including linear programming (Elmachtoub and Grigas 2022), quadratic programming (Amos and Kolter 2017, Agrawal et al. 2019a), and general nonlinear optimization (Shah et al. 2022). Recent work develops scalable, generalpurpose techniques, such as directional gradients (Huang and Gupta 2024) and landscape surrogates (Zharmagambetov et al. 2023). These methods differ primarily in the training loss and in how gradients are computed for backpropagation; we refer to Mandi et al. (2024) for a comprehensive survey. Theoretical comparisons under the stochastic view show that PTO dominates DFL in regret when the predictor model class is well-specified and data are sufficient, with the reverse result holding in the misspecified setting (Elmachtoub et al. 2023). Elmachtoub et al. (2025) refined this characterization with finite-sample regret bounds depending on the degree of misspecification. Empirically, DFL has demonstrated gains over PTO in multi-period inventory management (Qi et al. 2023) and healthcare inventory allocation (Chung et al. 2022). Together, these findings indicate that the integrated view is valuable when prediction and decision performance are misaligned.

2.2

Multi-Task Learning

Multi-task learning (MTL), also referred to as multi-objective learning, trains a single shared model against several objectives. Sener and Koltun (2018) cast MTL explicitly as multi-objective optimization and formalizes two targets: Pareto optimality across all objectives, and the weaker Pareto stationarity that holds when no common descent direction improves all objectives simultaneously. MTL training algorithms weigh objectives either statically or dynamically. Static weighting applies a fixed scalarization, replacing the vector-valued loss with a linear combination of the objectives (Kendall et al. 2018), which reduces MTL to standard scalar loss minimization. Hu et al. (2023) showed that scalarization is in general incapable of tracing out the full Pareto front. Dynamic weighting adapts the weights on the objective gradients during training to reduce conflicts; examples include MGDA (Désidéri 2012), PCGrad (Yu et al. 2020), and Nash-MTL (Navon et al. 2022). There is mixed empirical evidence comparing the weighting schemes: simple scalarization with standard regularization has been found competitive with, or superior to, more elaborate dynamic rules (Xin et al. 2022, Kurin et al. 2022). Closest to our work, Jeon et al. (2025) applies MTL within DFL, treating the prediction loss as a secondary objective alongside the decision-quality loss and combining the two gradients into an update rule whose iterates converge to a Pareto-stationary point. We extend this idea to three objectives by adding prediction fairness. Our work contributes no new gradient-combination method; we treat MTL techniques as interchangeable components of FDFL for balancing the training objectives. 3

2.3

Fairness in Prediction and Optimization

Fairness has been studied extensively in both machine learning and optimization, though the two communities pursue different goals. Fair machine learning seeks parity across groups or individuals in predictive models to reduce discriminatory bias; Mehrabi et al. (2021) survey the fairness definitions and techniques. Fairness in optimization instead emphasizes the fairness of decision outcomes, typically measured by how utilities are distributed among individuals or groups; Chen and Hooker (2023) survey utility-based fairness metrics and their use in fairness-aware optimization. Recent work in both communities increasingly emphasizes outcome-aware fairness, motivated by the finding that fair predictions do not guarantee equitable outcomes (Liu et al. 2018, Corbett-Davies et al. 2023). This perspective supports embedding fairness throughout the decision pipeline rather than in the prediction stage alone (Scantamburlo et al. 2024). One line of research pursues outcome-aware fairness by directly learning fair decision policies (Chohlas-Wood et al. 2024), aligning with the decision-ruleoptimization paradigm of Sadana et al. (2025). A related line extends DFL to fairness-aware settings by embedding fairness in the decision model. Kotary et al. (2022) imposes fairness constraints in the learning-to-rank problem, leveraging the linearity of the problem to develop a DFL method for fair rankings. Dinh et al. (2024b) also studies fair ranking, defining fairness through an ordered weighted average (OWA) of group exposures and developing a DFL algorithm with customized forward and backward passes; Dinh et al. (2024a) generalizes the OWA approach to fairness among multiple decision objectives under uncertainty. These extensions address decision fairness alone. A persistent gap remains: no integrated framework has addressed fairness in both prediction and decision stages. Our work fills this gap, with fair resource allocation as the decision problem.

3

Problem Formulation: End-to-End Fairness Optimization

We formalize E2EFO for resource allocation. A decision maker distributes limited resources among stakeholders who derive beneficial impacts from the resources, such as improved access, health, or opportunity. These impacts are not observable before allocation and must be predicted. In an allocation instance, a decision maker considers m stakeholders, indexed by i ∈ [m] = {1, . . . , m}, for the allocation of R resource types. In the single-resource case (R = 1), we let d ∈ Rm ≥0 denote the allocation, where di is the amount given to stakeholder i, and let r ∈ Rm >0 denote the allocation outcomes, where ri represents the positive impact of one unit of resource on i. For the general case with R > 1, the allocation becomes d ∈ Rm×R with entries dij , and each stakeholder carries an impact vector ≥0 ri = (ri1 , . . . , riR ) ∈ RR , where r is the impact of one unit of resource j on i. In this section, we ij >0 present the single-resource case (R = 1) for notational ease. The experiments in Section 5 consider both a single-resource setting and a multiple-resource setting (R = 3). Each stakeholder i is described by a feature vector xi . The m feature vectors of an instance form the matrix X = [x1 , . . . , xm ]⊤ . The prediction task estimates each stakeholder’s impact from its features, r̂i = fθ (xi ) parameterized by θ ∈ Θ. Applying the predictor to a full instance gives the estimate r̂ = fθ (X) ∈ Rm . Since true impacts are positive, we require positive predictor output. Prediction quality is evaluated along two dimensions. A prediction accuracy loss L(r̂; r) measures thePdiscrepancy between predicted and true impacts. We adopt the mean squared error L(r̂; r) = m 1 2 i=1 (ri − r̂i ) . To mitigate potential biases in historical data or structural inequities in the featurem to-impact mapping, we consider a prediction fairness criterion F (r̂; r), which quantifies disparities in prediction errors across stakeholders. We specify F in Section 3.1. For both metrics, lower values are desirable. We abbreviate them as L(r̂) and F (r̂) to simplify notation. Given predictions r̂, each stakeholder derives utility ui = Ui (di ; r̂i ) from the received allocation. Thus a decision d induces a utility distribution u = (u1 , . . . , um ). A decision fairness measure W : Rm → R evaluates the fairness of this distribution, W (d; r̂) = W (U1 (d1 ; r̂1 ), . . . , Um (dm ; r̂m )), with higher values reflecting more equitable outcomes. The decision task solves max d

W (d; r̂) s.t.

d ∈ S.

(1)

We assume that the feasible set S ⊆ Rm is nonempty, compact, convex, and independent of r̂. S may vary across instances through instance-specific data such as resource budgets. Let d∗ (r̂) denote an optimal solution to (1), and d∗ (r) the solution under the true impacts r. We refer to this prediction-informed decision process as the E2EFO problem. The predictor fθ is trained on historical instances {Xs , rs , Ss }N s=1 drawn from a population P. In deployment, facing a new 4

instance Xtest ∼ P with unknown impacts, the decision maker uses r̂ = fθ (Xtest ) in (1). Prediction and decision jointly shape the outcome, which motivates evaluating the predictor on decision quality as well as prediction quality. We measure decision quality by decision regret, the gap in the decisionfairness objective between the prediction-based decision and the full-information decision when both are evaluated under the true impacts: Regret(r̂; r) := W (d∗ (r); r) − W (d∗ (r̂); r) .

(2)

Since d∗ (r) maximizes W (· ; r) over S and d∗ (r̂) ∈ S, the regret is nonnegative and vanishes when the prediction-based decision attains the same fairness as the full-information decision. The first term of (2) is the optimal value of (1) and is unique; the second term is unambiguous when d∗ (·) is single-valued, which holds under the fairness measure specified in Section 3.1. E2EFO thus evaluates a predictor along three dimensions, all to be minimized: L(r̂), F (r̂), and Regret(r̂; r). Remark 1. Our formulation adopts a deterministic view of the decision task: the predictor produces a point estimate r̂, which is plugged into (1), and regret is measured against the impacts realized after the decision. An alternative stochastic view models r as a random vector and estimates its conditional distribution, requiring a stochastic or robust decision model. We leave the extension of E2EFO to this stochastic view for future work.

3.1

Fairness Definitions

E2EFO accommodates a wide range of fairness definitions. We specify the fairness metrics used in our study. All metrics are defined within a single instance: prediction fairness over the prediction errors of the m stakeholders, and decision fairness over their utility distribution. We focus on group-based fairness, where groups G1 , . . . , GK partition the m stakeholders and fairness is assessed across these groups in both the prediction and decision tasks. Prediction Fairness. The prediction task should not systematically disadvantage any group. We measure prediction fairness with an accuracy-disparity metric, which applies a dispersion measure to the group-level prediction errors, taking smaller values when the errors are more evenly distributed. Such measures are widely used in fair regression (Berk et al. 2017, Agarwal et al. 2019). Our running example is the mean absolute deviation (MAD) of group-level mean squared error (MSE). With mk = |Gk |, the PK P 1 group-k error is MSEk = m1k i∈Gk (ri −r̂i )2 . The average error over all groups is MSE = K k=1 MSEk . The group-based accuracy disparity is: K

F (r̂) =

1 X |MSEk − MSE|. K

(3)

k=1

Decision Fairness. We suppose ri captures i’s utility gain per unit of resource, and specify the stakeholder utility to be linear in the allocation: Ui (di ; ri ) = ri di , with Ui (di ; r̂i ) = r̂i di when utilities are evaluated under predicted impacts. With multiple resources (R > 1), utility aggregates across PR resource types: Ui (di ; ri ) = j=1 rij dij . A fairness measure W aggregates utility values into a scalar score. We adopt the widely used αfairness (e.g., Mo and Walrand 2000), a family of measures offering a tunable trade-off between efficiency (P 1−α m ui if α ≥ 0 and α ̸= 1; 1−α , . At α = 0, this measure simplifies to the and equity: Wα (u) = Pi=1 m log(u ), if α = 1 i i=1 utility sum, a pure efficiency goal. As α increases, the measure places greater emphasis on equity, with α → ∞ recovering the Rawlsian max-min criterion. Applied directly to the utility vector u, this measure captures individual-level decision fairness. To assess fairness across groups, we extend it to a two-level construction that applies the α-fair kernel within and across groups. Let gk (u) denote the raw α-fair score of group Gk , that is, Wα applied to the utility distribution in Gk . The outer aggregation applies the α-fair kernel to these group scores, which requires each to be strictly positive. The raw score is guaranteed to be positive only for 0 ≤ α < 1. When α ≥ 1, we replace gk by a strictly increasing positive transformation, which preserves the ordering of groups by intra-group

5

fairness while making the outer aggregation well defined. This yields the intra-group fairness score: P u1−α  i  if 0 ≤ α < 1;  i∈G 1−α = gk (u), k Q
if α = 1; (4) hk (u) = i∈Gk ui = exp gk (u) ,   α−1 P  = −1/g (u), if α > 1, 1−α k i∈Gk ui

where a higher hk (u) indicates a more equitable utility distribution within Gk . Aggregating the intragroup scores with the same α-fair kernel gives group-based α-fairness: (P h1−α (u) K k if α ≥ 0, α ̸= 1; g k=1 1−α , Wα (u) = PK (5) k=1 log(hk (u)), if α = 1. This formulation accounts for both intra-group fairness, via hk , and inter-group fairness, via the outer aggregation. Like the individual measure Wα , Wαg reduces to the utility sum at α = 0 and weighs equity more heavily as α grows; we therefore require α > 0 in the decision objective, excluding the pure-efficiency case. To our knowledge, this two-level α-fair composition has not appeared in prior work. The group-based measure Wαg (u) is strictly concave in u for every α > 0 (Lemma 6). In the singleresource case, the utility map ui = r̂i di is injective, so the decision objective is strictly concave in d and P d∗ (r̂) is unique. For R > 1, ithe map ui = j r̂ij dij is not injective; the uniqueness of d∗ (r̂) requires the conic nondegeneracy condition of Assumption 5 in Appendix D.3. Remark 2 (Coincidence with the individual measure). The coincides with the Q two-level measure (5) P P individual measure Wα at two values of α. At α = 1, hk = i∈Gk ui gives W1g (u) = k i∈Gk log ui =
P P P −1 −1 W1 (u). At α = 2, hk = gives W2g (u) = − k i∈Gk u−1 = W2 (u). At all other α > 0, i i∈Gk ui the two measures differ, and the group structure affects the allocation.

4

Fair Decision-Focused Learning Algorithms

To solve an E2EFO problem, we first learn a predictor that produces the estimate r̂ and then compute d∗ (r̂) by solving (1). The decision task is a convex program that off-the-shelf solvers handle directly. The predictor training should account for the three E2EFO evaluation dimensions of Section 3. We formalize this training problem as Fair Decision-Focused Learning (FDFL). We consider a differentiable parametric predictor fθ (e.g., a neural network) shared across instances, generating r̂s = fθ (Xs ) and informing the decision d∗ (r̂s ). The three training objectives are empirical avPN erages over the N training instances: the prediction-accuracy loss Lpred (θ) = N1 s=1 L(fθ (Xs )), the prePN PN diction disparity F (θ) = N1 s=1 F (fθ (Xs )), and the decision regret Lregret (θ) = N1 s=1 Regret(fθ (Xs ); rs ). They form the vector-valued FDFL objective:
LFDFL (θ) = Lpred (θ), F (θ), Lregret (θ) . (6) FDFL generalizes the two established training paradigms: PTO methods train on the prediction objectives and omit Lregret , whereas DFL methods train on Lregret alone. To enable gradient-based training, we apply the chain rule to decompose the three objective gradients. The instance subscript s is dropped since the gradient chain is identical across instances. ∇θ Lpred (θ) =

∂L(r̂) ∂r̂ · , ∂r̂ ∂θ

∇θ Lregret (θ) = −

∇θ F (θ) =

∂F (r̂) ∂r̂ · ∂r̂ ∂θ

∂W (d∗ (r̂); r) ∂d∗ (r̂) ∂r̂ · · ∂d∗ (r̂) ∂r̂ ∂θ

(7) (8)

∂F (r̂) These components differ in computational difficulty. The objective-gradient terms, ∂L(r̂) ∂r̂ , ∂r̂ , ∗ ∂W (d (r̂);r) , differentiate explicit metrics: L and W are differentiable on their domains, and the MAD∂d∗ (r̂) based disparity F is piecewise differentiable admitting closed-form subgradients at the kinks of its ∂r̂ absolute-value terms. The predictor Jacobian ∂θ is available by automatic differentiation. The deci∗ ∂d (r̂) sion Jacobian ∂r̂ poses the main technical challenge, since d∗ (r̂) generally lacks an explicit form and need not be differentiable at every r̂; Section 4.1 addresses its computation in our setup.

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Algorithm 1 Gradient-based Training for FDFL N N Require: features {Xs }N s=1 ; true impacts {rs }s=1 ; decision feasible sets {Ss }s=1 ; prediction-accuracy loss L; prediction disparity F ; decision fairness objective W ; learning rate η; combination rule Combine. 1: Initialize predictor parameters θ ∈ Θ for fθ . 2: for each training epoch do 3: for each instance s in a sampled batch B do 4: r̂s ← fθ (Xs ). ▷ Predict impacts 5: d∗ (r̂s ) ← arg maxd∈Ss W (d; r̂s ). ▷ Solve prediction-informed decisions 6: end for P P 1 1 7: Lpred (θ) ← |B| s∈B L(r̂s ; rs ), F (θ) ← |B| s∈B F (r̂s ; rs ), P ∗ ∗ 1 8: Lregret (θ) ← |B| [W (d (r ); r ) − W (d (r̂s ); rs )]. ▷ Evaluate objectives s s s∈B 9: gpred ← ∇θ Lpred (θ), gfair ← ∇θ F (θ), gdec ← ∇θ Lregret (θ). ▷ Compute gradients 10: g ← Combine(gpred , gfair , gdec ). ▷ Static or dynamic combination 11: Update θ ← θ − η g. 12: end for 13: return θ

FDFL is an MTL problem in which the three objective gradients are combined into a single update direction, either by static scalarization or by dynamic weighting that adapts the combination in each iteration to avoid conflicts. Section 4.2 provides details for implementing both MTL strategies. Algorithm 1 summarizes the training procedure. Remark 3. The decision-regret loss is generally nonconvex in θ, so gradient-based training targets stationary points rather than global optima. Section 4.2 states the convergence properties of each combination rule and Section 4.3 gives statistical guarantees for the scalarized objective. A structural regret ordering between decision-focused and prediction-focused methods in the spirit of Elmachtoub et al. (2023) remains open for the E2EFO setup.

4.1

Decision Jacobian Computation

The decision Jacobian is computed in the same manner for each training instance: each instance s ∗ contributes ∂d∂r̂(r̂s s ) to the average regret gradient; we continue to drop the instance subscript s for notational ease. When the optimal decision is available in closed form, the Jacobian follows by direct differentiation. Otherwise, the general technique is to differentiate the optimality conditions of the decision problem (Mandi et al. 2024). Section 4.1.1 derives the closed-form Jacobian for α-fairness maximization over a nonnegative knapsack, and Section 4.1.2 applies differentiable convex optimization layers (Agrawal et al. 2019a) to a general convex feasible region. 4.1.1

Closed-Form Jacobian

We consider allocation under a single budget constraint. The decision maker has budget Q > 0 to distribute among m stakeholders, partitioned into K groups. Each unit of resource allocated to stakeholder i incurs a known cost ci > 0 and leads to impact ri (or its prediction r̂i ). The feasible set Pm S = {d ≥ 0 : i=1 ci di ≤ Q} is a nonnegative knapsack, varying across instances through the instancespecific (c, Q). The decision task maximizes the group-based α-fairness of the induced utilities: max Wαg (u) d

s.t. ui = r̂i di , di ≥ 0, ∀i;

m X

ci di ≤ Q.

(9)

i=1

In Proposition 1, we use the Karush-Kuhn-Tucker (KKT) conditions to derive the closed-form optimal solution to (9). Differentiating the solution yields the exact decision Jacobian, whose components we state in Proposition 2 of Appendix A. We give all formulas with the true impacts r as input. During training, ∗ the Jacobian ∂d∂r̂(r̂) is obtained by substituting r̂ for r. We present the general case α ∈ (0, 1) ∪ (1, ∞); the cases α ∈ {0, 1} and α → ∞ and all proofs appear in Appendix A. Proposition  1 (Closed-Form Decisions). The optimal solution to (9) with given  by: for all   parameter r is−α+2 1 1 1−α 1−α 1 1 1+ −α+2 P PK 1+ − − 2 2 K α−2 −α +2α−2 i ∈ [m], d∗i = Qci α ri α Sk(i) / k=1 Sk α−2 at 0 < α < 1, or d∗i = Qci α ri α Sk(i) / k=1 Sk −α +2α−2  1−α P −1/α 1/α at α > 1, where k(i) is the group of stakeholder i, and for each group k ∈ [K], Sk = i∈Gk ci ri . 7

4.1.2

Differentiation through Optimality Conditions

For a general convex, compact feasible region S, the optimal allocation d∗ (r̂) admits no closed form, but the decision Jacobian remains computable by implicitly differentiating the optimality conditions of the decision model. The DFL literature offers several such methods (Mandi et al. 2024) that differentiate the KKT conditions of a quadratic program (OptNet; Amos and Kolter 2017), the optimality conditions of a conic program (cvxpylayers; Agrawal et al. 2019a), or the fixed-point condition of an iterative algorithm (FoldOpt; Kotary et al. 2023). We adopt cvxpylayers, which differentiates through any convex program expressible as a disciplined parametrized programming (DPP) form. DPP restricts how parameters enter an optimization model, so that the canonicalization into a cone program depends affinely on the parameters. Cvxpylayers computes the decision Jacobian by implicitly differentiating the optimality conditions of the resulting conic program, returning a least-squares approximation when the conic program is not differentiable (Appendix B, Agrawal et al. 2019a). Our decision model is DPP-compliant with d and u as the variables and r̂ as the only parameter. The objective P Wαg (u) is concave in u and parameter-free. The utility coupling ui = r̂i di or ui = j r̂ij dij is a sum of products, each pairing a factor affine in the parameter with a parameter-free variable. S is convex and independent of r̂. Hence cvxpylayers applies.

4.2

Gradient Combination

At each training iteration, FDFL combines the three objective gradients into one update direction, g = Combine(gpred , gfair , gdec ). Two objectives conflict at θ when their gradients form an obtuse angle, ⟨gi , gj ⟩ < 0, so that descending one objective increases the other. MTL aims to attain Pareto stationarity, that is, learn a predictor parameter θ at which no common descent direction improves all three objectives. We next describe three combination rules that we implement in FDFL. Static scalarization. The simplest rule is a fixed weighted sum. Normalizing the decision-regret weight to one, we place µ ≥ 0 on prediction accuracy and λ ≥ 0 on prediction disparity. We refer to FDFL adopting the following update direction as FDFL-Scal. gFDFL-Scal = µ gpred + λ gfair + gdec ,

(10)

FDFL-Scal trains a predictor by gradient descent on the scalarized loss µLpred (θ) + λF (θ) + Lregret (θ), targeting convergence to a first-order stationary point of the combined loss, which is Pareto stationary by definition. Scalarization is simple and interpretable, but fixed weights may lead to undesirable trade-offs when the gradients conflict during training. Dynamic conflict-avoidant combination. The alternative strategy re-computes a combination at each iteration from the geometry of the objective gradients. From the MTL literature, we select dynamic rules with convergence guarantee to Pareto stationarity. FDFL-PCGrad applies the technique from Yu et al. (2020) to remove conflict by projection: each gradient is projected onto the normal plane of any other gradient with which it has a negative inner product, and the de-conflicted gradients are summed into the update direction. Following Yu et al. (2020), at each iteration and for each objective i ∈ {pred, fair, dec}, we initialize giPC = gi and traverse the other two objectives j in a random order, projecting the running gradient onto the normal plane of gj whenever the two conflict:  min ⟨giPC , gj ⟩, 0 PC PC PC PC PC gj ; g = gpred + gfair + gdec . (11) gi ← gi − ∥gj ∥2 FDFL-NashMTL applies the Nash bargaining combination from Navon et al. (2022). With combination treated as a bargaining game among the objectives, we use the update direction g = ω1 gpred + ω2 gfair + ω3 gdec where the weights ω > 0 solve G⊤ G ω = [

1 1 1 ⊤ , , ] , ω1 ω2 ω3

G = [ gpred , gfair , gdec ].

(12)

Two additional rules – multiple-gradient-descent algorithm (Désidéri 2012) taking the minimumnorm combination from the convex hull of the objective gradients, and a prediction-disparity-regularized extension of prediction-loss-guided DFL (Jeon et al. 2025) – are implemented and evaluated in Section 5. Appendix B.2 explains both. 8

4.3

Generalization Theory

We now provide a finite-sample guarantee for FDFL-Scal. We collect the data of one allocation instance as z = (X, a, r, DS ), where X and r are the feature matrix and true impacts, a ∈ [K]m records group memberships, and DS specifies S (e.g., the costs c and budget Q in Section 4.1.1). The training set D = {zs }N s=1 contains independent and identically distributed (i.i.d.) samples from a population P on the instance space Z. This sampling formulation and the notations v0 and v̂0 used below follow the integrated estimation-optimization framework of Elmachtoub et al. (2025). The per-instance composite loss aggregates three terms, ℓfull (θ; z) = ℓreg (θ; z) + λ ℓfair (θ; z) + pred µℓ (θ; z), with ℓreg (θ; z) := Regret(fθ (X); r), ℓfair (θ; z) := F (fθ (X)), and ℓpred (θ; z) := L(fθ (X)). We denote the population loss as v0full (θ) := Ez∼P [ℓfull (θ; z)], and its empirical counterpart as v̂0full (θ) := P full 1 (θ; zs ). We analyze the empirical loss minimizer θ̂ ∈ arg minθ∈Θ v̂0full (θ), the target of FDFLsℓ N Scal training, by benchmarking it against θ∗ ∈ arg minθ∈Θ v0full (θ) through the composite excess risk Rfull (θ̂) := v0full (θ̂) − v0full (θ∗ ). We require two standard conditions in the analysis: the loss components are bounded and Lipschitz in θ. Appendix D verifies these conditions for the prediction losses and for the single-resource decision oracle, and gives a sufficient conic-nondegeneracy condition for decision-oracle Lipschitzness in the noninjective multi-resource setting. Assumption 1 (Parameter-space geometry). The parameter space Θ ⊂ Rq is compact and contained in a Euclidean ball of radius EΘ := inf θ0 ∈Rq supθ∈Θ ∥θ − θ0 ∥ centered at some θ0 ∈ Rq . Assumption 2 (Boundedness of losses). The per-instance loss components are uniformly bounded: |ℓreg (θ; z)| ≤ Breg , |ℓfair (θ; z)| ≤ BF , and |ℓpred (θ; z)| ≤ BL for all (θ, z). Assumption 3 (Lipschitzness of decision fairness measure and decision oracle). There exists LW > 0 such that, for every instance z and all d1 , d2 in a convex set containing the oracle-reachable decisions, the decision fairness measure satisfies |W (d1 ; r) − W (d2 ; r)| ≤ LW ∥d1 − d2 ∥. There exists Lor > 0 such that, for every instance z and all θ1 , θ2 ∈ Θ, ∥d∗z (fθ1 (X)) − d∗z (fθ2 (X))∥ ≤ Lor ∥θ1 − θ2 ∥. Assumption 4 (Lipschitzness of prediction-related losses and predictor). There exist Lf , LL , LF > 0 such that, for every instance z and all θ1 , θ2 ∈ Θ, the predictor satisfies ∥fθ1 (X) − fθ2 (X)∥ ≤ Lf ∥θ1 − θ2 ∥, and the prediction-related losses satisfy |ℓpred (θ1 ; z)−ℓpred (θ2 ; z)| ≤ LL ∥θ1 −θ2 ∥ and |ℓfair (θ1 ; z)−ℓfair (θ2 ; z)| ≤ LF ∥θ1 − θ2 ∥. Assumptions 3 and 4 are the analogs of the standard regularity conditions of the DFL literature (Elmachtoub et al. 2025, Donti et al. 2017), where oracle Lipschitzness typically follows from strong convexity of the decision objective. Under all four assumptions, Θ is compact and both v̂0full , v0full are Lipschitz in θ (Appendix D), so the minimizers θ̂ and θ∗ exist. Theorem 1 (Composite-loss generalization for FDFL-Scal). Under Assumptions 1–4, for any δ ∈ (0, 1), there exists a constant Cabs such that, with probability at least 1 − δ: q q   q full . (13) R (θ̂) ≤ 2 Cabs LW Lor + λ LF + µ LL EΘ N + 2 (Breg + λ BF + µ BL ) 2 log(2/δ) N Theorem 1 is related to the generalization bound of Elmachtoub et al. (2025, Theorem 1), which bounds the decision regret in a stochastic setting that fits a distribution to estimate unknown parameters. Because our predictor feeds a point estimate to the decision oracle instead (Remark 1), we control the loss class through Lipschitzness of the decision map and Dudley chaining on the parameter space, whereas their analysis uses total variation between fitted distributions and a vector-contraction inequality. We incorporate prediction-quality components by standard generalization arguments for bounded Lipschitz loss classes. Theorem 1 is proved in Appendix D. We note three limitations. First, the bound in (13) is specific to a fixed scalarization with constants depending p on the scalarization weights, and does not hold uniformly over all λ, µ. Second, the complexity term q/N scales with the parameter count q of the predictor, so the bound is informative in the underparametrized regime (e.g., linear predictor). Once q exceeds N , the bound becomes vacuous and sharper p guarantees require different analysis. Third, we do not examine whether the q/N rate can be improved or certified as optimal by a matching minimax lower bound. These refinements are possible directions for future work.

9

5

Numerical Experiments

We evaluate FDFL on two resource allocation problems in which predicted benefits inform allocation under budget constraints. The first setup allocates a single healthcare resource among patients, using the patient-level data of Obermeyer et al. (2019); the second setup allocates three resources among stakeholders, using synthetic data in which group imbalance is varied by construction. Section 5.1 specifies the experimental setups. Section 5.2 describes all methods and the training configuration. Section 5.3 presents experimental results and discusses key findings.

5.1

Experimental Setups

5.1.1

Single Resource Allocation

This setup is motivated by healthcare applications in which predictions about patients guide the allocation of care. An allocation instance s is a set of patients considered together for one type of care resource in a single period (e.g., week, month): assigning one unit to patient i incurs a known cost ci and delivers a benefit (impact) ri , subject to budget Qs . Patients are partitioned into K = 2 groups by race. Data and instances. We use the dataset of Obermeyer et al. (2019), which contains 48784 patients with features about demographics, comorbidities, prior care utilization, and biomarkers, among whom 5582 (11.4%) are Black and the remaining 43202 are non-Black. We construct the ground-truth benefit ri from two features, the number of chronic conditions and the avoidable medical expenses from receiving care, and derive the allocation cost ci from the healthcare spending feature; Appendix B.1 gives the construction details. Excluding the three features used, there are 147 patient-level features for training a benefit predictor. We partition patients into training, validation, and test pools, each preserving the full dataset’s Black-patient proportion. Instances of m = 5000 patients are drawn from the corresponding pool. We train on N = 50 instances and use 30 validation instances for model selection and 30 test instances for evaluation. E2EFO tasks. The prediction task estimates each patient’s benefit r̂i from the 147 features, evaluated by MSE and by the MAD (3) of the two group-level MSEs. Given the predicted benefits of an instance, the decision task allocates a continuous amount di ≥ 0 of the resource to each patient, consuming ci di of the budget and yielding utility P ui = r̂i di . The allocation maximizes the group-based α-fairness (5) subject to the budgetPconstraint i ci di ≤ Q, where the instance budget Q equals 30% of the instance’s aggregate cost i ci . Decision quality is measured by the regret (2) of d∗ (r̂) evaluated under the constructed benefits r. This decision problem is the single-budget allocation (9) from Section 4.1.1, so its decision Jacobian is available in closed form (Proposition 2). 5.1.2

Multiple Resource Allocation

We next consider allocation of R = 3 resource types under a fully synthetic design that allows the inherent imbalance across groups to be controlled. In an instance, the three resources are distributed among m = 200 stakeholders partitioned into K equal-size groups, subject to one budget per resource; stakeholder i gains benefit (impact) rij and incurs known cost cij per unit of resource j. Data and instances. Each stakeholder has features xi drawn from a 5-dimensional standard Gaussian distribution and a group label gi ∈ {0, . . . , K − 1}, with smaller gi denoting a more advantaged group. We construct benefits by combining a degree-two polynomial of the features, which captures the noiseless signal shared by all groups, with additive group biases and per-group noise. The costs are generated by adding biases and noises to a fixed feature-independent baseline. A single scalar ℓ ∈ [0, 1] sets both the magnitude of the group biases and the noise inflation of the disadvantaged groups: at ℓ = 0, there is no imbalance and groups are exchangeable; increasing ℓ separates the group means while inflating the noise of the disadvantaged groups. Because the baseline signal-to-noise ratio is held fixed, imbalance is not confounded with signal strength. Appendix B.1 gives the generation formulas. We generate a pool of 4000 stakeholders and partition it into stakeholder-disjoint training (65%), validation (15%), and test (20%) subpopulations. We sample instances of m = 200 stakeholders without replacement from the corresponding subpopulation. We use N = 50 training, 30 validation, and 30 test instances.

10

E2EFO tasks. The prediction task estimates the per-resource benefits r̂ij , again evaluated by MSE and m×R MAD.P The decision task chooses P d ∈ R≥0 maximizing the group-based α-fairness (5) of the utilities ui = j r̂ij dij , subject to i cij dij ≤ Qsj for each resource j, where Qsj is 35% of the instance’s aggregate cost for that resource. This model is DPP-compliant (Section 4.1.2), so its decision Jacobian is computed with cvxpylayers.

5.2

Methods and Implementation

All methods share the same downstream allocation model and differ only in how the predictor is trained. A method is specified by which of the three objectives – decision regret, prediction accuracy (MSE), and prediction fairness (MAD) – are included, and how their gradients are combined when more than one objective is included. Table 1 lists the method pool. The names in its first column are used throughout the paper, including in all figures. Table 1: Method pool, grouped by the number of included training objectives. Entries give the weight on an included objective, or ✓ where an objective is included without a fixed weight. Name

Regret

MSE

MAD

PTO SAA

— —

1 ✓

— —

WDRO

—

✓

—

DFL

1

—

—

Predict-then-optimize baseline Sample average approximation, feature-free baseline estimating r̂ with the mean impact of training samples instead of fitting a predictor from data Wasserstein distributionally robust optimization, robust regularization of the MSE objective (Gao et al. 2024) Decision-focused learning baseline

Description

FPTO Regret-and-MAD Regret-and-MSE

— 1 1

1 — {0.1, 0.5, 1}

1 1 —

Prediction-fairness-embedded PTO Decision fairness with prediction fairness Decision fairness with prediction accuracy

FDFL-Scal FDFL-PCGrad FDFL-NashMTL FDFL-MGDA FDFL-FPLG

1 ✓ ✓ ✓ ✓

{0.1, 0.5, 1} ✓ ✓ ✓ ✓

1 ✓ ✓ ✓ {0, 1}

All three objectives; static scalarization (10) All three objectives; gradient projection (11) All three objectives; Nash bargaining weights (12) All three objectives; minimum-norm combination All three objectives; hybrid static-dynamic combination

Training configuration. We use three predictor architectures of increasing capacity, which refers to their power to represent the feature-to-impact mapping: a linear model, a neural network model with two 16-unit hidden layers (MLP-16), and a neural network model with two 64-unit hidden layers (MLP64), all with a Softplus output layer enforcing nonnegative predictions. Predictors are trained with Adam, with the learning rate and stopping step selected on the validation split, by validation MSE for methods without the regret objective and by validation regret for all other methods that include the regret objective. Each configuration is run with five seeds, and we report means and standard deviations across seeds. In both setups, we vary the decision fairness parameter α ∈ {0.5, 1.5, 2} and the number of training instances N ∈ {10, 20, 50}. In the synthetic multi-resource setup, we additionally vary the imbalance level ℓ ∈ {0, 0.2, 0.4, 0.6, 0.8} and the number of groups K ∈ {2, 4}. Evaluation. We report three performance metrics on test instances: prediction MSE, the MAD of group prediction errors, and normalized decision regret Regret(r̂; r)/|W (d∗ (r); r)|. Implementation environment. All methods are implemented in PyTorch, with automatic differentiation computing every component of the gradient chain except the decision Jacobian. In the singleresource setup, the decision model is both solved and differentiated using the closed-form formulas; in the multi-resource setup, the decision model is solved by MOSEK (with CLARABEL/SCS fallbacks) and differentiated with cvxpylayers. All experiments run on a Google Colab standard CPU / high-RAM runtime. All data, code and results are available in https://github.com/DennisWang2488/fair-dfl.

5.3

Results and Findings

We report the test performance of selected methods in Table 2 and Table 3. The full results, covering all methods and the complete grid over predictor capacity, α, ℓ, N , K, µ (the weight on the MSE objective) appear in Appendix C. 11

Table 2: Single resource allocation results across predictor capacity (α=2, N =50). Mean ± standard deviation over 5 seeds; bold marks the lowest mean in each column. Linear

MLP-16

MLP-64

Method

λ

µ

Regret

MAD

MSE

Regret

PTO SAA DFL

0 – 0

– – 0

0.193 ± 0.005 0.279 ± 0.003 0.229 ± 0.007

29.5 ± 5.1 59.2 ± 10.8 101.3 ± 11.8

136.5 ± 1.7 249.3 ± 4.5 346.5 ± 7.3

0.155 ± 0.035 0.279 ± 0.003 0.129 ± 0.003

FPTO 1 – 0.179 ± 0.016 Regret-and-MAD 1 0 0.132 ± 0.002 Regret-and-MSE 0 0.5 0.130 ± 0.002

27.8 ± 5.6 37.1 ± 3.0 41.7 ± 6.1

135.2 ± 2.8 0.160 ± 0.034 39.9 ± 42.3 171.2 ± 108.7 0.142 ± 0.002 20.6 ± 3.5 122.6 ± 1.9 199.7 ± 15.6 0.129 ± 0.002 25.3 ± 5.5 136.4 ± 4.4 0.129 ± 0.003 27.2 ± 5.5 144.5 ± 11.7 191.7 ± 13.5 0.129 ± 0.003 28.2 ± 2.6 125.8 ± 2.2 0.129 ± 0.003 27.3 ± 5.1 129.5 ± 2.7

0.131 ± 0.002 0.216 ± 0.002 0.172 ± 0.001

34.1 ± 2.7 63.7 ± 9.8 76.7 ± 8.4

175.5 ± 4.4 306.6 ± 4.8 298.4 ± 7.5

FDFL-Scal 1 FDFL-PCGrad – FDFL-NashMTL –

1 – –

MAD

0.131 ± 0.002 0.134 ± 0.002 0.130 ± 0.002

MSE

40.1 ± 41.3 167.8 ± 109.3 59.2 ± 10.8 249.3 ± 4.5 96.3 ± 14.7 342.9 ± 25.2

24.3 ± 3.0 22.4 ± 4.3 37.7 ± 21.3

123.9 ± 1.8 126.0 ± 6.4 163.0 ± 64.6

Regret 0.139 ± 0.003 0.279 ± 0.003 0.129 ± 0.003

0.130 ± 0.003 0.130 ± 0.003 0.129 ± 0.003

MAD

MSE

21.2 ± 3.0 118.9 ± 1.9 59.2 ± 10.8 249.3 ± 4.5 71.0 ± 11.0 270.3 ± 18.5

25.3 ± 3.4 24.8 ± 4.7 25.8 ± 4.4

126.8 ± 4.6 136.0 ± 10.1 126.4 ± 2.8

Table 3: Multiple resource allocation results across group imbalance ℓ (α=2, N = 50, MLP-64, K=2). Mean ± standard deviation over 5 seeds; bold marks the lowest mean in each column. Normalized regret

MAD

Method

λ

µ

ℓ=0

.2

.4

.6

.8

ℓ=0

.2

.4

.6

.8

PTO SAA DFL

0 – 0

– – 0

0.135 ± 0.038 0.664 ± 0.084 0.139 ± 0.039

0.164 ± 0.042 0.663 ± 0.082 0.165 ± 0.036

0.193 ± 0.039 0.660 ± 0.079 0.196 ± 0.045

0.215 ± 0.046 0.656 ± 0.080 0.216 ± 0.058

0.216 ± 0.044 0.656 ± 0.086 0.219 ± 0.056

0.042 ± 0.012 0.187 ± 0.085 0.068 ± 0.037

0.078 ± 0.040 0.207 ± 0.079 0.114 ± 0.035

0.149 ± 0.079 0.233 ± 0.075 0.177 ± 0.042

0.228 ± 0.117 0.260 ± 0.072 0.176 ± 0.035

0.311 ± 0.158 0.286 ± 0.072 0.393 ± 0.153

FPTO 1 – Regret-and-MAD 1 0 Regret-and-MSE 0 0.5

0.138 ± 0.038 0.137 ± 0.038 0.137 ± 0.038

0.167 ± 0.038 0.164 ± 0.036 0.166 ± 0.041

0.196 ± 0.037 0.198 ± 0.045 0.195 ± 0.043

0.228 ± 0.038 0.217 ± 0.053 0.214 ± 0.057

0.225 ± 0.039 0.218 ± 0.055 0.219 ± 0.057

0.042 ± 0.013 0.058 ± 0.028 0.049 ± 0.012

0.072 ± 0.036 0.095 ± 0.032 0.065 ± 0.021

0.117 ± 0.064 0.109 ± 0.027 0.103 ± 0.026

0.142 ± 0.082 0.174 ± 0.104 0.102 ± 0.019 0.132 ± 0.056 0.150 ± 0.047 0.160 ± 0.089

FDFL-Scal 1 FDFL-PCGrad – FDFL-NashMTL –

0.137 ± 0.039 0.164 ± 0.039 0.194 ± 0.044 0.215 ± 0.054 0.215 ± 0.057 0.046 ± 0.012 0.059 ± 0.024 0.084 ± 0.031 0.136 ± 0.038 0.162 ± 0.039 0.190 ± 0.040 0.211 ± 0.051 0.215 ± 0.057 0.042 ± 0.013 0.072 ± 0.034 0.105 ± 0.059 0.132 ± 0.036 0.156 ± 0.037 0.193 ± 0.048 0.209 ± 0.058 0.213 ± 0.055 0.041 ± 0.012 0.074 ± 0.038 0.150 ± 0.087

1 – –

0.113 ± 0.057 0.126 ± 0.064 0.230 ± 0.126

0.142 ± 0.083 0.144 ± 0.080 0.302 ± 0.155

Our results verify the expected pattern that including an objective in training improves the metric it targets: the decision-regret objective lowers regret relative to prediction-focused training and each prediction-side objective improves the prediction metric it captures, demonstrating the value of multiobjective training (Section 5.3.1). We further observe complementary roles of the two fairness objectives: the prediction fairness objective reduces prediction disparity more effectively under larger group imbalances (Section 5.3.2), while the decision regret objective leads to greater regret advantages under weaker predictors and larger α (Section 5.3.3). Robustness of these findings is discussed in Appendix C.1. 5.3.1

The Value of Multi-Objective Training in FDFL

In the single resource setup at α = 2 with an MLP-64 predictor, Table 2 shows that training on decision regret alone (DFL) attains the lowest regret in the method pool, but the highest prediction disparity (MAD 71.0) and the lowest prediction accuracy (MSE 270.3). Adding the prediction-fairness objective reduces the disparity at no decision cost. Regret-and-MAD lowers MAD to 27.2 while holding regret at 0.129. In this setup, the prediction accuracy objective has the same effect: Regret-and-MSE reaches a MAD of 27.3 without targeting disparity. Moreover, the FDFL methods, which integrate all three objectives, closely track the performance of Regret-and-MSE. These observations show that the contributions of the two prediction objectives cannot be separated in this single-resource allocation problem. In the multiple resource setup, we observe a clear separation between prediction accuracy and prediction fairness. We examine the results at imbalance ℓ = 0.6 with α = 2 and an MLP-64 predictor (Table 3). Adding the prediction fairness objective to PTO reduces MAD from 0.228 to 0.142 in FPTO. Compared to DFL with MAD 0.176, adding prediction fairness lowers MAD to 0.102 in Regret-and-MAD, while adding prediction accuracy only lowers MAD to 0.150 in Regret-and-MSE. Therefore, prediction fairness is not redundant with prediction accuracy. Section 5.3.2 shows that the prediction fairness advantage is more salient at higher imbalance between groups. We also note that PTO and DFL attain similarly low regrets (0.215 and 0.216) in this setting; Section 5.3.3 identifies when a decision-focused regret advantage emerges. 5.3.2

Group Imbalance

Table 3 reports performance at varying imbalance levels ℓ in the multiple resource setting with α = 2, an MLP-64 predictor and K = 2 groups. Recall from Section 5.1.2 that as ℓ rises, the group means separate more and the disadvantaged group (represented by g = 1) experiences larger noise inflation, which leads to greater disparities intrinsic to the allocation. The PTO performance illustrates this pattern: as ℓ

12

0.30

0.6

0.25

0.5

0.20

0.4

MAD

Normalized regret

increases from 0 to 0.8, MAD rises steeply from 0.042 to 0.311, while regret rises modestly from 0.135 to 0.216. The prediction-fairness gain widens monotonically with imbalance in both training paradigms. Among prediction-focused methods, the MAD reduction from PTO to FPTO grows from 0.006 (the gap between 0.078 and 0.072) at ℓ = 0.2 to 0.137 at ℓ = 0.8; among decision-focused methods, the MAD reduction from DFL to Regret-and-MAD grows from 0.010 at ℓ = 0 to 0.261 at ℓ = 0.8. The right panel of Figure 1 visualizes this trend.

0.15

0.3

0.10

0.2

0.05

0.1

0.00

0

0.2

0.4

Group imbalance

0.6

0.8

PTO

FPTO

0.0

DFL

0

0.2

0.4

Group imbalance

0.6

0.8

Regret-and-MAD

Figure 1: Normalized regret (left) and MAD (right) across imbalance levels ℓ and α = 2, MLP-64, K = 2 in multiple resource allocation setup. Bars are means over five seeds with 95% confidence intervals. The decision-regret objective has stable impacts across imbalance levels: with the MLP-64 predictor configuration, Table 3 shows similar regrets from all feature-based methods (only SAA is excluded) at every ℓ. With our chosen seeds, decision-focused methods can improve over the PTO baseline regret slightly, specifically, FDFL-NashMTL attains the lowest mean regret at four of the five imbalance levels and FDFL-PCGrad at ℓ = 0.4, but every improvement margin falls within the across-seed standard deviation. Therefore, in our setting, imbalance shifts the prediction fairness advantage without affecting the regret comparison among methods. With the MLP-64 predictor, prediction accuracy and decision quality are aligned, leaving little regret gap between accuracy-only training and the decision-focused methods. In the next subsection, we demonstrate that the predictor capacity and the decision fairness parameter α shape the decision regret advantage. 5.3.3

Predictor Capacity and Decision Fairness Parameter

Whether prediction accuracy and decision quality align depends on how well the predictor represents the feature-to-impact mapping and how sharply the allocation reacts to prediction errors. The predictor capacity affects the former, while the α choice affects the latter: at larger α, the α-fairness objective places more emphasis on prioritizing low-utility recipients, and small mispredictions can change who is prioritized. We compare three predictor types under α = 2 in the single resource setting in Table 2. PTO’s MSE falls from 136.5 with a linear predictor to 118.9 with an MLP-64 predictor, consistent with the capacity ordering 1 . The regret gap between PTO and the best decision-focused method contracts as capacity grows: regret reduction of 32.6% with a linear predictor (0.193 to 0.130), 16.8% with an MLP-16 predictor (0.155 to 0.129), and 7.2% with an MLP-64 predictor (0.139 to 0.129). We observe similar regret-gap contracting patterns in multiple resource allocation results from Table 6 in Appendix C. With limited predictor capacity, MSE-only training leaves prediction errors that are large enough to carry over to affect decision quality, creating the need for a decision regret objective to mitigate such error propagation. Notably, the regret objective alone may be insufficient at low capacity: with the linear predictor, DFL attains a worse regret than PTO (Table 2). We next examine the role of α under a fixed MLP-64 predictor in both allocation settings using the results in Tables 5 and 7 in Appendix C. At α = 2, the decision-focused methods reduce regret by 7.2% (single resource setting, Table 2) and 2.8% (multiple resource setting at ℓ = 0.6, Table 3) relative to 1 The across-seed mean MSE of PTO method with an MLP-16 predictor is inflated by high across-seed variance.

13

PTO. When α is 0.5 or 1.5, the regret advantage disappears, and some decision-focused methods deliver worse regrets than PTO: the allocation responds weakly to prediction errors, so an accurate predictor already induces low-regret decisions. At larger α, the allocation reacts more sharply to prediction errors, creating large decision regrets without an explicit decision objective. Our findings show that the two fairness objectives are useful in complementary regimes. The decisionregret advantage emerges under a capacity-limited predictor and a sufficiently large α, which cause prediction errors to propagate into decision regret, while the prediction-fairness advantage is driven by group imbalance. Training with the full FDFL objective is therefore desirable to cover both regimes. Appendix C.1 reports supporting analyses: the effect of the prediction accuracy weight µ, a comparison of gradient-combination rules, and confirmation that these findings persist at α = 0.5, across N ∈ {10, 20, 50}, and at K = 4.

6

Conclusion and Discussion

This work studies E2EFO as a unifying framework for fair prediction-informed decision-making, focusing on resource allocation among groups of stakeholders. We measure prediction disparity by the MAD of prediction errors across groups and decision fairness by the group-based α-fairness of allocation utilities. To train a predictor, we propose FDFL, a family of gradient-based algorithms that treat the three objectives—prediction error, prediction disparity, and decision regret—as a multi-task learning problem. The challenging gradient component, the decision Jacobian, is computed exactly from the closed-form solution we derive for the single-budget α-fair allocation and numerically through cvxpylayers for general convex feasible sets. We also establish a finite-sample generalization bound for the scalarized FDFL objective, with constants verified for the allocation tasks. We evaluate a pool of FDFL algorithms on a single-resource healthcare allocation and a synthetic multiple-resource allocation where group imbalance is varied by construction. Experimental results show that training on decision regret alone attains low regret but can leave large disparity in prediction errors; adding a prediction-side objective removes most of this disparity at no observed regret cost. The two fairness objectives serve complementary roles: the prediction-fairness objective reduces prediction disparity more effectively as group imbalance grows, while the decision-regret objective yields greater regret advantages under weaker predictors and larger α. Because either regime can arise in practice, these results support training on the full FDFL objective and monitoring all three metrics in deployment for end-to-end fairness. Several directions remain for future work. We restrict the scope of this work to specific fairness definitions and a deterministic resource allocation as the decision task. Future research could explore E2EFO and FDFL with alternative fairness criteria, in other decision contexts, or under the stochastic view of the decision task (Remark 1). On the empirical side, our experiments rely on constructed outcomes: the single resource setup builds benefits from patient records, and the multi-resource setup is fully synthetic. Formulating E2EFO problems with observed real-world outcomes and examining FDFL performance in such settings are natural next steps. On the theoretical side, the generalization bound holds for a fixed scalarization, but it does not explain when decision-focused training outperforms prediction-focused training under fairness considerations or characterize the Pareto frontier the dynamic combinations trace.

References Alekh Agarwal, Miroslav Dudı́k, and Zhiwei Steven Wu. Fair regression: Quantitative definitions and reductionbased algorithms. In International Conference on Machine Learning, pages 120–129. PMLR, 2019. Akshay Agrawal, Brandon Amos, Shane Barratt, Stephen Boyd, Steven Diamond, and J Zico Kolter. Differentiable convex optimization layers. Advances in neural information processing systems, 32, 2019a. Akshay Agrawal, Shane Barratt, Stephen Boyd, Enzo Busseti, and Walaa M. Moursi. Differentiating through a cone program. Journal of Applied and Numerical Optimization, 1(2):107–115, 2019b. Brandon Amos and J Zico Kolter. Optnet: Differentiable optimization as a layer in neural networks. In International conference on machine learning, pages 136–145. PMLR, 2017. Richard Berk, Hoda Heidari, Shahin Jabbari, Matthew Joseph, Michael Kearns, Jamie Morgenstern, Seth Neel, and Aaron Roth. A convex framework for fair regression. arXiv preprint arXiv:1706.02409, 2017. Dimitris Bertsimas and Nathan Kallus. From predictive to prescriptive analytics. Management Science, 66(3): 1025–1044, 2020.

14

Enzo Busseti, Walaa M. Moursi, and Stephen Boyd. Solution refinement at regular points of conic problems. Computational Optimization and Applications, 74:627–643, 2019. Lisha Chen, Heshan Fernando, Yiming Ying, and Tianyi Chen. Three-way trade-off in multi-objective learning: Optimization, generalization and conflict-avoidance. Journal of Machine Learning Research, 25(193):1–53, 2024. Violet Xinying Chen and J. N. Hooker. A guide to formulating fairness in an optimization model. Annals of Operations Research, 326(1):581–619, July 2023. Alex Chohlas-Wood, Madison Coots, Henry Zhu, Emma Brunskill, and Sharad Goel. Learning to be fair: A consequentialist approach to equitable decision making. Management Science, 2024. Alexandra Chouldechova, Diana Benavides-Prado, Oleksandr Fialko, and Rhema Vaithianathan. A case study of algorithm-assisted decision making in child maltreatment hotline screening decisions. In Conference on fairness, accountability and transparency, pages 134–148. PMLR, 2018. Tsai-Hsuan Chung, Vahid Rostami, Hamsa Bastani, and Osbert Bastani. Decision-aware learning for optimizing health supply chains. arXiv preprint arXiv:2211.08507, 2022. Sam Corbett-Davies, Johann D Gaebler, Hamed Nilforoshan, Ravi Shroff, and Sharad Goel. The measure and mismeasure of fairness. The Journal of Machine Learning Research, 24(1):14730–14846, 2023. Jean-Antoine Désidéri. Multiple-gradient descent algorithm (mgda) for multiobjective optimization. Comptes Rendus. Mathématique, 350(5-6):313–318, 2012. My H Dinh, James Kotary, and Ferdinando Fioretto. End-to-end learning for fair multiobjective optimization under uncertainty. In Uncertainty in Artificial Intelligence, pages 1129–1145. PMLR, 2024a. My H Dinh, James Kotary, and Ferdinando Fioretto. Learning fair ranking policies via differentiable optimization of ordered weighted averages. In The 2024 ACM Conference on Fairness, Accountability, and Transparency, pages 2508–2517, 2024b. Priya Donti, Brandon Amos, and J Zico Kolter. Task-based end-to-end model learning in stochastic optimization. Advances in neural information processing systems, 30, 2017. Adam N Elmachtoub and Paul Grigas. Smart “predict, then optimize”. Management Science, 68(1):9–26, 2022. Adam N Elmachtoub, Henry Lam, Haofeng Zhang, and Yunfan Zhao. Estimate-then-optimize versus integratedestimation-optimization versus sample average approximation: A stochastic dominance perspective. arXiv preprint arXiv:2304.06833, 2023. Adam N Elmachtoub, Henry Lam, Haixiang Lan, and Haofeng Zhang. Dissecting the impact of model misspecification in data-driven optimization. In International Conference on Artificial Intelligence and Statistics, pages 1594–1602. PMLR, 2025. Rui Gao, Xi Chen, and Anton J. Kleywegt. Wasserstein distributionally robust optimization and variation regularization. Operations Research, 72(3):1177–1191, 2024. Yuzheng Hu, Ruicheng Xian, Qilong Wu, Qiuling Fan, Lang Yin, and Han Zhao. Revisiting scalarization in multi-task learning: A theoretical perspective. Advances in Neural Information Processing Systems, 36: 48510–48533, 2023. Michael Huang and Vishal Gupta. Decision-focused learning with directional gradients. Advances in Neural Information Processing Systems, 37:79194–79220, 2024. Haeun Jeon, Hyunglip Bae, Chanyeong Kim, Yongjae Lee, and Woo Chang Kim. Prediction loss guided decisionfocused learning. arXiv preprint arXiv:2509.08359, 2025. Rohit Kannan, Güzin Bayraksan, and James R Luedtke. Residuals-based distributionally robust optimization with covariate information. Mathematical Programming, 207(1-2):369–425, 2024. Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7482–7491, 2018. James Kotary, Ferdinando Fioretto, Pascal Van Hentenryck, and Ziwei Zhu. End-to-end learning for fair ranking systems. In Proceedings of the ACM Web Conference 2022, pages 3520–3530, 2022. James Kotary, My H Dinh, and Ferdinando Fioretto. Backpropagation of unrolled solvers with folded optimization. In Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI ’23, 2023. ISBN 978-1-956792-03-4. Matthias Kuppler, Christoph Kern, Ruben L Bach, and Frauke Kreuter. From fair predictions to just decisions? conceptualizing algorithmic fairness and distributive justice in the context of data-driven decision-making. Frontiers in sociology, 7:883999, 2022. Vitaly Kurin, Alessandro De Palma, Ilya Kostrikov, Shimon Whiteson, and Pawan K Mudigonda. In defense of the unitary scalarization for deep multi-task learning. Advances in Neural Information Processing Systems, 35:12169–12183, 2022.

15

Lydia T Liu, Sarah Dean, Esther Rolf, Max Simchowitz, and Moritz Hardt. Delayed impact of fair machine learning. In International Conference on Machine Learning, pages 3150–3158. PMLR, 2018. Jayanta Mandi, James Kotary, Senne Berden, Maxime Mulamba, Victor Bucarey, Tias Guns, and Ferdinando Fioretto. Decision-focused learning: Foundations, state of the art, benchmark and future opportunities. Journal of Artificial Intelligence Research, 80:1623–1701, 2024. Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan. A survey on bias and fairness in machine learning. ACM computing surveys (CSUR), 54(6):1–35, 2021. Jeonghoon Mo and Jean Walrand. Fair end-to-end window-based congestion control. IEEE/ACM Transactions on networking, 8(5):556–567, 2000. Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT Press, 2nd edition, 2018. Aviv Navon, Aviv Shamsian, Idan Achituve, Haggai Maron, Kenji Kawaguchi, Gal Chechik, and Ethan Fetaya. Multi-task learning as a bargaining game. In International Conference on Machine Learning, pages 16428– 16446. PMLR, 2022. Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan. Dissecting racial bias in an algorithm used to manage the health of populations. Science, 366(6464):447–453, 2019. Jessica K Paulus and David M Kent. Predictably unequal: understanding and addressing concerns that algorithmic clinical prediction may increase health disparities. NPJ digital medicine, 3(1):99, 2020. Meng Qi, Yuanyuan Shi, Yongzhi Qi, Chenxin Ma, Rong Yuan, Di Wu, and Zuo-Jun Shen. A practical end-to-end inventory management model with deep learning. Management Science, 69(2):759–773, 2023. Utsav Sadana, Abhilash Chenreddy, Erick Delage, Alexandre Forel, Emma Frejinger, and Thibaut Vidal. A survey of contextual optimization methods for decision-making under uncertainty. European Journal of Operational Research, 320(2):271–289, 2025. Teresa Scantamburlo, Joachim Baumann, and Christoph Heitz. On prediction-modelers and decision-makers: why fairness requires more than a fair prediction model. AI & SOCIETY, pages 1–17, 2024. Bodhisattva Sen. A gentle introduction to empirical process theory and applications, 2018. Lecture notes, Columbia University. Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization. In Advances in Neural Information Processing Systems, volume 31, 2018. Sanket Shah, Kai Wang, Bryan Wilder, Andrew Perrault, and Milind Tambe. Decision-focused learning without decision-making: Learning locally optimized decision losses. Advances in Neural Information Processing Systems, 35:1320–1332, 2022. Aman Sinha, Hongseok Namkoong, and John Duchi. Certifying some distributional robustness with principled adversarial training. In International Conference on Learning Representations, 2018. Bryan Wilder, Bistra Dilkina, and Milind Tambe. Melding the data-decisions pipeline: Decision-focused learning for combinatorial optimization. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 1658–1665, 2019. Derrick Xin, Behrooz Ghorbani, Justin Gilmer, Ankush Garg, and Orhan Firat. Do current multi-task optimization methods in deep learning even help? Advances in neural information processing systems, 35: 13597–13609, 2022. Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. In Advances in Neural Information Processing Systems, volume 33, pages 5824–5836, 2020. Arman Zharmagambetov, Brandon Amos, Aaron Ferber, Taoan Huang, Bistra Dilkina, and Yuandong Tian. Landscape surrogate: Learning decision losses for mathematical optimization under partial information. Advances in Neural Information Processing Systems, 36:27332–27350, 2023.

A

Closed-Form Decisions and Decision Jacobian

Proposition( 2 (Decision Jacobian). Let d∗ denote the optimal solution to (9) with parameter r, and  1−α 1 P , if 0 < α < 1; −1/α 1/α define β = α−2 . For k ∈ [K], S = c r . For i ∈ [m], k(i) k i i i∈Gk −α+2 −α2 +2α−2 , if α > 1 denote the group it belongs to. ∗ The Jacobian ∂d∂r(r) consists of the following components: for all i ∈ [m],

16

• Derivative of decision di with respect to i’s own impact: β Sk(i) 1 1 ∂Sk(i) 1 − α − α1 α1 −2 ∂d∗i −α α −1 r = Qci ri + Qc PK i i 1+β ∂ri α ∂ri k=1 Sk

β−1 2β βSk(i) (1 + β)Sk(i) − PK PK 1+β ( k=1 Sk1+β )2 k=1 Sk

! .

• Derivative of decision di with respect to the impact of another member ℓ of the same group: β−1 2β ! β Sk(i) (1 + β) Sk(i) 1 1 ∂Sk(i) ∂d∗i −α α −1 , ∀ℓ ̸= i, ℓ ∈ Gk(i) . − PK = Qci ri PK 1+β ∂rℓ ∂rℓ ( k=1 Sk1+β )2 k=1 Sk • Derivative of decision di with respect to the impact of a member j of a different group: β ∂d∗i − 1 1 −1 β ∂Sk(j) (β + 1)Sk(j) = −Qci α riα Sk(i) , ∀j ∈ / Gk(i) . PK ∂rj ∂rj ( k=1 Sk1+β )2

In these formulas, the gradient of Sk is given by ∂Sk 1 − α 1− α1 α1 −2 ∂Sk = c ri = 0, ∀i ∈ / Gk . , ∀i ∈ Gk ; ∂ri α i ∂ri Proposition 3 (Closed-Form Decisions and Decision Jacobian, Special Cases of α). For special cases ∗ of α = 0, 1, ∞, the optimal solution d∗ to (9) with parameter r and its Jacobian ∂d∂r(r) are, respectively, given by: r

• α = 0: d∗i = cQi if i = arg maxj cjj and d∗i = 0 otherwise, allocating the entire budget to the stakeholder with the highest reward-to-cost ratio. d∗ (r) is not differentiable at the maximizer i∗ = ∂d∗ r arg maxj cjj , and has ∂rji = 0 for all i ̸= i∗ and all j. ∂d∗

Q , ∀i ∈ [m]; ∂rki = 0 for all i, k ∈ [m] as d∗ (r) does not depend on r. • α = 1: d∗i = mc i ∂d∗

Q i P • α → ∞: d∗i = cj , ∀i ∈ [m], equalizing the utilities of all stakeholders, with ∂rk = ri m j=1 rj  − 2 PmQ + r3 (PmQckcj /rj )2 , if i = k; rk ( j=1 cj /rj ) j=1 k . k  r r2 (PQc , if i ̸= k m cj /rj )2 i k

j=1

Proof for closed-form decisions and decision Jacobian: • Special case α = 0: The objective function simplifies to the linear utility sum, Wαg (u) = PK P k=1 i∈Gk ri di . Maximizing a linear function under a knapsack constraint yields a greedy solution: allocate the full budget to the stakeholder with the highest reward-to-cost ratio, rcii . Therefore, r the optimal solution assigns d∗i = cQi if rcii = maxj∈[m] { cjj }, and d∗i = 0 for all other stakeholders. This function d∗ (r) is piecewise constant, so its derivative is zero everywhere except at points where the identity of i∗ changes, at those non-differentiable points the zero matrix is a valid subgradient selection for optimization algorithms. • General case α > 0 and α ̸= 1: When α > 0, the objective function is concave in d, and we need a different strategy to conclude the optimal solution. For notation ease, we denote Wαg (u) with W (g(d)), where g(d) = (g1 (d), . . . , gK (d)) computes defined in (4). The Lagrangian Pm all gk (u) as P m function of (9) is L(d, ν, ξ) = −W (g(d)) + ν( i=1 ci di − Q) − i=1 ξi di , where ν ∈ R≥0 and m ξ ∈ R≥0 are Lagrangian multipliers. The Karush-Kuhn-Tucker (KKT) stationarity condition ∂gk ∂L ∂W −α requires ∂d = − ∂W . ∂gk · ∂di + νci − ξi = 0 for each i ∈ Gk (by the chain rule), with ∂gk = gk (d) i
2 1−α −α k We compute ∂g di with Ak = 1 for 0 < α ≤ 1 and Ak = P α−1 for ∂di = Ak ri (ri di )1−α i∈Gk

α > 1. Complementary slackness (ξi di = 0) requires ξi = 0, di > 0 or ξi > 0, di = 0, and the latter conflicts with stationarity; hence ξi = 0 for all i, and solving the stationarity condition gk−α Ak ri1−α d−α = νci for di gives, with k(i) the group index of i, i −1

1−α

di = Dk(i) ci α ri α , 17

Dk := νgkα /Ak

−1/α

,

(14)

where Dk collects the factors common to all members of group k. 1 Dk1−α Sk for 0 < α < 1 and gk = Substituting (14) into the definition of gk (d) gives gk = 1−α P −1/α 1/α 1−α (α − 1)(Dk1−α Sk )−1 for α > 1, where Sk := i∈Gk (ci ri ) . Substituting these, together with the regime-specific Ak , back into the definition of Dk in (14) and rearranging (the resulting exponents α2 − 2α and −α2 + 2α − 2 are nonzero on the respective ranges of α) yields, in both regimes, the common form Dk = ν e Pk , where the exponent e and the group prefactor Pk depend only on α and Sk : 
 1 Sk αe  , 1−α , if 0 < α < 1;  2 (e, Pk ) =  α − 2α 
1  Sk (2−α)e  , α−1 , if α > 1. 2 −α + 2α − 2

Pm Lastly, we plug the di formula (14) into the budget constraint i=1 ci di = Q. For each group k, 1−α α−1 P P recall Sk = i∈Gk ci α ri α , so i∈Gk ci di = Dk Sk , and the constraint eliminates ν in one step, identically in both regimes: K X

Dk Sk = ν

k=1

e

K X k=1

Sk Pk = Q ⇒ Dk = PK

−1

Q Pk

m=1 Sm Pm

1−α

Q Pk(i) c α ri α ⇒ di = PK i , m=1 Sm Pm

which, after substituting the regime-specific Pk , is the stated closed form. The decision Jacobian formula in Proposition 2 follow from direct differentiation of the decision formulas. Q • Special case α = 1: With hk = i∈Gk ui from (4), the outer aggregation gives W1g (u) = P Pm Pm k∈[K] log hk = i=1 log ui = W1 (u), so the objective is max i=1 log(ri di ), whose logarithmic 1 ∗ nature ensures an interior solution di > 0. The P KKT stationarity condition∗ di − νci = 0 implies di = 1/(νci ), and the budget constraint gives i ci /(νci ) = m/ν = Q, so di = Q/(mci ), which is independent of r with zero gradient. • Special case α → ∞: The objective is max mink gk (d), and for large α each harmonic-meanlike gk (d) is dominated by the smallest utility ri di within the group, so the optimum equalizes all across groups: ri di P = C, i.e., di = C/ri . The budget constraint gives P individual utilities P P Q c (C/r ) = C (c /r ) = Q, so C = Q/ i i i i i i j (cj /rj ) and di = ri j (cj /rj ) . The decision Jacobian formula follows from direct differentiation of the decisions.

A.1

Individual-Based Fairness

The individual-based counterpart of (9) replaces the group welfare Wαg with the individual measure Wα of (??): m X max Wα (u) s.t. ui = r̂i di , di ≥ 0, ∀i; ci di ≤ Q. (15) d

i=1

Proposition 4 (Individual-based Closed-Form Decisions and Gradients for Special Cases of α). For the individual-based problem (15) with parameter r, the optimal solutions and Jacobian for α = 0 and α → ∞ coincide with the group-based formulas of Proposition 3. For α = 1, the optimal solution is ∂d∗ Q d∗i = mc for all i ∈ [m], with ∂rki = 0 for all i, k ∈ [m] as d∗ (r) does not depend on r. i Proof for closed-form decisions: For α = 0 P and α → ∞, the individual-based objective coincides with the group-based one—both reduce to maxd i ri di and to maxd mini ri di , respectively—so the group-based derivations and gradients apply verbatim, and it remains to treat the following two cases. Pm di )1−α • General case α > 0 and α ̸= 1: The Lagrangian of problem (15) is L(d, ν, ξ) = i=1 (ri1−α − Pm Pm ν( i=1 ci di − Q) + i=1 ξi di . For any α > 0, the objective penalizes zero allocations with either an infinite marginal utility (0 < α < 1) or an infinite penalty (α ≥ 1), ensuring an interior solution d∗i > 0 given Q, ci , ri > 0; by complementary slackness, ξi = 0 for all i. The KKT stationarity

18

1−α

∂L = 0 reads (ri )1−α d−α − νci = 0, i.e., di = (νci )−1/α ri α . At the optimum, the condition ∂d i i budget constraint holds with equality. Substituting di into the constraint gives m X

m  X 1−α  1− 1 1−α cj (νcj )−1/α rj α = ν −1/α cj α rj α = Q,

j=1

j=1

 Pm 1− α1 1−α so ν −1/α = Q rj α , and substituting this back into the expression for di gives the j=1 cj closed form solution formula −1

1

−1

c α · riα · Q di = P i , 1 1 1− α m α −1 · r c j j=1 j

∀i ∈ [m].

The corresponding Jacobian follows from direct differentiation of this closed form. Pm • Special case α = 1: The objective becomes max i=1 log(ri di ), whose logarithmic nature ensures 1 an interior solution d∗i > 0. The di − νci = 0 implies di = 1/(νci ), and P KKT stationarity condition ∗ the budget constraint gives i ci /(νci ) = m/ν = Q, so di = Q/(mci ), which is independent of r with zero gradient.

B

Additional Experiment Details

B.1

Data Generation Details

B.1.1

Single Resource Allocation.

We construct a benefit score ri for each patient i by combining two components: (1) potential for health improvement, and (2) cost savings. We treat a patient’s number of active chronic illnesses as a proxy for the health improvement potential, denoted hi , reflecting that the care resource helps patients manage chronic conditions. We apply min-max normalization to scale the chronic illness count from dataset to [0, 1]. Cost savings, denoted si , are estimated from avoidable healthcare costs from program enrollment. The original avoidable costs range from $0 to $642,700. To reduce skewness, we apply the log transformation log(1 + si ) adopted in Obermeyer et al. (2019) followed by min-max normalization to scale the values to [0, 1]. The unscaled benefit score is computed as a weighted average, bi = 0.5hi +0.5si . To ensure numerical stability, we rescale these values to lie in the interval [2, 101] as the ground-truth benefit score ri = max{100 bi , 1} + 1. The allocation cost ci is computed from ci = max{10 c̃i , 1}, where c̃i is the min-max normalized healthcare spending for patient i from the dataset. B.1.2

Multiple Resource Allocation.

For each stakeholder i with features xi ∼ N (0d , Id ) and group label gi ∈ {0, . . . , K − 1}, we generate ground-truth benefits by combining a noiseless feature-to-benefit signal, groupP bias, and
per-group noise. P The noiseless benefit signal is a degree-P polynomial of the features, f (x) = p=1 x⊙p Wp with Wp ∈ Rd×R , (Wp )kj ∼ N (0, 1/p2 ) and P = 2, where x⊙p denotes the element-wise p-th power. This noiseless prediction target is nonlinear in x. For resource type j, we fix its standard deviation in the associated σj with SNR=5. The benefit is benefit signal σj and set the baseline benefit-noise scale as ηj = √SNR constructed as:

rij = softplus fj (xi ) + ρ(gi ) βb + εij + 0.05, εij ∼ N 0, (ηj sb (gi ))2 , where ρ(gi ) βb is the additive benefit bias for group i, which applies a scaling function ρ(gi ) to the benefit bias factor βb , and εij is the noise accounting for group-specific scaling factor sb (gi ). We define ρ(g) = 1 − 2g/(K − 1) and sb (g) = 1 + (νb − 1) g/(K − 1). The parameters βb and νb are explained below with the imbalance parameter ℓ. We generate costs in a similar structure by combining a noiseless signal with group-specific bias and noise. The noiseless cost signal is feature-independent and fixed as µc . We generate the cost cij as:
cij = max µc + ρ(gi ) βc + ζij sc (gi ), 10−3 , ζij ∼ N (0, σc2 ), with µc =1.0, σc =0.2, cost group-bias βc , and per-group noise scale sc (g) = 1 + (νc − 1) g/(K − 1). The parameters βc and νc depend on the imbalance parameter ℓ. 19

We use one parameter ℓ ∈ [0, 1] to adjust the per-group scaling on benefit and cost bias and noise. The scaling is controlled via the four imbalance factors βb = βc = 0.9 ℓ and νb = νc = 1 + ℓ. At ℓ = 0, the groups are exchangeable; increasing ℓ shifts the group means apart and inflates the disadvantaged group’s noise on both benefit and cost. Because the signal-to-noise ratio is fixed independently of ℓ, imbalance is decoupled from overall signal strength.

B.2

Additional Methods in the Method Pool

This appendix describes the methods not specified in the main text. The training objective gradients are gdec , gpred , gfair . FDFL-FPLG. This method adapts the prediction-loss guided (PLG) DFL of Jeon et al. (2025) to include a prediction fairness objective. We combine gpred and gdec dynamically using the rule from Jeon et al. (2025) and includes gfair with a fixed weight as in FDFL-Scal. Let û• := g• /∥g• ∥, the p û +γt ûpred ∥gdec ∥ ∥gpred ∥ ∥ûdec + λ gfair , where the guidance weight update direction is: gFPLG = dec +γt ûpred ∥ γt = κ0 /(1 + κ t) decays over training, so the prediction loss stabilizes the early iterates and the decision objective dominates later ones. FDFL-MGDA. This method applies the multiple gradient descent algorithm (MGDA) from Désidéri (2012), where each iteration chooses the minimum-norm combination from n the convex hull of individual

objective gradients. The update direction is solved from: gM GDA = arg min ∥ω1 gpred + ω2 gfair + ω3 gdec ∥2 : The solution g is either a nonzero common descent direction or equal to 0, which occurs exactly at Pareto-stationary points. WDRO.

The variation-regularized Wasserstein-1 objective is P P min n1 i ℓθ (xi , ri ) + ε n1 i ∇x ℓθ (xi , ri ) , θ

(16)

the MSE loss augmented with an input-gradient-norm penalty. We adopt this regularized form rather than the exact worst-case formulation because the latter is tractable only for restricted model classes: with a nonconvex neural network predictor, the inner supremum over the Wasserstein ball admits no exact finite reformulation, whereas for smooth losses the worst-case objective agrees with the empirical loss plus the gradient-norm variation penalty of (16) to first order in the ball radius ε (Gao et al. 2024); Sinha et al. (2018) give certified smoothed relaxations of the same form. WDRO thus serves as a robustness-regularized prediction baseline rather than a full distributionally robust program.

C

Additional Results and Findings

This appendix reports the full experiment results: all methods (including WDRO, FDFL-FPLG, FDFLMGDA, described in Appendix B.2) and the complete grid over predictor capacity, the decision fairness parameter α, the imbalance level ℓ, the number of training instances N , the number of groups K, and the prediction-accuracy weight µ (carried by the Regret-and-MSE and FDFL-Scal rows). The aggregation is identical to the main-text tables: 5 seeds, mean ± standard deviation, best column mean highlighted in bold.

C.1

Robustness and Additional Analyses

We report complementary analyses that support and extend the main findings. First, we discuss the effect of the weight µ on the prediction accuracy objective. In single resource allocation, Table 4 shows that with MLP-64 predictor, relative to DFL, increasing µ in Regret-and-MSE steadily lowers prediction MSE while maintaining stable regret at 0.129, which fits the expected gain from targeting the prediction accuracy metric. Because the prediction disparity metric (MAD) compares prediction errors, improving accuracy is aligned with reducing disparity: over the same range of µ, Regret-and-MSE attains lower MAD with a larger µ even though the training does not include the prediction fairness objective (λ = 0). In multiple resource allocation results, Table 6 shows consistent patterns that support targeting the prediction accuracy objective to enhance prediction quality.

20

P3

i=1 ωi = 1

Table 4: Single resource allocation, full method pool across predictor capacity (α=2, N =50). Linear

MLP-16

MLP-64

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.193 ± 0.005 0.279 ± 0.003 0.172 ± 0.004 0.229 ± 0.007

29.5 ± 5.1 59.2 ± 10.8 29.2 ± 4.0 101.3 ± 11.8

136.5 ± 1.7 249.3 ± 4.5 138.1 ± 1.8 346.5 ± 7.3

0.155 ± 0.035 0.279 ± 0.003 0.241 ± 0.033 0.129 ± 0.003

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 0.179 ± 0.016 1 0 0.132 ± 0.002 0 0.1 0.171 ± 0.009 0 0.5 0.130 ± 0.002 0 1 0.131 ± 0.002 0 – 0.205 ± 0.001

27.8 ± 5.6 37.1 ± 3.0 82.2 ± 9.4 41.7 ± 6.1 35.7 ± 2.7 94.5 ± 11.6

135.2 ± 2.8 0.160 ± 0.034 39.9 ± 42.3 171.2 ± 108.7 0.142 ± 0.002 20.6 ± 3.5 122.6 ± 1.9 199.7 ± 15.6 0.129 ± 0.002 25.3 ± 5.5 136.4 ± 4.4 0.129 ± 0.003 27.2 ± 5.5 144.5 ± 11.7 304.5 ± 7.4 0.129 ± 0.002 55.9 ± 27.4 209.0 ± 102.6 0.129 ± 0.003 40.4 ± 16.7 166.7 ± 52.3 191.7 ± 13.5 0.129 ± 0.003 28.2 ± 2.6 125.8 ± 2.2 0.129 ± 0.003 27.3 ± 5.1 129.5 ± 2.7 168.9 ± 3.7 0.129 ± 0.003 24.5 ± 3.6 123.1 ± 2.0 0.129 ± 0.003 24.8 ± 4.4 126.0 ± 3.0 332.1 ± 6.2 0.129 ± 0.002 47.6 ± 30.0 194.7 ± 77.8 0.130 ± 0.003 26.4 ± 4.7 127.8 ± 3.5

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 1 0.5 1 1 1 – – – – – – –

0.131 ± 0.002 0.131 ± 0.002 0.131 ± 0.002 0.188 ± 0.003 0.216 ± 0.002 0.218 ± 0.001 0.172 ± 0.001

36.0 ± 4.5 34.3 ± 3.7 34.1 ± 2.7 77.3 ± 10.2 63.7 ± 9.8 67.5 ± 8.8 76.7 ± 8.4

193.0 ± 5.8 179.2 ± 5.0 175.5 ± 4.4 312.6 ± 6.5 306.6 ± 4.8 313.6 ± 3.9 298.4 ± 7.5

0.130 ± 0.001 0.130 ± 0.002 0.131 ± 0.002 0.129 ± 0.002 0.134 ± 0.002 0.141 ± 0.006 0.130 ± 0.002

MSE

Reg

40.1 ± 41.3 167.8 ± 109.3 59.2 ± 10.8 249.3 ± 4.5 105.9 ± 9.1 376.0 ± 27.1 96.3 ± 14.7 342.9 ± 25.2

25.7 ± 3.5 25.6 ± 3.5 24.3 ± 3.0 26.2 ± 5.8 22.4 ± 4.3 32.7 ± 3.3 37.7 ± 21.3

139.7 ± 10.2 127.8 ± 2.9 123.9 ± 1.8 144.5 ± 14.8 126.0 ± 6.4 181.9 ± 24.5 163.0 ± 64.6

MAD

0.139 ± 0.003 0.279 ± 0.003 0.138 ± 0.002 0.129 ± 0.003

0.129 ± 0.002 0.130 ± 0.003 0.130 ± 0.003 0.130 ± 0.002 0.130 ± 0.003 0.134 ± 0.002 0.129 ± 0.003

MSE

21.2 ± 3.0 118.9 ± 1.9 59.2 ± 10.8 249.3 ± 4.5 22.1 ± 2.5 118.2 ± 1.5 71.0 ± 11.0 270.3 ± 18.5

26.3 ± 6.9 25.8 ± 6.1 25.3 ± 3.4 24.6 ± 4.1 24.8 ± 4.7 28.1 ± 5.1 25.8 ± 4.4

145.9 ± 9.4 135.0 ± 8.1 126.8 ± 4.6 129.3 ± 4.0 136.0 ± 10.1 134.2 ± 5.1 126.4 ± 2.8

Table 5: Single resource allocation, full method pool across decision fairness parameter α (MLP-64, N =50). α=0.5 Method

λ

µ

PTO SAA WDRO DFL

0 – – 0

– – – 0

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

Reg

α=1.5

MAD

MSE

Reg

α=2

MAD

MSE

Reg

0.044 ± 0.000 21.1 ± 3.4 119.0 ± 1.8 0.075 ± 0.001 59.2 ± 10.8 249.3 ± 4.5 0.044 ± 0.001 21.2 ± 2.5 116.6 ± 2.9 0.053 ± 0.001 105.9 ± 10.9 380.5 ± 10.6

0.013 ± 0.000 0.026 ± 0.000 0.012 ± 0.000 0.012 ± 0.000

21.8 ± 3.4 120.8 ± 1.4 59.2 ± 10.8 249.3 ± 4.5 22.2 ± 2.8 118.2 ± 1.5 62.3 ± 16.2 233.0 ± 50.6

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.045 ± 0.001 0.061 ± 0.001 0.044 ± 0.001 0.044 ± 0.000 0.044 ± 0.000 0.053 ± 0.001

20.2 ± 2.8 120.4 ± 3.5 103.7 ± 10.5 371.6 ± 8.1 20.2 ± 3.6 118.5 ± 2.8 20.6 ± 3.4 119.5 ± 1.9 21.1 ± 3.5 119.0 ± 1.8 94.1 ± 9.9 356.7 ± 17.7

0.013 ± 0.000 0.017 ± 0.001 0.012 ± 0.000 0.013 ± 0.000 0.012 ± 0.000 0.012 ± 0.000

21.9 ± 3.3 20.8 ± 4.5 21.5 ± 4.0 21.4 ± 3.5 21.1 ± 3.5 21.6 ± 3.5

1 0.1 1 0.5 1 1 1 – – – – – – –

0.048 ± 0.001 0.046 ± 0.001 0.045 ± 0.001 0.061 ± 0.001 0.046 ± 0.000 0.053 ± 0.001 0.045 ± 0.001

20.6 ± 2.5 21.4 ± 3.4 20.1 ± 2.8 103.7 ± 10.4 21.4 ± 3.6 105.8 ± 11.0 21.7 ± 6.6

0.014 ± 0.000 24.2 ± 3.2 136.6 ± 10.2 0.013 ± 0.000 21.8 ± 2.9 124.1 ± 2.1 0.013 ± 0.000 20.6 ± 3.9 122.0 ± 2.0 0.012 ± 0.000 24.5 ± 4.3 141.1 ± 2.7 0.012 ± 0.000 23.8 ± 5.1 129.7 ± 5.7 0.012 ± 0.000 34.9 ± 8.2 147.0 ± 9.3 0.012 ± 0.000 25.4 ± 6.3 122.9 ± 4.0

132.2 ± 3.9 124.2 ± 3.6 120.3 ± 3.4 371.5 ± 8.1 121.4 ± 1.8 380.1 ± 9.9 123.1 ± 2.5

α=4

MAD

0.139 ± 0.003 0.279 ± 0.003 0.138 ± 0.002 0.129 ± 0.003

MSE

Reg

129.9 ± 5.8 0.142 ± 0.002 20.6 ± 3.5 122.6 ± 1.9 183.4 ± 28.2 0.129 ± 0.003 27.2 ± 5.5 144.5 ± 11.7 118.5 ± 1.7 0.129 ± 0.003 40.4 ± 16.7 166.7 ± 52.3 119.3 ± 1.5 0.129 ± 0.003 27.3 ± 5.1 129.5 ± 2.7 119.1 ± 1.8 0.129 ± 0.003 24.8 ± 4.4 126.0 ± 3.0 118.7 ± 2.0 0.130 ± 0.003 26.4 ± 4.7 127.8 ± 3.5 0.129 ± 0.002 0.130 ± 0.003 0.130 ± 0.003 0.130 ± 0.002 0.130 ± 0.003 0.134 ± 0.002 0.129 ± 0.003

MAD

MSE

21.2 ± 3.0 118.9 ± 1.9 40.127 ± 7.164 23.9 ± 2.2 137.6 ± 3.3 59.2 ± 10.8 249.3 ± 4.5 153.145 ± 6.777 59.2 ± 10.8 249.3 ± 4.5 22.1 ± 2.5 118.2 ± 1.5 41.325 ± 7.755 24.2 ± 2.1 136.9 ± 3.2 71.0 ± 11.0 270.3 ± 18.5 32.410 ± 14.715 109.1 ± 11.7 377.9 ± 8.1

26.3 ± 6.9 25.8 ± 6.1 25.3 ± 3.4 24.6 ± 4.1 24.8 ± 4.7 28.1 ± 5.1 25.8 ± 4.4

145.9 ± 9.4 135.0 ± 8.1 126.8 ± 4.6 129.3 ± 4.0 136.0 ± 10.1 134.2 ± 5.1 126.4 ± 2.8

39.491 ± 6.286 32.410 ± 14.715 32.410 ± 14.715 32.410 ± 14.715 32.410 ± 14.716 27.557 ± 4.904

22.0 ± 1.9 109.1 ± 11.7 109.1 ± 11.7 109.1 ± 11.7 109.1 ± 11.7 24.7 ± 2.7

144.8 ± 2.8 377.9 ± 8.1 377.9 ± 8.1 377.9 ± 8.1 377.9 ± 8.1 139.3 ± 4.6

32.410 ± 14.715 32.410 ± 14.716 32.410 ± 14.716 27.452 ± 4.967 29.039 ± 3.943 33.158 ± 15.415 27.406 ± 4.945

109.1 ± 11.7 377.9 ± 8.1 109.1 ± 11.7 377.9 ± 8.1 109.1 ± 11.7 377.9 ± 8.1 25.0 ± 2.4 141.7 ± 6.8 23.8 ± 5.4 143.3 ± 10.0 106.8 ± 10.7 373.5 ± 10.0 24.4 ± 2.2 139.1 ± 4.6

Table 6: Multiple resource allocation, full method pool across predictor capacity (α=2, imbalance 0.6, K=2). Linear

MLP-16

MLP-64

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MSE

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.345 ± 0.063 0.656 ± 0.080 0.343 ± 0.061 0.287 ± 0.067

0.189 ± 0.110 0.260 ± 0.072 0.197 ± 0.131 0.420 ± 0.269

1.707 ± 0.761 2.772 ± 1.169 1.731 ± 0.760 2.535 ± 1.250

0.248 ± 0.057 0.656 ± 0.080 0.255 ± 0.054 0.224 ± 0.049

0.222 ± 0.128 0.260 ± 0.072 0.179 ± 0.114 0.405 ± 0.191

1.063 ± 0.428 2.772 ± 1.169 1.090 ± 0.445 2.380 ± 1.018

0.215 ± 0.046 0.656 ± 0.080 0.215 ± 0.049 0.216 ± 0.058

0.228 ± 0.117 0.260 ± 0.072 0.194 ± 0.115 0.176 ± 0.035

0.953 ± 0.368 2.772 ± 1.169 0.961 ± 0.375 1.567 ± 0.763

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.351 ± 0.063 0.287 ± 0.067 0.287 ± 0.067 0.287 ± 0.067 0.288 ± 0.067 0.296 ± 0.065

0.181 ± 0.100 0.412 ± 0.264 0.417 ± 0.266 0.406 ± 0.257 0.391 ± 0.246 0.283 ± 0.203

1.711 ± 0.762 2.509 ± 1.239 2.524 ± 1.244 2.482 ± 1.220 2.431 ± 1.193 2.074 ± 0.937

0.257 ± 0.054 0.222 ± 0.054 0.224 ± 0.050 0.223 ± 0.052 0.225 ± 0.052 0.221 ± 0.054

0.144 ± 0.082 0.318 ± 0.137 0.367 ± 0.152 0.249 ± 0.086 0.188 ± 0.075 0.120 ± 0.067

1.111 ± 0.448 2.075 ± 0.924 2.246 ± 0.972 1.837 ± 0.784 1.547 ± 0.653 1.242 ± 0.497

0.228 ± 0.038 0.217 ± 0.053 0.215 ± 0.055 0.214 ± 0.057 0.215 ± 0.057 0.214 ± 0.051

0.142 ± 0.082 0.102 ± 0.019 0.202 ± 0.021 0.150 ± 0.047 0.143 ± 0.070 0.131 ± 0.086

0.998 ± 0.402 1.351 ± 0.652 1.722 ± 0.900 1.334 ± 0.638 1.122 ± 0.495 1.066 ± 0.435

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 0.287 ± 0.067 0.409 ± 0.262 1 0.5 0.287 ± 0.067 0.398 ± 0.253 1 1 0.288 ± 0.067 0.384 ± 0.242 1 – 0.295 ± 0.066 0.268 ± 0.196 – – 0.305 ± 0.068 0.173 ± 0.102 – – 0.475 ± 0.077 0.270 ± 0.156 – – 0.296 ± 0.070 0.210 ± 0.159

2.499 ± 1.233 2.457 ± 1.210 2.406 ± 1.183 2.040 ± 0.923 1.747 ± 0.771 2.593 ± 1.076 1.831 ± 0.818

0.223 ± 0.051 0.283 ± 0.119 1.971 ± 0.888 0.216 ± 0.057 0.124 ± 0.038 0.226 ± 0.050 0.185 ± 0.080 1.652 ± 0.762 0.213 ± 0.058 0.110 ± 0.029 0.228 ± 0.056 0.171 ± 0.086 1.530 ± 0.703 0.215 ± 0.054 0.113 ± 0.057 0.219 ± 0.052 0.108 ± 0.041 1.329 ± 0.582 0.217 ± 0.051 0.089 ± 0.036 0.216 ± 0.050 0.120 ± 0.056 1.082 ± 0.465 0.211 ± 0.051 0.126 ± 0.064 0.431 ± 0.079 0.129 ± 0.061 1.864 ± 0.775 0.238 ± 0.045 0.098 ± 0.048 0.216 ± 0.049 0.225 ± 0.121 0.999 ± 0.390 0.209 ± 0.058 0.230 ± 0.126

1.467 ± 0.787 1.285 ± 0.591 1.140 ± 0.497 1.174 ± 0.531 1.040 ± 0.437 1.122 ± 0.504 0.963 ± 0.382

Table 7: Multiple resource allocation, full method pool across decision fairness parameter α (MLP-64, imbalance 0.6, K=2). α=0.5

α=1.5

α=2

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MSE

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.016 ± 0.004 0.053 ± 0.014 0.016 ± 0.004 0.021 ± 0.007

0.222 ± 0.118 0.260 ± 0.072 0.194 ± 0.115 0.562 ± 0.247

0.960 ± 0.369 2.772 ± 1.169 0.961 ± 0.375 3.088 ± 1.453

0.035 ± 0.005 0.150 ± 0.030 0.035 ± 0.006 0.038 ± 0.009

0.228 ± 0.117 0.260 ± 0.072 0.194 ± 0.115 0.406 ± 0.223

0.953 ± 0.368 2.772 ± 1.169 0.961 ± 0.375 2.424 ± 1.271

0.215 ± 0.046 0.656 ± 0.080 0.215 ± 0.049 0.216 ± 0.058

0.228 ± 0.117 0.260 ± 0.072 0.194 ± 0.115 0.176 ± 0.035

0.953 ± 0.368 2.772 ± 1.169 0.961 ± 0.375 1.567 ± 0.763

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.017 ± 0.004 0.021 ± 0.007 0.018 ± 0.006 0.019 ± 0.006 0.018 ± 0.005 0.020 ± 0.006

0.135 ± 0.076 0.090 ± 0.029 0.193 ± 0.062 0.147 ± 0.088 0.166 ± 0.095 0.308 ± 0.221

1.017 ± 0.405 1.368 ± 0.593 1.619 ± 0.738 1.183 ± 0.440 1.079 ± 0.403 1.972 ± 0.829

0.038 ± 0.004 0.037 ± 0.008 0.037 ± 0.008 0.036 ± 0.008 0.036 ± 0.008 0.036 ± 0.007

0.142 ± 0.082 0.102 ± 0.035 0.166 ± 0.095 0.200 ± 0.112 0.214 ± 0.117 0.140 ± 0.075

0.998 ± 0.402 1.288 ± 0.603 1.092 ± 0.449 1.012 ± 0.395 0.989 ± 0.385 1.055 ± 0.439

0.228 ± 0.038 0.217 ± 0.053 0.215 ± 0.055 0.214 ± 0.057 0.215 ± 0.057 0.214 ± 0.051

0.142 ± 0.082 0.102 ± 0.019 0.202 ± 0.021 0.150 ± 0.047 0.143 ± 0.070 0.131 ± 0.086

0.998 ± 0.402 1.351 ± 0.652 1.722 ± 0.900 1.334 ± 0.638 1.122 ± 0.495 1.066 ± 0.435

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 1 0.5 1 1 1 – – – – – – –

0.019 ± 0.006 0.019 ± 0.006 0.017 ± 0.006 0.019 ± 0.006 0.019 ± 0.005 0.018 ± 0.006 0.017 ± 0.005

0.092 ± 0.030 0.116 ± 0.058 0.119 ± 0.064 0.099 ± 0.034 0.129 ± 0.072 0.119 ± 0.072 0.212 ± 0.111

1.266 ± 0.541 1.216 ± 0.466 1.106 ± 0.436 1.384 ± 0.646 1.174 ± 0.491 1.091 ± 0.452 0.999 ± 0.376

0.037 ± 0.008 0.093 ± 0.033 0.036 ± 0.008 0.108 ± 0.049 0.036 ± 0.007 0.135 ± 0.076 0.036 ± 0.007 0.086 ± 0.035 0.037 ± 0.008 0.118 ± 0.056 0.038 ± 0.006 0.118 ± 0.066 0.035 ± 0.007 0.223 ± 0.118

1.186 ± 0.527 1.081 ± 0.447 1.022 ± 0.409 1.156 ± 0.532 1.062 ± 0.462 1.081 ± 0.476 0.955 ± 0.379

0.216 ± 0.057 0.124 ± 0.038 0.213 ± 0.058 0.110 ± 0.029 0.215 ± 0.054 0.113 ± 0.057 0.217 ± 0.051 0.089 ± 0.036 0.211 ± 0.051 0.126 ± 0.064 0.238 ± 0.045 0.098 ± 0.048 0.209 ± 0.058 0.230 ± 0.126

1.467 ± 0.787 1.285 ± 0.591 1.140 ± 0.497 1.174 ± 0.531 1.040 ± 0.437 1.122 ± 0.504 0.963 ± 0.382

21

Table 8: Multiple resource allocation, full method pool across group imbalance ℓ (α=2, MLP-64, K=2). ℓ=0

ℓ=0.2

ℓ=0.4

ℓ=0.6

ℓ=0.8

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MSE

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.135 ± 0.038 0.664 ± 0.084 0.133 ± 0.034 0.139 ± 0.039

0.042 ± 0.012 0.187 ± 0.085 0.043 ± 0.014 0.068 ± 0.037

0.545 ± 0.231 2.408 ± 1.055 0.550 ± 0.233 0.952 ± 0.538

0.164 ± 0.042 0.663 ± 0.082 0.166 ± 0.041 0.165 ± 0.036

0.078 ± 0.040 0.207 ± 0.079 0.070 ± 0.042 0.114 ± 0.035

0.660 ± 0.275 2.506 ± 1.088 0.664 ± 0.278 1.399 ± 0.849

0.193 ± 0.039 0.660 ± 0.079 0.196 ± 0.045 0.196 ± 0.045

0.149 ± 0.079 0.233 ± 0.075 0.127 ± 0.077 0.177 ± 0.042

0.797 ± 0.321 2.628 ± 1.127 0.802 ± 0.326 1.577 ± 0.863

0.215 ± 0.046 0.656 ± 0.080 0.215 ± 0.049 0.216 ± 0.058

0.228 ± 0.117 0.260 ± 0.072 0.194 ± 0.115 0.176 ± 0.035

0.953 ± 0.368 2.772 ± 1.169 0.961 ± 0.375 1.567 ± 0.763

0.216 ± 0.044 0.656 ± 0.086 0.227 ± 0.043 0.219 ± 0.056

0.311 ± 0.158 0.286 ± 0.072 0.257 ± 0.149 0.393 ± 0.153

1.130 ± 0.419 2.936 ± 1.215 1.149 ± 0.432 2.197 ± 0.988

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.138 ± 0.038 0.137 ± 0.038 0.137 ± 0.033 0.137 ± 0.038 0.135 ± 0.037 0.136 ± 0.035

0.042 ± 0.013 0.058 ± 0.028 0.056 ± 0.024 0.049 ± 0.012 0.046 ± 0.012 0.047 ± 0.013

0.546 ± 0.230 0.788 ± 0.404 0.809 ± 0.418 0.674 ± 0.301 0.650 ± 0.295 0.606 ± 0.257

0.167 ± 0.038 0.164 ± 0.036 0.168 ± 0.045 0.166 ± 0.041 0.165 ± 0.040 0.160 ± 0.040

0.072 ± 0.036 0.095 ± 0.032 0.075 ± 0.024 0.065 ± 0.021 0.063 ± 0.028 0.061 ± 0.029

0.665 ± 0.278 1.237 ± 0.750 0.977 ± 0.514 0.836 ± 0.381 0.771 ± 0.342 0.726 ± 0.309

0.196 ± 0.037 0.198 ± 0.045 0.199 ± 0.041 0.195 ± 0.043 0.194 ± 0.044 0.197 ± 0.044

0.117 ± 0.064 0.109 ± 0.027 0.100 ± 0.020 0.103 ± 0.026 0.103 ± 0.050 0.093 ± 0.050

0.821 ± 0.338 1.318 ± 0.756 1.178 ± 0.623 1.109 ± 0.569 0.997 ± 0.459 0.891 ± 0.381

0.228 ± 0.038 0.217 ± 0.053 0.215 ± 0.055 0.214 ± 0.057 0.215 ± 0.057 0.214 ± 0.051

0.142 ± 0.082 0.102 ± 0.019 0.202 ± 0.021 0.150 ± 0.047 0.143 ± 0.070 0.131 ± 0.086

0.998 ± 0.402 1.351 ± 0.652 1.722 ± 0.900 1.334 ± 0.638 1.122 ± 0.495 1.066 ± 0.435

0.225 ± 0.039 0.218 ± 0.055 0.218 ± 0.056 0.219 ± 0.057 0.214 ± 0.057 0.217 ± 0.058

0.174 ± 0.104 0.132 ± 0.056 0.243 ± 0.037 0.160 ± 0.089 0.212 ± 0.116 0.158 ± 0.117

1.189 ± 0.467 1.634 ± 0.774 1.864 ± 0.840 1.327 ± 0.539 1.284 ± 0.518 1.272 ± 0.492

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 0.136 ± 0.034 0.053 ± 0.021 1 0.5 0.134 ± 0.036 0.048 ± 0.013 1 1 0.137 ± 0.039 0.046 ± 0.012 1 – 0.135 ± 0.034 0.046 ± 0.013 – – 0.136 ± 0.038 0.042 ± 0.013 – – 0.181 ± 0.066 0.044 ± 0.016 – – 0.132 ± 0.036 0.041 ± 0.012

0.767 ± 0.381 0.656 ± 0.285 0.630 ± 0.269 0.590 ± 0.247 0.555 ± 0.229 0.765 ± 0.366 0.545 ± 0.233

0.164 ± 0.041 0.069 ± 0.020 0.166 ± 0.042 0.064 ± 0.022 0.164 ± 0.039 0.059 ± 0.024 0.165 ± 0.040 0.061 ± 0.029 0.162 ± 0.039 0.072 ± 0.034 0.210 ± 0.081 0.075 ± 0.029 0.156 ± 0.037 0.074 ± 0.038

0.927 ± 0.452 0.809 ± 0.357 0.772 ± 0.339 0.759 ± 0.349 0.686 ± 0.286 0.838 ± 0.410 0.663 ± 0.280

0.194 ± 0.043 0.087 ± 0.014 0.194 ± 0.042 0.089 ± 0.024 0.194 ± 0.044 0.084 ± 0.031 0.198 ± 0.046 0.073 ± 0.032 0.190 ± 0.040 0.105 ± 0.059 0.223 ± 0.049 0.105 ± 0.043 0.193 ± 0.048 0.150 ± 0.087

1.109 ± 0.541 1.081 ± 0.538 1.003 ± 0.468 0.960 ± 0.448 0.849 ± 0.370 0.939 ± 0.443 0.803 ± 0.329

0.216 ± 0.057 0.124 ± 0.038 0.213 ± 0.058 0.110 ± 0.029 0.215 ± 0.054 0.113 ± 0.057 0.217 ± 0.051 0.089 ± 0.036 0.211 ± 0.051 0.126 ± 0.064 0.238 ± 0.045 0.098 ± 0.048 0.209 ± 0.058 0.230 ± 0.126

1.467 ± 0.787 1.285 ± 0.591 1.140 ± 0.497 1.174 ± 0.531 1.040 ± 0.437 1.122 ± 0.504 0.963 ± 0.382

0.220 ± 0.054 0.127 ± 0.048 0.216 ± 0.057 0.109 ± 0.052 0.215 ± 0.057 0.142 ± 0.083 0.217 ± 0.055 0.103 ± 0.041 0.215 ± 0.057 0.144 ± 0.080 0.238 ± 0.060 0.143 ± 0.064 0.213 ± 0.055 0.302 ± 0.155

1.586 ± 0.724 1.370 ± 0.581 1.304 ± 0.535 1.405 ± 0.638 1.255 ± 0.528 1.285 ± 0.564 1.136 ± 0.427

Next, we compare the gradient-combination rules in training with the full FDFL objective. In both allocation setups, the combination handlers, including scalarization, PCGrad, and NashMTL, attain comparable performances under an MLP-64 predictor (Table 4 and Table 6). The handlers separate more at limited predictor capacity. For example, in the multiple resource allocation, Table 6 shows that with the MLP-16 predictor, FDFL-PCGrad attains a superior balance over the other FDFL methods, yielding low regrets, MAD and MSE simultaneously. Finally, our findings are robust to several design choices. The imbalance-driven prediction fairness trend in Section 5.3.2 is not specific to α = 2: Table 11 demonstrates similar trends from varying imbalance levels at α = 0.5, with prediction disparity worsening with ℓ increase in methods without the prediction fairness objective and effectively reduced from including prediction fairness. In single resource allocation, Table 9 validates that all results remain stable as the number of training instances grows from N = 10 to 50. In the multiple resource allocation, Table 10 shows that increasing the number of groups from K = 2 to K = 4 leaves the regret comparison unchanged and the prediction fairness objective continues to reduce disparity within each setting, though the disparity magnitudes are not directly comparable across a different number of groups. Table 9: Single resource allocation, full method pool across number of training instances N (MLP-64, α=2). N =10 µ

Reg

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.139 ± 0.003 0.279 ± 0.002 0.139 ± 0.003 0.131 ± 0.003

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 0.143 ± 0.002 18.1 ± 6.0 126.2 ± 7.6 0.142 ± 0.002 17.9 ± 6.0 127.0 ± 9.6 0.142 ± 0.002 20.6 ± 3.5 122.6 ± 1.9 1 0 0.130 ± 0.004 29.8 ± 6.8 139.7 ± 10.7 0.130 ± 0.003 27.7 ± 4.4 141.5 ± 7.4 0.129 ± 0.003 27.2 ± 5.5 144.5 ± 11.7 0 0.1 0.147 ± 0.038 41.2 ± 14.3 179.2 ± 45.3 0.226 ± 0.216 44.5 ± 17.2 189.4 ± 61.6 0.129 ± 0.003 40.4 ± 16.7 166.7 ± 52.3 0 0.5 0.130 ± 0.003 27.0 ± 5.7 130.1 ± 4.7 0.136 ± 0.016 26.3 ± 5.5 127.9 ± 3.3 0.129 ± 0.003 27.3 ± 5.1 129.5 ± 2.7 0 1 0.130 ± 0.003 24.7 ± 4.4 126.4 ± 2.9 0.129 ± 0.003 24.1 ± 4.0 125.4 ± 3.4 0.129 ± 0.003 24.8 ± 4.4 126.0 ± 3.0 0 – 0.130 ± 0.002 24.8 ± 4.9 127.0 ± 3.9 0.129 ± 0.003 25.8 ± 5.5 128.3 ± 2.8 0.130 ± 0.003 26.4 ± 4.7 127.8 ± 3.5

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 1 0.5 1 1 1 – – – – – – –

D

MSE

19.9 ± 3.3 119.7 ± 1.3 57.2 ± 11.2 248.4 ± 4.7 20.5 ± 3.2 118.8 ± 1.5 65.4 ± 11.0 260.3 ± 25.5

26.1 ± 6.4 23.4 ± 4.0 23.6 ± 5.6 24.0 ± 3.4 24.3 ± 4.5 26.1 ± 6.3 24.7 ± 2.6

137.6 ± 6.4 129.8 ± 4.5 129.4 ± 7.4 130.1 ± 7.5 129.8 ± 3.9 131.9 ± 6.7 126.7 ± 3.0

Reg 0.139 ± 0.003 0.279 ± 0.002 0.138 ± 0.003 0.130 ± 0.004

0.143 ± 0.030 0.131 ± 0.003 0.130 ± 0.003 0.130 ± 0.003 0.131 ± 0.003 0.134 ± 0.005 0.129 ± 0.003

MAD

N =50

λ

0.130 ± 0.003 0.131 ± 0.003 0.131 ± 0.004 0.131 ± 0.003 0.131 ± 0.004 0.133 ± 0.002 0.130 ± 0.003

MAD

N =20

Method

MSE

19.9 ± 3.0 119.1 ± 1.2 57.2 ± 11.4 248.4 ± 4.8 20.6 ± 2.7 118.2 ± 1.4 68.9 ± 6.3 267.6 ± 23.1

25.9 ± 5.3 23.9 ± 3.0 22.4 ± 3.0 24.5 ± 3.6 22.5 ± 4.6 22.8 ± 3.7 24.0 ± 3.5

138.3 ± 8.3 129.7 ± 4.6 126.1 ± 2.4 131.1 ± 7.3 132.1 ± 3.9 131.9 ± 9.1 126.8 ± 3.9

Reg 0.139 ± 0.003 0.279 ± 0.003 0.138 ± 0.002 0.129 ± 0.003

0.129 ± 0.002 0.130 ± 0.003 0.130 ± 0.003 0.130 ± 0.002 0.130 ± 0.003 0.134 ± 0.002 0.129 ± 0.003

MAD

MSE

21.2 ± 3.0 118.9 ± 1.9 59.2 ± 10.8 249.3 ± 4.5 22.1 ± 2.5 118.2 ± 1.5 71.0 ± 11.0 270.3 ± 18.5

26.3 ± 6.9 25.8 ± 6.1 25.3 ± 3.4 24.6 ± 4.1 24.8 ± 4.7 28.1 ± 5.1 25.8 ± 4.4

145.9 ± 9.4 135.0 ± 8.1 126.8 ± 4.6 129.3 ± 4.0 136.0 ± 10.1 134.2 ± 5.1 126.4 ± 2.8

Proof of Generalization Bound

This appendix contains the proof of Theorem 1 stated in Section 4.3, together with verification remarks for the Lipschitzness assumptions in our experiment settings. Recall the notation of Section 4.3: z = (X, a, r, DS ) collects the instance features, group memberships, true impacts, and feasible-set data (we suppress a and DS where they play no active role); D = {zs }N s=1 is the set of i.i.d. training samples, Θ ⊂ Rq is the parameter space, the per-instance composite loss ℓfull (θ; z) from Section 4.3, and we abbreviate the instance-dependent oracle d∗z as d∗ when the instance is clear from context. Because our predictor feeds point predictions to the decision oracle, instead of a parametric distribution family as in Elmachtoub et al. (2025), we control the loss classes by a different set of standard tools, including Lipschitz transfer from the parameter space, metric-entropy bounds, and Dudley chaining.

22

Table 10: Multiple resource allocation, full method pool across the number of groups K (MLP-64, α=2, imbalance 0.6; MAD values are not comparable across K). K=2

K=4

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

MSE

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.215 ± 0.046 0.656 ± 0.080 0.215 ± 0.049 0.216 ± 0.058

0.228 ± 0.117 0.260 ± 0.072 0.194 ± 0.115 0.176 ± 0.035

0.953 ± 0.368 2.772 ± 1.169 0.961 ± 0.375 1.567 ± 0.763

0.217 ± 0.045 0.662 ± 0.076 0.222 ± 0.052 0.221 ± 0.051

0.194 ± 0.083 0.404 ± 0.119 0.171 ± 0.080 0.228 ± 0.104

0.929 ± 0.364 2.743 ± 1.166 0.935 ± 0.374 1.440 ± 0.726

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.228 ± 0.038 0.217 ± 0.053 0.215 ± 0.055 0.214 ± 0.057 0.215 ± 0.057 0.214 ± 0.051

0.142 ± 0.082 0.102 ± 0.019 0.202 ± 0.021 0.150 ± 0.047 0.143 ± 0.070 0.131 ± 0.086

0.998 ± 0.402 1.351 ± 0.652 1.722 ± 0.900 1.334 ± 0.638 1.122 ± 0.495 1.066 ± 0.435

0.222 ± 0.044 0.218 ± 0.056 – – 0.219 ± 0.053 0.213 ± 0.048

0.177 ± 0.076 0.201 ± 0.092 – – 0.191 ± 0.097 0.179 ± 0.081

0.942 ± 0.375 1.317 ± 0.623 – – 1.124 ± 0.512 1.025 ± 0.427

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 0.216 ± 0.057 0.124 ± 0.038 1 0.5 0.213 ± 0.058 0.110 ± 0.029 1 1 0.215 ± 0.054 0.113 ± 0.057 1 – 0.217 ± 0.051 0.089 ± 0.036 – – 0.211 ± 0.051 0.126 ± 0.064 – – 0.238 ± 0.045 0.098 ± 0.048 – – 0.209 ± 0.058 0.230 ± 0.126

1.467 ± 0.787 1.285 ± 0.591 1.140 ± 0.497 1.174 ± 0.531 1.040 ± 0.437 1.122 ± 0.504 0.963 ± 0.382

– – 0.218 ± 0.047 0.217 ± 0.050 0.220 ± 0.043 0.249 ± 0.072 0.214 ± 0.048

– – 0.185 ± 0.092 0.166 ± 0.081 0.174 ± 0.076 0.177 ± 0.078 0.197 ± 0.087

– – 1.135 ± 0.524 1.112 ± 0.515 0.981 ± 0.396 1.112 ± 0.465 0.937 ± 0.377

Table 11: Multiple resource allocation, full method pool across group imbalance ℓ at α=0.5 (MLP-64, K=2). ℓ=0

ℓ=0.2

ℓ=0.4

ℓ=0.6 MSE

Reg

ℓ=0.8

Method

λ

µ

Reg

MAD

MSE

Reg

MAD

MSE

Reg

MAD

MAD

MSE

Reg

MAD

MSE

PTO SAA WDRO DFL

0 – – 0

– – – 0

0.013 ± 0.001 0.098 ± 0.019 0.012 ± 0.001 0.013 ± 0.001

0.043 ± 0.012 0.187 ± 0.085 0.043 ± 0.014 0.077 ± 0.032

0.567 ± 0.233 2.408 ± 1.055 0.550 ± 0.233 0.945 ± 0.526

0.014 ± 0.001 0.089 ± 0.013 0.014 ± 0.002 0.014 ± 0.002

0.075 ± 0.038 0.207 ± 0.079 0.070 ± 0.042 0.106 ± 0.055

0.677 ± 0.276 2.506 ± 1.088 0.664 ± 0.278 1.094 ± 0.658

0.011 ± 0.003 0.047 ± 0.012 0.011 ± 0.003 0.011 ± 0.003

0.144 ± 0.078 0.233 ± 0.075 0.127 ± 0.077 0.116 ± 0.050

0.810 ± 0.322 0.016 ± 0.004 2.628 ± 1.127 0.053 ± 0.014 0.802 ± 0.326 0.016 ± 0.004 1.410 ± 0.893 0.021 ± 0.007

0.222 ± 0.118 0.260 ± 0.072 0.194 ± 0.115 0.562 ± 0.247

0.960 ± 0.369 2.772 ± 1.169 0.961 ± 0.375 3.088 ± 1.453

0.023 ± 0.006 0.065 ± 0.015 0.023 ± 0.006 0.030 ± 0.009

0.303 ± 0.157 0.286 ± 0.072 0.263 ± 0.151 0.904 ± 0.297

1.129 ± 0.418 2.936 ± 1.215 1.140 ± 0.430 4.004 ± 1.644

FPTO Regret-and-MAD Regret-and-MSE Regret-and-MSE Regret-and-MSE FDFL-FPLG

1 – 1 0 0 0.1 0 0.5 0 1 0 –

0.013 ± 0.001 0.013 ± 0.002 0.013 ± 0.001 0.012 ± 0.002 0.013 ± 0.002 0.013 ± 0.002

0.043 ± 0.013 0.043 ± 0.011 0.055 ± 0.016 0.042 ± 0.011 0.044 ± 0.013 0.053 ± 0.017

0.569 ± 0.232 0.597 ± 0.259 0.663 ± 0.299 0.565 ± 0.241 0.585 ± 0.245 0.640 ± 0.289

0.014 ± 0.001 0.014 ± 0.002 0.014 ± 0.002 0.014 ± 0.002 0.014 ± 0.002 0.014 ± 0.002

0.069 ± 0.035 0.059 ± 0.023 0.072 ± 0.035 0.071 ± 0.037 0.074 ± 0.039 0.069 ± 0.038

0.685 ± 0.282 0.773 ± 0.353 0.805 ± 0.387 0.684 ± 0.291 0.689 ± 0.285 0.819 ± 0.445

0.011 ± 0.003 0.012 ± 0.003 0.012 ± 0.004 0.011 ± 0.003 0.012 ± 0.004 0.012 ± 0.004

0.105 ± 0.056 0.074 ± 0.029 0.095 ± 0.056 0.131 ± 0.074 0.134 ± 0.081 0.087 ± 0.031

0.840 ± 0.342 1.001 ± 0.471 1.008 ± 0.469 0.858 ± 0.343 0.854 ± 0.343 1.146 ± 0.657

0.017 ± 0.004 0.021 ± 0.007 0.018 ± 0.006 0.019 ± 0.006 0.018 ± 0.005 0.020 ± 0.006

0.135 ± 0.076 0.090 ± 0.029 0.193 ± 0.062 0.147 ± 0.088 0.166 ± 0.095 0.308 ± 0.221

1.017 ± 0.405 1.368 ± 0.593 1.619 ± 0.738 1.183 ± 0.440 1.079 ± 0.403 1.972 ± 0.829

0.024 ± 0.006 0.027 ± 0.008 0.024 ± 0.006 0.026 ± 0.007 0.024 ± 0.006 0.025 ± 0.007

0.158 ± 0.092 0.121 ± 0.034 0.347 ± 0.119 0.149 ± 0.058 0.192 ± 0.121 0.333 ± 0.187

1.213 ± 0.473 1.668 ± 0.712 2.063 ± 0.922 1.452 ± 0.502 1.237 ± 0.448 1.985 ± 0.803

FDFL-Scal FDFL-Scal FDFL-Scal FDFL-FPLG FDFL-PCGrad FDFL-MGDA FDFL-NashMTL

1 0.1 1 0.5 1 1 1 – – – – – – –

0.013 ± 0.002 0.013 ± 0.001 0.013 ± 0.002 0.013 ± 0.001 0.012 ± 0.002 0.013 ± 0.002 0.012 ± 0.001

0.047 ± 0.012 0.618 ± 0.263 0.014 ± 0.002 0.042 ± 0.011 0.562 ± 0.238 0.014 ± 0.001 0.044 ± 0.013 0.582 ± 0.246 0.014 ± 0.002 0.044 ± 0.011 0.599 ± 0.255 0.014 ± 0.002 0.042 ± 0.013 0.553 ± 0.233 0.014 ± 0.002 0.042 ± 0.017 0.635 ± 0.287 0.015 ± 0.002 0.041 ± 0.012 0.548 ± 0.235 0.014 ± 0.002

0.063 ± 0.026 0.063 ± 0.029 0.070 ± 0.037 0.064 ± 0.031 0.072 ± 0.038 0.070 ± 0.036 0.075 ± 0.039

0.758 ± 0.339 0.696 ± 0.298 0.694 ± 0.291 0.795 ± 0.387 0.686 ± 0.295 0.715 ± 0.313 0.661 ± 0.282

0.012 ± 0.004 0.011 ± 0.003 0.012 ± 0.004 0.012 ± 0.004 0.012 ± 0.004 0.012 ± 0.004 0.011 ± 0.003

0.082 ± 0.034 0.087 ± 0.049 0.103 ± 0.062 0.074 ± 0.030 0.089 ± 0.052 0.098 ± 0.049 0.141 ± 0.075

0.977 ± 0.438 0.885 ± 0.364 0.871 ± 0.357 1.036 ± 0.518 0.891 ± 0.377 0.905 ± 0.386 0.814 ± 0.335

0.019 ± 0.006 0.019 ± 0.006 0.017 ± 0.006 0.019 ± 0.006 0.019 ± 0.005 0.018 ± 0.006 0.017 ± 0.005

0.092 ± 0.030 0.116 ± 0.058 0.119 ± 0.064 0.099 ± 0.034 0.129 ± 0.072 0.119 ± 0.072 0.212 ± 0.111

1.266 ± 0.541 1.216 ± 0.466 1.106 ± 0.436 1.384 ± 0.646 1.174 ± 0.491 1.091 ± 0.452 0.999 ± 0.376

0.024 ± 0.006 0.104 ± 0.025 0.026 ± 0.008 0.119 ± 0.041 0.024 ± 0.006 0.126 ± 0.069 0.025 ± 0.006 0.125 ± 0.045 0.026 ± 0.009 0.154 ± 0.095 0.024 ± 0.007 0.146 ± 0.082 0.023 ± 0.006 0.288 ± 0.151

1.467 ± 0.614 1.458 ± 0.565 1.283 ± 0.501 1.574 ± 0.718 1.355 ± 0.569 1.273 ± 0.516 1.154 ± 0.432

D.1

Supporting Lemmas

Lemma 1 (The regret class is Lipschitz in θ). Under Assumption 3, the map θ 7→ ℓreg (θ; z) is LW Lor Lipschitz uniformly in z with LW and Lor the Lipschitz constants of Assumption 3.: |ℓreg (θ1 ; z) − ℓreg (θ2 ; z)| ≤ LW Lor ∥θ1 − θ2 ∥, Proof. By definition ℓreg (θ; z) = W (d∗ (r); r) − W (d∗ (fθ (X)); r). The first term is independent of θ, so the θ-dependence enters only through W (d∗ (fθ (X)); r). Applying the two Lipschitz properties of Assumption 3 in turn, |ℓreg (θ1 ; z) − ℓreg (θ2 ; z)| ≤ LW ∥d∗ (fθ1 (X)) − d∗ (fθ2 (X))∥ ≤ LW Lor ∥θ1 − θ2 ∥ follows. □ Lemma 2 (Parametric Rademacher bound for uniformly Lipschitz classes). Let F := {φθ : θ ∈ Θ} be a class of real-valued functions on Z indexed by Θ such that, for some L > 0, supz∈Z |φθ1 (z) − φθ2 (z)| ≤ L ∥θ1 − θ2 ∥ for all θ1 , θ2 ∈ Θ. Under Assumption 1, there exists an absolute constant Cabs > 0, independent of L, Θ, and N , such that p RN (F) ≤ Cabs L EΘ q/N . Proof. Fix a sample D = {zs }N s=1 , let σ1 , . . . , σN be i.i.d. Rademacher variables independent of D, and define on Θ the scaled Euclidean metric dN (θ1 , θ2 ) := √LN ∥θ1 − θ2 ∥. For the centered proPN PN cess Xθ := N1 s=1 σs φθ (zs ), each increment Xθ1 − Xθ2 = s=1 σs as is a weighted Rademacher sum with weights as := (φθ1 (zs ) − φθ2 (zs ))/N , whose norm the assumed Lipschitzness controls by
2 2 2 PN ∥a∥2 = N12 s=1 φθ1 (zs ) − φθ2 (zs ) ≤ L ∥θ1N−θ2 ∥ = dN (θ1 , θ2 )2 , so Hoeffding’s lemma (Mohri et al. 2018, Lemma D.1) gives, for all s ∈ R,   2 2 2 2 Eσ es(Xθ1 −Xθ2 ) ≤ es ∥a∥ /2 ≤ es dN (θ1 ,θ2 ) /2 ,  
and hence, by a Chernoff bound, Pr |Xθ1 − Xθ2 | ≥ u ≤ 2 exp −u2 /(2 dN (θ1 , θ2 )2 ) for all u ≥ 0: the increments of (Xθ )θ∈Θ are sub-Gaussian with respect to dN . The process is also separable in the sense of Sen (2018, Definition 4.4), since its sample paths are Lipschitz in θ and Θ ⊆ Rq is separable. 23

Let D(ϵ, T, d) denote the ϵ-packing number of a metric space (T, d), the maximum number of points of T that are pairwise more than ϵ apart (Sen 2018, Definitions 2.4–2.5). Covering the ball of radius EΘ that encloses Θ (Assumption 1) by at most (6EΘ /η)q balls of radius η/2 (Mohri et al. 2018, Lemma 6.27), each of which contains at most one point of any η-packing, ∥) ≤ q log(6EΘ /η) for √ yields log D(η, Θ, ∥ · √ η ∈ (0, 2EΘ ]; since dN rescales the Euclidean metric by L/ N , for ϵ ∈ (0, 2LEΘ / N ],  ϵ√N  6LEΘ log D(ϵ, Θ, dN ) = log D , Θ, ∥ · ∥ ≤ q log √ , L ϵ N √ and the dN -diameter of Θ is at most D̄ := 2LEΘ / N . Dudley’s entropy-integral bound for separable sub-Gaussian processes (Sen 2018, Theorem 4.5) then provides an absolute constant c0 > 0 such that, for any fixed θ0 ∈ Θ (using Eσ [Xθ0 ] = 0), Z D̄/4 p h i h i b RD (F) = Eσ sup Xθ ≤ Eσ sup Xθ − Xθ0 ≤ c0 log D(ϵ, Θ, dN ) dϵ θ∈Θ θ∈Θ 0 s √ Z 2LEΘ / N Z r 6LEΘ 6 √ √ LEΘ 2 log √ dϵ = c0 q √ log ≤ c0 q u ϵ N N 0 0 r r Z ∞ √ −v q q √ LEΘ √ = c0 q v e dv ≤ 6 Γ(3/2) c0 L EΘ ·6 = Cabs L EΘ , N N N log 3 √ where the two equalities use the changes of variables u = ϵ N /(LEΘ ) and v = log(6/u), and Cabs := √ 3 π c0 . The right-hand side is deterministic in D, so taking ED yields the claimed bound. □ Lemma 3 (Rademacher decomposition). For any λ, µ ≥ 0, RN (Ffull ) ≤ RN (Freg ) + λ RN (Ffair ) + µ RN (Fpred ). h
i PN 1 reg fair pred b D (Ffull ) = Eσ sup Proof. For any fixed sample D, R σ ℓ (θ; z ) + λℓ (θ; z ) + µℓ (θ; z ) ≤ s s s s θ∈Θ N s=1 b D (Freg ) + λR b D (Ffair ) + µR b D (Fpred ), where the inequality uses subadditivity of the supremum and R λ, µ ≥ 0. Taking expectation over D proves the claim. □ p Lemma 4 (Rademacher bound pfor Ffair and Fpred ). Under Assumptions 1 and 4, RN (Ffair ) ≤ Cabs LF EΘ q/N and RN (Fpred ) ≤ Cabs LL EΘ q/N , with the same absolute constant Cabs as in Lemma 2. Proof. By Assumption 4, |ℓfair (θ1 ; z) − ℓfair (θ2 ; z)| ≤ LF ∥θ1 − θ2 ∥ and |ℓpred (θ1 ; z) − ℓpred (θ2 ; z)| ≤ LL ∥θ1 − θ2 ∥ for every z ∈ Z and θ1 , θ2 ∈ Θ. The classes Ffair and Fpred therefore satisfy the required conditions of Lemma 2 with L = LF and L = LL , respectively, and the claimed bounds follow. □ Lemma 5 (Uniform deviation for the composite training objective). Under Assumptions 1–4, for any δ ∈ (0, 1), with probability at least 1 − δ, q
sup v0full (θ) − v̂0full (θ) ≤ 2 RN (Freg ) + 2λ RN (Ffair ) + 2µ RN (Fpred ) + (Breg + λBF + µBL ) 2 log(1/δ) . N θ∈Θ

fair pred Proof. Write B := Breg + λBF + µBL , so |ℓfull (θ; z)| ≤ |ℓreg (θ; (θ; z)| ≤ B
z)| + λ|ℓ (θ; z)| + µ|ℓ full full by Assumption 2, and define Φ(D) := supθ∈Θ v0 (θ) − v̂0 (θ) , measurable by the same separability N be a ghost sample independent of D, argument as in the proof of Lemma 2. Let D′ = {zs′ }N s=1 ∼ P ′ ′ P N full,D 1 full ′ with empirical objective v̂0 (θ) := N s=1 ℓ (θ; zs ) satisfying v0full (θ) = ED′ [v̂0full,D (θ)], and let σ1 , . . . , σN be i.i.d. Rademacher variables independent of (D, D′ ). Then h  i ′ ED [Φ(D)] = ED sup ED′ v̂0full,D (θ) − v̂0full (θ) θ∈Θ

N h
i 1 X full ≤ ED,D′ sup ℓ (θ; zs′ ) − ℓfull (θ; zs ) θ∈Θ N s=1

=E

D,D ′ ,σ

h

(Jensen; Fubini)

N

i 1 X sup σs ℓfull (θ; zs′ ) − ℓfull (θ; zs ) θ∈Θ N s=1

(swap zs ↔ zs′ )

N N h i h i 1 X 1 X ≤ ED′ ,σ sup σs ℓfull (θ; zs′ ) + ED,σ sup (−σs ) ℓfull (θ; zs ) θ∈Θ N s=1 θ∈Θ N s=1 d

d

= 2 RN (Ffull )

(−σ = σ, D′ = D)

≤ 2 RN (Freg ) + 2λ RN (Ffair ) + 2µ RN (Fpred ).

(Lemma 3)

24

Moreover, replacing the s-th coordinate zs of D with an arbitrary z̃ ∈ Z yields a sample D(s) with 2B 1 full ℓ (θ; zs ) − ℓfull (θ; z̃) ≤ , N N θ∈Θ θ∈Θ  so McDiarmid’s bounded-difference inequality (Mohri et al. 2018, Theorem D.8) gives Pr Φ(D) ≥ p  ED [Φ(D)]+t ≤ exp(−N t2 /(2B 2 )) for all t > 0; setting the right-hand side to δ, i.e., t = B 2 log(1/δ)/N , and combining with the display above proves the claim. □ (s)

|Φ(D) − Φ(D(s) )| ≤ sup v̂0full,D (θ) − v̂0full,D (θ) = sup

D.2

Proof of Theorem 1

Write B := Breg + λBF + µBL . Adding and subtracting v̂0full (θ̂) and v̂0full (θ∗ ),


Rfull (θ̂) := v0full (θ̂) − v0full (θ∗ ) = v0full (θ̂) − v̂0full (θ̂) + v̂0full (θ̂) − v̂0full (θ∗ ) + v̂0full (θ∗ ) − v0full (θ∗ ) , {z } | {z } | {z } | (A)

(M )

(B)

where (M ) ≤ 0 deterministically since θ̂ ∈ arg minθ∈Θ v̂0full (θ), and (A) ≤ supθ∈Θ v0full (θ) − v̂0full (θ) . Each of the following two bounds holds with probability at least 1 − δ/2 over D ∼ P N : q (A) ≤ 2 RN (Freg ) + 2λ RN (Ffair ) + 2µ RN (Fpred ) + B 2 log(2/δ) , (17) N q N   1 X full ∗ (B) = ℓ (θ ; zs ) − E ℓfull (θ∗ ; z) ≤ B 2 log(2/δ) , (18) N N s=1 where (17) applies Lemma 5 at confidence level δ/2, and (18) applies Hoeffding’s inequality (Mohri et al. 2018, Theorem D.2) to the i.i.d. variables ℓfull (θ∗ ; zs ) ∈ [−B, B] (Assumption 2; θ∗ is non-random). On the intersection of the two events, whose probability is at least 1 − δ by the union bound, q full R (θ̂) ≤ 2 RN (Freg ) + 2λ RN (Ffair ) + 2µ RN (Fpred ) + 2B 2 log(2/δ) N q
qq 2 log(2/δ) ≤ 2Cabs LW Lor + λ LF + µ LL EΘ N + 2B , N where the last inequality substitutes the parametric Rademacher bounds of Lemma 2, applied to Freg with L = LW Lor (Lemma 1) and to Ffair and Fpred with L = LF and L = LL (Lemma 4). This is exactly the bound stated in (13). □

D.3

Verification remarks

This subsection verifies Assumptions 3 and 4 for the settings of Section 5.1. Remark 4 (Verification of Assumption 3). We verify Assumption 3 for the two allocation tasks via the following two lemmas. Lemma 6 (Concavity of α-fairness). Suppose utilities are all positive. (i) For every α > 0, the group-based measure Wαg in (5) is strictly concave in u. (ii) If the utility map d 7→ u(d) is affine, d 7→ Wαg (u(d)) is concave in d; if the map is injective on the feasible set, Wαg (u(d)) is strictly concave in d with a negative-definite Hessian. P Proof. Part (i), case α > 1. Write β := α − 1 > 0 and Sk := i∈Gk u−β i . The intra-group score is −β−1 β hk = β/Sk , so the kth group term of (5) equals h1−α /(1 − α) = −β S ; k therefore, it suffices to show k β −β−1 that Sk is strictly convex in the group’s coordinates. With wi := ui , the Hessian of Skβ is:
∇2 Skβ = β 3 (β − 1)Skβ−2 ww⊤ + β 2 (β + 1)Skβ−1 diag u−β−2 . i P Fix a direction v ̸= 0 and set T := i∈Gk u−β−2 vi2 > 0. If β ≥ 1, the rank-one term β 3 (β − 1)Skβ−2 ww⊤ i is positive semidefinite, so v ⊤ ∇2 Skβ v ≥ β 2 (β +1)Skβ−1 T > 0. If 0 < β < 1, the rank-one term is negative; P −β−1
2 vi ≤ Sk T then gives v ⊤ ∇2 Skβ v ≥ β 2 (β 2 + 1)Skβ−1 T > 0. In the Cauchy–Schwarz bound i∈Gk ui β both cases Sk is strictly convex, so each group term is strictly concave in its group’s coordinates.

25

P Part (i), case 0 < α < 1. Write γ := 1 − α ∈ (0, 1) and Tk := i∈Gk uγi . The intra-group score is hk = Tk /γ, so the kth group term of (5) equals hγk /γ = γ −γ−1 Tkγ , and it suffices to show that Tkγ is strictly concave in the group’s coordinates. Its Hessian is
∇2 Tkγ = γ 3 (γ − 1)Tkγ−2 ww⊤ + γ 2 (γ − 1)Tkγ−1 diag uγ−2 , wi := uγ−1 . i i Since γ ∈ (0, 1), both coefficients are negative, so the Hessian is negative definite and the group term is strictly concave in the group’s coordinates. g In either case, the groups partition [m], so the Hessian of WQ α is block-diagonal with negative-definite Pm g blocks and Wα is strictly concave in u. (At α = 1, hk = i∈Gk ui gives W1g (u) = i=1 log ui by −2 Remark 2, whose Hessian − diag(ui ) is negative definite, so the claim holds there as well.) Part (ii). Concavity is preserved by composition with an affine map. If the map is injective on the feasible set, its linear part A has trivial kernel, so the composition’s Hessian A⊤ (∇2u Wαg )A is negative definite and the composition is strictly concave; on a compact set of decisions with strictly positive utilities, the Hessian’s eigenvalues are bounded away from zero by continuity, so −Wαg (u(·)) is strongly convex there. □ Lemma 7 (Decision oracle Lipschitzness from strong convexity). Suppose, in all instances (i) −W (·; r̂) is ρc -strongly convex in d on a convex set containing the oracle-reachable allocations; (ii) ∇d W (d; r̂) is LW,r -Lipschitz in r̂; and (iii) the prediction map θ 7→ fθ (X) ∈ Rm is Lf -Lipschitz in θ. Then the decision map θ 7→ d∗ (fθ (X)) satisfies Assumption 3 with Lor ≤ (LW,r /ρc ) Lf . Proof. Denote ψ(·; r̂) := −W (·; r̂) and dj := d∗ (r̂j ) for j = 1, 2. The first-order optimality conditions on the convex feasible set give ⟨∇ψ(d1 ; r̂1 ), d2 − d1 ⟩ ≥ 0 and ⟨∇ψ(d2 ; r̂2 ), d2 − d1 ⟩ ≤ 0. Combining them with the strong monotonicity of ∇ψ(·; r̂1 ) and the Cauchy–Schwarz inequality, ρc ∥d2 − d1 ∥2 ≤ ⟨∇ψ(d2 ; r̂1 ) − ∇ψ(d1 ; r̂1 ), d2 − d1 ⟩ ≤ ⟨∇ψ(d2 ; r̂1 ) − ∇ψ(d2 ; r̂2 ), d2 − d1 ⟩ ≤ LW,r ∥r̂1 − r̂2 ∥ ∥d2 − d1 ∥, so ∥d∗ (r̂1 ) − d∗ (r̂2 )∥ ≤ (LW,r /ρc ) ∥r̂1 − r̂2 ∥; composing with the Lf -Lipschitz map θ 7→ fθ (X) gives the bound. □ In the single resource allocation task, the predicted benefits are positive and bounded, with r̂i ∈ [2, 101] for all i. By Proposition 1, d∗ (r̂) is a continuous and strictly positive function of r̂, so the oracle-reachable allocations admit uniform bounds 0 < dmin ≤ d∗i ≤ dmax = Q/cmin . In addition, W , ∇d W , and ∇2d W are continuous with negative-definite ∇2d W (Lemma 6); hence finite Breg , LW , LW,r and a strong-concavity modulus ρc > 0 exist, and Lemma 7 gives a uniform Lor ≤ (LW,r /ρc )Lf on Θ. PD Next, we examine the multiple resource allocation setting. The utility function ui = j=1 r̂ij dij has Jacobian A = ∂u/∂d ∈ Rm×mD of rank m, so ∇2d W = A⊤ (∇2u W )A is rank-deficient by m(D − 1): the α-fairness objective function is strictly concave in u but only concave in d. This has two consequences. First, Lemma 7 does not apply, since the required strong-convexity does not hold. Second, because W is constant along ker A within the feasible set, the optimal utility u∗ is unique but the optimal allocation d∗ is not guaranteed to be unique. To obtain decision-oracle Lipschitzness, we require an additional nondegeneracy assumption as follows. Assumption 5 (Uniform conic nondegeneracy for non-injective utility maps). At every r̂, the conic reformulation of (1) admits a unique primal–dual optimal solution satisfying strict complementarity; by Agrawal et al. (2019b), Busseti et al. (2019), this solution is a regular point at which the normalized residual map N is differentiable with invertible derivative DN . We assume in addition that the smallest singular value of DN at these solutions is bounded below by a uniform constant σmin > 0. Decision-oracle Lipschitzness then follows from the differentiable cone-program framework of Agrawal et al. (2019a) underlying our cvxpylayers backend: the α-fair allocation admits a power-cone representation, and under Assumption 5 the implicit function theorem applied to the residual map of the homogeneous self-dual embedding (Agrawal et al. 2019b, Busseti et al. 2019, §2) gives ∥Dr̂ d∗ ∥ ≤ Cdata /σmin throughout the prediction range, where Cdata bounds the canonicalization and extraction maps under the bounded problem data. The mean-value inequality over the convex, compact prediction range lifts this pointwise bound to a Lipschitz constant for r̂ 7→ d∗ , which composes with the Lf -Lipschitz predictor to yield Lor on Θ. Remark 5 (Verification of LF for the MAD-based prediction disparity). Let I index the predicted coordinates of an instance (I = [m] in the single-resource task; I = [m] × [R] in the multiple-resource task), SK and let I = k=1 Ik be the partition induced by the group memberships a. The MAD-based prediction
2 disparity measure is a function of the squared prediction errors e(θ; z) ∈ RI with ei (θ; z) := ri −fθ (xi ) .

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Since true impacts r and predictions lie in a common bounded interval, the prediction errors obey |ri − fθ (xi )| ≤ Be for all i ∈ I, θ ∈ Θ, and z ∈ Z, so e(θ; z) ∈ [0, Be2 ]I . The prediction map θ 7→ fθ (X) is Lf -Lipschitz with respect to ∥ · ∥∞ (coordinate-wise projection onto a fixed interval is 1-Lipschitz, so the projected predictor retains Lf ). Let ŷij := fθj (xi ), ei (θ1 ; z)−ei (θ2 ; z) = ŷi2 −ŷi1 · (ri −ŷi1 )+(ri −ŷi2 ) ≤ 2Be ŷi1 − ŷi2 . Taking the maximum over i ∈ I gives e(θ1 ; z) − e(θ2 ; z) ∞ ≤ 2Be Lf ∥θ1 − θ2 ∥

for all θ1 , θ2 ∈ Θ.

(19)

The group-based disparity passes this vector and then a MAD function,
P through a group1 average PK ℓfair (θ; z) = p m(e(θ; z)) , where mk (e) := |I1k | i∈Ik ei , p(m) := K k=1 mk − m̄ , and m̄ is the mean of m; the loss is thus a single scalar per instance and is per-instance decomposable. EachP of the two maps is 1-Lipschitz with respect to ∥ · ∥∞ . For the group average, |mk (e) − mk (e′ )| ≤ |Ik |−1 i∈Ik |ei − e′i | ≤ ∥e−e′ ∥∞ for every k. For the MAD map, linearity p a seminorm, so the reverse triangle  of m 7→ m̄ makes  inequality gives |p(m)−p(m′ )| ≤ p(m−m′ ) ≤ sup∥x∥∞ ≤1 p(x) ∥m−m′ ∥∞ ; by convexity the supremum is attained at a vertex of the cube, which yields maxK1 +K2 =K 4K1 K2 /K 2 ≤ 1, so p is 1-Lipschitz in ∥ · ∥∞ . The same vertex argument over [0, Be2 ]K bounds the range of p by Be2 /2. Combining with (19) gives BF ≤ Be2 /2 and LF ≤ 2Be Lf . Remark 6 (Verification of LL for the MSE prediction loss). The prediction loss shares the structure analyzed in Remark 5, applying a simpler P outer map to the same vector of squared prediction errors: it is the plain average ℓpred (θ; z) = |I|−1 i∈I ei (θ; z), without the group-averaging and centering steps, and is likewise per-instance decomposable. Averaging is 1-Lipschitz with respect to ∥ · ∥∞ by the same estimate used there for the group average, and it maps [0, Be2 ]I into [0, Be2 ]. Combining with (19) gives BL ≤ Be2 and LL ≤ 2Be Lf . Both prediction-side losses therefore satisfy Assumptions 2 and 4 with the same Lipschitz constant 2Be Lf , differing only in their range bounds.

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