FedRSPO+：一种面向决策聚焦联邦学习的异构感知算法

# FedRSPO+: A Heterogeneity-aware Algorithm for Decision-focused Federated Learning
Source: [https://arxiv.org/html/2610.09091](https://arxiv.org/html/2610.09091)
Alexander VinelAffiliation:Auburn UniversityEmail:[alexander\.vinel@auburn\.edu](mailto:)Jiaqi Wang††thanks:Corresponding author\.Affiliation:Auburn UniversityEmail:[jqwang@auburn\.edu](mailto:)

###### Abstract

Decision\-focused learning \(DFL\) trains predictive models for downstream optimization, but existing methods largely assume centralized data\. In cross\-silo settings, federated learning is a natural alternative, yet standard federated pipelines optimize prediction quality rather than decision quality and do not account for client heterogeneity in downstream objectives or feasible sets\. Heterogeneity is especially problematic for DFL methods: under heterogeneous polyhedral decision problems, small perturbations can cause discontinuous changes in optimal decisions, leading to unstable client updates and aggregation\. We proposeFedRSPO\+, a heterogeneity\-aware framework for decision\-focused federated learning\. Our approach is built onRSPO\+, a regularized predict\-then\-optimize surrogate that smooths the decision map through projection, enabling stable optimization even when clients face different objectives and constraints\. We show that RSPO\+ upper bounds downstream decision error and regret of the regularized decision and, under exact regularization and consistent LP solution selection, the original LP decision\. We also demonstrate that FedRSPO\+ has cross\-client heterogeneity bounds that\(i\)scale with both objective and feasible\-set heterogeneity,\(ii\)vanish at homogeneity, and\(iii\)do not require strong convexity\. We develop an annealed and modular training procedure that is compatible with standard federated personalization and aggregation methods\. We evaluate FedRSPO\+ on three experimental settings: controlled synthetic knapsack instances that isolate multiple axes of heterogeneity, a shortest path DFL benchmark, and an energy pricing case study based on real data\. Across these experiments, we compare against prediction\-only federated learning and DFL frameworks under varying heterogeneity and communication budgets\. Together, our results suggest that smoothing is a useful ingredient for stable collaborative decision learning and provide a heterogeneity\-aware foundation for federated DFL111The code for the paper is available at[https://github\.com/kziliask/RSPO\_Plus](https://github.com/kziliask/RSPO_Plus)\.\.

## 1Introduction

Decision\-focused learning \(DFL\) has emerged as a common approach to tackle data\-driven contextual optimization\[[1](https://arxiv.org/html/2610.09091#bib.bib1),[2](https://arxiv.org/html/2610.09091#bib.bib23),[3](https://arxiv.org/html/2610.09091#bib.bib22),[4](https://arxiv.org/html/2610.09091#bib.bib14)\]\. Among DFL methods, surrogate loss functions allow practitioners to take advantage of deep learning innovations in operational contexts, with the Smart Predict\-then\-Optimize \(SPO\) framework\[[5](https://arxiv.org/html/2610.09091#bib.bib15)\]being widely adopted\. Its convex surrogate loss, SPO\+, provides a tractable upper bound on the downstream decision loss and has seen broad use\[[6](https://arxiv.org/html/2610.09091#bib.bib12),[7](https://arxiv.org/html/2610.09091#bib.bib8),[8](https://arxiv.org/html/2610.09091#bib.bib7),[9](https://arxiv.org/html/2610.09091#bib.bib13)\]\. However, cross\-silo applications of DFL face a key constraint: collaborative analytics are often prohibited due to regulatory, privacy, and competitive considerations\.

Federated learning \(FL\), which enables collaborative model training without centralizing data, offers a natural training procedure for addressing these constraints\[[10](https://arxiv.org/html/2610.09091#bib.bib11),[11](https://arxiv.org/html/2610.09091#bib.bib10),[12](https://arxiv.org/html/2610.09091#bib.bib9)\]\. In cross\-silo FL, a global model is trained across multiple organizations without centralizing data: each client keeps its local dataset and shares only model updates \(or related statistics\), which are aggregated to update a global model\[[13](https://arxiv.org/html/2610.09091#bib.bib24)\]\. There exists an extensive FL literature on mitigating data, model, system, and task heterogeneity\[[14](https://arxiv.org/html/2610.09091#bib.bib25),[15](https://arxiv.org/html/2610.09091#bib.bib5),[16](https://arxiv.org/html/2610.09091#bib.bib16),[17](https://arxiv.org/html/2610.09091#bib.bib4),[18](https://arxiv.org/html/2610.09091#bib.bib18),[19](https://arxiv.org/html/2610.09091#bib.bib17),[20](https://arxiv.org/html/2610.09091#bib.bib6)\]\.

![Refer to caption](https://arxiv.org/html/2610.09091v1/figs/background_dffl_arxiv.png)Figure 1:DFFL setting: a central server broadcasts a global model to heterogeneous clients, each with its own objective function and feasible region\.In the DFL setting, though, when organizations collaboratively train DFL models while keeping data local, heterogeneity in client objectives and constraints creates a fundamental challenge: standard federated aggregation becomes brittle because the optimal solution map can be discontinuous across clients\. Let clientiihave covariatesx∼𝒳ix\\sim\\mathcal\{X\}\_\{i\}, a client\-specific true objective mapci:𝒳i→ℝdc\_\{i\}:\\mathcal\{X\}\_\{i\}\\to\\mathbb\{R\}^\{d\}, and a shared federated predictorg∈ℋg\\in\\mathcal\{H\}used in the downstream optimization modelminw∈Si⁡g​\(x\)T​w\\min\_\{w\\in S\_\{i\}\}g\(x\)^\{T\}w\. Here, the objective mapci​\(⋅\)c\_\{i\}\(\\cdot\)and feasible regionSiS\_\{i\}can differ across clients, which induces*objective*and*constraint*heterogeneity that direct naive implementation of a decision\-focused federated learning \(DFFL\) does not address \([Fig\.1](https://arxiv.org/html/2610.09091#S1.F1)\)\. For example, SPO\+, the most prominent DFL approach, relies on an optimal solution map that can be discontinuous when clients have different linear objectives over polyhedral feasible sets\. Near boundaries where the optimal basis changes, arbitrarily small perturbations can switch the optimizer to a different extreme point, producing aO⁡\(diam⁡\(Si\)\)O\(\\mathrm\{diam\}\(S\_\{i\}\)\)change in the optimal decision\.

We propose RSPO\+, a smoothed surrogate that replaces the discontinuous optimal solution map with a projection\-based decision, and FedRSPO\+, a federated training framework built on RSPO\+ with per\-client calibration and co\-annealed scheduling\. Our contributions are summarized as follows:

- •We introduceRSPO\+andFedRSPO\+\. RSPO\+ is a regularized surrogate loss that smooths the decision map while retaining a provable connection to the original downstream regret\. FedRSPO\+ builds on RSPO\+ to provide a modular federated algorithm compatible with standard FL personalization and aggregation methods\[[21](https://arxiv.org/html/2610.09091#bib.bib31),[22](https://arxiv.org/html/2610.09091#bib.bib32),[14](https://arxiv.org/html/2610.09091#bib.bib25),[16](https://arxiv.org/html/2610.09091#bib.bib16)\]\.
- •We formally show why naive federation of SPO\+ can fail and how FedRSPO\+ avoids the same issues, developing point\-wise cross\-client loss bounds that degrade gracefully with heterogeneity, vanish at homogeneity, and require no strong convexity\.
- •We evaluate the approach on three tasks\[[23](https://arxiv.org/html/2610.09091#bib.bib20),[24](https://arxiv.org/html/2610.09091#bib.bib35),[25](https://arxiv.org/html/2610.09091#bib.bib36)\]: synthetic knapsack, shortest path, and PJM energy pricing\. We compare against FedAvg, naive SPO\+ federation and perturbation methods across communication budgets and controlled heterogeneity, demonstrating strong improvements in convex resource\-allocation settings and competitive performance under structural mismatch\.

## 2Related Work

#### Decision\-Focused Learning and Contextual Optimization

Decision\-focused federated learning is largely unexplored\. The closest prior work is\[[26](https://arxiv.org/html/2610.09091#bib.bib67)\], which analyzed federating SPO\+ under client\-level objective and feasible\-set heterogeneity\. Their results showed that naive SPO\+\-based federation can be brittle in heterogeneous polyhedral problems: optimizer instability induces point\-wise heterogeneity bounds with an irreducible diameter term, while stronger geometric assumptions such as strong convexity yield tighter stability guarantees\. They also proposed Interp\-DFFL, which interpolates between local and federated models using client validation data, which is complementary to our method, and we compare against it in[Section4](https://arxiv.org/html/2610.09091#S4)\. We build on this foundation by introducing a smoothed surrogate, RSPO\+, and a modular federated training procedure that replaces discontinuous optimizer jumps with stable projection\-based decisions, thereby removing the diameter penalty and yielding bounds that vanish at homogeneity\.

More broadly, DFL is a sub\-topic of the broader area of contextual optimization\[[2](https://arxiv.org/html/2610.09091#bib.bib23),[3](https://arxiv.org/html/2610.09091#bib.bib22)\]\. SPO\+ is among the most widely used surrogates in this literature, with strong theoretical support, including generalization bounds\[[27](https://arxiv.org/html/2610.09091#bib.bib19)\]and calibration guarantees\[[28](https://arxiv.org/html/2610.09091#bib.bib21),[23](https://arxiv.org/html/2610.09091#bib.bib20)\]\. Other decision\-aware surrogates include perturbation\-based gradient losses\[[29](https://arxiv.org/html/2610.09091#bib.bib41)\]and Fenchel–Young losses\[[30](https://arxiv.org/html/2610.09091#bib.bib37)\], while black\-box and implicit differentiation methods differentiate through the solver directly rather than introducing an explicit surrogate\[[24](https://arxiv.org/html/2610.09091#bib.bib35),[31](https://arxiv.org/html/2610.09091#bib.bib38),[32](https://arxiv.org/html/2610.09091#bib.bib40),[33](https://arxiv.org/html/2610.09091#bib.bib39)\]\. Recent work has also begun to study DFL across multiple downstream tasks and changing objectives, suggesting a growing interest in DFL beyond the single\-problem setting, though federated DFL still remains underexplored\[[34](https://arxiv.org/html/2610.09091#bib.bib66),[35](https://arxiv.org/html/2610.09091#bib.bib64),[36](https://arxiv.org/html/2610.09091#bib.bib65)\]\.

#### Federated Learning under Heterogeneity

FL, in turn, includes extensive literature on heterogeneity, including proximal penalties\[[14](https://arxiv.org/html/2610.09091#bib.bib25)\], control variates\[[37](https://arxiv.org/html/2610.09091#bib.bib26)\], localized or partially personalized models\[[38](https://arxiv.org/html/2610.09091#bib.bib27),[39](https://arxiv.org/html/2610.09091#bib.bib2),[40](https://arxiv.org/html/2610.09091#bib.bib28),[41](https://arxiv.org/html/2610.09091#bib.bib3),[42](https://arxiv.org/html/2610.09091#bib.bib29),[43](https://arxiv.org/html/2610.09091#bib.bib30)\], and regularized bilevel formulations\[[21](https://arxiv.org/html/2610.09091#bib.bib31)\]\. However, these methods are designed primarily for heterogeneity in data distributions, model heterogeneity, or predictive objectives, not for heterogeneity in downstream optimization structure\. This distinction is especially important in DFL, where small perturbations can induce large changes in the optimal decision\. Notably,\[[21](https://arxiv.org/html/2610.09091#bib.bib31)\]use Moreau envelopes as a personalization regularizer, whereas we use Moreau smoothing on the decision\-focused surrogate itself to stabilize the decision map\. More generally, Bregman divergences are widely used in machine learning and optimization as geometry\-aware discrepancy measures, supporting methods such as mirror descent, clustering, and structured prediction\[[44](https://arxiv.org/html/2610.09091#bib.bib61),[45](https://arxiv.org/html/2610.09091#bib.bib62),[46](https://arxiv.org/html/2610.09091#bib.bib60)\]\. We leverage this to construct an adaptive surrogate aligned with each feasible set\. Taken together, prior work addresses either decision\-aware learning in centralized settings or heterogeneity\-aware federated learning for predictive objectives, but not federated DFL with heterogeneous downstream feasible sets\.

## 3Method

Figure 2:FedRSPO\+: A heterogeneity\-aware federated decision\-focused learning pipeline\.The server broadcasts a global model, clients perform personalized calibration, local prediction, and projection\-based smoothing \(RSPO\+\), and updates are aggregated\. During personalization, co\-annealed regularization enables stable training under heterogeneous objectives and feasible sets\.Table 1:Overview of the four stages of FedRSPO\+, along with the frequency and setting of each stage\. We elaborate on each component throughout[Section3](https://arxiv.org/html/2610.09091#S3)\.ComponentExecutionOverviewCommunication\(1\) Calibrationclient, oncesetλj,ηj\\lambda\_\{j\},\\eta\_\{j\}\(Eq\. \([11](https://arxiv.org/html/2610.09091#S3.E11)\)\)–\(2\) Federated trainingserver,RfedR\_\{\\mathrm\{fed\}\}roundsSGD onFj​\(θ,λj\)F\_\{j\}\(\\theta;\\lambda\_\{j\}\), FedAvg \(∝nj\\propto n\_\{j\}\)model weightsθ\\theta\(3\) Personalizationclient,RperR\_\{\\mathrm\{per\}\}roundsSGD onFj​\(θ,λj,t\)F\_\{j\}\(\\theta;\\lambda\_\{j,t\}\), co\-annealλ,η\\lambda,\\eta–\(4\) Freezingclientstop annealing whenU~j,t≤τ\\tilde\{U\}\_\{j,t\}\\leq\\tau\(Eq\. \([6](https://arxiv.org/html/2610.09091#S3.E6)\)\)–[Fig\.2](https://arxiv.org/html/2610.09091#S3.F2)and[Table1](https://arxiv.org/html/2610.09091#S3.T1)provide an overview of the four components of the FedRSPO\+ pipeline\. We begin by defining the notation used throughout this paper, the tools we use to derive our methods, and by stating the assumptions for the client feasible sets and ground truth objectives, which are typical in DFL literature\[[47](https://arxiv.org/html/2610.09091#bib.bib46),[48](https://arxiv.org/html/2610.09091#bib.bib45),[5](https://arxiv.org/html/2610.09091#bib.bib15),[33](https://arxiv.org/html/2610.09091#bib.bib39)\]\. All proofs are given in the Appendix\.

We use∥⋅∥=∥⋅∥2\\\|\\cdot\\\|=\\\|\\cdot\\\|\_\{2\}throughout this study\. Each feasible setSShas a diameter and radius,D=supw,w′∈S‖w−w′‖,R=supw∈S‖w‖\.\\ D=\\sup\_\{w,w^\{\\prime\}\\in S\}\\\|w\-w^\{\\prime\}\\\|,\\ R=\\sup\_\{w\\in S\}\\\|w\\\|\.The downstream optimization problem is defined aszLP=minw∈S⁡cT​wz\_\{\\mathrm\{LP\}\}=\\min\_\{w\\in S\}c^\{T\}w, and the quadratically regularized linear program \(QLP\) aszQLP=minw∈S⁡cT​w\+λ2​‖w‖2\.z\_\{\\mathrm\{QLP\}\}=\\min\_\{w\\in S\}c^\{T\}w\+\\frac\{\\lambda\}\{2\}\\\|w\\\|^\{2\}\.The support functionξ\\xiof feasible setSSis defined asξS​\(u\)=supv∈S⟨u,v⟩\.\\xi\_\{S\}\(u\)=\\sup\_\{v\\in S\}\\langle u,v\\rangle\.Recall the generalized Bregman divergence is defined asDϕs​\(x,y\)=ϕ⁡\(x\)−ϕ⁡\(y\)−sT​\(x−y\)D^\{s\}\_\{\\phi\}\(x,y\)=\\phi\(x\)\-\\phi\(y\)\-s^\{T\}\(x\-y\), wheres∈∂ϕ⁡\(y\)s\\in\\partial\\phi\(y\)\. The Moreau envelope of a function with smoothing parameterλ\\lambdais defined asMλ​f​\(v\)=infu∈ℝd\{f⁡\(u\)\+12​λ​‖u−v‖2\}\.M\_\{\\lambda f\}\(v\)=\\inf\_\{u\\in\\mathbb\{R\}^\{d\}\}\\left\\\{f\(u\)\+\\frac\{1\}\{2\\lambda\}\\\|u\-v\\\|^\{2\}\\right\\\}\.The SPO\+ loss function\[[5](https://arxiv.org/html/2610.09091#bib.bib15)\]is defined asℓSPO\+​\(c,c^\)=ξS​\(c−2​c^\)−2​w⋆​\(c\)​\(c−c^\)\+z⋆​\(c\),\\ell\_\{\\mathrm\{SPO\+\}\}\(c,\\hat\{c\}\)=\\xi\_\{S\}\(c\-2\\hat\{c\}\)\-2w^\{\\star\}\(c\)\(c\-\\hat\{c\}\)\+z^\{\\star\}\(c\),whereSSis the feasible set of the downstream optimization problem,c^=g⁡\(x\)\\hat\{c\}=g\(x\)is the model output,ccthe ground truth cost vector,w⋆=arg​minw∈S⁡cT​ww^\{\\star\}=\\argmin\_\{w\\in S\}c^\{T\}wandz⋆z^\{\\star\}the optimal objective function value\.

###### Assumption 1\(Bounded, non\-empty feasible sets\)\.

The feasible sets of interest are nonempty and compact, soξS​\(⋅\)\\xi\_\{S\}\(\\cdot\)is finite and maxima/minima in the sets are attained\.

###### Assumption 2\(Convex feasible sets\)\.

For each instance, the feasible sets of interestS⊆ℝdS\\subseteq\\mathbb\{R\}^\{d\}are convex\.

###### Assumption 3\(Bounded, non\-zero objective\)\.

The objective function vector’s norm0<Cmin≤‖c‖≤Cmax0<C\_\{\\min\}\\leq\\\|c\\\|\\leq C\_\{\\max\}is bounded above and strictly bounded away from zero, while the prediction vector’s norm‖c^‖≤B\\\|\\hat\{c\}\\\|\\leq Bis also bounded above\.

### 3\.1RSPO\+

###### Proposition 1\(SPO\+ as a Bregman divergence\)\.

The SPO\+ loss admits the representationℓSPO\+​\(c,c^\)=Dξs​\(c−2​c^,−c\),where​s=w⋆​\(c\)∈∂ξS​\(−c\)\\ell\_\{\\mathrm\{SPO\+\}\}\(c,\\hat\{c\}\)=D^\{s\}\_\{\\xi\}\(c\-2\\hat\{c\},\-c\),\\text\{where\}\\ s=w^\{\\star\}\(c\)\\in\\partial\\xi\_\{S\}\(\-c\)\[[26](https://arxiv.org/html/2610.09091#bib.bib67)\]\.

Motivated by Proposition[1](https://arxiv.org/html/2610.09091#Thmproposition1), we define a smoothed variant by replacingξS\\xi\_\{S\}with itsλ\\lambda\-Moreau envelope,Mλ​ξS​\(u\)=12​λ​‖u‖2−λ2​dS2​\(uλ\)M\_\{\\lambda\\xi\_\{S\}\}\(u\)=\\tfrac\{1\}\{2\\lambda\}\\\|u\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\,d\_\{S\}^\{2\}\\\!\\Big\(\\tfrac\{u\}\{\\lambda\}\\Big\), wheredS2​\(u\)=infs∈S‖u−s‖2d\_\{S\}^\{2\}\(u\)=\\inf\_\{s\\in S\}\\\|u\-s\\\|^\{2\}\. We call the resulting lossRSPO\+:

ℓRSPO\+​\(c,c^,λ\):=DMλ​ξS​\(c−2​c^,−c\)\.\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\):=D\_\{M\_\{\\lambda\\xi\_\{S\}\}\}\(c\-2\\hat\{c\},\-c\)\.
###### Proposition 2\(Closed form of RSPO\+\)\.

Letu=\(c−2​c^\)/λu=\(c\-2\\hat\{c\}\)/\\lambda,v=−c/λv=\-c/\\lambda, andΠS\\Pi\_\{S\}denote projection ontoSS\. Then

ℓRSPO\+​\(c,c^,λ\)=λ2​\(‖u−ΠS​\(v\)‖2−‖u−ΠS​\(u\)‖2\)\.\\boxed\{\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=\\tfrac\{\\lambda\}\{2\}\\Big\(\\\|u\-\\Pi\_\{S\}\(v\)\\\|^\{2\}\-\\\|u\-\\Pi\_\{S\}\(u\)\\\|^\{2\}\\Big\)\.\}

###### Corollary 1\(Gradient of Smoothed Loss\)\.

From[Proposition2](https://arxiv.org/html/2610.09091#Thmproposition2), the gradient of RSPO\+ is

∇c^ℓRSPO\+​\(c,c^,λ\)=−2​\(ΠS​\(u\)−ΠS​\(v\)\)\.\\boxed\{\\nabla\_\{\\hat\{c\}\}\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=\-2\\Big\(\\Pi\_\{S\}\(u\)\-\\Pi\_\{S\}\(v\)\\Big\)\.\}

###### Proposition 3\(Characteristics of RSPO\+\)\.

DefineℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}from[Proposition2](https://arxiv.org/html/2610.09091#Thmproposition2)\. It has the following characteristics:

- •ℓRSPO\+≥0\\ell\_\{\\mathrm\{RSPO\+\}\}\\geq 0as a divergence\.
- •ℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}is globally convex with respect toc^\\hat\{c\}\.
- •ℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}is 2D\-Lipschitz continuous under compactness ofSS\.
- •ℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}is4/λ−4/\\lambda\-smooth, and thereforeλ/4\\lambda/4\-cocoercive\.

###### Theorem 1\.

Definef⁡\(c,w\)=cT​w\+λ/2​‖w‖2f\(c,w\)=c^\{T\}w\+\\lambda/2\\\|w\\\|^\{2\},w⋆=ΠS\(−c/λ\)w^\{\\star\}=\\Pi\_\{S\}\(\-c/\\lambda\)the optimal action, andw^=ΠS\(−c^/λ\)\\hat\{w\}=\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\)the predicted action\. The true regularized regret isR⁡\(c,w^\)=f⁡\(c,w^\)−f⁡\(c,w⋆\),R\(c,\\hat\{w\}\)=f\(c,\\hat\{w\}\)\-f\(c,w^\{\\star\}\),therefore,

ℓRSPO\+≥R⁡\(c,w^\)\+λ​‖w^−w⋆‖2\.\\ell\_\{\\mathrm\{RSPO\+\}\}\\geq R\(c,\\hat\{w\}\)\+\\lambda\\\|\\hat\{w\}\-w^\{\\star\}\\\|^\{2\}\.\(1\)Furthermore, becauseR⁡\(c,w^\)≥0R\(c,\\hat\{w\}\)\\geq 0, we also have,

‖w^−w⋆‖≤ℓRSPO\+λ\.\\\|\\hat\{w\}\-w^\{\\star\}\\\|\\leq\\sqrt\{\\frac\{\\ell\_\{\\mathrm\{RSPO\+\}\}\}\{\\lambda\}\}\.\(2\)

###### Corollary 2\(Unregularized Regret Bound for RSPO\+\)\.

Letz⋆=minw∈S⁡cT​wz^\{\\star\}=\\min\_\{w\\in S\}c^\{T\}wbe the unregularized objective, andwLP⋆w^\{\\star\}\_\{\\mathrm\{LP\}\}be the minimizer\. For any predictionc^\\hat\{c\}and regularization parameterλ\\lambda, the true unregularized regretRunreg​\(c,w^\)=cT​w^−z⋆R\_\{\\mathrm\{unreg\}\}\(c,\\hat\{w\}\)=c^\{T\}\\hat\{w\}\-z^\{\\star\}of the predicted decisionw^=ΠS\(−c^/λ\)\\hat\{w\}=\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\)satisfies:

Runreg​\(c,w^\)≤ℓRSPO\+​\(c,c^,λ\)\+λ2​R2\.R\_\{\\mathrm\{unreg\}\}\(c,\\hat\{w\}\)\\leq\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)\+\\frac\{\\lambda\}\{2\}R^\{2\}\.\(3\)

### 3\.2FedRSPO\+

[Section3\.1](https://arxiv.org/html/2610.09091#S3.SS1)established bounds on the*regularized*regret and decision error, with[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2)connecting the RSPO\+ loss to the unregularized regret of the regularized decisionw^λ=ΠS\(−c^/λ\)\.\\hat\{w\}\_\{\\lambda\}=\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\)\.In our experiments, however, evaluation is performed using the original downstream decision rule,w^LP​\(c^\)∈arg⁡minw∈S​c^⊤​w,\\hat\{w\}\_\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\in\\arg\\min\_\{w\\in S\}\\hat\{c\}^\{\\top\}w,rather than the regularized decision\. Thus,[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2)should be interpreted as a direct guarantee for the regularized decision and as a surrogate\-level guarantee for the evaluated LP decision\. For polyhedral feasible sets, this distinction disappears onceλ\\lambdais sufficiently small under a problem\-dependent exact\-regularization threshold: the quadratically regularized solution is then also LP\-optimal, with the regularizer selecting a particular LP optimum\[[49](https://arxiv.org/html/2610.09091#bib.bib33),[50](https://arxiv.org/html/2610.09091#bib.bib34)\]\. Under this condition, the RSPO\+ bound also applies to the LP decision used at evaluation\. In practice, our freeze rule in[Eq\.6](https://arxiv.org/html/2610.09091#S3.E6)stops annealing beforeλ\\lambdareaches this threshold, and we quantify this residual gap in[SectionC\.6](https://arxiv.org/html/2610.09091#A3.SS6)in the Appendix\.

More generally, asλ→0\\lambda\\rightarrow 0, the bias termλ2​R2\\frac\{\\lambda\}\{2\}R^\{2\}in[Eq\.3](https://arxiv.org/html/2610.09091#S3.E3)vanishes, motivating an annealing schedule that gradually transitions from a smooth, stable surrogate toward the original downstream LP objective\. In this section, we build on this connection by introducing a regularization parameter scheduler, which we refer to as an annealing schedule\. Finally, we derive bounds on cross\-client loss discrepancy under optimization heterogeneity and compare them with bounds on SPO\+ loss to show why FedRSPO\+ can mitigate discontinuities from vertex jumping\.

#### Annealing Schedule\.

Corollary 2 shows that RSPO\+ controls the original unregularized regret up to anO⁡\(λ​R2\)O\(\\lambda R^\{2\}\)bias\. Thus,λ\\lambdainduces a bias\-stability tradeoff: largerλ\\lambdasmooths the decision map and stabilizes optimization, while smallerλ\\lambdabetter aligns the surrogate with the original LP objective\. This motivates annealingλ\\lambdaduring local training\. Specifically, we gradually decreaseλ\\lambdaso that the regularized decision tracks the original downstream optimum \(i\.e\., as a homotopy continuation from the regularized to the non\-regularized objective\)\. We justify this schedule by analyzing how the RSPO\+ loss and gradient change under perturbations ofλ\\lambdain[Proposition4](https://arxiv.org/html/2610.09091#Thmproposition4)\.

###### Proposition 4\(RSPO\+ Loss and Gradient under perturbation ofλ\\lambda\)\.

Forλ1\>λ2\>0\\lambda\_\{1\}\>\\lambda\_\{2\}\>0regularization parameters of RSPO\+ loss, we can bound the difference of loss under parameter shiftλ1→λ2\\lambda\_\{1\}\\rightarrow\\lambda\_\{2\}as,

\|ℓRSPO\+​\(λ1\)−ℓRSPO\+​\(λ2\)\|≤R2​\|λ1−λ2\|\+2​‖c‖​‖c−c^‖​\|1λ1−1λ2\|,\|\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{1\}\)\-\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{2\}\)\|\\leq R^\{2\}\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\+2\\\|c\\\|\\\|c\-\\hat\{c\}\\\|\\left\|\\frac\{1\}\{\\lambda\_\{1\}\}\-\\frac\{1\}\{\\lambda\_\{2\}\}\\right\|,\(4\)and difference in RSPO\+ gradients as,

‖∇ℓRSPO\+​\(λ1\)−∇ℓRSPO\+​\(λ2\)‖≤2​\(‖c−2​c^‖\+‖c‖\)​\|1λ1−1λ2\|\.\\\|\\nabla\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{1\}\)\-\\nabla\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{2\}\)\\\|\\leq 2\\Big\(\\\|c\-2\\hat\{c\}\\\|\+\\\|c\\\|\\Big\)\\left\|\\frac\{1\}\{\\lambda\_\{1\}\}\-\\frac\{1\}\{\\lambda\_\{2\}\}\\right\|\.\(5\)

Stopping condition\.As reducingλ\\lambdaweakens the smoothness properties of RSPO\+ in[Proposition3](https://arxiv.org/html/2610.09091#Thmproposition3), we devise a stopping condition for theλ\\lambda\-schedule in FedRSPO\+ that depends on the regularization biasciT​wi,t⋆,reg−zi⋆c\_\{i\}^\{T\}w\_\{i,t\}^\{\\star,\\mathrm\{reg\}\}\-z\_\{i\}^\{\\star\}\.

For a subset of the client dataset𝒱c\\mathcal\{V\}\_\{c\}, at local stepttwithλt\\lambda\_\{t\}, definew^i,t=Π𝒮i\(−c^i,t/λt\)\\hat\{w\}\_\{i,t\}=\\Pi\_\{\\mathcal\{S\}\_\{i\}\}\(\-\\hat\{c\}\_\{i,t\}/\\lambda\_\{t\}\)andwi,t⋆,reg=Π𝒮i\(−ci/λt\)w\_\{i,t\}^\{\\star,\\mathrm\{reg\}\}=\\Pi\_\{\\mathcal\{S\}\_\{i\}\}\(\-c\_\{i\}/\\lambda\_\{t\}\)\. Letzi⋆=minw∈𝒮i⁡ciT​wz\_\{i\}^\{\\star\}=\\min\_\{w\\in\\mathcal\{S\}\_\{i\}\}c\_\{i\}^\{T\}w\(pre\-solved once on𝒱c\\mathcal\{V\}\_\{c\}\)\. We stop annealing when the normalized average regularization bias over𝒱c\\mathcal\{V\}\_\{c\}falls below a toleranceτ\\tau:

U~t=1\|𝒱c\|​∑i∈𝒱c\(ciT​wi,t⋆,reg−zi⋆\)1\|𝒱c\|​∑i∈𝒱c\|zi⋆\|\+ε≤τ\.\\tilde\{U\}\_\{t\}=\\frac\{\\frac\{1\}\{\|\\mathcal\{V\}\_\{c\}\|\}\\sum\_\{i\\in\\mathcal\{V\}\_\{c\}\}\\left\(c\_\{i\}^\{T\}w\_\{i,t\}^\{\\star,\\mathrm\{reg\}\}\-z\_\{i\}^\{\\star\}\\right\)\}\{\\frac\{1\}\{\|\\mathcal\{V\}\_\{c\}\|\}\\sum\_\{i\\in\\mathcal\{V\}\_\{c\}\}\|z\_\{i\}^\{\\star\}\|\+\\varepsilon\}\\leq\\tau\.\(6\)Intuitively, this criterion freezesλt\\lambda\_\{t\}once the surrogate regularized target is sufficiently aligned with the true downstream LP objective\. Thus,U~t\\tilde\{U\}\_\{t\}empirically tracks the QP–LP regularization bias discussed above and gives a data\-dependent stopping rule\. The full FedRSPO\+ algorithm is provided in pseudo\-code in[Algorithm1](https://arxiv.org/html/2610.09091#alg1)in the Appendix\.

###### Theorem 2\(FedRSPO\+ Convergence Rate with linear predictors\)\.

Under standard regularity assumptions, the weighted average iterateΘ^T\\hat\{\\Theta\}\_\{T\}of FedRSPO\+ satisfies

𝔼⁡\[FT​\(Θ^T\)−FT​\(Θ⋆\)\]=O⁡\(1∑t=0T−1ηt\+∑t=0T−1ηt2∑t=0T−1ηt\+∑t=0T−1ηt​Δt∑t=0T−1ηt\),\\mathbb\{E\}\\\!\\left\[F\_\{T\}\(\\hat\{\\Theta\}\_\{T\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\right\]=O\\\!\\left\(\\frac\{1\}\{\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\}\+\\frac\{\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}^\{2\}\}\{\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\}\+\\frac\{\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\\Delta\_\{t\}\}\{\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\}\\right\),with anE2E^\{2\}\-dependent constant from local updates\. For a fixed\-λ\\lambdaobjective andηt≍1/t\\eta\_\{t\}\\asymp 1/t, the bound reduces toO⁡\(1/log⁡T\)O\(1/\\log T\)\. For the actual annealed objective, the additional tracking term appears, which vanishes after freezing and the bound becomes the standard constant\-stepsize formO⁡\(1/\(T​ηf\)\+ηf\)O\(1/\(T\\eta\_\{f\}\)\+\\eta\_\{f\}\)\. The formal statement and proof are in the Appendix\.

We now characterize how the point\-wise RSPO\+ loss degrades under both objective shift and feasible set shift\. This parallels the SPO\+ heterogeneity analysis of\[[26](https://arxiv.org/html/2610.09091#bib.bib67)\], but has improved stability due to the smoothness of the RSPO\+\. We measure objective heterogeneity bycd=‖ci−cj‖c\_\{d\}=\\\|c\_\{\\mathrm\{i\}\}\-c\_\{\\mathrm\{j\}\}\\\|and feasible\-set heterogeneity by the Hausdorff distanceδ=dH​\(Si,Sj\)\\delta=d\_\{H\}\(S\_\{\\mathrm\{i\}\},S\_\{\\mathrm\{j\}\}\)\. Unlike SPO\+, the origin\-centered quadratic penalty in the Moreau envelope breaks translation invariance, so we use Hausdorff distance directly rather than the shape distanceδN\\delta\_\{N\}\. LetRmax=max⁡\{Ri,Rj\}R\_\{\\max\}=\\max\\\{R\_\{\\mathrm\{i\}\},R\_\{\\mathrm\{j\}\}\\\}denote the joint bounding radius\.

Table 2:Heterogeneity bound comparison\.cd=‖cref−ccl‖c\_\{d\}=\\\|c\_\{\\mathrm\{ref\}\}\-c\_\{\\mathrm\{cl\}\}\\\|is the objective shift,δ\\deltais the Hausdorff set distance,δN\\delta\_\{N\}is the translation\-invariant Hausdorff set distance, andDminD\_\{\\min\}is the minimum set diameter\.Q1: Rate?;Q2: Vanishes at homogeneity?;Q3: Requires strong convexity?SPO\+RSPO\+ \(Ours\)Q1O⁡\(cd\+δN\+Dmin\)O\\\!\\big\(c\_\{d\}\+\\delta\_\{N\}\+D\_\{\\min\}\\big\)O⁡\(cd/λ\+δ/λ\+δ\)O\\\!\\big\(c\_\{d\}/\\lambda\+\\sqrt\{\\delta/\\lambda\}\+\\delta\\big\)Q2No \(DminD\_\{\\min\}persists\)YesQ3Yes, for basic stabilityNoThe key difference from SPO\+ is in how decisions respond to perturbations\. In SPO\+, the optimal decisionw⋆​\(c\)∈arg⁡minw∈S​cT​ww^\{\\star\}\(c\)\\in\\arg\\min\_\{w\\in S\}\\,c^\{T\}wselects a vertex ofSS\. Under infinitesimal perturbations, this vertex can jump discontinuously across the entire feasible set, introducing a diameter penaltyDminD\_\{\\min\}that does not vanish even as heterogeneity tends to zero\. In RSPO\+, the optimal decision is the Euclidean projectionvλ,S⋆\(c\)=ΠS\(−c/λ\)v^\{\\star\}\_\{\\lambda,S\}\(c\)=\\Pi\_\{S\}\(\-c/\\lambda\), which is stable under small changes to the feasible set \(measured in Hausdorff distance\)\. For any compact convexS1,S2S\_\{1\},S\_\{2\}withdH​\(S1,S2\)≤δd\_\{H\}\(S\_\{1\},S\_\{2\}\)\\leq\\delta:

‖ΠS1​\(p\)−ΠS2​\(p\)‖≤2​δ​\(‖p‖\+Rmax\)\.\\big\\\|\\Pi\_\{S\_\{1\}\}\(p\)\-\\Pi\_\{S\_\{2\}\}\(p\)\\big\\\|\\;\\leq\\;\\sqrt\{2\\delta\\big\(\\\|p\\\|\+R\_\{\\max\}\\big\)\}\.\(7\)Thisδ\\sqrt\{\\delta\}stability holds for*all*convex sets, including polyhedrals, without any strong convexity assumption\.

###### Example 1\(Why SPO\+ struggles with polyhedral clients\)\.

Consider two clients with the same polyhedral feasible setSSand cost vectorsc\(i\),c\(j\)c^\{\(i\)\},c^\{\(j\)\}, differing byϵ\\epsilon\. For SPO\+, the optimizer can switch between extreme points ofSS, so the decision gap between clients remainsΩ⁡\(D⁡\(S\)\)\\Omega\(D\(S\)\)even as‖c\(i\)−c\(j\)‖→0\\\|c^\{\(i\)\}\-c^\{\(j\)\}\\\|\\rightarrow 0\. By contrast, RSPO\+ uses projected decisionsvλ​S⋆\(c\)=ΠS\(−c/λ\),v^\{\\star\}\_\{\\lambda S\}\(c\)=\\Pi\_\{S\}\(\-c/\\lambda\),and since projection is non\-expansive,

‖vλ​S⋆​\(c\(i\)\)−vλ​S⋆​\(c\(j\)\)‖≤1λ​‖c\(i\)−c\(j\)‖\.\\\|v^\{\\star\}\_\{\\lambda S\}\(c^\{\(i\)\}\)\-v^\{\\star\}\_\{\\lambda S\}\(c^\{\(j\)\}\)\\\|\\leq\\frac\{1\}\{\\lambda\}\\\|c^\{\(i\)\}\-c^\{\(j\)\}\\\|\.Thus, RSPO\+ removes the discontinuous vertex\-switching in polyhedral instances\.

###### Theorem 3\(RSPO\+ loss discrepancy under heterogeneity\)\.

Let\(ci,Si\)\(c\_\{\\mathrm\{i\}\},S\_\{\\mathrm\{i\}\}\)and\(cj,Sj\)\(c\_\{\\mathrm\{j\}\},S\_\{\\mathrm\{j\}\}\)satisfy our assumptions, withcd=‖ci−cj‖c\_\{d\}=\\\|c\_\{\\mathrm\{i\}\}\-c\_\{\\mathrm\{j\}\}\\\|,cmin=min⁡\{‖ci‖,‖cj‖\}c\_\{\\min\}=\\min\\\{\\\|c\_\{i\}\\\|,\\\|c\_\{j\}\\\|\\\}, andδ=dH​\(Si,Sj\)\\delta=d\_\{H\}\(S\_\{\\mathrm\{i\}\},S\_\{\\mathrm\{j\}\}\)\. Then for any predictionc^\\hat\{c\}:

\|ℓSi​\(c^,ci\)−ℓSj​\(c^,cj\)\|≤T⁡\(c^\)\+B⁡\(c^\),\\big\|\\ell\_\{S\_\{\\mathrm\{i\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{i\}\}\)\-\\ell\_\{S\_\{\\mathrm\{j\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{j\}\}\)\\big\|\\;\\leq\\;T\(\\hat\{c\}\)\+B\(\\hat\{c\}\),\(8\)where theenvelope shiftT⁡\(c^\)T\(\\hat\{c\}\)anddecision shiftB⁡\(c^\)B\(\\hat\{c\}\)are defined as follows\.

The envelope shift captures how the Moreau envelope values change due to combined objective and set perturbation:

T⁡\(c^\)=Di​cd\+δ⁡\(‖cj−2​c^‖\+‖cj‖\+2​λ​Rmax\),T\(\\hat\{c\}\)=D\_\{i\}c\_\{d\}\+\\delta\\big\(\\\|c\_\{\\mathrm\{j\}\}\-2\\hat\{c\}\\\|\+\\\|c\_\{\\mathrm\{j\}\}\\\|\+2\\lambda R\_\{\\max\}\\big\),\(9\)
and the decision shift captures how the projection\-based decisions drift, amplified by the prediction:

B⁡\(c^\)=\(2​‖c^‖\+2​λ​Rmax\)​cdλ\+2​δ​\(cminλ\+Rmax\)\.B\(\\hat\{c\}\)=\\big\(2\\\|\\hat\{c\}\\\|\+2\\lambda R\_\{\\max\}\\big\)\\frac\{c\_\{d\}\}\{\\lambda\}\+\\sqrt\{2\\delta\\\!\\bigg\(\\frac\{c\_\{\\min\}\}\{\\lambda\}\+R\_\{\\max\}\\bigg\)\}\.\(10\)

Combining the regularization bias in[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2)with the explicitλ\\lambda\-dependence in[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3)gives the following informal bias–stability tradeoff:

error⁡\(λ\)≲λ​R2\+cdλ\+δλ,\\mathrm\{error\}\(\\lambda\)\\lesssim\\lambda R^\{2\}\+\\frac\{c\_\{d\}\}\{\\lambda\}\+\\sqrt\{\\frac\{\\delta\}\{\\lambda\}\},up to constants depending on the bounded costs, predictions, and feasible\-set radii\. The first term is theO⁡\(λ​R2\)O\(\\lambda R^\{2\}\)regularization bias, while the latter two terms come from the objective and feasible\-set heterogeneity terms in[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3)\. Thus, decreasingλ\\lambdaimproves alignment with the original LP objective but worsens stability under client heterogeneity\. This motivates annealing only until the regularized objective is sufficiently aligned with the original downstream objective, rather than reducingλ\\lambdato zero and replacing RSPO\+ with SPO\+\. We formalize this relationship in the Appendix\.

#### Calibration ofλ\\lambdaandη\\eta\.

From the RSPO\+ loss structure,λ\\lambdascales the cost vectors relative to the feasible set: too large and the loss reduces to MSE, too small and it reverts to SPO\+ \(see[Remark3](https://arxiv.org/html/2610.09091#Thmremark3)\)\. For the loss to alternate between regimes,λ\\lambdashould place the scaled vector near the feasible\-set boundary\. Combined with the4/λ4/\\lambda\-smoothness of RSPO\+ \(Proposition[3](https://arxiv.org/html/2610.09091#Thmproposition3)\), which suggests a step size of orderλ/4\\lambda/4\[[51](https://arxiv.org/html/2610.09091#bib.bib42)\], we initialize per clientjj:

λj:=‖c¯j‖Rj,ηj:=α​‖c¯j‖4​Rj,\\lambda\_\{j\}:=\\frac\{\\\|\\bar\{c\}\_\{j\}\\\|\}\{R\_\{j\}\},\\qquad\\eta\_\{j\}:=\\frac\{\\alpha\\,\\\|\\bar\{c\}\_\{j\}\\\|\}\{4\\,R\_\{j\}\},\(11\)where‖c¯j‖\\\|\\bar\{c\}\_\{j\}\\\|is the average cost\-vector magnitude in clientjj’s dataset,RjR\_\{j\}is the feasible\-set radius, andα\\alphais a small positive scalar \(e\.g\.α=0\.01\\alpha=0\.01\)\. When annealingλ\\lambda, we co\-annealη\\etawithκLR∝κλ\\kappa^\{\\mathrm\{LR\}\}\\propto\\kappa^\{\\lambda\}to maintain this step\-size relationship\. This is also validated by the convergence analysis in[SectionA\.1](https://arxiv.org/html/2610.09091#A1.SS1)\. A sensitivity analysis overκ\\kappais in Appendix[AppendixC](https://arxiv.org/html/2610.09091#A3)\.

## 4Experiment

With each experiment, we test a different aspect of FedRSPO\+\. Our synthetic fractional knapsack experiment uses controlled heterogeneity across all three axes \(feasible set, objective, and data\) to test[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3)empirically and the performance of our method against the state\-of\-the\-art\. Our PJM experiment uses real price dynamics and heterogeneous feasible sets to test our method on real datasets\. Finally, our Warcraft experiment has heterogeneous starts and destinations to vary feasible sets and datasets across clients to evaluate how FedRSPO\+ performs when the optimization task relies on the structure of the unregularized objective, and how RSPO\+’s regularization affects performance\.

For all experiments, we use a federated warm\-start followed by local personalization\. Our model is a shallow neural network with 64 hidden units for the fractional knapsack and PJM tasks, and a truncated ResNet\-18 for the Warcraft task following prior work\[[1](https://arxiv.org/html/2610.09091#bib.bib1)\]\. We report relative regret in[Table3](https://arxiv.org/html/2610.09091#S4.T3)and the optimality gap in[Table6](https://arxiv.org/html/2610.09091#S4.T6)\. While FedRSPO\+ is trained on a regularized objective, we report on the actual downstream regret\. We also compare with popular perturbation DFL methods, perturbed Fenchel\-Young Loss \(PFYL,\[[30](https://arxiv.org/html/2610.09091#bib.bib37)\]\) and Differentiable Perturbation Optimizers \(DPO,\[[52](https://arxiv.org/html/2610.09091#bib.bib63)\]\)\. We also compare with Interp\-DFFL\[[26](https://arxiv.org/html/2610.09091#bib.bib67)\]and an OptNet\-style baseline\[[48](https://arxiv.org/html/2610.09091#bib.bib45)\]\. Implementation details, hyperparameters, and solver configurations are provided in the Appendix[C](https://arxiv.org/html/2610.09091#A3), and additional centralized and local\-only baselines in Appendix[SectionC\.5](https://arxiv.org/html/2610.09091#A3.SS5)\.

Table 3:Mean client relative regret \(%\)↓\\downarrowacross heterogeneity configurations\. Each axis \(ηo\\eta\_\{\\mathrm\{o\}\}:objective,ηc,\\eta\_\{\\mathrm\{c\}\}:constraint,ηd,\\eta\_\{\\mathrm\{d\}\}:data\) is varied independently while others are fixed at 0\. “Fed”: global federated model; “\+P”: after local personalization\. Results: mean±\\pmstd over 5 seeds\.MSEPFYLDPOSPO\+RSPO\+ \(Ours\)\(ηo,ηc,ηd\)\(\\eta\_\{\\mathrm\{o\}\},\\eta\_\{\\mathrm\{c\}\},\\eta\_\{\\mathrm\{d\}\}\)Fed\+PFed\+PFed\+PFed\+PFed\+P\(0,0,0\)\(0,0,0\)15\.9±\\pm3\.310\.5±\\pm2\.88\.5±\\pm1\.56\.1±\\pm1\.116\.3±\\pm2\.712\.6±\\pm2\.29\.7±\\pm2\.15\.6±\\pm1\.42\.3±\\pm0\.41\.9±\\pm0\.3\(0\.5,0,0\)\(0\.5,0,0\)16\.0±\\pm3\.410\.6±\\pm2\.88\.8±\\pm1\.56\.3±\\pm1\.216\.7±\\pm2\.512\.8±\\pm2\.310\.0±\\pm2\.15\.8±\\pm1\.52\.5±\\pm0\.31\.9±\\pm0\.3\(1,0,0\)\(1,0,0\)16\.4±\\pm3\.410\.8±\\pm2\.99\.9±\\pm1\.66\.9±\\pm1\.317\.4±\\pm2\.513\.4±\\pm2\.210\.9±\\pm2\.26\.3±\\pm1\.63\.0±\\pm0\.32\.0±\\pm0\.3\(0,0\.5,0\)\(0,0\.5,0\)13\.2±\\pm3\.59\.8±\\pm2\.713\.3±\\pm2\.48\.1±\\pm1\.919\.6±\\pm3\.613\.1±\\pm3\.111\.9±\\pm2\.57\.3±\\pm2\.13\.9±\\pm0\.62\.3±\\pm0\.4\(0,1,0\)\(0,1,0\)13\.3±\\pm2\.310\.5±\\pm2\.015\.0±\\pm2\.39\.1±\\pm1\.520\.8±\\pm3\.314\.1±\\pm2\.412\.9±\\pm2\.08\.1±\\pm1\.54\.6±\\pm0\.52\.6±\\pm0\.4\(0,0,0\.5\)\(0,0,0\.5\)17\.5±\\pm3\.09\.6±\\pm2\.214\.0±\\pm1\.88\.2±\\pm1\.220\.6±\\pm3\.214\.4±\\pm2\.414\.1±\\pm2\.06\.9±\\pm1\.36\.5±\\pm0\.83\.8±\\pm0\.2\(0,0,1\)\(0,0,1\)19\.1±\\pm2\.88\.4±\\pm1\.718\.0±\\pm1\.98\.8±\\pm0\.722\.9±\\pm2\.914\.1±\\pm1\.817\.7±\\pm2\.16\.9±\\pm0\.89\.3±\\pm1\.23\.4±\\pm0\.2

### 4\.1Fractional Knapsack

To test FedRSPO\+ against different axes of heterogeneity in the client dataset, we implement a fractional knapsack experiment, using a synthetic dataset for feature sizep=8p=8, cost vector sized=50d=50, and 20 clients\. We test 3 different levels of heterogeneity, represented by\(ηo,ηc,ηd\)\(\\eta\_\{\\mathrm\{o\}\},\\eta\_\{\\mathrm\{c\}\},\\eta\_\{\\mathrm\{d\}\}\)in[Table3](https://arxiv.org/html/2610.09091#S4.T3): no heterogeneity, moderate heterogeneity, and extreme heterogeneity\. Overall, RSPO\+ has the lowest relative regret across the board\. The largest gap is during federation, meaning RSPO\+ succeeds in developing an effective model for all clients despite heterogeneity\. Across all methods, data heterogeneity was most detrimental to model performance, with constraint heterogeneity moderately affecting regret and objective heterogeneity minimally\. RSPO\+ was found resilient to all forms of heterogeneity compared to the baseline methods\.

We also test additional baselines\. As shown in[Table4](https://arxiv.org/html/2610.09091#S4.T4), FedRSPO\+ has lower regret than Interp\-DFFL in the homogeneous, objective\-heterogeneous, and feasible\-set\-heterogeneous settings\. Under pure data heterogeneity\(0,0,1\)\(0,0,1\), however, Interp\-DFFL is stronger \(2\.47% vs\. 3\.37%\), consistent with its explicit local–federated interpolation, which targets statistical heterogeneity\. As RSPO\+ is a loss and Interp\-DFFL an aggregation rule, they are not mutually exclusive\. The OptNet\-style baseline, which uses the same regularized QP with a different training signal, is substantially weaker \(9\.4–11\.3%\), suggesting that the gains come from the RSPO\+ surrogate, not just quadratic regularization\.

Table 4:Comparison of FedRSPO\+ against Interp\-DFFL and OptNet across fractional knapsack heterogeneity settings and PJM cases\. Values are mean±\\pmstandard deviation across training seeds of mean absolute regret\.SettingInterp\-DFFLOptNet\-styleFedRSPO\+\(0,0,0\)\(0,0,0\)2\.42±0\.302\.42\\pm 0\.309\.4±1\.99\.4\\pm 1\.91\.86±0\.25\\mathbf\{1\.86\\pm 0\.25\}\(1,0,0\)\(1,0,0\)2\.42±0\.332\.42\\pm 0\.339\.7±2\.09\.7\\pm 2\.02\.02±0\.26\\mathbf\{2\.02\\pm 0\.26\}\(0,1,0\)\(0,1,0\)3\.51±0\.773\.51\\pm 0\.7711\.3±2\.011\.3\\pm 2\.02\.58±0\.38\\mathbf\{2\.58\\pm 0\.38\}\(0,0,1\)\(0,0,1\)2\.47±0\.39\\mathbf\{2\.47\\pm 0\.39\}11\.3±1\.911\.3\\pm 1\.93\.37±0\.223\.37\\pm 0\.22PJM \($\)69\.6±1\.769\.6\\pm 1\.769\.8±1\.369\.8\\pm 1\.360\.7±0\.8\\mathbf\{60\.7\\pm 0\.8\}We also run a sweep across relevant parameters, shown in[Fig\.3](https://arxiv.org/html/2610.09091#S4.F3), to test the performance of our method against SPO\+ through different fractions of active clients per round, number of clients, and different values ofα\\alpha\. We see that the communication budget has a modest impact on the performance of FedRSPO\+, with a general improvement over SPO\+ as the fraction increases\. The parameterα\\alphawas moderately more impactful on FedRSPO\+ performance, with relative improvement ranging from 1\.1% forα=10−3\\alpha=10^\{\-3\}to 5\.4% forα=0\.1\\alpha=0\.1\. However, the most surprising result is for increasing number of clients\. There FedRSPO\+ dramatically surpasses the federated SPO\+ setup, with a difference of 12\.5% at 50 clients\. Extra sweep results forκλ,κLR,\\kappa^\{\\lambda\},\\kappa^\{\\mathrm\{LR\}\},andτ\\tauare available in Appendix[AppendixC](https://arxiv.org/html/2610.09091#A3)\. In all cases, FedRSPO\+ outperforms the SPO\+ implementation\.

0\.0010\.0050\.010\.100%22%44%66%88%1\.1%1\.1\\%2\.8%2\.8\\%3\.6%3\.6\\%5\.4%5\.4\\%Learning rate scaleα\\alphaRelative Regret

510205000%55%1010%1515%0\.9%0\.9\\%1\.4%1\.4\\%3\.6%3\.6\\%12\.5%12\.5\\%Number of clientsBestBaselineOurs

10%40%70%100%00%55%1010%5\.7%5\.7\\%5\.8%5\.8\\%6\.0%6\.0\\%6\.1%6\.1\\%Fraction of active clients

Figure 3:Relative regret \(%\) of the best baseline SPO\+ and our method RSPO\+ across different values of the learning rate scaleα\\alphafrom[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11), number of clients, and communication budgets\.Lower is better \(↓\\downarrow\)\. Numbers above the bars show the improvement in relative regret of RSPO\+ relative to SPO\+\.
### 4\.2PJM Case Study

We evaluate our approach on hourly day\-ahead electricity prices from the PJM Interconnection for the year 2025\[[25](https://arxiv.org/html/2610.09091#bib.bib36)\]\. Each client represents an aggregator or a large consumer with a fixed daily electricity demandDjD\_\{j\}that forecasts pricing in advance to inform their bids\[[53](https://arxiv.org/html/2610.09091#bib.bib69)\]\. Their goal is to minimize total procurement cost\. In effect, the downstream optimization task is a top\-kkselection problem, where the cheapestk=Djk=D\_\{j\}hours are chosen to fulfill demand\. We use weather data from the Open\-Meteo API\[[54](https://arxiv.org/html/2610.09091#bib.bib54)\]as input data\. We use public price and weather data for reproducibility, which provides naturally occurring client\-level price variation and input data heterogeneity\. Additional experimental info can be found in the Appendix[SectionC\.4](https://arxiv.org/html/2610.09091#A3.SS4)\.

[Table5](https://arxiv.org/html/2610.09091#S4.T5)presents an ablation of the FedRSPO\+ components on this task\. Starting from MSE \(175\.5 mean absolute regret\), switching to SPO\+ reduces regret to 81\.7, and RSPO\+ further improves to 72\.9\.

Table 5:Ablation study for FedRSPO\+ on PJM data\. Values are mean±\\pmstandard deviation across training seeds of mean absolute regret\. “w/ P” denotes the personalization phase,λj/ηj\\lambda\_\{j\}/\\eta\_\{j\}denotes per\-client parameters, and Full additionally includes time\-inverse scheduling\.MethodMean absolute regret↓\\downarrowMSE175\.453±7\.260175\.453\\pm 7\.260SPO\+81\.676±1\.76881\.676\\pm 1\.768RSPO\+72\.890±4\.23172\.890\\pm 4\.231RSPO\+w/ P68\.136±1\.40268\.136\\pm 1\.402RSPO\+w/ P \+λj/ηj\\lambda\_\{j\}/\\eta\_\{j\}63\.006±1\.05963\.006\\pm 1\.059RSPO\+\(Full\)60\.654±0\.796\\mathbf\{60\.654\}\\pm 0\.796RSPO\+\(Full\)\+Ditto69\.482±2\.93269\.482\\pm 2\.932RSPO\+\(Full\)\+FedProx60\.602±0\.623\\mathbf\{60\.602\}\\pm 0\.623Each additional component: personalization, per\-clientλ/η\\lambda/\\etacalibration \([Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\), and co\-annealed scheduling yields consistent gains, with the full FedRSPO\+ pipeline achieving 60\.7 \(±\\pm0\.8\), a 65% reduction over MSE and 26% over naive SPO\+ federation\. PFYL \(67\.8±1\.267\.8\\pm 1\.2regret\) and DPO \(74\.7±2\.274\.7\\pm 2\.2regret\) performed well in this case study compared to SPO\+\. This extends to the Interp\-DFFL and OptNet\-style baselines, as shown in[Table4](https://arxiv.org/html/2610.09091#S4.T4), where FedRSPO\+ outperforms both\. We also incorporate popular FL algorithms to showcase FedRSPO\+’s modularity\. FedProx achieves similar mean regret, with a modest reduction in variance\. Ditto performs similarly to RSPO\+ with personalization, which coincides with our expectations\. Ditto introduces proximal regularization towards the global model, which conflicts with RSPO\+’s gradual scheduling and its transition from the regularized to the unregularized objective\. This suggests that FL methods adding regularization at the personalization stage require careful coordination with RSPO\+’s own scheduling, which is an interesting direction for future work\. Finally, we perform additional experiments, including fixed demand across the federation and data\-scarcity regimes, and showcase the results in the Appendix\.

### 4\.3Warcraft Shortest Path

To characterize RSPO\+’s behavior under structural mismatch, we evaluate on the 12x12 Warcraft shortest\-path benchmark\[[24](https://arxiv.org/html/2610.09091#bib.bib35)\]with 10,000 training maps\. We define 44 heterogeneous clients by assigning each border cell as an origin and its 180°\-rotated opposite border cell as the destination, with additional terrain\-cost perturbations\. We use Dijkstra’s algorithm\[[55](https://arxiv.org/html/2610.09091#bib.bib53),[56](https://arxiv.org/html/2610.09091#bib.bib48)\]as the linear oracle\.

We explicitly chose shortest path because regularization alters the downstream optimization problem\. Importantly, adding the quadratic term destroys the shortest path’s total unimodularity and allows fractional flows across the grid\. RSPO\+ still achieves a 2\.571% optimality gap \([Table6](https://arxiv.org/html/2610.09091#S4.T6)\) and outperforms perturbation methods and MSE, losing out marginally to SPO\+\. Annealing narrows the gap between the regularized flow objective and the true shortest\-path objective, yielding a∼\\sim66% improvement during personalization, the largest personalization gain across all three experiments\.

Table 6:Mean optimality gap \(↓\\downarrow\) on the Warcraft Shortest Path benchmark\. Results are mean±\\pmstd over 5 seeds\.MethodFederated \(%\)\+P \(%\)DPO11\.309±2\.24411\.309\\pm 2\.24411\.018±1\.71811\.018\\pm 1\.718MSE14\.784±2\.49114\.784\\pm 2\.4917\.241±1\.3367\.241\\pm 1\.336PFYL8\.959±1\.6888\.959\\pm 1\.68811\.449±1\.04811\.449\\pm 1\.048SPO\+5\.978±1\.8425\.978\\pm 1\.8421\.850±0\.3771\.850\\pm 0\.377Ours7\.504±1\.9297\.504\\pm 1\.9292\.571±0\.9012\.571\\pm 0\.901Overall, this highlights a limitation of the proposed FedRSPO\+ method, as it is most effective when the downstream problem structure is compatible with quadratic regularization\.

### 4\.4Computational overhead

While FedRSPO\+ has no additional communication costs compared with FedAvg, it has additional computational overhead, since solving QLPs is more expensive than LP or shortest path problems\. We observed 1\.35×\\times–1\.77×\\timesslower solves than LPs with Gurobi and up to∼\\sim50×\\timesslower solves than Dijkstra\. However, oracle\-only timings overstate the end\-to\-end cost\. In the Warcraft experiment, MSE and SPO\+ both took about 67 seconds per round, whereas RSPO\+ took 133 seconds, i\.e\., a 2\.0×\\timesincrease in training time\. The difference is mainly due to solving the QLP and computing the freezing criterion\. The remaining model training is shared across methods and dominates most of the runtime\. Moreover, the additional QLP oracle calls are parallelizable across clients and samples, suggesting that further engineering could reduce the observed wall\-clock overhead\.

## 5Conclusions

In this paper, we connected the fields of FL and DFL by addressing the brittleness of current methods under optimization heterogeneity through our proposed method, FedRSPO\+\. We showed how smoothing via Moreau envelopes replaces discontinuous vertex\-switching withδ\\sqrt\{\\delta\}\-stable projected decisions and achieves heterogeneity bounds that vanish at homogeneity without strong convexity\. Our principled per\-client calibration and co\-annealed scheduling make the method practical without expensive tuning, and we demonstrated compatibility with existing FL methods such as FedProx and Ditto\. Across our synthetic knapsack and PJM energy pricing experiments, RSPO\+ consistently outperformed both prediction\-only FL and naive SPO\+ federation, with the largest gains appearing during the federation phase\. It also outperformed Interp\-DFFL and an OptNet\-style baseline under objective and feasible\-set heterogeneity\. Even under the mismatched downstream objective in the Warcraft experiment, FedRSPO\+ achieved close to optimal performance for clients, with significant gains over MSE and popular DFL methods\. Together, these results position projection\-based smoothing as a key ingredient for stable collaborative decision learning\.

#### Limitations\.

Our goal is to introduce a heterogeneity\-aware surrogate and training methodology for DFFL, and to evaluate it against representative prediction\-only FL, SPO\+, personalization and aggregation variants, and standard DFL baselines where they are most applicable\. A broader benchmark spanning additional FL optimizers, privacy mechanisms, combinatorial DFL methods, and how they fit in the field of DFFL is an important direction for future work\. Our regret and decision\-error guarantees are for the regularized decision, and apply to the LP decision only under exact regularization, which our stopping criterion does not enforce\. Furthermore, our convergence analysis applies to linear estimators, while our main experiments use neural networks\. We examine FedRSPO\+ performance with linear estimators in the Appendix\. RSPO\+ replaces each LP solve with a QP of the same dimension\. Specialized QP solvers can match simplex\-based solvers on this structure\[[57](https://arxiv.org/html/2610.09091#bib.bib44),[58](https://arxiv.org/html/2610.09091#bib.bib43)\], and warm\-starting can further reduce overhead\. Furthermore, RSPO\+ introduces hyperparametersα,κLR,κλ\\alpha,\\kappa^\{\\mathrm\{LR\}\},\\kappa^\{\\lambda\}beyond SPO\+ and MSE\. While we provide principled defaults \([Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\), tuning for new domains may add overhead\. Our convexity assumptions \([Assumptions3](https://arxiv.org/html/2610.09091#Thmassumption3),[2](https://arxiv.org/html/2610.09091#Thmassumption2)and[1](https://arxiv.org/html/2610.09091#Thmassumption1)\) are standard in DFL but exclude mixed\-integer programs\. In[Section4\.3](https://arxiv.org/html/2610.09091#S4.SS3)we demonstrate that FedRSPO\+ is not appropriate for all downstream objectives, and extending FedRSPO\+ to other domains and problem types via relaxations or decompositions is a natural direction\.

#### Broader Impact\.

FedRSPO\+ enables privacy\-preserving collaboration for data\-driven decision\-making in settings where data sharing is restricted by regulation or competition, potentially lowering barriers for smaller organizations that lack the data scale to train effective models independently\. However, participants with higher\-quality data or more favorable constraint structures may benefit disproportionately from federation, and model updates can still leak information about local objectives, particularly in small\-coalition settings\. Practitioners should consider pairing FedRSPO\+ with differential privacy or secure aggregation mechanisms in sensitive domains\.

#### Acknowledgments

We thank the anonymous reviewers and area chair for their feedback, which helped us refine and substantially strengthen this work\.

#### Funding\.

This work was made possible in part by high\-performance computing resources and technical support provided by the Alabama Supercomputer Authority\.

#### Competing interests\.

The authors declare no competing interests\.

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## Appendix

## Appendix AAdditional Results

### A\.1Convergence of FedRSPO\+

###### Assumption 4\(Model Regularity\)\.

We analyze predictorsc^=g⁡\(x,Θ\)\\hat\{c\}=g\(x;\\Theta\), including the linear caseg⁡\(x,Θ\)=Θ​xg\(x;\\Theta\)=\\Theta x, with bounded features‖x‖≤X\\\|x\\\|\\leq X, bounded initialization‖Θ0−Θ⋆‖≤DΘ\\\|\\Theta\_\{0\}\-\\Theta^\{\\star\}\\\|\\leq D\_\{\\Theta\}, and the prediction/cost bounds of[Assumption3](https://arxiv.org/html/2610.09091#Thmassumption3):‖c^‖≤B\\\|\\hat\{c\}\\\|\\leq B,‖c‖≤Cmax\\\|c\\\|\\leq C\_\{\\max\}\. For nonlinear predictors,g⁡\(x,Θ\)g\(x;\\Theta\)is differentiable inΘ\\Thetaand, for allxxand allΘ,Θ′\\Theta,\\Theta^\{\\prime\}on the training trajectory, the Jacobian satisfies‖JΘ​g​\(x,Θ\)‖op≤J,\\\|J\_\{\\Theta\}g\(x;\\Theta\)\\\|\_\{\\mathrm\{op\}\}\\leq J,and‖JΘ​g​\(x,Θ\)−JΘ​g​\(x,Θ′\)‖op≤ρ​‖Θ−Θ′‖\\\|J\_\{\\Theta\}g\(x;\\Theta\)\-J\_\{\\Theta\}g\(x;\\Theta^\{\\prime\}\)\\\|\_\{\\mathrm\{op\}\}\\leq\\rho\\\|\\Theta\-\\Theta^\{\\prime\}\\\|, where∥⋅∥op\\\|\\cdot\\\|\_\{\\mathrm\{op\}\}is the operator norm\[[33](https://arxiv.org/html/2610.09091#bib.bib39)\]\. For the linear model, these hold withJ≤XJ\\leq Xandρ=0\\rho=0\. Under this assumption, the convexity and4/λt4/\\lambda\_\{t\}\-smoothness of RSPO\+ \([Proposition3](https://arxiv.org/html/2610.09091#Thmproposition3)\) imply thatFi,t​\(Θ\)F\_\{i,t\}\(\\Theta\)is convex andLtL\_\{t\}\-smooth withLt=4​X2/λtL\_\{t\}=4X^\{2\}/\\lambda\_\{t\}in the linear case\. Mini\-batch stochastic gradients are unbiased estimators of∇Fi,t​\(Θ\)\\nabla F\_\{i,t\}\(\\Theta\)\.

###### Assumption 5\(Federated Schedule\)\.

Clients performEElocal SGD steps per synchronization round, with co\-annealed learning rateηt=α​λt/4\\eta\_\{t\}=\\alpha\\lambda\_\{t\}/4for a constantα\>0\\alpha\>0as in[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\.

We establish that FedRSPO\+ converges under FedAvg aggregation and validate the co\-annealed schedule of[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\. LetFi,t​\(Θ\):=𝔼\(x,c\)∼𝒟i​\[ℓRSPO\+​\(c,g⁡\(x,Θ\),λt\)\]F\_\{i,t\}\(\\Theta\):=\\mathbb\{E\}\_\{\(x,c\)\\sim\\mathcal\{D\}\_\{i\}\}\[\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\,g\(x;\\Theta\);\\lambda\_\{t\}\)\]denote the local objective for clientiiat roundtt, andFt=1N​∑i=1NFi,tF\_\{t\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}F\_\{i,t\}the global objective\. Because the annealedλt\\lambda\_\{t\}schedule makesFtF\_\{t\}time\-varying, we track the deviation from the target objective at the frozen parameterλT\\lambda\_\{T\}using[Proposition4](https://arxiv.org/html/2610.09091#Thmproposition4):

Δt:=R2​\(λt−λT\)\+2​Cmax​\(Cmax\+B\)​\(1λT−1λt\)\.\\textstyle\\Delta\_\{t\}:=R^\{2\}\(\\lambda\_\{t\}\-\\lambda\_\{T\}\)\+2\\,C\_\{\\max\}\(C\_\{\\max\}\+B\)\\\!\\left\(\\tfrac\{1\}\{\\lambda\_\{T\}\}\-\\tfrac\{1\}\{\\lambda\_\{t\}\}\\right\)\.\(12\)
###### Theorem 4\(FedRSPO\+ Convergence, restatement of[Theorem2](https://arxiv.org/html/2610.09091#Thmtheorem2)\)\.

Under[Assumptions4](https://arxiv.org/html/2610.09091#Thmassumption4)and[5](https://arxiv.org/html/2610.09091#Thmassumption5), running FedRSPO\+ withEElocal SGD steps per round produces global iteratesΘ¯t=1N​∑iΘt\(i\)\\bar\{\\Theta\}\_\{t\}=\\frac\{1\}\{N\}\\sum\_\{i\}\\Theta\_\{t\}^\{\(i\)\}\. The weighted averageΘ^T=∑tηt​Θ¯t/∑tηt\\hat\{\\Theta\}\_\{T\}=\\sum\_\{t\}\\eta\_\{t\}\\bar\{\\Theta\}\_\{t\}/\\sum\_\{t\}\\eta\_\{t\}satisfies

𝔼⁡\[FT​\(Θ^T\)−FT​\(Θ⋆\)\]≤DΘ2\+∑t=0T−1ηt2​G2​\(1\+4​γ​E2\)\+4​∑t=0T−1ηt​Δt2​∑t=0T−1ηt,\\mathbb\{E\}\\\!\\big\[F\_\{T\}\(\\hat\{\\Theta\}\_\{T\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\big\]\\;\\leq\\;\\frac\{D\_\{\\Theta\}^\{2\}\+\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}^\{2\}\\,G^\{2\}\(1\+4\\gamma E^\{2\}\)\+4\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\\,\\Delta\_\{t\}\}\{2\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\},\(13\)whereG=2​D​XG=2DXis the uniform gradient bound \([Corollary1](https://arxiv.org/html/2610.09091#Thmcorollary1)\), andγ=ηt​Lt=α​X2\\gamma=\\eta\_\{t\}L\_\{t\}=\\alpha X^\{2\}is the*constant*effective condition number enforced by the co\-annealed scheduleηt=α​λt/4\\eta\_\{t\}=\\alpha\\lambda\_\{t\}/4\.

We follow a similar proof strategy to\[[59](https://arxiv.org/html/2610.09091#bib.bib58)\]\. Letg¯t=1N​∑i=1Ngi,t\\bar\{g\}\_\{t\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}g\_\{i,t\}denote the aggregated update direction, soΘ¯t\+1=Θ¯t−ηt​g¯t\\bar\{\\Theta\}\_\{t\+1\}=\\bar\{\\Theta\}\_\{t\}\-\\eta\_\{t\}\\bar\{g\}\_\{t\}\. Expanding the squared distance toΘ⋆\\Theta^\{\\star\}:

𝔼​‖Θ¯t\+1−Θ⋆‖2=𝔼​‖Θ¯t−Θ⋆‖2−2​ηt​𝔼​⟨g¯t,Θ¯t−Θ⋆⟩\+ηt2​𝔼​‖g¯t‖2\.\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\+1\}\-\\Theta^\{\\star\}\\\|^\{2\}=\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\\|^\{2\}\-2\\eta\_\{t\}\\,\\mathbb\{E\}\\langle\\bar\{g\}\_\{t\},\\,\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\rangle\+\\eta\_\{t\}^\{2\}\\,\\mathbb\{E\}\\\|\\bar\{g\}\_\{t\}\\\|^\{2\}\.\(14\)Since𝔼⁡\[‖g¯t‖2\]≤G2\\mathbb\{E\}\[\\\|\\bar\{g\}\_\{t\}\\\|^\{2\}\]\\leq G^\{2\}by[Assumption4](https://arxiv.org/html/2610.09091#Thmassumption4), it remains to lower\-bound the inner product\. By the convexity andLtL\_\{t\}\-smoothness ofFi,tF\_\{i,t\}, we decompose the per\-client term:

⟨∇Fi,t​\(Θt\(i\)\),Θ¯t−Θ⋆⟩\\displaystyle\\langle\\nabla F\_\{i,t\}\(\\Theta\_\{t\}^\{\(i\)\}\),\\,\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\rangle=⟨∇Fi,t​\(Θt\(i\)\),Θt\(i\)−Θ⋆⟩\+⟨∇Fi,t​\(Θt\(i\)\),Θ¯t−Θt\(i\)⟩\\displaystyle=\\langle\\nabla F\_\{i,t\}\(\\Theta\_\{t\}^\{\(i\)\}\),\\,\\Theta\_\{t\}^\{\(i\)\}\-\\Theta^\{\\star\}\\rangle\+\\langle\\nabla F\_\{i,t\}\(\\Theta\_\{t\}^\{\(i\)\}\),\\,\\bar\{\\Theta\}\_\{t\}\-\\Theta\_\{t\}^\{\(i\)\}\\rangle≥\(Fi,t​\(Θt\(i\)\)−Fi,t​\(Θ⋆\)\)\+\(Fi,t​\(Θ¯t\)−Fi,t​\(Θt\(i\)\)−Lt2​‖Θ¯t−Θt\(i\)‖2\)\\displaystyle\\geq\\big\(F\_\{i,t\}\(\\Theta\_\{t\}^\{\(i\)\}\)\-F\_\{i,t\}\(\\Theta^\{\\star\}\)\\big\)\+\\big\(F\_\{i,t\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{i,t\}\(\\Theta\_\{t\}^\{\(i\)\}\)\-\\tfrac\{L\_\{t\}\}\{2\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta\_\{t\}^\{\(i\)\}\\\|^\{2\}\\big\)=Fi,t​\(Θ¯t\)−Fi,t​\(Θ⋆\)−Lt2​‖Θ¯t−Θt\(i\)‖2\.\\displaystyle=F\_\{i,t\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{i,t\}\(\\Theta^\{\\star\}\)\-\\tfrac\{L\_\{t\}\}\{2\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta\_\{t\}^\{\(i\)\}\\\|^\{2\}\.Averaging over clients yields:

𝔼⁡⟨g¯t,Θ¯t−Θ⋆⟩≥𝔼⁡\[Ft​\(Θ¯t\)−Ft​\(Θ⋆\)\]−Lt2​N​∑i=1N𝔼​‖Θ¯t−Θt\(i\)‖2\.\\mathbb\{E\}\\langle\\bar\{g\}\_\{t\},\\,\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\rangle\\geq\\mathbb\{E\}\\big\[F\_\{t\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{t\}\(\\Theta^\{\\star\}\)\\big\]\-\\frac\{L\_\{t\}\}\{2N\}\\sum\_\{i=1\}^\{N\}\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta\_\{t\}^\{\(i\)\}\\\|^\{2\}\.\(15\)The client drift is bounded by the accumulation of at mostEElocal gradient steps with bounded normGG:

1N​∑i=1N𝔼​‖Θ¯t−Θt\(i\)‖2≤4​E2​ηt2​G2\.\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta\_\{t\}^\{\(i\)\}\\\|^\{2\}\\leq 4E^\{2\}\\eta\_\{t\}^\{2\}G^\{2\}\.\(16\)Substituting[Eqs\.15](https://arxiv.org/html/2610.09091#A1.E15)and[16](https://arxiv.org/html/2610.09091#A1.E16)into[Eq\.14](https://arxiv.org/html/2610.09091#A1.E14)and isolating the objective gap:

2​ηt​𝔼​\[Ft​\(Θ¯t\)−Ft​\(Θ⋆\)\]≤𝔼​‖Θ¯t−Θ⋆‖2−𝔼​‖Θ¯t\+1−Θ⋆‖2\+ηt2​G2\+4​Lt​E2​ηt3​G2\.2\\eta\_\{t\}\\,\\mathbb\{E\}\\big\[F\_\{t\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{t\}\(\\Theta^\{\\star\}\)\\big\]\\leq\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\\|^\{2\}\-\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\+1\}\-\\Theta^\{\\star\}\\\|^\{2\}\+\\eta\_\{t\}^\{2\}G^\{2\}\+4L\_\{t\}E^\{2\}\\eta\_\{t\}^\{3\}G^\{2\}\.\(17\)To relate the transient objectiveFtF\_\{t\}to the targetFTF\_\{T\}, we apply the tracking boundΔt\\Delta\_\{t\}from[Eq\.12](https://arxiv.org/html/2610.09091#A1.E12):

Ft​\(Θ¯t\)−Ft​\(Θ⋆\)≥FT​\(Θ¯t\)−FT​\(Θ⋆\)−2​Δt\.F\_\{t\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{t\}\(\\Theta^\{\\star\}\)\\geq F\_\{T\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\-2\\Delta\_\{t\}\.\(18\)Substituting into[Eq\.17](https://arxiv.org/html/2610.09091#A1.E17)and applying the co\-annealed invariantηt​Lt=α​X2=:γ\\eta\_\{t\}L\_\{t\}=\\alpha X^\{2\}=:\\gamma:

2​ηt​𝔼​\[FT​\(Θ¯t\)−FT​\(Θ⋆\)\]\\displaystyle 2\\eta\_\{t\}\\,\\mathbb\{E\}\\big\[F\_\{T\}\(\\bar\{\\Theta\}\_\{t\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\big\]≤𝔼​‖Θ¯t−Θ⋆‖2−𝔼​‖Θ¯t\+1−Θ⋆‖2\\displaystyle\\leq\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\}\-\\Theta^\{\\star\}\\\|^\{2\}\-\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{t\+1\}\-\\Theta^\{\\star\}\\\|^\{2\}\+ηt2​G2​\(1\+4​γ​E2\)\+4​ηt​Δt\.\\displaystyle\\quad\+\\eta\_\{t\}^\{2\}G^\{2\}\(1\+4\\gamma E^\{2\}\)\+4\\eta\_\{t\}\\Delta\_\{t\}\.\(19\)Summing fromt=0t=0toT−1T\{\-\}1telescopes the distance terms\. Dropping the non\-positive terminal distance−𝔼​‖Θ¯T−Θ⋆‖2\-\\mathbb\{E\}\\\|\\bar\{\\Theta\}\_\{T\}\-\\Theta^\{\\star\}\\\|^\{2\}, dividing by2​∑tηt2\\sum\_\{t\}\\eta\_\{t\}, and applying Jensen’s inequality for the convexFTF\_\{T\}over the weighted average iterate completes the proof\.

###### Corollary 3\(Frozen schedule:O⁡\(1/\(T​ηf\)\+ηf\)O\(1/\(T\\eta\_\{f\}\)\+\\eta\_\{f\}\)rate\)\.

Under the conditions of[Theorem4](https://arxiv.org/html/2610.09091#Thmtheorem4), suppose the schedule is frozen at roundtft\_\{f\}, so thatλt=λf\\lambda\_\{t\}=\\lambda\_\{f\}andηt=ηf\\eta\_\{t\}=\\eta\_\{f\}for allt≥tft\\geq t\_\{f\}\. Then forT≫tfT\\gg t\_\{f\}:

𝔼⁡\[FT​\(Θ^T\)−FT​\(Θ⋆\)\]=O⁡\(1T​ηf\+ηf\)\.\\mathbb\{E\}\\\!\\big\[F\_\{T\}\(\\hat\{\\Theta\}\_\{T\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\big\]=O\\\!\\left\(\\frac\{1\}\{T\\,\\eta\_\{f\}\}\+\\eta\_\{f\}\\right\)\.

###### Proof\.

After freezing,λt=λT=λf\\lambda\_\{t\}=\\lambda\_\{T\}=\\lambda\_\{f\}for allt≥tft\\geq t\_\{f\}, soΔt=0\\Delta\_\{t\}=0fort≥tft\\geq t\_\{f\}\. Splitting the sums in[Eq\.13](https://arxiv.org/html/2610.09091#A1.E13)attft\_\{f\}and noting that the pre\-freeze contributions are bounded by constantsA1,A2,A3A\_\{1\},A\_\{2\},A\_\{3\}depending on\(tf,λ0,η0\)\(t\_\{f\},\\lambda\_\{0\},\\eta\_\{0\}\):

∑t=0T−1ηt\\displaystyle\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}=∑t=0tf−1ηt⏟A1\+\(T−tf\)​ηf,\\displaystyle=\\underbrace\{\\sum\_\{t=0\}^\{t\_\{f\}\-1\}\\eta\_\{t\}\}\_\{A\_\{1\}\}\+\(T\-t\_\{f\}\)\\,\\eta\_\{f\},∑t=0T−1ηt2​G2​\(1\+4​γ​E2\)\\displaystyle\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}^\{2\}\\,G^\{2\}\(1\+4\\gamma E^\{2\}\)=∑t=0tf−1ηt2​G2​\(1\+4​γ​E2\)⏟A2\+\(T−tf\)​ηf2​G2​\(1\+4​γ​E2\),\\displaystyle=\\underbrace\{\\sum\_\{t=0\}^\{t\_\{f\}\-1\}\\eta\_\{t\}^\{2\}\\,G^\{2\}\(1\+4\\gamma E^\{2\}\)\}\_\{A\_\{2\}\}\+\(T\-t\_\{f\}\)\\,\\eta\_\{f\}^\{2\}\\,G^\{2\}\(1\+4\\gamma E^\{2\}\),∑t=0T−1ηt​Δt\\displaystyle\\sum\_\{t=0\}^\{T\-1\}\\eta\_\{t\}\\,\\Delta\_\{t\}=∑t=0tf−1ηt​Δt=:A3\.\\displaystyle=\\sum\_\{t=0\}^\{t\_\{f\}\-1\}\\eta\_\{t\}\\,\\Delta\_\{t\}=:A\_\{3\}\.Substituting into[Eq\.13](https://arxiv.org/html/2610.09091#A1.E13),

𝔼⁡\[FT​\(Θ^T\)−FT​\(Θ⋆\)\]≤DΘ2\+A2\+A3\(T−tf\)​ηf\+ηf​G2​\(1\+4​γ​E2\)2\.\\mathbb\{E\}\\\!\\big\[F\_\{T\}\(\\hat\{\\Theta\}\_\{T\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\big\]\\;\\leq\\;\\frac\{D\_\{\\Theta\}^\{2\}\+A\_\{2\}\+A\_\{3\}\}\{\(T\-t\_\{f\}\)\\eta\_\{f\}\}\+\\frac\{\\eta\_\{f\}\\,G^\{2\}\(1\+4\\gamma E^\{2\}\)\}\{2\}\.ForTTsufficiently large relative totft\_\{f\}, the transient constants are absorbed into theO⁡\(1/\(T​ηf\)\)O\(1/\(T\\eta\_\{f\}\)\)term, yielding

𝔼⁡\[FT​\(Θ^T\)−FT​\(Θ⋆\)\]=O⁡\(1T​ηf\+ηf\)\.\\mathbb\{E\}\\\!\\big\[F\_\{T\}\(\\hat\{\\Theta\}\_\{T\}\)\-F\_\{T\}\(\\Theta^\{\\star\}\)\\big\]=O\\\!\\left\(\\frac\{1\}\{T\\,\\eta\_\{f\}\}\+\\eta\_\{f\}\\right\)\.∎

###### Lemma 1\(Neural composition constants\)\.

Supposeg⁡\(x,θ\)g\(x;\\theta\)is differentiable and satisfies[Assumption4](https://arxiv.org/html/2610.09091#Thmassumption4)\. Then

‖∇θFi,t​\(θ\)‖≤2​D​J,Fi,t​is​\(4​J2λt\+2​D​ρ\)​\-smooth\.\\\|\\nabla\_\{\\theta\}F\_\{i,t\}\(\\theta\)\\\|\\leq 2DJ,\\qquad F\_\{i,t\}\\text\{ is \}\\left\(\\frac\{4J^\{2\}\}\{\\lambda\_\{t\}\}\+2D\\rho\\right\)\\text\{\-smooth\}\.

###### Proof\.

By the chain rule,∇θℓ=Jθ​g​\(x,θ\)T​∇c^ℓ\\nabla\_\{\\theta\}\\ell=J\_\{\\theta\}g\(x;\\theta\)^\{T\}\\nabla\_\{\\hat\{c\}\}\\ell\.[Proposition3](https://arxiv.org/html/2610.09091#Thmproposition3)gives‖∇c^ℓ‖≤2​D\\\|\\nabla\_\{\\hat\{c\}\}\\ell\\\|\\leq 2Dand∇c^ℓ\\nabla\_\{\\hat\{c\}\}\\ellis4/λt4/\\lambda\_\{t\}\-Lipschitz\. Combining these with the Jacobian bound and Jacobian Lipschitzness gives

‖∇θℓ​\(θ\)−∇θℓ​\(θ′\)‖≤\(4​J2λt\+2​D​ρ\)​‖θ−θ′‖\.\\\|\\nabla\_\{\\theta\}\\ell\(\\theta\)\-\\nabla\_\{\\theta\}\\ell\(\\theta^\{\\prime\}\)\\\|\\leq\\left\(\\frac\{4J^\{2\}\}\{\\lambda\_\{t\}\}\+2D\\rho\\right\)\\\|\\theta\-\\theta^\{\\prime\}\\\|\.Taking expectations preserves both bounds and concludes the proof\. ∎

###### Proposition 5\(Post\-freeze nonconvex FedRSPO\+\)\.

Suppose[Assumptions1](https://arxiv.org/html/2610.09091#Thmassumption1),[3](https://arxiv.org/html/2610.09091#Thmassumption3),[2](https://arxiv.org/html/2610.09091#Thmassumption2),[4](https://arxiv.org/html/2610.09091#Thmassumption4)and[5](https://arxiv.org/html/2610.09091#Thmassumption5)hold\. After the annealing schedule freezes atλf\\lambda\_\{f\}, define

Ff​\(θ\):=1N​∑i=1NFi,f​\(θ\),LfNN:=4​J2λf\+2​D​ρ\.F\_\{f\}\(\\theta\):=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}F\_\{i,f\}\(\\theta\),\\qquad L\_\{f\}^\{\\mathrm\{NN\}\}:=\\frac\{4J^\{2\}\}\{\\lambda\_\{f\}\}\+2D\\rho\.Assume unbiased stochastic gradients with variance at mostσ2\\sigma^\{2\}and post\-freeze gradient dissimilarity

1N​∑i=1N‖∇Fi,f​\(θ\)−∇Ff​\(θ\)‖2≤ζ2for all​θ\.\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\\|\\nabla F\_\{i,f\}\(\\theta\)\-\\nabla F\_\{f\}\(\\theta\)\\\|^\{2\}\\leq\\zeta^\{2\}\\quad\\text\{for all \}\\theta\.Running FedRSPO\+ withEElocal SGD steps and constant post\-freeze step sizeηf≤c/\(E​LfNN\)\\eta\_\{f\}\\leq c/\(EL\_\{f\}^\{\\mathrm\{NN\}\}\)gives, fors∼Unif⁡\{tf,…,T−1\}s\\sim\\mathrm\{Unif\}\\\{t\_\{f\},\\ldots,T\-1\\\},

𝔼​‖∇Ff​\(θ¯s\)‖2≤O⁡\(Ff​\(θ¯tf\)−Ffinfηf​E​\(T−tf\)\+LfNN​ηf​σ2N\+\(LfNN\)2​ηf2​E2​\(σ2\+ζ2\)\)\.\\mathbb\{E\}\\\|\\nabla F\_\{f\}\(\\bar\{\\theta\}\_\{s\}\)\\\|^\{2\}\\leq O\\\!\\left\(\\frac\{F\_\{f\}\(\\bar\{\\theta\}\_\{t\_\{f\}\}\)\-F\_\{f\}^\{\\inf\}\}\{\\eta\_\{f\}E\(T\-t\_\{f\}\)\}\+L\_\{f\}^\{\\mathrm\{NN\}\}\\eta\_\{f\}\\frac\{\\sigma^\{2\}\}\{N\}\+\(L\_\{f\}^\{\\mathrm\{NN\}\}\)^\{2\}\\eta\_\{f\}^\{2\}E^\{2\}\(\\sigma^\{2\}\+\\zeta^\{2\}\)\\right\)\.Thus, choosingηf=Θ\(\(T−tf\)−1/2\)\\eta\_\{f\}=\\Theta\(\(T\-t\_\{f\}\)^\{\-1/2\}\)yields𝔼∥∇Ff\(θ¯s\)∥2=O\(\(T−tf\)−1/2\)\\mathbb\{E\}\\\|\\nabla F\_\{f\}\(\\bar\{\\theta\}\_\{s\}\)\\\|^\{2\}=O\(\(T\-t\_\{f\}\)^\{\-1/2\}\), up to constants depending onJ,ρ,D,λf,σ,ζJ,\\rho,D,\\lambda\_\{f\},\\sigma,\\zeta, andEE\.

Additionally, the co\-annealed scheduleηt=α​λt/4\\eta\_\{t\}=\\alpha\\lambda\_\{t\}/4satisfies

ηt​LtNN=α​J2\+α​D​ρ​λt2≤α⁡\(J2\+D​ρ​λmax2\),\\eta\_\{t\}L\_\{t\}^\{\\mathrm\{NN\}\}=\\alpha J^\{2\}\+\\frac\{\\alpha D\\rho\\lambda\_\{t\}\}\{2\}\\leq\\alpha\\left\(J^\{2\}\+\\frac\{D\\rho\\lambda\_\{\\max\}\}\{2\}\\right\),so choosingα≤c/\(E⁡\(J2\+D​ρ​λmax/2\)\)\\alpha\\leq c/\(E\(J^\{2\}\+D\\rho\\lambda\_\{\\max\}/2\)\)keeps the effective step size controlled during annealing\.

###### Proof sketch\.

By Lemma[1](https://arxiv.org/html/2610.09091#Thmlemma1), after freezingλt=λf\\lambda\_\{t\}=\\lambda\_\{f\}the objectiveFfF\_\{f\}isLfNNL\_\{f\}^\{\\mathrm\{NN\}\}\-smooth, withLfNN=4​J2/λf\+2​D​ρL\_\{f\}^\{\\mathrm\{NN\}\}=4J^\{2\}/\\lambda\_\{f\}\+2D\\rho\. The stated bound then follows by applying the standard nonconvex FedAvg descent analysis in the literature\[[60](https://arxiv.org/html/2610.09091#bib.bib68),[59](https://arxiv.org/html/2610.09091#bib.bib58)\]under unbiased stochastic gradients and bounded client gradient dissimilarity\. Summing the one\-round descent inequality fromtft\_\{f\}toT−1T\-1and samplings∼Unif⁡\{tf,…,T−1\}s\\sim\\mathrm\{Unif\}\\\{t\_\{f\},\\ldots,T\-1\\\}gives the displayed stationary\-point bound\. The co\-annealed stability claim follows by substitutingηt=α​λt/4\\eta\_\{t\}=\\alpha\\lambda\_\{t\}/4intoηt​LtNN\\eta\_\{t\}L\_\{t\}^\{\\mathrm\{NN\}\}\. ∎

### A\.2Unified bias–heterogeneity tradeoff

The results in[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3),[Theorem1](https://arxiv.org/html/2610.09091#Thmtheorem1), and[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2)can be combined into a single certificate for the unregularized regret of the regularized decision\.

###### Proposition 6\.

For clientii, define

w^i,λ\(c^\):=ΠSi\(−c^/λ\),Regi,λLP\(c^\):=ciTw^i,λ\(c^\)−minw∈SiciTw\.\\widehat\{w\}\_\{i,\\lambda\}\(\\hat\{c\}\):=\\Pi\_\{S\_\{i\}\}\(\-\\hat\{c\}/\\lambda\),\\qquad\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\):=c\_\{i\}^\{T\}\\widehat\{w\}\_\{i,\\lambda\}\(\\hat\{c\}\)\-\\min\_\{w\\in S\_\{i\}\}c\_\{i\}^\{T\}w\.Letℓiλ​\(c^\)\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)denote the RSPO\+ loss on clientii, and letHi​jλ​\(c^\):=Ti​jλ​\(c^\)\+Bi​jλ​\(c^\)H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\):=T\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\+B\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)be the heterogeneity bound from Theorem 3\. Then for any aggregation weightsq∈ΔNq\\in\\Delta\_\{N\},

Regi,λLP​\(c^\)≤∑j=1Nqj​ℓjλ​\(c^\)\+∑j=1Nqj​Hi​jλ​\(c^\)\+λ2​Ri2\.\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+\\sum\_\{j=1\}^\{N\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}R\_\{i\}^\{2\}\.Consequently,

∑i=1Nqi​Regi,λLP​\(c^\)≤∑i=1Nqi​ℓiλ​\(c^\)\+λ2​∑i=1Nqi​Ri2\+∑i,j=1Nqi​qj​Hi​jλ​\(c^\)\.\\sum\_\{i=1\}^\{N\}q\_\{i\}\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\sum\_\{i=1\}^\{N\}q\_\{i\}\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}\\sum\_\{i=1\}^\{N\}q\_\{i\}R\_\{i\}^\{2\}\+\\sum\_\{i,j=1\}^\{N\}q\_\{i\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\.Under uniform bounds‖ci‖≤C\\\|c\_\{i\}\\\|\\leq C,‖c^‖≤B\\\|\\hat\{c\}\\\|\\leq B,Ri≤RR\_\{i\}\\leq R,Di≤DD\_\{i\}\\leq D,‖ci−cj‖≤cΔ\\\|c\_\{i\}\-c\_\{j\}\\\|\\leq c\_\{\\Delta\}, anddH​\(Si,Sj\)≤δd\_\{H\}\(S\_\{i\},S\_\{j\}\)\\leq\\delta, this gives

Reg¯λLP≤ℓ¯λ\+O⁡\(λ​R2\+cΔλ\+δλ\+δ\),\\overline\{\\mathrm\{Reg\}\}\_\{\\lambda\}^\{\\mathrm\{LP\}\}\\leq\\overline\{\\ell\}\_\{\\lambda\}\+O\\\!\\left\(\\lambda R^\{2\}\+\\frac\{c\_\{\\Delta\}\}\{\\lambda\}\+\\sqrt\{\\frac\{\\delta\}\{\\lambda\}\}\+\\delta\\right\),for fixed bounded constants and0<λ≤10<\\lambda\\leq 1\. Thus smallerλ\\lambdareduces regularization bias but amplifies objective and feasible\-set heterogeneity\.

The proposition can be viewed as the bias–stability tradeoff for the regularized decisionw^i,λ\\widehat\{w\}\_\{i,\\lambda\}targeted by the annealing and freezing rule\. When the exact\-regularization condition discussed above holds, the same certificate applies to the LP decision selected at evaluation\.

###### Proof\.

For each clientii, write

ℓiλ​\(c^\):=ℓRSPO\+​\(ci,c^,Si,λ\)=DMλ​ξSi​\(ci−2​c^,−ci\),\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\):=\\ell\_\{\\mathrm\{RSPO\+\}\}\(c\_\{i\},\\hat\{c\};S\_\{i\},\\lambda\)=D\_\{M\_\{\\lambda\}\\xi\_\{S\_\{i\}\}\}\(c\_\{i\}\-2\\hat\{c\},\-c\_\{i\}\),and define the regularized decision

w^i,λ\(c^\):=ΠSi\(−c^/λ\)\.\\widehat\{w\}\_\{i,\\lambda\}\(\\hat\{c\}\):=\\Pi\_\{S\_\{i\}\}\(\-\\hat\{c\}/\\lambda\)\.The unregularized regret of this regularized decision is

Regi,λLP​\(c^\):=ciT​w^i,λ​\(c^\)−zi⋆,zi⋆:=minw∈Si⁡ciT​w\.\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\):=c\_\{i\}^\{T\}\\widehat\{w\}\_\{i,\\lambda\}\(\\hat\{c\}\)\-z\_\{i\}^\{\\star\},\\qquad z\_\{i\}^\{\\star\}:=\\min\_\{w\\in S\_\{i\}\}c\_\{i\}^\{T\}w\.
For two clientsi,ji,j, define

ci​j:=‖ci−cj‖,δi​j:=dH​\(Si,Sj\),Ri​j:=max⁡\{Ri,Rj\}\.c\_\{ij\}:=\\\|c\_\{i\}\-c\_\{j\}\\\|,\\qquad\\delta\_\{ij\}:=d\_\{H\}\(S\_\{i\},S\_\{j\}\),\\qquad R\_\{ij\}:=\\max\\\{R\_\{i\},R\_\{j\}\\\}\.Let

Hi​jλ​\(c^\):=Ti​jλ​\(c^\)\+Bi​jλ​\(c^\),H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\):=T\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\+B\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\),where the two terms are the symmetric versions of the quantities in Theorem 3:

Ti​jλ​\(c^\):=min⁡\{Di​ci​j\+δi​j​\(‖cj−2​c^‖\+‖cj‖\+2​λ​Ri​j\),Dj​ci​j\+δi​j​\(‖ci−2​c^‖\+‖ci‖\+2​λ​Ri​j\)\},T\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\):=\\min\\Big\\\{D\_\{i\}c\_\{ij\}\+\\delta\_\{ij\}\\big\(\\\|c\_\{j\}\-2\\hat\{c\}\\\|\+\\\|c\_\{j\}\\\|\+2\\lambda R\_\{ij\}\\big\),\\;D\_\{j\}c\_\{ij\}\+\\delta\_\{ij\}\\big\(\\\|c\_\{i\}\-2\\hat\{c\}\\\|\+\\\|c\_\{i\}\\\|\+2\\lambda R\_\{ij\}\\big\)\\Big\\\},and

Bi​jλ​\(c^\):=\(2​‖c^‖\+2​λ​Ri​j\)​\[ci​jλ\+min⁡\{2​δi​j​\(‖cj‖λ\+Ri​j\),2​δi​j​\(‖ci‖λ\+Ri​j\)\}\]\.B\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\):=\(2\\\|\\hat\{c\}\\\|\+2\\lambda R\_\{ij\}\)\\left\[\\frac\{c\_\{ij\}\}\{\\lambda\}\+\\min\\left\\\{\\sqrt\{2\\delta\_\{ij\}\\left\(\\frac\{\\\|c\_\{j\}\\\|\}\{\\lambda\}\+R\_\{ij\}\\right\)\},\\sqrt\{2\\delta\_\{ij\}\\left\(\\frac\{\\\|c\_\{i\}\\\|\}\{\\lambda\}\+R\_\{ij\}\\right\)\}\\right\\\}\\right\]\.By[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3),

\|ℓiλ​\(c^\)−ℓjλ​\(c^\)\|≤Hi​jλ​\(c^\)\.\|\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)\-\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\|\\leq H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\.Equivalently,

ℓiλ​\(c^\)≤ℓjλ​\(c^\)\+Hi​jλ​\(c^\)for every​j\.\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)\\leq\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\\qquad\\text\{for every \}j\.
Letq∈ΔNq\\in\\Delta\_\{N\}be any distribution over clients\. Multiplying the last inequality byqjq\_\{j\}and summing overjjyields

ℓiλ​\(c^\)=∑j=1Nqj​ℓiλ​\(c^\)≤∑j=1Nqj​ℓjλ​\(c^\)\+∑j=1Nqj​Hi​jλ​\(c^\)\.\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)=\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)\\leq\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+\\sum\_\{j=1\}^\{N\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\.By[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2), the unregularized regret of the regularized decision satisfies

Regi,λLP​\(c^\)≤ℓiλ​\(c^\)\+λ2​Ri2\.\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\ell\_\{i\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}R\_\{i\}^\{2\}\.Combining the previous two displays gives, for every target clientii,

Regi,λLP​\(c^\)≤∑j=1Nqj​ℓjλ​\(c^\)\+∑j=1Nqj​Hi​jλ​\(c^\)\+λ2​Ri2\.\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+\\sum\_\{j=1\}^\{N\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}R\_\{i\}^\{2\}\.This proves the client\-wise statement\.

For the aggregate statement, multiply the client\-wise inequality byqiq\_\{i\}and sum overii:

∑i=1Nqi​Regi,λLP​\(c^\)≤∑i=1Nqi​∑j=1Nqj​ℓjλ​\(c^\)\+∑i=1Nqi​∑j=1Nqj​Hi​jλ​\(c^\)\+λ2​∑i=1Nqi​Ri2\.\\sum\_\{i=1\}^\{N\}q\_\{i\}\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\sum\_\{i=1\}^\{N\}q\_\{i\}\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+\\sum\_\{i=1\}^\{N\}q\_\{i\}\\sum\_\{j=1\}^\{N\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}\\sum\_\{i=1\}^\{N\}q\_\{i\}R\_\{i\}^\{2\}\.Since∑iqi=1\\sum\_\{i\}q\_\{i\}=1, the first term simplifies:

∑i=1Nqi​∑j=1Nqj​ℓjλ​\(c^\)=∑j=1Nqj​ℓjλ​\(c^\)\.\\sum\_\{i=1\}^\{N\}q\_\{i\}\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)=\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\.Therefore,

∑i=1Nqi​Regi,λLP​\(c^\)≤∑j=1Nqj​ℓjλ​\(c^\)\+λ2​∑i=1Nqi​Ri2\+∑i,j=1Nqi​qj​Hi​jλ​\(c^\)\.\\sum\_\{i=1\}^\{N\}q\_\{i\}\\mathrm\{Reg\}\_\{i,\\lambda\}^\{\\mathrm\{LP\}\}\(\\hat\{c\}\)\\leq\\sum\_\{j=1\}^\{N\}q\_\{j\}\\ell\_\{j\}^\{\\lambda\}\(\\hat\{c\}\)\+\\frac\{\\lambda\}\{2\}\\sum\_\{i=1\}^\{N\}q\_\{i\}R\_\{i\}^\{2\}\+\\sum\_\{i,j=1\}^\{N\}q\_\{i\}q\_\{j\}H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\.
It remains only to derive the simplified asymptotic form\. Suppose

‖ci‖≤C,‖c^‖≤B,Ri≤R,Di≤D,\\\|c\_\{i\}\\\|\\leq C,\\qquad\\\|\\hat\{c\}\\\|\\leq B,\\qquad R\_\{i\}\\leq R,\\qquad D\_\{i\}\\leq D,and for all client pairs,

‖ci−cj‖≤cΔ,dH​\(Si,Sj\)≤δ\.\\\|c\_\{i\}\-c\_\{j\}\\\|\\leq c\_\{\\Delta\},\\qquad d\_\{H\}\(S\_\{i\},S\_\{j\}\)\\leq\\delta\.ThenRi​j≤RR\_\{ij\}\\leq R, and the envelope term satisfies

Ti​jλ​\(c^\)≤D​cΔ\+δ⁡\(2​C\+2​B\+2​λ​R\)\.T\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\\leq Dc\_\{\\Delta\}\+\\delta\(2C\+2B\+2\\lambda R\)\.Similarly, the decision term satisfies

Bi​jλ​\(c^\)≤\(2​B\+2​λ​R\)​\[cΔλ\+2​δ​\(Cλ\+R\)\]\.B\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\\leq\(2B\+2\\lambda R\)\\left\[\\frac\{c\_\{\\Delta\}\}\{\\lambda\}\+\\sqrt\{2\\delta\\left\(\\frac\{C\}\{\\lambda\}\+R\\right\)\}\\right\]\.Thus

Hi​jλ​\(c^\)≤D​cΔ\+δ⁡\(2​C\+2​B\+2​λ​R\)\+\(2​B\+2​λ​R\)​\[cΔλ\+2​δ​\(Cλ\+R\)\]\.H\_\{ij\}^\{\\lambda\}\(\\hat\{c\}\)\\leq Dc\_\{\\Delta\}\+\\delta\(2C\+2B\+2\\lambda R\)\+\(2B\+2\\lambda R\)\\left\[\\frac\{c\_\{\\Delta\}\}\{\\lambda\}\+\\sqrt\{2\\delta\\left\(\\frac\{C\}\{\\lambda\}\+R\\right\)\}\\right\]\.Substituting this bound into the aggregate inequality and using∑iqi​Ri2≤R2\\sum\_\{i\}q\_\{i\}R\_\{i\}^\{2\}\\leq R^\{2\}gives

Reg¯λLP≤ℓ¯λ\+λ2​R2\+D​cΔ\+δ⁡\(2​C\+2​B\+2​λ​R\)\+\(2​B\+2​λ​R\)​\[cΔλ\+2​δ​\(Cλ\+R\)\]\.\\overline\{\\mathrm\{Reg\}\}\_\{\\lambda\}^\{\\mathrm\{LP\}\}\\leq\\overline\{\\ell\}\_\{\\lambda\}\+\\frac\{\\lambda\}\{2\}R^\{2\}\+Dc\_\{\\Delta\}\+\\delta\(2C\+2B\+2\\lambda R\)\+\(2B\+2\\lambda R\)\\left\[\\frac\{c\_\{\\Delta\}\}\{\\lambda\}\+\\sqrt\{2\\delta\\left\(\\frac\{C\}\{\\lambda\}\+R\\right\)\}\\right\]\.For fixed boundedB,C,D,RB,C,D,Rand0<λ≤10<\\lambda\\leq 1, this is summarized as

Reg¯λLP≤ℓ¯λ\+O⁡\(λ​R2\+cΔλ\+δλ\+δ\)\.\\overline\{\\mathrm\{Reg\}\}\_\{\\lambda\}^\{\\mathrm\{LP\}\}\\leq\\overline\{\\ell\}\_\{\\lambda\}\+O\\\!\\left\(\\lambda R^\{2\}\+\\frac\{c\_\{\\Delta\}\}\{\\lambda\}\+\\sqrt\{\\frac\{\\delta\}\{\\lambda\}\}\+\\delta\\right\)\.∎

## Appendix BProofs and Derivations

### Proof of[Proposition1](https://arxiv.org/html/2610.09091#Thmproposition1): SPO\+ as a Bregman Divergence

###### Proof\.

Recall the SPO\+ loss from[Section3](https://arxiv.org/html/2610.09091#S3):

ℓSPO\+​\(c,c^\)=ξS​\(c−2​c^\)−2​w⋆​\(c\)T​\(c−c^\)\+z⋆​\(c\)\.\\ell\_\{\\mathrm\{SPO\+\}\}\(c,\\hat\{c\}\)=\\xi\_\{S\}\(c\-2\\hat\{c\}\)\-2w^\{\\star\}\(c\)^\{T\}\(c\-\\hat\{c\}\)\+z^\{\\star\}\(c\)\.Sincew⋆​\(c\)∈arg⁡minw∈S​cT​w=arg⁡maxw∈S​\(−c\)T​w=∂ξS​\(−c\)w^\{\\star\}\(c\)\\in\\arg\\min\_\{w\\in S\}c^\{T\}w=\\arg\\max\_\{w\\in S\}\\,\(\-c\)^\{T\}w=\\partial\\xi\_\{S\}\(\-c\)andz⋆​\(c\)=minw∈S⁡cT​w=−ξS​\(−c\)z^\{\\star\}\(c\)=\\min\_\{w\\in S\}c^\{T\}w=\-\\xi\_\{S\}\(\-c\), and moreover if the optimizer is unique∇ξS​\(−c\)=w⋆​\(c\)\\nabla\\xi\_\{S\}\(\-c\)=w^\{\\star\}\(c\)\(by Danskin’s theorem, asSSis compact\), we can write the linearization term as

−2w⋆\(c\)T\(c−c^\)\+z⋆\(c\)=−ξS\(−c\)−∇ξS\(−c\)T\[\(c−2c^\)−\(−c\)\]\.\-2w^\{\\star\}\(c\)^\{T\}\(c\-\\hat\{c\}\)\+z^\{\\star\}\(c\)=\-\\xi\_\{S\}\(\-c\)\-\\nabla\\xi\_\{S\}\(\-c\)^\{T\}\\bigl\[\(c\-2\\hat\{c\}\)\-\(\-c\)\\bigr\]\.Substituting into the loss yields

ℓSPO\+\(c,c^\)=ξS\(c−2c^\)−ξS\(−c\)−∇ξS\(−c\)T\[\(c−2c^\)−\(−c\)\]=DξS\(c−2c^,−c\),\\ell\_\{\\mathrm\{SPO\+\}\}\(c,\\hat\{c\}\)=\\xi\_\{S\}\(c\-2\\hat\{c\}\)\-\\xi\_\{S\}\(\-c\)\-\\nabla\\xi\_\{S\}\(\-c\)^\{T\}\\bigl\[\(c\-2\\hat\{c\}\)\-\(\-c\)\\bigr\]=D\_\{\\xi\_\{S\}\}\(c\-2\\hat\{c\},\\,\-c\),which is the Bregman divergence of the support function evaluated atx=c−2​c^x=c\-2\\hat\{c\},y=−cy=\-c\. ∎

### Proof of[Proposition2](https://arxiv.org/html/2610.09091#Thmproposition2): Smoothed SPO\+ Loss

###### Proof\.

Step 1: Moreau envelope of the support function\.By convex duality, the conjugate ofξS\\xi\_\{S\}is the indicator functionISI\_\{S\}of the setSS\. The Moreau envelope can therefore be written as the biconjugate of the regularized indicator:

Fλ,S\(u\)=Mλ​ξS\(u\)=\(IS\+λ2∥⋅∥2\)∗\(u\)=sups∈S\{⟨u,s⟩−λ2∥s∥2\}\.F\_\{\\lambda,S\}\(u\)=M\_\{\\lambda\\xi\_\{S\}\}\(u\)=\\Bigl\(I\_\{S\}\+\\tfrac\{\\lambda\}\{2\}\\\|\\cdot\\\|^\{2\}\\Bigr\)^\{\*\}\(u\)=\\sup\_\{s\\in S\}\\Bigl\\\{\\langle u,s\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|s\\\|^\{2\}\\Bigr\\\}\.Completing the square inside the supremum gives

⟨u,s⟩−λ2​‖s‖2=12​λ​‖u‖2−λ2​‖s−u/λ‖2\.\\langle u,s\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|s\\\|^\{2\}=\\tfrac\{1\}\{2\\lambda\}\\\|u\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\bigl\\\|s\-u/\\lambda\\bigr\\\|^\{2\}\.The first term is independent ofss, so taking the supremum overs∈Ss\\in Sis equivalent to minimizing‖s−u/λ‖2\\\|s\-u/\\lambda\\\|^\{2\}overSS:

Fλ,S​\(u\)=12​λ​‖u‖2−λ2​dS2​\(u/λ\),F\_\{\\lambda,S\}\(u\)=\\tfrac\{1\}\{2\\lambda\}\\\|u\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\,d\_\{S\}^\{2\}\\\!\\bigl\(u/\\lambda\\bigr\),\(20\)wheredS2​\(v\)=infs∈S‖v−s‖2d\_\{S\}^\{2\}\(v\)=\\inf\_\{s\\in S\}\\\|v\-s\\\|^\{2\}\. The supremum is attained ats⋆=ΠS​\(u/λ\)s^\{\\star\}=\\Pi\_\{S\}\(u/\\lambda\), and Danskin’s theorem yields

∇Fλ,S​\(u\)=ΠS​\(u/λ\)\.\\nabla F\_\{\\lambda,S\}\(u\)=\\Pi\_\{S\}\(u/\\lambda\)\.\(21\)
Step 2: Bregman divergence\.We evaluateDFλ,S​\(x,y\)=Fλ,S​\(x\)−Fλ,S​\(y\)−⟨∇Fλ,S​\(y\),x−y⟩D\_\{F\_\{\\lambda,S\}\}\(x,y\)=F\_\{\\lambda,S\}\(x\)\-F\_\{\\lambda,S\}\(y\)\-\\langle\\nabla F\_\{\\lambda,S\}\(y\),\\,x\-y\\ranglewithx=c−2​c^x=c\-2\\hat\{c\}andy=−cy=\-c\. Writev¯=ΠS\(y/λ\)=ΠS\(−c/λ\)\\bar\{v\}=\\Pi\_\{S\}\(y/\\lambda\)=\\Pi\_\{S\}\(\-c/\\lambda\)\. Expanding theyy\-dependent terms using \([20](https://arxiv.org/html/2610.09091#A2.E20)\) and \([21](https://arxiv.org/html/2610.09091#A2.E21)\):

Fλ,S​\(y\)\+⟨∇Fλ,S​\(y\),x−y⟩\\displaystyle F\_\{\\lambda,S\}\(y\)\+\\langle\\nabla F\_\{\\lambda,S\}\(y\),\\,x\-y\\rangle=12​λ​‖y‖2−λ2​‖y/λ−v¯‖2\+⟨v¯,x−y⟩\\displaystyle=\\tfrac\{1\}\{2\\lambda\}\\\|y\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|y/\\lambda\-\\bar\{v\}\\\|^\{2\}\+\\langle\\bar\{v\},\\,x\-y\\rangle=⟨y,v¯⟩−λ2​‖v¯‖2\+⟨v¯,x−y⟩\\displaystyle=\\langle y,\\bar\{v\}\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|\\bar\{v\}\\\|^\{2\}\+\\langle\\bar\{v\},\\,x\-y\\rangle=⟨x,v¯⟩−λ2​‖v¯‖2,\\displaystyle=\\langle x,\\bar\{v\}\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|\\bar\{v\}\\\|^\{2\},where the second equality uses12​λ​‖y‖2−λ2​‖y/λ−v¯‖2=⟨y,v¯⟩−λ2​‖v¯‖2\\tfrac\{1\}\{2\\lambda\}\\\|y\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|y/\\lambda\-\\bar\{v\}\\\|^\{2\}=\\langle y,\\bar\{v\}\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|\\bar\{v\}\\\|^\{2\}, which follows from expanding the squared norm\. Subtracting fromFλ,S​\(x\)F\_\{\\lambda,S\}\(x\):

DFλ,S​\(x,y\)\\displaystyle D\_\{F\_\{\\lambda,S\}\}\(x,y\)=12​λ​‖x‖2−λ2​dS2​\(x/λ\)−⟨x,v¯⟩\+λ2​‖v¯‖2\\displaystyle=\\tfrac\{1\}\{2\\lambda\}\\\|x\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\,d\_\{S\}^\{2\}\(x/\\lambda\)\-\\langle x,\\bar\{v\}\\rangle\+\\tfrac\{\\lambda\}\{2\}\\\|\\bar\{v\}\\\|^\{2\}=λ2​‖x/λ−v¯‖2−λ2​dS2​\(x/λ\)\.\\displaystyle=\\tfrac\{\\lambda\}\{2\}\\bigl\\\|x/\\lambda\-\\bar\{v\}\\bigr\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\,d\_\{S\}^\{2\}\(x/\\lambda\)\.Settingu=\(c−2​c^\)/λu=\(c\-2\\hat\{c\}\)/\\lambdaandv=−c/λv=\-c/\\lambda, notingv¯=ΠS​\(v\)\\bar\{v\}=\\Pi\_\{S\}\(v\)anddS2​\(u\)=‖u−ΠS​\(u\)‖2d\_\{S\}^\{2\}\(u\)=\\\|u\-\\Pi\_\{S\}\(u\)\\\|^\{2\}, we obtain

ℓRSPO\+​\(c,c^,λ\)=DFλ,S​\(c−2​c^,−c\)=λ2​‖u−ΠS​\(v\)‖2−λ2​‖u−ΠS​\(u\)‖2\.\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=D\_\{F\_\{\\lambda,S\}\}\(c\-2\\hat\{c\},\\,\-c\)=\\tfrac\{\\lambda\}\{2\}\\bigl\\\|u\-\\Pi\_\{S\}\(v\)\\bigr\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\bigl\\\|u\-\\Pi\_\{S\}\(u\)\\bigr\\\|^\{2\}\.∎

### Proof of[Corollary1](https://arxiv.org/html/2610.09091#Thmcorollary1): Gradient of RSPO\+

###### Proof\.

From Proposition 3\.2,ℓRSPO\+​\(c,c^,λ\)=DFλ,S​\(x,y\)\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=D\_\{F\_\{\\lambda,S\}\}\(x,y\)withx⁡\(c^\)=c−2​c^x\(\\hat\{c\}\)=c\-2\\hat\{c\}andy=−cy=\-c\(independent ofc^\\hat\{c\}\)\. The gradient of the Bregman divergence with respect to its first argument is∇xDF​\(x,y\)=∇F​\(x\)−∇F​\(y\)\\nabla\_\{x\}D\_\{F\}\(x,y\)=\\nabla F\(x\)\-\\nabla F\(y\)\. By the chain rule,∇c^x=−2​I\\nabla\_\{\\hat\{c\}\}\\,x=\-2I, so

∇c^ℓRSPO\+​\(c,c^,λ\)=−2​\(∇Fλ,S​\(c−2​c^\)−∇Fλ,S​\(−c\)\)\.\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=\-2\\bigl\(\\nabla F\_\{\\lambda,S\}\(c\-2\\hat\{c\}\)\-\\nabla F\_\{\\lambda,S\}\(\-c\)\\bigr\)\.Substituting∇Fλ,S​\(u\)=ΠS​\(u/λ\)\\nabla F\_\{\\lambda,S\}\(u\)=\\Pi\_\{S\}\(u/\\lambda\)from \([21](https://arxiv.org/html/2610.09091#A2.E21)\) yields

∇c^ℓRSPO\+​\(c,c^,λ\)=−2​\(ΠS​\(c−2​c^λ\)−ΠS​\(−cλ\)\)\.\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=\-2\\Bigl\(\\Pi\_\{S\}\\\!\\bigl\(\\tfrac\{c\-2\\hat\{c\}\}\{\\lambda\}\\bigr\)\-\\Pi\_\{S\}\\\!\\bigl\(\-\\tfrac\{c\}\{\\lambda\}\\bigr\)\\Bigr\)\.
∎

### Proof of[Proposition3](https://arxiv.org/html/2610.09091#Thmproposition3): Characteristics of RSPO\+

###### Proof\.

We prove each property in turn\.

#### \(i\) Non\-negativity\.

Non\-negativity is immediate sinceℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}is a Bregman divergence of the convex functionFλ,SF\_\{\\lambda,S\}\.

#### \(ii\) Convexity inc^\\hat\{c\}\.

The Moreau envelopeFλ,SF\_\{\\lambda,S\}is convex \(as the Moreau envelope of a convex function\)\. A Bregman divergenceDF​\(x,y\)D\_\{F\}\(x,y\)is convex inxxfor any convexFF, sinceDF​\(x,y\)=F⁡\(x\)−F⁡\(y\)−⟨∇F​\(y\),x−y⟩D\_\{F\}\(x,y\)=F\(x\)\-F\(y\)\-\\langle\\nabla F\(y\),\\,x\-y\\rangleand the last two terms are affine inxx\. Becausex=c−2​c^x=c\-2\\hat\{c\}is an affine mapping ofc^\\hat\{c\}, the compositionℓRSPO\+​\(c,c^,λ\)=DFλ,S​\(c−2​c^,−c\)\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=D\_\{F\_\{\\lambda,S\}\}\(c\-2\\hat\{c\},\\,\-c\)is globally convex with respect toc^\\hat\{c\}\.

#### \(iii\)2​D2D\-Lipschitz continuity\.

By Corollary 3\.1, for any\(c,c^\)\(c,\\hat\{c\}\),

‖∇c^ℓRSPO\+‖=2​‖ΠS​\(u\)−ΠS​\(v\)‖≤2​supa,b∈S‖a−b‖=2​D,\\\|\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}\\\|=2\\bigl\\\|\\Pi\_\{S\}\(u\)\-\\Pi\_\{S\}\(v\)\\bigr\\\|\\leq 2\\sup\_\{a,b\\in S\}\\\|a\-b\\\|=2D,since both projections land inSS\. A function whose gradient norm is uniformly bounded byLLisLL\-Lipschitz\.

#### \(iv\)4/λ4/\\lambda\-smoothness andλ/4\\lambda/4\-cocoercivity\.

The Euclidean projectionΠS\\Pi\_\{S\}onto a closed convex set is*firmly non\-expansive*:

⟨ΠS​\(a\)−ΠS​\(b\),a−b⟩≥‖ΠS​\(a\)−ΠS​\(b\)‖2∀a,b\.\\langle\\Pi\_\{S\}\(a\)\-\\Pi\_\{S\}\(b\),\\,a\-b\\rangle\\geq\\\|\\Pi\_\{S\}\(a\)\-\\Pi\_\{S\}\(b\)\\\|^\{2\}\\quad\\forall\\,a,b\.Letc^1,c^2\\hat\{c\}\_\{1\},\\hat\{c\}\_\{2\}be two prediction vectors and definexk=\(c−2​c^k\)/λx\_\{k\}=\(c\-2\\hat\{c\}\_\{k\}\)/\\lambdafork=1,2k=1,2\. By Corollary 3\.1,∇c^ℓ​\(c^k\)=−2​\(ΠS​\(xk\)−ΠS​\(v\)\)\\nabla\_\{\\hat\{c\}\}\\,\\ell\(\\hat\{c\}\_\{k\}\)=\-2\(\\Pi\_\{S\}\(x\_\{k\}\)\-\\Pi\_\{S\}\(v\)\)wherev=−c/λv=\-c/\\lambdais common\. Thus

∇ℓ​\(c^1\)−∇ℓ​\(c^2\)=−2​\(ΠS​\(x1\)−ΠS​\(x2\)\),\\nabla\\ell\(\\hat\{c\}\_\{1\}\)\-\\nabla\\ell\(\\hat\{c\}\_\{2\}\)=\-2\\bigl\(\\Pi\_\{S\}\(x\_\{1\}\)\-\\Pi\_\{S\}\(x\_\{2\}\)\\bigr\),andx1−x2=−2\(c^1−c^2\)/λx\_\{1\}\-x\_\{2\}=\-2\(\\hat\{c\}\_\{1\}\-\\hat\{c\}\_\{2\}\)/\\lambda\. Evaluating the inner product:

⟨∇ℓ​\(c^1\)−∇ℓ​\(c^2\),c^1−c^2⟩\\displaystyle\\langle\\nabla\\ell\(\\hat\{c\}\_\{1\}\)\-\\nabla\\ell\(\\hat\{c\}\_\{2\}\),\\,\\hat\{c\}\_\{1\}\-\\hat\{c\}\_\{2\}\\rangle=−2​⟨ΠS​\(x1\)−ΠS​\(x2\),c^1−c^2⟩\\displaystyle=\-2\\langle\\Pi\_\{S\}\(x\_\{1\}\)\-\\Pi\_\{S\}\(x\_\{2\}\),\\,\\hat\{c\}\_\{1\}\-\\hat\{c\}\_\{2\}\\rangle=λ⁡⟨ΠS​\(x1\)−ΠS​\(x2\),x1−x2⟩\\displaystyle=\\lambda\\langle\\Pi\_\{S\}\(x\_\{1\}\)\-\\Pi\_\{S\}\(x\_\{2\}\),\\,x\_\{1\}\-x\_\{2\}\\rangle≥λ​‖ΠS​\(x1\)−ΠS​\(x2\)‖2\\displaystyle\\geq\\lambda\\,\\\|\\Pi\_\{S\}\(x\_\{1\}\)\-\\Pi\_\{S\}\(x\_\{2\}\)\\\|^\{2\}=λ4​‖∇ℓ​\(c^1\)−∇ℓ​\(c^2\)‖2,\\displaystyle=\\tfrac\{\\lambda\}\{4\}\\,\\\|\\nabla\\ell\(\\hat\{c\}\_\{1\}\)\-\\nabla\\ell\(\\hat\{c\}\_\{2\}\)\\\|^\{2\},where the inequality uses firm non\-expansiveness and the last equality uses‖∇ℓ​\(c^1\)−∇ℓ​\(c^2\)‖=2​‖ΠS​\(x1\)−ΠS​\(x2\)‖\\\|\\nabla\\ell\(\\hat\{c\}\_\{1\}\)\-\\nabla\\ell\(\\hat\{c\}\_\{2\}\)\\\|=2\\\|\\Pi\_\{S\}\(x\_\{1\}\)\-\\Pi\_\{S\}\(x\_\{2\}\)\\\|\. This establishesλ/4\\lambda/4\-cocoercivity\. By the Baillon–Haddad theorem\[[61](https://arxiv.org/html/2610.09091#bib.bib59)\], a convex function with aβ\\beta\-cocoercive gradient is1/β1/\\beta\-smooth, soℓRSPO\+\\ell\_\{\\mathrm\{RSPO\+\}\}is4/λ4/\\lambda\-smooth\. ∎

### Proof of[Theorem1](https://arxiv.org/html/2610.09091#Thmtheorem1): Regret and Decision\-Error Bounds

###### Proof\.

Define the regularized objectivef⁡\(c,w\)=cT​w\+λ2​‖w‖2f\(c,w\)=c^\{T\}w\+\\tfrac\{\\lambda\}\{2\}\\\|w\\\|^\{2\}and letw⋆=ΠS\(−c/λ\)w^\{\\star\}=\\Pi\_\{S\}\(\-c/\\lambda\),w^=ΠS\(−c^/λ\)\\hat\{w\}=\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\)denote the optimal and predicted regularized decisions, respectively\. The regularized regret isR⁡\(c,w^\)=f⁡\(c,w^\)−f⁡\(c,w⋆\)R\(c,\\hat\{w\}\)=f\(c,\\hat\{w\}\)\-f\(c,w^\{\\star\}\)\.

Withu=\(c−2​c^\)/λu=\(c\-2\\hat\{c\}\)/\\lambda, Proposition 3\.2 gives

ℓRSPO\+=λ2​‖u−w⋆‖2−λ2​dS2​\(u\)\.\\ell\_\{\\mathrm\{RSPO\+\}\}=\\tfrac\{\\lambda\}\{2\}\\\|u\-w^\{\\star\}\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\,d\_\{S\}^\{2\}\(u\)\.Sincew^∈S\\hat\{w\}\\in S, the pointw^\\hat\{w\}is feasible for the distance minimization, sodS2​\(u\)≤‖u−w^‖2d\_\{S\}^\{2\}\(u\)\\leq\\\|u\-\\hat\{w\}\\\|^\{2\}\. Therefore

ℓRSPO\+≥λ2​‖u−w⋆‖2−λ2​‖u−w^‖2\.\\ell\_\{\\mathrm\{RSPO\+\}\}\\geq\\tfrac\{\\lambda\}\{2\}\\\|u\-w^\{\\star\}\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|u\-\\hat\{w\}\\\|^\{2\}\.\(22\)Expanding the difference of squares:

λ2​‖u−w⋆‖2−λ2​‖u−w^‖2\\displaystyle\\tfrac\{\\lambda\}\{2\}\\\|u\-w^\{\\star\}\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|u\-\\hat\{w\}\\\|^\{2\}=λ⁡⟨u,w^−w⋆⟩\+λ2​‖w⋆‖2−λ2​‖w^‖2\\displaystyle=\\lambda\\langle u,\\,\\hat\{w\}\-w^\{\\star\}\\rangle\+\\tfrac\{\\lambda\}\{2\}\\\|w^\{\\star\}\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}=⟨c−2​c^,w^−w⋆⟩\+λ2​‖w⋆‖2−λ2​‖w^‖2\.\\displaystyle=\\langle c\-2\\hat\{c\},\\,\\hat\{w\}\-w^\{\\star\}\\rangle\+\\tfrac\{\\lambda\}\{2\}\\\|w^\{\\star\}\\\|^\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\.Rearranging the linear terms:

=\(cT​w^\+λ2​‖w^‖2\)−\(cT​w⋆\+λ2​‖w⋆‖2\)⏟R⁡\(c,w^\)\+λ​‖w⋆‖2−λ​‖w^‖2−2​c^T​\(w^−w⋆\)\\displaystyle=\\underbrace\{\\bigl\(c^\{T\}\\hat\{w\}\+\\tfrac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\\bigr\)\-\\bigl\(c^\{T\}w^\{\\star\}\+\\tfrac\{\\lambda\}\{2\}\\\|w^\{\\star\}\\\|^\{2\}\\bigr\)\}\_\{R\(c,\\,\\hat\{w\}\)\}\+\\lambda\\\|w^\{\\star\}\\\|^\{2\}\-\\lambda\\\|\\hat\{w\}\\\|^\{2\}\-2\\hat\{c\}^\{T\}\(\\hat\{w\}\-w^\{\\star\}\)=R⁡\(c,w^\)\+2​\[\(c^T​w⋆\+λ2​‖w⋆‖2\)−\(c^T​w^\+λ2​‖w^‖2\)\]\\displaystyle=R\(c,\\hat\{w\}\)\+2\\Bigl\[\\bigl\(\\hat\{c\}^\{T\}w^\{\\star\}\+\\tfrac\{\\lambda\}\{2\}\\\|w^\{\\star\}\\\|^\{2\}\\bigr\)\-\\bigl\(\\hat\{c\}^\{T\}\\hat\{w\}\+\\tfrac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\\bigr\)\\Bigr\]=R⁡\(c,w^\)\+2​\[f⁡\(c^,w⋆\)−f⁡\(c^,w^\)\]\.\\displaystyle=R\(c,\\hat\{w\}\)\+2\\bigl\[f\(\\hat\{c\},w^\{\\star\}\)\-f\(\\hat\{c\},\\hat\{w\}\)\\bigr\]\.Sincef⁡\(c^,⋅\)f\(\\hat\{c\},\\cdot\)isλ\\lambda\-strongly convex inwwand is minimized overSSatw^=ΠS\(−c^/λ\)\\hat\{w\}=\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\), the suboptimality gap satisfies

f⁡\(c^,w⋆\)−f⁡\(c^,w^\)≥λ2​‖w⋆−w^‖2\.f\(\\hat\{c\},w^\{\\star\}\)\-f\(\\hat\{c\},\\hat\{w\}\)\\geq\\tfrac\{\\lambda\}\{2\}\\\|w^\{\\star\}\-\\hat\{w\}\\\|^\{2\}\.Combining with \([22](https://arxiv.org/html/2610.09091#A2.E22)\) yields[Eq\.1](https://arxiv.org/html/2610.09091#S3.E1):

ℓRSPO\+​\(c,c^,λ\)≥R⁡\(c,w^\)\+λ​‖w^−w⋆‖2\.\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)\\geq R\(c,\\hat\{w\}\)\+\\lambda\\\|\\hat\{w\}\-w^\{\\star\}\\\|^\{2\}\.SinceR⁡\(c,w^\)≥0R\(c,\\hat\{w\}\)\\geq 0\(by optimality ofw⋆w^\{\\star\}\), the decision\-error bound[Eq\.2](https://arxiv.org/html/2610.09091#S3.E2)follows immediately:

‖w^−w⋆‖≤ℓRSPO\+​\(c,c^,λ\)λ\.\\\|\\hat\{w\}\-w^\{\\star\}\\\|\\leq\\sqrt\{\\frac\{\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)\}\{\\lambda\}\}\.
∎

### Proof of[Corollary2](https://arxiv.org/html/2610.09091#Thmcorollary2): Unregularized Regret Bound for RSPO\+

###### Proof\.

By definition, the regularized regret from[Theorem1](https://arxiv.org/html/2610.09091#Thmtheorem1)expands asR⁡\(c,w^\)=\(cT​w^\+λ2​‖w^‖2\)−\(cT​wreg⋆\+λ2​‖wreg⋆‖2\)R\(c,\\hat\{w\}\)=\\big\(c^\{T\}\\hat\{w\}\+\\frac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\\big\)\-\\big\(c^\{T\}w^\{\\star\}\_\{\\mathrm\{reg\}\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\\big\)\. Rearranging to isolate the unregularized cost of the prediction gives:

cT​w^=R⁡\(c,w^\)\+cT​wreg⋆\+λ2​‖wreg⋆‖2−λ2​‖w^‖2\.c^\{T\}\\hat\{w\}=R\(c,\\hat\{w\}\)\+c^\{T\}w^\{\\star\}\_\{\\mathrm\{reg\}\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\-\\frac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\.Becausewreg⋆w^\{\\star\}\_\{\\mathrm\{reg\}\}minimizes the regularized objective overSS, its regularized objective is lower than that of the true unregularized optimumwLP⋆∈Sw^\{\\star\}\_\{\\mathrm\{LP\}\}\\in S:

cT​wreg⋆\+λ2​‖wreg⋆‖2≤cT​wLP⋆\+λ2​‖wLP⋆‖2=z⋆\+λ2​‖wLP⋆‖2\.c^\{T\}w^\{\\star\}\_\{\\mathrm\{reg\}\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\\leq c^\{T\}w^\{\\star\}\_\{\\mathrm\{LP\}\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|^\{2\}=z^\{\\star\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|^\{2\}\.Substituting this upper bound into our expression forcT​w^c^\{T\}\\hat\{w\}and subtractingz⋆z^\{\\star\}from both sides:

Runreg​\(c,w^\)≤R⁡\(c,w^\)\+λ2​‖wLP⋆‖2−λ2​‖w^‖2\.R\_\{\\mathrm\{unreg\}\}\(c,\\hat\{w\}\)\\leq R\(c,\\hat\{w\}\)\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|^\{2\}\-\\frac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\.SubstitutingR⁡\(c,w^\)≤ℓRSPO\+​\(c,c^,λ\)−λ​‖w^−wreg⋆‖2R\(c,\\hat\{w\}\)\\leq\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)\-\\lambda\\\|\\hat\{w\}\-w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}from[Theorem1](https://arxiv.org/html/2610.09091#Thmtheorem1)yields:

Runreg​\(c,w^\)≤ℓRSPO\+\+λ2​‖wLP⋆‖2−λ​‖w^−wreg⋆‖2−λ2​‖w^‖2\.R\_\{\\mathrm\{unreg\}\}\(c,\\hat\{w\}\)\\leq\\ell\_\{\\mathrm\{RSPO\+\}\}\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|^\{2\}\-\\lambda\\\|\\hat\{w\}\-w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\-\\frac\{\\lambda\}\{2\}\\\|\\hat\{w\}\\\|^\{2\}\.To maximize the negative quadratic components, letx=w^−wreg⋆x=\\hat\{w\}\-w^\{\\star\}\_\{\\mathrm\{reg\}\}, so‖w^‖2=‖x\+wreg⋆‖2\\\|\\hat\{w\}\\\|^\{2\}=\\\|x\+w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\. Thexx\-dependent terms become−λ​‖x‖2−λ2​‖x\+wreg⋆‖2=−3​λ2​‖x‖2−λ​xT​wreg⋆−λ2​‖wreg⋆‖2\-\\lambda\\\|x\\\|^\{2\}\-\\frac\{\\lambda\}\{2\}\\\|x\+w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}=\-\\frac\{3\\lambda\}\{2\}\\\|x\\\|^\{2\}\-\\lambda x^\{T\}w^\{\\star\}\_\{\\mathrm\{reg\}\}\-\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\. This unconstrained concave quadratic is strictly maximized atx=−w⋆reg/3x=\-w^\{\\star\}\_\{\\mathrm\{reg\}\}/3, yielding a maximum value of−λ3​‖wreg⋆‖2\-\\frac\{\\lambda\}\{3\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\. Substituting this maximum gives us,

Runreg​\(c,w^\)≤ℓRSPO\+​\(c,c^,λ\)\+λ2​‖wLP⋆‖2−λ3​‖wreg⋆‖2\.R\_\{\\mathrm\{unreg\}\}\(c,\\hat\{w\}\)\\leq\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)\+\\frac\{\\lambda\}\{2\}\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|^\{2\}\-\\frac\{\\lambda\}\{3\}\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\.\(23\)Since‖wreg⋆‖2≥0\\\|w^\{\\star\}\_\{\\mathrm\{reg\}\}\\\|^\{2\}\\geq 0and‖wLP⋆‖≤R\\\|w^\{\\star\}\_\{\\mathrm\{LP\}\}\\\|\\leq R, this yields the proof\. ∎

### Proof of[Proposition4](https://arxiv.org/html/2610.09091#Thmproposition4): RSPO\+ Stability under Perturbation ofλ\\lambda

###### Proof\.

We prove the two bounds separately\.

#### Gradient bound[Eq\.5](https://arxiv.org/html/2610.09091#S3.E5)

By[Corollary1](https://arxiv.org/html/2610.09091#Thmcorollary1),∇c^ℓRSPO\+​\(λ\)=−2​\(ΠS​\(uλ\)−ΠS​\(vλ\)\)\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\)=\-2\\bigl\(\\Pi\_\{S\}\(u\_\{\\lambda\}\)\-\\Pi\_\{S\}\(v\_\{\\lambda\}\)\\bigr\), whereuλ=\(c−2​c^\)/λu\_\{\\lambda\}=\(c\-2\\hat\{c\}\)/\\lambdaandvλ=−c/λv\_\{\\lambda\}=\-c/\\lambda\. For two parametersλ1,λ2\>0\\lambda\_\{1\},\\lambda\_\{2\}\>0:

‖∇ℓ​\(λ1\)−∇ℓ​\(λ2\)‖\\displaystyle\\\|\\nabla\\ell\(\\lambda\_\{1\}\)\-\\nabla\\ell\(\\lambda\_\{2\}\)\\\|≤2​\(‖ΠS​\(uλ1\)−ΠS​\(uλ2\)‖\+‖ΠS​\(vλ1\)−ΠS​\(vλ2\)‖\),\\displaystyle\\leq 2\\bigl\(\\\|\\Pi\_\{S\}\(u\_\{\\lambda\_\{1\}\}\)\-\\Pi\_\{S\}\(u\_\{\\lambda\_\{2\}\}\)\\\|\+\\\|\\Pi\_\{S\}\(v\_\{\\lambda\_\{1\}\}\)\-\\Pi\_\{S\}\(v\_\{\\lambda\_\{2\}\}\)\\\|\\bigr\),by the triangle inequality\. SinceΠS\\Pi\_\{S\}is11\-Lipschitz \(non\-expansive\),

‖ΠS​\(uλ1\)−ΠS​\(uλ2\)‖≤‖uλ1−uλ2‖=‖c−2​c^‖​\|1λ1−1λ2\|,\\\|\\Pi\_\{S\}\(u\_\{\\lambda\_\{1\}\}\)\-\\Pi\_\{S\}\(u\_\{\\lambda\_\{2\}\}\)\\\|\\leq\\\|u\_\{\\lambda\_\{1\}\}\-u\_\{\\lambda\_\{2\}\}\\\|=\\\|c\-2\\hat\{c\}\\\|\\,\\bigl\|\\tfrac\{1\}\{\\lambda\_\{1\}\}\-\\tfrac\{1\}\{\\lambda\_\{2\}\}\\bigr\|,and similarly‖ΠS​\(vλ1\)−ΠS​\(vλ2\)‖≤‖c‖​\|1λ1−1λ2\|\\\|\\Pi\_\{S\}\(v\_\{\\lambda\_\{1\}\}\)\-\\Pi\_\{S\}\(v\_\{\\lambda\_\{2\}\}\)\\\|\\leq\\\|c\\\|\\,\|\\frac\{1\}\{\\lambda\_\{1\}\}\-\\frac\{1\}\{\\lambda\_\{2\}\}\|\. Combining gives

‖∇ℓ​\(λ1\)−∇ℓ​\(λ2\)‖≤2​\(‖c−2​c^‖\+‖c‖\)​\|1λ1−1λ2\|\.\\\|\\nabla\\ell\(\\lambda\_\{1\}\)\-\\nabla\\ell\(\\lambda\_\{2\}\)\\\|\\leq 2\\bigl\(\\\|c\-2\\hat\{c\}\\\|\+\\\|c\\\|\\bigr\)\\Bigl\|\\frac\{1\}\{\\lambda\_\{1\}\}\-\\frac\{1\}\{\\lambda\_\{2\}\}\\Bigr\|\.

#### Loss bound[Eq\.4](https://arxiv.org/html/2610.09091#S3.E4)

Write the RSPO\+ loss via its Bregman divergence representation:

ℓ\(λ\)=Fλ,S\(c−2c^\)−Fλ,S\(−c\)−⟨ΠS\(−c/λ\),2c−2c^⟩\.\\ell\(\\lambda\)=F\_\{\\lambda,S\}\(c\-2\\hat\{c\}\)\-F\_\{\\lambda,S\}\(\-c\)\-\\langle\\Pi\_\{S\}\(\-c/\\lambda\),\\,2c\-2\\hat\{c\}\\rangle\.We bound the variation of each component\.

*Envelope terms\.*Using the supremum formFλ,S​\(u\)=sups∈S\{⟨u,s⟩−λ2​‖s‖2\}F\_\{\\lambda,S\}\(u\)=\\sup\_\{s\\in S\}\\\{\\langle u,s\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|s\\\|^\{2\}\\\}, for any fixeduuands∈Ss\\in S:

\|⟨u,s⟩−λ12​‖s‖2\|−\|⟨u,s⟩−λ22​‖s‖2\|≤\|λ1−λ2\|2​‖s‖2≤R22​\|λ1−λ2\|\.\\bigl\|\\langle u,s\\rangle\-\\tfrac\{\\lambda\_\{1\}\}\{2\}\\\|s\\\|^\{2\}\\bigr\|\-\\bigl\|\\langle u,s\\rangle\-\\tfrac\{\\lambda\_\{2\}\}\{2\}\\\|s\\\|^\{2\}\\bigr\|\\leq\\tfrac\{\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\}\{2\}\\\|s\\\|^\{2\}\\leq\\tfrac\{R^\{2\}\}\{2\}\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\.Taking suprema overs∈Ss\\in Son both sides yields\|Fλ1,S​\(u\)−Fλ2,S​\(u\)\|≤R22​\|λ1−λ2\|\|F\_\{\\lambda\_\{1\},S\}\(u\)\-F\_\{\\lambda\_\{2\},S\}\(u\)\|\\leq\\tfrac\{R^\{2\}\}\{2\}\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\. Applying this tou=c−2​c^u=c\-2\\hat\{c\}andu=−cu=\-cand summing:

\|Fλ1,S​\(c−2​c^\)−Fλ2,S​\(c−2​c^\)\|\+\|Fλ1,S​\(−c\)−Fλ2,S​\(−c\)\|≤R2​\|λ1−λ2\|\.\\bigl\|F\_\{\\lambda\_\{1\},S\}\(c\-2\\hat\{c\}\)\-F\_\{\\lambda\_\{2\},S\}\(c\-2\\hat\{c\}\)\\bigr\|\+\\bigl\|F\_\{\\lambda\_\{1\},S\}\(\-c\)\-F\_\{\\lambda\_\{2\},S\}\(\-c\)\\bigr\|\\leq R^\{2\}\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\.
*Linearization term\.*The variation in the inner\-product term is

\|⟨ΠS\(−c/λ1\)−ΠS\(−c/λ2\),2c−2c^⟩\|\\displaystyle\\bigl\|\\langle\\Pi\_\{S\}\(\-c/\\lambda\_\{1\}\)\-\\Pi\_\{S\}\(\-c/\\lambda\_\{2\}\),\\,2c\-2\\hat\{c\}\\rangle\\bigr\|≤2∥c−c^∥∥ΠS\(−c/λ1\)−ΠS\(−c/λ2\)∥\\displaystyle\\leq 2\\\|c\-\\hat\{c\}\\\|\\,\\\|\\Pi\_\{S\}\(\-c/\\lambda\_\{1\}\)\-\\Pi\_\{S\}\(\-c/\\lambda\_\{2\}\)\\\|≤2​‖c−c^‖⋅‖c‖​\|1λ1−1λ2\|,\\displaystyle\\leq 2\\\|c\-\\hat\{c\}\\\|\\cdot\\\|c\\\|\\,\\bigl\|\\tfrac\{1\}\{\\lambda\_\{1\}\}\-\\tfrac\{1\}\{\\lambda\_\{2\}\}\\bigr\|,by non\-expansiveness ofΠS\\Pi\_\{S\}\. Combining the envelope and linearization bounds by the triangle inequality:

\|ℓRSPO\+​\(λ1\)−ℓRSPO\+​\(λ2\)\|≤R2​\|λ1−λ2\|\+2​‖c‖​‖c−c^‖​\|1λ1−1λ2\|\.\|\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{1\}\)\-\\ell\_\{\\mathrm\{RSPO\+\}\}\(\\lambda\_\{2\}\)\|\\leq R^\{2\}\|\\lambda\_\{1\}\-\\lambda\_\{2\}\|\+2\\\|c\\\|\\,\\\|c\-\\hat\{c\}\\\|\\,\\Bigl\|\\frac\{1\}\{\\lambda\_\{1\}\}\-\\frac\{1\}\{\\lambda\_\{2\}\}\\Bigr\|\.∎

### B\.1Proof of[Remark3](https://arxiv.org/html/2610.09091#Thmremark3):MSE\-SPO\+ homotopy

###### Proof\.

Since0∈int⁡\(S\)0\\in\\operatorname\{int\}\(S\), there existsr\>0r\>0such thatB⁡\(0,r\):=\{x∈ℝd:‖x‖<r\}⊆SB\(0,r\):=\\\{x\\in\\mathbb\{R\}^\{d\}:\\\|x\\\|<r\\\}\\subseteq S\. For anyppwith‖p‖<r\\\|p\\\|<rwe havep∈Sp\\in S, and sinceSSis convex and closed the projection satisfiesΠS​\(p\)=p\\Pi\_\{S\}\(p\)=p\. Setu=c−2​c^λu=\\tfrac\{c\-2\\hat\{c\}\}\{\\lambda\}andv=−cλv=\-\\tfrac\{c\}\{\\lambda\}\. Under the boundedness assumptions,

‖u‖=‖c−2​c^‖λ≤Cmax\+2​Bλ≤Cmax\+2​Bλ¯=r,\\\|u\\\|=\\frac\{\\\|c\-2\\hat\{c\}\\\|\}\{\\lambda\}\\leq\\frac\{C\_\{\\max\}\+2B\}\{\\lambda\}\\leq\\frac\{C\_\{\\max\}\+2B\}\{\\bar\{\\lambda\}\}=r,and

‖v‖=‖c‖λ≤Cmaxλ≤Cmaxλ¯<r\.\\\|v\\\|=\\frac\{\\\|c\\\|\}\{\\lambda\}\\leq\\frac\{C\_\{\\max\}\}\{\\lambda\}\\leq\\frac\{C\_\{\\max\}\}\{\\bar\{\\lambda\}\}<r\.Henceu,v∈Su,v\\in S, soΠS​\(u\)=u\\Pi\_\{S\}\(u\)=uandΠS​\(v\)=v\\Pi\_\{S\}\(v\)=v\. From[Proposition2](https://arxiv.org/html/2610.09091#Thmproposition2), the RSPO\+ loss is

ℓRSPO\+​\(c,c^,λ\)=λ2​\(‖u−ΠS​\(v\)‖2−‖u−ΠS​\(u\)‖2\)\.\\ell\_\{\\mathrm\{RSPO\+\}\}\(c,\\hat\{c\};\\lambda\)=\\frac\{\\lambda\}\{2\}\\bigl\(\\\|u\-\\Pi\_\{S\}\(v\)\\\|^\{2\}\-\\\|u\-\\Pi\_\{S\}\(u\)\\\|^\{2\}\\bigr\)\.SubstitutingΠS​\(u\)=u\\Pi\_\{S\}\(u\)=uandΠS​\(v\)=v\\Pi\_\{S\}\(v\)=vyields

ℓRSPO\+=λ2​\(‖u−v‖2−0\)=λ2​‖c−2​c^λ\+cλ‖2=λ2⋅4​‖c−c^‖2λ2=2λ​‖c−c^‖2\.\\ell\_\{\\mathrm\{RSPO\+\}\}=\\frac\{\\lambda\}\{2\}\\bigl\(\\\|u\-v\\\|^\{2\}\-0\\bigr\)=\\frac\{\\lambda\}\{2\}\\left\\\|\\frac\{c\-2\\hat\{c\}\}\{\\lambda\}\+\\frac\{c\}\{\\lambda\}\\right\\\|^\{2\}=\\frac\{\\lambda\}\{2\}\\cdot\\frac\{4\\\|c\-\\hat\{c\}\\\|^\{2\}\}\{\\lambda^\{2\}\}=\\frac\{2\}\{\\lambda\}\\\|c\-\\hat\{c\}\\\|^\{2\}\.From[Corollary1](https://arxiv.org/html/2610.09091#Thmcorollary1),∇c^ℓRSPO\+=−2​\(ΠS​\(u\)−ΠS​\(v\)\)\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}=\-2\\bigl\(\\Pi\_\{S\}\(u\)\-\\Pi\_\{S\}\(v\)\\bigr\)\. Substituting the identity projections:

∇c^ℓRSPO\+=−2​\(u−v\)=−2​\(c−2​c^λ\+cλ\)=−4λ​\(c−c^\)\.\\nabla\_\{\\hat\{c\}\}\\,\\ell\_\{\\mathrm\{RSPO\+\}\}=\-2\(u\-v\)=\-2\\left\(\\frac\{c\-2\\hat\{c\}\}\{\\lambda\}\+\\frac\{c\}\{\\lambda\}\\right\)=\-\\frac\{4\}\{\\lambda\}\(c\-\\hat\{c\}\)\.This is exactly∇c^\[2λ​‖c−c^‖2\]\\nabla\_\{\\hat\{c\}\}\\bigl\[\\tfrac\{2\}\{\\lambda\}\\\|c\-\\hat\{c\}\\\|^\{2\}\\bigr\]\. ∎

### B\.2Proof of[Theorem3](https://arxiv.org/html/2610.09091#Thmtheorem3): RSPO\+ Heterogeneity Bound

We prove Theorem[3](https://arxiv.org/html/2610.09091#Thmtheorem3)through three supporting lemmas and an algebraic decomposition of the RSPO\+ loss\. Throughout, we use∥⋅∥=∥⋅∥\\\|\\cdot\\\|=\\\|\\cdot\\\|and writevλ,S⋆\(c\):=ΠS\(−c/λ\)v^\{\\star\}\_\{\\lambda,S\}\(c\):=\\Pi\_\{S\}\(\-c/\\lambda\)for the optimal regularized decision\.

#### Loss decomposition\.

Recall from Proposition[2](https://arxiv.org/html/2610.09091#Thmproposition2)that the RSPO\+ loss is the generalized Bregman divergence of the Moreau envelope of the support function:

ℓS​\(c^,c\)=Fλ,S​\(c−2​c^\)−Fλ,S​\(−c\)−⟨∇Fλ,S​\(−c\),\(c−2​c^\)−\(−c\)⟩\.\\ell\_\{S\}\(\\hat\{c\},c\)\\;=\\;F\_\{\\lambda,S\}\(c\-2\\hat\{c\}\)\\;\-\\;F\_\{\\lambda,S\}\(\-c\)\\;\-\\;\\bigl\\langle\\nabla F\_\{\\lambda,S\}\(\-c\),\\;\(c\-2\\hat\{c\}\)\-\(\-c\)\\bigr\\rangle\.\(24\)Since the Moreau envelope of the support function satisfies∇Fλ,S\(−c\)=ΠS\(−c/λ\)=vλ,S⋆\(c\)\\nabla F\_\{\\lambda,S\}\(\-c\)=\\Pi\_\{S\}\(\-c/\\lambda\)=v^\{\\star\}\_\{\\lambda,S\}\(c\)and evaluates toFλ,S​\(−c\)=⟨−c,vλ,S⋆​\(c\)⟩−λ2​‖vλ,S⋆​\(c\)‖2F\_\{\\lambda,S\}\(\-c\)=\\langle\-c,v^\{\\star\}\_\{\\lambda,S\}\(c\)\\rangle\-\\tfrac\{\\lambda\}\{2\}\\\|v^\{\\star\}\_\{\\lambda,S\}\(c\)\\\|^\{2\}, we expand the linearization term as

−⟨vλ,S⋆​\(c\),2​c−2​c^⟩\\displaystyle\-\\bigl\\langle v^\{\\star\}\_\{\\lambda,S\}\(c\),\\;2c\-2\\hat\{c\}\\bigr\\rangle=−2​cT​vλ,S⋆​\(c\)\+2​c^T​vλ,S⋆​\(c\)\\displaystyle=\-2c^\{T\}v^\{\\star\}\_\{\\lambda,S\}\(c\)\+2\\hat\{c\}^\{T\}v^\{\\star\}\_\{\\lambda,S\}\(c\)=2​\(Fλ,S​\(−c\)\+λ2​‖vλ,S⋆​\(c\)‖2\)\+2​c^T​vλ,S⋆​\(c\)\.\\displaystyle=2\\Bigl\(F\_\{\\lambda,S\}\(\-c\)\+\\tfrac\{\\lambda\}\{2\}\\\|v^\{\\star\}\_\{\\lambda,S\}\(c\)\\\|^\{2\}\\Bigr\)\+2\\hat\{c\}^\{T\}v^\{\\star\}\_\{\\lambda,S\}\(c\)\.Substituting back into \([24](https://arxiv.org/html/2610.09091#A2.E24)\) yields

ℓS​\(c^,c\)=Fλ,S​\(c−2​c^\)\+Fλ,S​\(−c\)⏟Mλ,S​\(c^,c\)\+2​c^T​vλ,S⋆​\(c\)\+λ​‖vλ,S⋆​\(c\)‖2⏟Bλ,S​\(c^,c\)\.\\ell\_\{S\}\(\\hat\{c\},c\)\\;=\\;\\underbrace\{F\_\{\\lambda,S\}\(c\-2\\hat\{c\}\)\+F\_\{\\lambda,S\}\(\-c\)\}\_\{\\displaystyle M\_\{\\lambda,S\}\(\\hat\{c\},\\,c\)\}\\;\+\\;\\underbrace\{2\\hat\{c\}^\{T\}v^\{\\star\}\_\{\\lambda,S\}\(c\)\+\\lambda\\\|v^\{\\star\}\_\{\\lambda,S\}\(c\)\\\|^\{2\}\}\_\{\\displaystyle B\_\{\\lambda,S\}\(\\hat\{c\},\\,c\)\}\.\(25\)This decomposition mirrors the SPO\+ structure and allows us to bound the envelope termMMand the decision termBBindependently under perturbations toccandSS\.

#### Hausdorff distance\.

Unlike SPO\+, the origin\-centered quadratic penaltyλ2​‖w‖2\\frac\{\\lambda\}\{2\}\\\|w\\\|^\{2\}in the Moreau envelope breaks translation invariance\. We therefore measure set perturbations via the standard Hausdorff distanceδ:=dH​\(Sref,Scl\)\\delta:=d\_\{H\}\(S\_\{\\mathrm\{ref\}\},S\_\{\\mathrm\{cl\}\}\), and define the joint bounding radiusRmax:=max⁡\{R⁡\(Sref\),R⁡\(Scl\)\}R\_\{\\max\}:=\\max\\\{R\(S\_\{\\mathrm\{ref\}\}\),\\,R\(S\_\{\\mathrm\{cl\}\}\)\\\}\.

###### Lemma 2\(Sensitivity of the Moreau envelope to set perturbations\)\.

LetS1,S2⊆ℝdS\_\{1\},S\_\{2\}\\subseteq\\mathbb\{R\}^\{d\}be nonempty compact convex sets withmax⁡\{R⁡\(S1\),R⁡\(S2\)\}≤Rmax\\max\\\{R\(S\_\{1\}\),R\(S\_\{2\}\)\\\}\\leq R\_\{\\max\}andδ=dH​\(S1,S2\)\\delta=d\_\{H\}\(S\_\{1\},S\_\{2\}\)\. Then for anyu∈ℝdu\\in\\mathbb\{R\}^\{d\}:

\|Fλ,S1​\(u\)−Fλ,S2​\(u\)\|≤δ⁡\(‖u‖\+λ​Rmax\)\.\\bigl\|F\_\{\\lambda,S\_\{1\}\}\(u\)\-F\_\{\\lambda,S\_\{2\}\}\(u\)\\bigr\|\\;\\leq\\;\\delta\\bigl\(\\\|u\\\|\+\\lambda R\_\{\\max\}\\bigr\)\.

###### Proof\.

Letw1=arg⁡maxw∈S1⁡\(uT​w−λ2​‖w‖2\)w\_\{1\}=\\arg\\max\_\{w\\in S\_\{1\}\}\\bigl\(u^\{T\}w\-\\tfrac\{\\lambda\}\{2\}\\\|w\\\|^\{2\}\\bigr\)\. By definition of the Hausdorff distance, there existsw~2∈S2\\tilde\{w\}\_\{2\}\\in S\_\{2\}with‖w1−w~2‖≤δ\\\|w\_\{1\}\-\\tilde\{w\}\_\{2\}\\\|\\leq\\delta\. Sincew1w\_\{1\}is optimal forS1S\_\{1\}andw~2\\tilde\{w\}\_\{2\}is feasible forS2S\_\{2\},

Fλ,S1​\(u\)−Fλ,S2​\(u\)\\displaystyle F\_\{\\lambda,S\_\{1\}\}\(u\)\-F\_\{\\lambda,S\_\{2\}\}\(u\)≤\(uT​w1−λ2​‖w1‖2\)−\(uT​w~2−λ2​‖w~2‖2\)\\displaystyle\\leq\\bigl\(u^\{T\}w\_\{1\}\-\\tfrac\{\\lambda\}\{2\}\\\|w\_\{1\}\\\|^\{2\}\\bigr\)\-\\bigl\(u^\{T\}\\tilde\{w\}\_\{2\}\-\\tfrac\{\\lambda\}\{2\}\\\|\\tilde\{w\}\_\{2\}\\\|^\{2\}\\bigr\)=⟨u−λ2​\(w1\+w~2\),w1−w~2⟩\\displaystyle=\\bigl\\langle u\-\\tfrac\{\\lambda\}\{2\}\(w\_\{1\}\+\\tilde\{w\}\_\{2\}\),\\;w\_\{1\}\-\\tilde\{w\}\_\{2\}\\bigr\\rangle≤\(‖u‖\+λ2​\(‖w1‖\+‖w~2‖\)\)​‖w1−w~2‖\\displaystyle\\leq\\bigl\(\\\|u\\\|\+\\tfrac\{\\lambda\}\{2\}\(\\\|w\_\{1\}\\\|\+\\\|\\tilde\{w\}\_\{2\}\\\|\)\\bigr\)\\\|w\_\{1\}\-\\tilde\{w\}\_\{2\}\\\|≤\(‖u‖\+λ​Rmax\)​δ,\\displaystyle\\leq\\bigl\(\\\|u\\\|\+\\lambda R\_\{\\max\}\\bigr\)\\,\\delta,where the second line uses the identityaT​a−bT​b=⟨a\+b,a−b⟩a^\{T\}a\-b^\{T\}b=\\langle a\+b,a\-b\\rangleapplied to the quadratic terms, and the last line uses‖w1‖,‖w~2‖≤Rmax\\\|w\_\{1\}\\\|,\\\|\\tilde\{w\}\_\{2\}\\\|\\leq R\_\{\\max\}\. Swapping the roles ofS1S\_\{1\}andS2S\_\{2\}and taking the maximum of both directions yields the absolute\-value bound\. ∎

###### Lemma 3\(Diameter\-Lipschitzness of the joint envelope term\)\.

Under the compactness assumption of[Assumption1](https://arxiv.org/html/2610.09091#Thmassumption1), for anyc1,c2,c^∈ℝdc\_\{1\},c\_\{2\},\\hat\{c\}\\in\\mathbb\{R\}^\{d\}and fixed compact convex setSS:

\|Mλ,S​\(c^,c1\)−Mλ,S​\(c^,c2\)\|≤D⁡\(S\)​‖c1−c2‖\.\\bigl\|M\_\{\\lambda,S\}\(\\hat\{c\},c\_\{1\}\)\-M\_\{\\lambda,S\}\(\\hat\{c\},c\_\{2\}\)\\bigr\|\\;\\leq\\;D\(S\)\\,\\\|c\_\{1\}\-c\_\{2\}\\\|\.

###### Proof\.

By the maximization definition,Fλ,S​\(x\)−Fλ,S​\(y\)≤maxw∈S⁡\(x−y\)T​wF\_\{\\lambda,S\}\(x\)\-F\_\{\\lambda,S\}\(y\)\\leq\\max\_\{w\\in S\}\(x\-y\)^\{T\}wfor anyx,yx,y\(the quadratic terms cancel in the upper bound by using the same maximizer for both terms\)\. Applying this to both components ofMλ,SM\_\{\\lambda,S\}:

Fλ,S​\(c1−2​c^\)−Fλ,S​\(c2−2​c^\)\\displaystyle F\_\{\\lambda,S\}\(c\_\{1\}\-2\\hat\{c\}\)\-F\_\{\\lambda,S\}\(c\_\{2\}\-2\\hat\{c\}\)≤maxw∈S⁡\(c1−c2\)T​w,\\displaystyle\\leq\\max\_\{w\\in S\}\(c\_\{1\}\-c\_\{2\}\)^\{T\}w,Fλ,S​\(−c1\)−Fλ,S​\(−c2\)\\displaystyle F\_\{\\lambda,S\}\(\-c\_\{1\}\)\-F\_\{\\lambda,S\}\(\-c\_\{2\}\)≤maxv∈S⁡\(c2−c1\)T​v=maxv∈S⁡\(c1−c2\)T​\(−v\)\.\\displaystyle\\leq\\max\_\{v\\in S\}\(c\_\{2\}\-c\_\{1\}\)^\{T\}v\\;=\\;\\max\_\{v\\in S\}\(c\_\{1\}\-c\_\{2\}\)^\{T\}\(\-v\)\.Summing these inequalities:

Mλ,S​\(c^,c1\)−Mλ,S​\(c^,c2\)\\displaystyle M\_\{\\lambda,S\}\(\\hat\{c\},c\_\{1\}\)\-M\_\{\\lambda,S\}\(\\hat\{c\},c\_\{2\}\)≤maxw,v∈S⁡\(c1−c2\)T​\(w−v\)\\displaystyle\\leq\\max\_\{w,v\\in S\}\(c\_\{1\}\-c\_\{2\}\)^\{T\}\(w\-v\)≤‖c1−c2‖⋅supw,v∈S‖w−v‖=‖c1−c2‖​D​\(S\)\.\\displaystyle\\leq\\\|c\_\{1\}\-c\_\{2\}\\\|\\cdot\\sup\_\{w,v\\in S\}\\\|w\-v\\\|\\;=\\;\\\|c\_\{1\}\-c\_\{2\}\\\|\\,D\(S\)\.Swappingc1c\_\{1\}andc2c\_\{2\}and taking the maximum gives the absolute\-value bound\. ∎

###### Lemma 4\(Set stability of Euclidean projections\)\.

LetS1,S2⊆ℝdS\_\{1\},S\_\{2\}\\subseteq\\mathbb\{R\}^\{d\}be nonempty compact convex sets withmax⁡\{R⁡\(S1\),R⁡\(S2\)\}≤Rmax\\max\\\{R\(S\_\{1\}\),R\(S\_\{2\}\)\\\}\\leq R\_\{\\max\}andδ=dH​\(S1,S2\)\\delta=d\_\{H\}\(S\_\{1\},S\_\{2\}\)\. For anyp∈ℝdp\\in\\mathbb\{R\}^\{d\}:

‖ΠS1​\(p\)−ΠS2​\(p\)‖≤2​δ​\(‖p‖\+Rmax\)\.\\\|\\Pi\_\{S\_\{1\}\}\(p\)\-\\Pi\_\{S\_\{2\}\}\(p\)\\\|\\;\\leq\\;\\sqrt\{2\\delta\\bigl\(\\\|p\\\|\+R\_\{\\max\}\\bigr\)\}\.

###### Proof\.

Letzi=ΠSi​\(p\)z\_\{i\}=\\Pi\_\{S\_\{i\}\}\(p\)fori=1,2i=1,2\. By the Hausdorff distance bound, there exist proxy pointsz~1∈S1\\tilde\{z\}\_\{1\}\\in S\_\{1\}andz~2∈S2\\tilde\{z\}\_\{2\}\\in S\_\{2\}satisfying‖z2−z~1‖≤δ\\\|z\_\{2\}\-\\tilde\{z\}\_\{1\}\\\|\\leq\\deltaand‖z1−z~2‖≤δ\\\|z\_\{1\}\-\\tilde\{z\}\_\{2\}\\\|\\leq\\delta\.

The characterization of Euclidean projections onto closed convex sets gives the variational inequalities

⟨p−z1,z~1−z1⟩≤0,⟨p−z2,z~2−z2⟩≤0\.\\langle p\-z\_\{1\},\\;\\tilde\{z\}\_\{1\}\-z\_\{1\}\\rangle\\leq 0,\\qquad\\langle p\-z\_\{2\},\\;\\tilde\{z\}\_\{2\}\-z\_\{2\}\\rangle\\leq 0\.We expand‖z1−z2‖2\\\|z\_\{1\}\-z\_\{2\}\\\|^\{2\}by inserting the proxy points:

‖z1−z2‖2\\displaystyle\\\|z\_\{1\}\-z\_\{2\}\\\|^\{2\}=⟨z1−z2,z1−z2⟩\\displaystyle=\\langle z\_\{1\}\-z\_\{2\},\\;z\_\{1\}\-z\_\{2\}\\rangle=⟨p−z2,z1−z2⟩\+⟨p−z1,z2−z1⟩\\displaystyle=\\langle p\-z\_\{2\},\\;z\_\{1\}\-z\_\{2\}\\rangle\+\\langle p\-z\_\{1\},\\;z\_\{2\}\-z\_\{1\}\\rangle=\[⟨p−z2,z1−z~2⟩\+⟨p−z2,z~2−z2⟩\]\\displaystyle=\\bigl\[\\langle p\-z\_\{2\},\\;z\_\{1\}\-\\tilde\{z\}\_\{2\}\\rangle\+\\langle p\-z\_\{2\},\\;\\tilde\{z\}\_\{2\}\-z\_\{2\}\\rangle\\bigr\]\+\[⟨p−z1,z2−z~1⟩\+⟨p−z1,z~1−z1⟩\]\\displaystyle\\quad\+\\bigl\[\\langle p\-z\_\{1\},\\;z\_\{2\}\-\\tilde\{z\}\_\{1\}\\rangle\+\\langle p\-z\_\{1\},\\;\\tilde\{z\}\_\{1\}\-z\_\{1\}\\rangle\\bigr\]≤⟨p−z2,z1−z~2⟩\+0\+⟨p−z1,z2−z~1⟩\+0\\displaystyle\\leq\\langle p\-z\_\{2\},\\;z\_\{1\}\-\\tilde\{z\}\_\{2\}\\rangle\+0\+\\langle p\-z\_\{1\},\\;z\_\{2\}\-\\tilde\{z\}\_\{1\}\\rangle\+0≤‖p−z2‖​‖z1−z~2‖\+‖p−z1‖​‖z2−z~1‖\\displaystyle\\leq\\\|p\-z\_\{2\}\\\|\\,\\\|z\_\{1\}\-\\tilde\{z\}\_\{2\}\\\|\+\\\|p\-z\_\{1\}\\\|\\,\\\|z\_\{2\}\-\\tilde\{z\}\_\{1\}\\\|≤δ⁡\(‖p−z1‖\+‖p−z2‖\)\.\\displaystyle\\leq\\delta\\bigl\(\\\|p\-z\_\{1\}\\\|\+\\\|p\-z\_\{2\}\\\|\\bigr\)\.Sincezi∈Siz\_\{i\}\\in S\_\{i\}with‖zi‖≤Rmax\\\|z\_\{i\}\\\|\\leq R\_\{\\max\}, the triangle inequality gives‖p−zi‖≤‖p‖\+Rmax\\\|p\-z\_\{i\}\\\|\\leq\\\|p\\\|\+R\_\{\\max\}for eachii\. Therefore,

‖z1−z2‖2≤2​δ​\(‖p‖\+Rmax\)\.\\\|z\_\{1\}\-z\_\{2\}\\\|^\{2\}\\;\\leq\\;2\\delta\\bigl\(\\\|p\\\|\+R\_\{\\max\}\\bigr\)\.Taking the square root completes the proof\. ∎

###### Proof of Theorem[3](https://arxiv.org/html/2610.09091#Thmtheorem3)\.

Using decomposition \([25](https://arxiv.org/html/2610.09091#A2.E25)\), the total discrepancy satisfies the triangle inequality:

\|ℓSref​\(c^,cref\)−ℓScl​\(c^,ccl\)\|≤\|Mλ,Sref​\(c^,cref\)−Mλ,Scl​\(c^,ccl\)\|\+\|Bλ,Sref​\(c^,cref\)−Bλ,Scl​\(c^,ccl\)\|\.\\bigl\|\\ell\_\{S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-\\ell\_\{S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\\;\\leq\\;\\bigl\|M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-M\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\\;\+\\;\\bigl\|B\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-B\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\.

#### Step 1: BoundingT⁡\(c^\)T\(\\hat\{c\}\)\.

We add and subtract the intermediate termMλ,Sref​\(c^,ccl\)M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\):

\|Mλ,Sref​\(c^,cref\)−Mλ,Scl​\(c^,ccl\)\|\\displaystyle\\bigl\|M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-M\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|≤\|Mλ,Sref​\(c^,cref\)−Mλ,Sref​\(c^,ccl\)\|⏟objective shift at fixed​Sref\+\|Mλ,Sref​\(c^,ccl\)−Mλ,Scl​\(c^,ccl\)\|⏟set shift at fixed​ccl\.\\displaystyle\\quad\\leq\\underbrace\{\\bigl\|M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\}\_\{\\text\{objective shift at fixed \}S\_\{\\mathrm\{ref\}\}\}\\;\+\\;\\underbrace\{\\bigl\|M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\-M\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\}\_\{\\text\{set shift at fixed \}c\_\{\\mathrm\{cl\}\}\}\.By Lemma[3](https://arxiv.org/html/2610.09091#Thmlemma3), the objective\-shift term is bounded byDref​cdD\_\{\\mathrm\{ref\}\}\\,c\_\{d\}\.

For the set\-shift term, recall thatMλ,S​\(c^,c\)=Fλ,S​\(c−2​c^\)\+Fλ,S​\(−c\)M\_\{\\lambda,S\}\(\\hat\{c\},c\)=F\_\{\\lambda,S\}\(c\-2\\hat\{c\}\)\+F\_\{\\lambda,S\}\(\-c\)\. Applying Lemma[2](https://arxiv.org/html/2610.09091#Thmlemma2)to each envelope evaluation independently at argumentsu1=ccl−2​c^u\_\{1\}=c\_\{\\mathrm\{cl\}\}\-2\\hat\{c\}andu2=−cclu\_\{2\}=\-c\_\{\\mathrm\{cl\}\}:

\|Mλ,Sref​\(c^,ccl\)−Mλ,Scl​\(c^,ccl\)\|\\displaystyle\\bigl\|M\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\-M\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|≤δ⁡\(‖ccl−2​c^‖\+λ​Rmax\)\+δ⁡\(‖ccl‖\+λ​Rmax\)\\displaystyle\\leq\\delta\\bigl\(\\\|c\_\{\\mathrm\{cl\}\}\-2\\hat\{c\}\\\|\+\\lambda R\_\{\\max\}\\bigr\)\+\\delta\\bigl\(\\\|c\_\{\\mathrm\{cl\}\}\\\|\+\\lambda R\_\{\\max\}\\bigr\)=δ⁡\(‖ccl−2​c^‖\+‖ccl‖\+2​λ​Rmax\)\.\\displaystyle=\\delta\\bigl\(\\\|c\_\{\\mathrm\{cl\}\}\-2\\hat\{c\}\\\|\+\\\|c\_\{\\mathrm\{cl\}\}\\\|\+2\\lambda R\_\{\\max\}\\bigr\)\.Combining these gives:

\|Mref−Mcl\|≤Dref​cd\+δ⁡\(‖ccl−2​c^‖\+‖ccl‖\+2​λ​Rmax\)\.\\bigl\|M\_\{\\mathrm\{ref\}\}\-M\_\{\\mathrm\{cl\}\}\\bigr\|\\;\\leq\\;D\_\{\\mathrm\{ref\}\}\\,c\_\{d\}\+\\delta\\bigl\(\\\|c\_\{\\mathrm\{cl\}\}\-2\\hat\{c\}\\\|\+\\\|c\_\{\\mathrm\{cl\}\}\\\|\+2\\lambda R\_\{\\max\}\\bigr\)\.Alternatively, one may pivot throughMλ,Scl​\(c^,cref\)M\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)instead, yielding the analogous bound withDclD\_\{\\mathrm\{cl\}\},‖cref−2​c^‖\\\|c\_\{\\mathrm\{ref\}\}\-2\\hat\{c\}\\\|, and‖cref‖\\\|c\_\{\\mathrm\{ref\}\}\\\|\. Taking the minimum of both pivots givesT⁡\(c^\)T\(\\hat\{c\}\)\.

#### Step 2: BoundingB⁡\(c^\)B\(\\hat\{c\}\)\.

Writevi:=vλ,Si⋆\(ci\)=ΠSi\(−ci/λ\)v\_\{i\}:=v^\{\\star\}\_\{\\lambda,S\_\{i\}\}\(c\_\{i\}\)=\\Pi\_\{S\_\{i\}\}\(\-c\_\{i\}/\\lambda\)fori∈\{ref,cl\}i\\in\\\{\\mathrm\{ref\},\\mathrm\{cl\}\\\}\. The decision\-term gap factors as:

\|Bλ,Sref​\(c^,cref\)−Bλ,Scl​\(c^,ccl\)\|\\displaystyle\\bigl\|B\_\{\\lambda,S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-B\_\{\\lambda,S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|=\|2​c^T​vref\+λ​‖vref‖2−2​c^T​vcl−λ​‖vcl‖2\|\\displaystyle=\\bigl\|2\\hat\{c\}^\{T\}v\_\{\\mathrm\{ref\}\}\+\\lambda\\\|v\_\{\\mathrm\{ref\}\}\\\|^\{2\}\-2\\hat\{c\}^\{T\}v\_\{\\mathrm\{cl\}\}\-\\lambda\\\|v\_\{\\mathrm\{cl\}\}\\\|^\{2\}\\bigr\|=\|2​c^T​\(vref−vcl\)\+λ⁡⟨vref\+vcl,vref−vcl⟩\|\\displaystyle=\\bigl\|2\\hat\{c\}^\{T\}\(v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\)\+\\lambda\\langle v\_\{\\mathrm\{ref\}\}\+v\_\{\\mathrm\{cl\}\},\\;v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\rangle\\bigr\|=\|⟨2​c^\+λ⁡\(vref\+vcl\),vref−vcl⟩\|\\displaystyle=\\bigl\|\\langle 2\\hat\{c\}\+\\lambda\(v\_\{\\mathrm\{ref\}\}\+v\_\{\\mathrm\{cl\}\}\),\\;v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\rangle\\bigr\|≤\(2​‖c^‖\+λ​‖vref\+vcl‖\)​‖vref−vcl‖\\displaystyle\\leq\\bigl\(2\\\|\\hat\{c\}\\\|\+\\lambda\\\|v\_\{\\mathrm\{ref\}\}\+v\_\{\\mathrm\{cl\}\}\\\|\\bigr\)\\,\\\|v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\\|≤\(2​‖c^‖\+2​λ​Rmax\)​‖vref−vcl‖,\\displaystyle\\leq\\bigl\(2\\\|\\hat\{c\}\\\|\+2\\lambda R\_\{\\max\}\\bigr\)\\,\\\|v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\\|,where the last line uses‖vref\+vcl‖≤‖vref‖\+‖vcl‖≤2​Rmax\\\|v\_\{\\mathrm\{ref\}\}\+v\_\{\\mathrm\{cl\}\}\\\|\\leq\\\|v\_\{\\mathrm\{ref\}\}\\\|\+\\\|v\_\{\\mathrm\{cl\}\}\\\|\\leq 2R\_\{\\max\}since eachvi∈Siv\_\{i\}\\in S\_\{i\}\.

It remains to bound‖vref−vcl‖\\\|v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\\|\. We decompose via the triangle inequality, pivoting through the intermediate projectionΠSref\(−ccl/λ\)\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\):

‖vref−vcl‖\\displaystyle\\\|v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\\|=∥ΠSref\(−cref/λ\)−ΠScl\(−ccl/λ\)∥\\displaystyle=\\bigl\\\|\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{ref\}\}/\\lambda\)\-\\Pi\_\{S\_\{\\mathrm\{cl\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\\bigr\\\|≤∥ΠSref\(−cref/λ\)−ΠSref\(−ccl/λ\)∥⏟objective shift, fixed set\+∥ΠSref\(−ccl/λ\)−ΠScl\(−ccl/λ\)∥⏟set shift, fixed objective\.\\displaystyle\\leq\\underbrace\{\\bigl\\\|\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{ref\}\}/\\lambda\)\-\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\\bigr\\\|\}\_\{\\text\{objective shift, fixed set\}\}\\;\+\\;\\underbrace\{\\bigl\\\|\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\-\\Pi\_\{S\_\{\\mathrm\{cl\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\\bigr\\\|\}\_\{\\text\{set shift, fixed objective\}\}\.The first term is bounded by‖cref−ccl‖/λ=cd/λ\\\|c\_\{\\mathrm\{ref\}\}\-c\_\{\\mathrm\{cl\}\}\\\|/\\lambda=c\_\{d\}/\\lambdasince Euclidean projection onto a convex set is11\-Lipschitz \(firmly non\-expansive\)\. The second term is bounded by Lemma[4](https://arxiv.org/html/2610.09091#Thmlemma4)evaluated atp=−ccl/λp=\-c\_\{\\mathrm\{cl\}\}/\\lambda:

∥ΠSref\(−ccl/λ\)−ΠScl\(−ccl/λ\)∥≤2​δ​\(‖ccl‖λ\+Rmax\)\.\\bigl\\\|\\Pi\_\{S\_\{\\mathrm\{ref\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\-\\Pi\_\{S\_\{\\mathrm\{cl\}\}\}\(\-c\_\{\\mathrm\{cl\}\}/\\lambda\)\\bigr\\\|\\;\\leq\\;\\sqrt\{2\\delta\\Bigl\(\\frac\{\\\|c\_\{\\mathrm\{cl\}\}\\\|\}\{\\lambda\}\+R\_\{\\max\}\\Bigr\)\}\.Combining yields the decision bound:

‖vref−vcl‖≤cdλ\+2​δ​\(‖ccl‖λ\+Rmax\)\.\\\|v\_\{\\mathrm\{ref\}\}\-v\_\{\\mathrm\{cl\}\}\\\|\\;\\leq\\;\\frac\{c\_\{d\}\}\{\\lambda\}\+\\sqrt\{2\\delta\\Bigl\(\\frac\{\\\|c\_\{\\mathrm\{cl\}\}\\\|\}\{\\lambda\}\+R\_\{\\max\}\\Bigr\)\}\.Alternatively, pivoting throughΠScl\(−cref/λ\)\\Pi\_\{S\_\{\\mathrm\{cl\}\}\}\(\-c\_\{\\mathrm\{ref\}\}/\\lambda\)gives the bound with‖cref‖\\\|c\_\{\\mathrm\{ref\}\}\\\|replacing‖ccl‖\\\|c\_\{\\mathrm\{cl\}\}\\\|\. Taking the minimum of both pivots and multiplying by the prefactor givesB⁡\(c^\)B\(\\hat\{c\}\)\.

#### Combining\.

Summing the bounds from Steps 1 and 2 completes the proof:

\|ℓSref​\(c^,cref\)−ℓScl​\(c^,ccl\)\|≤T⁡\(c^\)\+B⁡\(c^\)\.\\bigl\|\\ell\_\{S\_\{\\mathrm\{ref\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{ref\}\}\)\-\\ell\_\{S\_\{\\mathrm\{cl\}\}\}\(\\hat\{c\},c\_\{\\mathrm\{cl\}\}\)\\bigr\|\\;\\leq\\;T\(\\hat\{c\}\)\+B\(\\hat\{c\}\)\.∎

#### Asymptotic comparison with SPO\+\.

We verify the claims in Table[2](https://arxiv.org/html/2610.09091#S3.T2)by examining the limiting behavior\. Ascd→0c\_\{d\}\\to 0andδ→0\\delta\\to 0:

- •RSPO\+:BothT⁡\(c^\)T\(\\hat\{c\}\)andB⁡\(c^\)B\(\\hat\{c\}\)vanish sinceT=𝒪⁡\(cd\+δ\)T=\\mathcal\{O\}\(c\_\{d\}\+\\delta\)andB=𝒪⁡\(cd/λ\+δ\)B=\\mathcal\{O\}\(c\_\{d\}/\\lambda\+\\sqrt\{\\delta\}\), giving a combined rate of𝒪⁡\(cd\+δ\)\\mathcal\{O\}\(c\_\{d\}\+\\sqrt\{\\delta\}\)\. No geometric assumptions onSSare required\.
- •SPO\+:The heterogeneity bound takes the formH⁡\(c^\)=𝒪⁡\(cd\+δN\+Dmin\)H\(\\hat\{c\}\)=\\mathcal\{O\}\(c\_\{d\}\+\\delta\_\{N\}\+D\_\{\\min\}\), whereDmin:=min⁡\{D⁡\(Sref\),D⁡\(Scl\)\}D\_\{\\min\}:=\\min\\\{D\(S\_\{\\mathrm\{ref\}\}\),D\(S\_\{\\mathrm\{cl\}\}\)\\\}persists as an irreducible bias even whencd=δN=0c\_\{d\}=\\delta\_\{N\}=0, unless the feasible sets are strongly convex \(in which caseDminD\_\{\\min\}can be replaced by𝒪⁡\(δN\)\\mathcal\{O\}\(\\sqrt\{\\delta\_\{N\}\}\)\)\.

## Appendix CExperimental Details

This section provides the full experimental details of the setup described in[Section4](https://arxiv.org/html/2610.09091#S4)\.

Experiments were conducted on AMD EPYC Milan CPU nodes with 128 cores per node and 528 GB RAM\. The total wall\-clock runtime for the reported experiments was approximately 250 hours\. For the Warcraft case study, each seed\-method run took approximately 4 hours for MSE and SPO\+ and approximately 6 hours for RSPO\+ and the other DFL methods\. For the synthetic knapsack experiments, one full heterogeneity\-design run covering the three main methods \(MSE, SPO\+, RSPO\+\) and three heterogeneity levels across three axes and five seeds took approximately 90 minutes\. The PJM experiments took approximately 2 minutes for five seeds\.

### C\.1Shared Configuration

For all experiments and baselines, we use a federated warm\-start then local personalization\. We use a shallow neural network with 64 neurons for the fractional knapsack and PJM experiments\. For the Warcraft experiment, we use a truncated ResNet\-18\[[62](https://arxiv.org/html/2610.09091#bib.bib47)\], as per the literature\[[24](https://arxiv.org/html/2610.09091#bib.bib35)\]\. We implement our method and baselines using theInferOpt\.jlframework\[[56](https://arxiv.org/html/2610.09091#bib.bib48)\]inJulia\[[63](https://arxiv.org/html/2610.09091#bib.bib50)\]with theLux\.jl\[[64](https://arxiv.org/html/2610.09091#bib.bib49)\]deep learning package\. We use theGurobi\[[65](https://arxiv.org/html/2610.09091#bib.bib52)\]through theJuMPframework\[[66](https://arxiv.org/html/2610.09091#bib.bib51)\]\.

While FedRSPO\+ uses the per\-clientλ\\lambdaand learning rates defined in[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11), we use a learning rate of10−310^\{\-3\}for the baselines\. We useκLR=0\.1,κλ=1\\kappa^\{\\mathrm\{LR\}\}=0\.1,\\kappa^\{\\lambda\}=1, andα=0\.01\\alpha=0\.01for the fractional knapsack experiments, andα=10−3\\alpha=10^\{\-3\}for the PJM and Warcraft experiments, selected to set initial learning rates within the same magnitude as the baseline learning rate10−310^\{\-3\}\. We use a stopping thresholdτ=1%\\tau=1\\%\. We report absolute regretR=zLP​\(c,w^\)−zLP​\(c,w⋆\)R=z\_\{\\mathrm\{LP\}\}\(c,\\hat\{w\}\)\-z\_\{\\mathrm\{LP\}\}\(c,w^\{\\star\}\)and relative regretR/zLPAvg​\(c,w⋆\)R/z^\{\\mathrm\{Avg\}\}\_\{\\mathrm\{LP\}\}\(c,w^\{\\star\}\)as metrics, wherezLPAvgz^\{\\mathrm\{Avg\}\}\_\{\\mathrm\{LP\}\}is the per\-client average ground\-truth objective functions\.

#### Optimizer\.

All experiments use Adam\[[67](https://arxiv.org/html/2610.09091#bib.bib57)\]with gradient clipping at norm1\.01\.0and no weight decay\. Tabular experiments \(knapsack, PJM\) useFloat64precision; the Warcraft experiment usesFloat32\.

#### Freeze mechanism\.

The annealing schedules forλ\\lambdaandη\\etain[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)use a scheduler: once the stopping criterion \([Eq\.6](https://arxiv.org/html/2610.09091#S3.E6)\) is satisfied, the schedule freezes and training continues\. For SPO\+ and MSE baselines, all schedules are held constant \(κλ=κLR=0\\kappa^\{\\lambda\}=\\kappa^\{\\mathrm\{LR\}\}=0\)\. We also tested early stopping a few rounds after freezing, and found that while it did help mitigate computational overhead, it did not significantly influence training performance, due to the already annealed learning rate slowing training in later rounds\. We implement an early stopping condition 3 rounds after the schedule freezing criterion is met\.

### C\.2Synthetic Knapsack

#### Optimization problem\.

We employ a fractional knapsack withd=50d=50items\. Each clientjjsolves

maxw∈Sj⁡cjT​w,Sj=\{w∈\[0,1\]d:αjT​w≤Bj\},\\max\_\{w\\in S\_\{j\}\}\\;c\_\{j\}^\{T\}w,\\qquad S\_\{j\}=\\bigl\\\{w\\in\[0,1\]^\{d\}:\\alpha\_\{j\}^\{T\}w\\leq B\_\{j\}\\bigr\\\},wherecjc\_\{j\}is the predicted cost \(objective\) vector,αj\\alpha\_\{j\}the item weights, andBjB\_\{j\}is the knapsack capacity\. Since the problem is a maximization problem, we use a sign transform in our loss function implementation,maxw∈Sj⁡cjT​w=minw∈Sj⁡\(−cj\)T​w\.\\max\_\{w\\in S\_\{j\}\}\\;c\_\{j\}^\{T\}w=\\min\_\{w\\in S\_\{j\}\}\\;\(\-c\_\{j\}\)^\{T\}w\.

#### Ground\-truth objective\.

A single shared basisWtrue∈ℝd×pW\_\{\\mathrm\{true\}\}\\\!\\in\\\!\\mathbb\{R\}^\{d\\times p\}is drawn once\. Each client receives a rotated copyWj=Wtrue​RjW\_\{j\}=W\_\{\\mathrm\{true\}\}\\,R\_\{j\}, where

Rj=exp⁡\(ηobj​Aj‖Aj‖\),Aj=Gj−GjT,Gj∼𝒩⁡\(0,I\)\.R\_\{j\}=\\exp\\\!\\Bigl\(\\eta\_\{\\mathrm\{obj\}\}\\,\\frac\{A\_\{j\}\}\{\\\|A\_\{j\}\\\|\}\\Bigr\),\\qquad A\_\{j\}=G\_\{j\}\-G\_\{j\}^\{T\},\\quad G\_\{j\}\\sim\\mathcal\{N\}\(0,I\)\.BecauseAjA\_\{j\}is skew\-symmetric,exp⁡\(⋅\)\\exp\(\\cdot\)is an orthogonal matrix, soRjR\_\{j\}is a rotation in feature space\. Whenηobj=0\\eta\_\{\\mathrm\{obj\}\}=0all clients share the same basis\. Largerηobj\\eta\_\{\\mathrm\{obj\}\}increases the spread of the per\-client bases\. Given a feature vectorx∈ℝpx\\in\\mathbb\{R\}^\{p\}, the true cost vector is

θtrue​\(x\)=−\(1\+\(1\+Wj​x/p\)4\),normalized by​mean⁡\(\|θtrue\|\)\.\\theta\_\{\\mathrm\{true\}\}\(x\)=\-\\bigl\(1\+\(1\+W\_\{j\}\\,x/\\sqrt\{p\}\\,\)^\{4\}\\bigr\),\\qquad\\text\{normalized by \}\\operatorname\{mean\}\(\|\\theta\_\{\\mathrm\{true\}\}\|\)\.\(26\)

#### Constraint heterogeneity \(ηconstr\\eta\_\{\\mathrm\{constr\}\}\)\.

A base capacity is set asB¯=0\.6×d=30\\bar\{B\}=0\.6\\times d=30\. Each client draws a multiplicative factorfj∼2​Beta​\(1/ηconstr,1/ηconstr\)f\_\{j\}\\sim 2\\,\\mathrm\{Beta\}\(1/\\eta\_\{\\mathrm\{constr\}\},\\,1/\\eta\_\{\\mathrm\{constr\}\}\)\(mean 1; forηconstr=0\\eta\_\{\\mathrm\{constr\}\}=0the factor degenerates to 1\) and receives capacityBj=B¯⋅fjB\_\{j\}=\\bar\{B\}\\cdot f\_\{j\}\. When the flageta\_constr\_affects\_weightsis set, each item weight also receives an independent per\-item Beta factor drawn from the same family, otherwise item weights are shared across clients\.

#### Covariate\-shift heterogeneity \(ηdata\\eta\_\{\\mathrm\{data\}\}\)\.

Each client draws a mean shiftμj=ηdata​zj\\mu\_\{j\}=\\eta\_\{\\mathrm\{data\}\}\\,z\_\{j\},zj∼𝒩⁡\(0,I\)z\_\{j\}\\sim\\mathcal\{N\}\(0,I\), and per\-feature positive scalesσj\\sigma\_\{j\}constructed with the same Beta\-factor mechanism used for constraints\. Features are then sampled as

x\|j∼𝒩⁡\(μj,diag⁡\(σj2\)\)\.x\\mid j\\;\\sim\\;\\mathcal\{N\}\\\!\\bigl\(\\mu\_\{j\},\\;\\operatorname\{diag\}\(\\sigma\_\{j\}^\{2\}\)\\bigr\)\.

#### Experimental protocol\.

Each of the 20 clients receives 100 training, 10 validation, and 100 test samples\. Each heterogeneity axisη∈\{0\.0,0\.5,1\.0\}\\eta\\in\\\{0\.0,0\.5,1\.0\\\}is varied independently while the others are held at zero\. Results are averaged over 5 seeds∈\{42,69,1993,67,2026\}\\in\\\{42,69,1993,67,2026\\\}and reported in[Table3](https://arxiv.org/html/2610.09091#S4.T3)\.

#### Hyperparameters\.

Table[7](https://arxiv.org/html/2610.09091#A3.T7)shows the full training configuration for both phases\.

#### Additional sensitivity analysis\.

We perform additional sensitivity analysis across more of the hyperparameters of FedRSPO\+ and demonstrate the results in[Fig\.4](https://arxiv.org/html/2610.09091#A3.F4)\. Overall FedRSPO\+ is fairly insensitive to changes in annealing parameterκ\\kappa, especially the learning rate parameterκLR\\kappa\_\{\\mathrm\{LR\}\}\. Reducingτ\\tauimproves the model slightly, but it reduces the fraction of clients that end up frozen in the 10 rounds of personalization\. To achieve even better results, a smallerτ\\tauwith more annealing rounds until the freezing criterion is met should be considered\. We also varied the number of calibration samples per client used to compute the freeze statisticU~t\\tilde\{U\}\_\{t\}\. Performance was largely insensitive to this: as\|Vc\|\|V\_\{c\}\|increased from 2 to 32, the relative\-regret improvement of RSPO\+ over SPO\+ remained nearly constant, varying only from 3\.61% to 3\.72%\. This suggests that the freezing rule is robust to the exact calibration subset size, so we use\|Vc\|=8\|V\_\{c\}\|=8in the main experiments\.

#### Additional ablation\.

We perform an additional ablation with the fractional knapsack experiment to complement the results of[Table5](https://arxiv.org/html/2610.09091#S4.T5)across different heterogeneity settings\. The results are shown in[Table9](https://arxiv.org/html/2610.09091#A3.T9)\.

Table 7:Synthetic knapsack hyperparameters\. RSPO\+\-only entries are marked; SPO\+ and MSE use the same configuration with constant schedules \(κ=0\\kappa=0\)\.HyperparameterWarm\-startPersonalizationCommunication rounds1010Local epochs per round31Batch size6464Client fraction0\.41\.0κλ\\kappa^\{\\lambda\}\(RSPO\+ only\)0\.01\.0η0\\eta\_\{0\}\(baseline\)10−310^\{\-3\}10−310^\{\-3\}κLR\\kappa^\{\\mathrm\{LR\}\}\(RSPO\+ only\)0\.00\.1Per\-clientλ/η\\lambda/\\eta\(RSPO\+ only\)✓✓α\\alpha\([Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\)0\.010\.01Freeze thresholdτ\\tau—0\.01Validation samples/client—8

### C\.3Warcraft Shortest Path

#### Projection model

While we use Dijkstra’s for the downstream LP oracle, we use a Gurobi QLP for the RSPO\+ projections\. For a Warcraft grid with vertex set𝒱\\mathcal\{V\}and canonical source–sink pair\(s,t\)\(s,t\), let𝒜⊆𝒱×𝒱\\mathcal\{A\}\\subseteq\\mathcal\{V\}\\times\\mathcal\{V\}denote the directed 8\-neighbor adjacency graph\. For each vertexvv, define incoming and outgoing arc sets

δ−\(v\)=\{a=\(u,v\)∈𝒜\},δ\+\(v\)=\{a=\(v,u\)∈𝒜\}\.\\delta^\{\-\}\(v\)=\\\{a=\(u,v\)\\in\\mathcal\{A\}\\\},\\qquad\\delta^\{\+\}\(v\)=\\\{a=\(v,u\)\\in\\mathcal\{A\}\\\}\.Given transformed costsc∈ℝ\|𝒱\|c\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}\|\}and regularization parameterλ\>0\\lambda\>0, the RSPO\+ projection oracle solves

ΠS​\(c\)∈arg⁡minx,e\\displaystyle\\Pi\_\{S\}\(c\)\\in\\arg\\min\_\{x,e\}\\quad∑v∈𝒱cv​wv\+λ2​∑v∈𝒱wv2\\displaystyle\\sum\_\{v\\in\\mathcal\{V\}\}c\_\{v\}w\_\{v\}\+\\frac\{\\lambda\}\{2\}\\sum\_\{v\\in\\mathcal\{V\}\}w\_\{v\}^\{2\}\(27a\)s\.t\.ws=1,\\displaystyle w\_\{s\}=1,\(27b\)wv=∑a∈δ−​\(v\)ea,\\displaystyle w\_\{v\}=\\sum\_\{a\\in\\delta^\{\-\}\(v\)\}e\_\{a\},∀v∈𝒱∖\{s\},\\displaystyle\\forall v\\in\\mathcal\{V\}\\setminus\\\{s\\\},\(27c\)wv=∑a∈δ\+​\(v\)ea,\\displaystyle w\_\{v\}=\\sum\_\{a\\in\\delta^\{\+\}\(v\)\}e\_\{a\},∀v∈𝒱∖\{t\},\\displaystyle\\forall v\\in\\mathcal\{V\}\\setminus\\\{t\\\},\(27d\)∑a∈δ−​\(s\)ea=0,\\displaystyle\\sum\_\{a\\in\\delta^\{\-\}\(s\)\}e\_\{a\}=0,\(27e\)∑a∈δ\+​\(t\)ea=0,\\displaystyle\\sum\_\{a\\in\\delta^\{\+\}\(t\)\}e\_\{a\}=0,\(27f\)0≤wv≤1,0≤ea≤1,\\displaystyle 0\\leq w\_\{v\}\\leq 1,\\quad 0\\leq e\_\{a\}\\leq 1,∀v∈𝒱,∀a∈𝒜\.\\displaystyle\\forall v\\in\\mathcal\{V\},\\quad\\forall a\\in\\mathcal\{A\}\.\(27g\)The returned relaxed Warcraft decision isw⋆w^\{\\star\}\. Equivalently,ΠS\(c\)=ΠS\(−c/λ\),\\Pi\_\{S\}\(c\)=\\Pi\_\{S\}\\\!\\left\(\-c/\\lambda\\right\),whereSSis the relaxed Warcraft path\-flow feasible set defined by the constraints above\. For a true cost vectorccand predictionc^\\hat\{c\}, RSPO\+ calls this oraclewu=ΠS​\(2​c^−c\),w\_\{u\}=\\Pi\_\{S\}\(2\\hat\{c\}\-c\),andwv=ΠS​\(c\),w\_\{v\}=\\Pi\_\{S\}\(c\),so thatwu=ΠS​\(\(c−2​c^\)/λ\)w\_\{u\}=\\Pi\_\{S\}\(\(c\-2\\hat\{c\}\)/\\lambda\)andwv=ΠS\(−c/λ\)w\_\{v\}=\\Pi\_\{S\}\(\-c/\\lambda\)\.

#### Dataset and federated partition\.

The Warcraft Shortest Path dataset\[[24](https://arxiv.org/html/2610.09091#bib.bib35)\]consists of 10,000 training, 1,000 validation, and 1,000 test 96×\\times96 RGB terrain maps on a12×1212\\times 12grid with 8\-connectivity\. In the benchmark\-aligned rerun, we form the federated benchmark from the original training split: we reserve 80% of the training set \(8,000 samples\) for federated training/validation and hold out the remaining 2,000 as a benchmark test set\. The 8,000 federated samples are distributed round\-robin \(after shuffling\) across 44 clients, yielding 182 samples per client before the80/2080/20intra\-client split, i\.e\., roughly 146 training and 36 validation samples per client\. Each client is evaluated on all 2,000 benchmark test maps\. We report mean optimality gap:\(predicted​cost−optimal​cost\)/optimal​cost\(\\mathrm\{predicted\}\\;\\mathrm\{cost\}\-\\mathrm\{optimal\}\\;\\mathrm\{cost\}\)/\\mathrm\{optimal\}\\;\\mathrm\{cost\}\), and its standard deviation across seeds\.

#### Client heterogeneity\.

Rather than 4 corner\-defined clients, we use a border\-antipodes construction on the12×1212\\times 12grid\. Since the grid border contains4⋅12−4=444\\cdot 12\-4=44cells, this yields 44 clients\. Each client’s source is one border cell and its sink is the antipodal border cell under a180∘180^\{\\circ\}rotation of the grid, inducing all four diagonal endpoint orientations\. We additionally enable objective heterogeneity: each client receives its own multiplicative reweighting of the 5 terrain valuesc=c0​exp⁡\(0\.6⋅x\)c=c\_\{0\}\\exp\(0\.6\\cdot x\), wherec0c\_\{0\}is the initial traversal cost of the terrain type andx∼𝒩⁡\(0,1\)x\\sim\\mathcal\{N\}\(0,1\)\. Therefore, clients differ in both endpoints and path costs\.

#### Architecture\.

The model uses a truncated ResNet\-18 backbone, followed by adaptive max\-pooling to a12×1212\\times 12grid and channel averaging to produce a12×1212\\times 12real\-valued weight matrix per sample, following the methodology of\[[1](https://arxiv.org/html/2610.09091#bib.bib1)\]\. Predicted cost grids are decoded into hard paths using the benchmark shortest\-path solver \(Dijkstra\)\. Float32 precision is used throughout\.

#### Additional baselines\.

Beyond RSPO\+, SPO\+, and MSE, the Warcraft experiment includes perturbed Fenchel–Young loss \(PFYL\), decision\-focused perturbed MSE \(DPO\)\. PFYL and DPO both use 10 perturbation samples with perturbation scaleϵ=1\.0\\epsilon=1\.0\.

#### Hyperparameters\.

Table[8](https://arxiv.org/html/2610.09091#A3.T8)reports the main configuration\. The Warcraft rerun uses five seeds rather than a single seed; reported variability therefore reflects both seed\-to\-seed and client\-level variation\.

Table 8:Warcraft shortest path hyperparameters\.HyperparameterWarm\-startPersonalizationCommunication rounds1010Local epochs per round32Batch size6464Client fraction0\.51\.0λ0\\lambda\_\{0\}\(RSPO\+ only\)2\.02\.0κλ\\kappa^\{\\lambda\}\(RSPO\+ only\)0\.01\.0η0\\eta\_\{0\}10−310^\{\-3\}10−310^\{\-3\}κLR\\kappa^\{\\mathrm\{LR\}\}\(RSPO\+ only\)0\.00\.1Per\-clientλ/η\\lambda/\\eta\(RSPO\+ only\)✓✓α\\alpha\([Eq\.11](https://arxiv.org/html/2610.09091#S3.E11), RSPO\+ only\)0\.0010\.001Perturbation samples \(PFYL/DPO\)1010Perturbation scaleϵ\\epsilon\(PFYL/DPO\)1\.01\.0Freeze thresholdτ\\tau\(RSPO\+ only\)0\.030\.01Table 9:Fractional\-knapsack component ablation\. Values are mean test relative regret \(%\)±\\pmsample standard deviation across five seeds, after averaging clients within each seed; lower is better\.Method\(0,0,0\)\(0,0,0\)↓\\downarrow\(1,0,0\)\(1,0,0\)↓\\downarrow\(0,1,0\)\(0,1,0\)↓\\downarrowLocal\-only RSPO\+9\.74±2\.739\.74\\pm 2\.739\.80±2\.789\.80\\pm 2\.786\.99±1\.896\.99\\pm 1\.89FedSPO\+9\.69±2\.089\.69\\pm 2\.0810\.93±2\.1910\.93\\pm 2\.1912\.89±1\.9912\.89\\pm 1\.99RSPO\+\(surrogate only\)2\.26±0\.402\.26\\pm 0\.403\.05±0\.253\.05\\pm 0\.254\.71±0\.494\.71\\pm 0\.49RSPO\+w/λj/ηj\\lambda\_\{j\}/\\eta\_\{j\}2\.26±0\.402\.26\\pm 0\.403\.05±0\.253\.05\\pm 0\.254\.63±0\.464\.63\\pm 0\.46RSPO\+w/λj/ηj\\lambda\_\{j\}/\\eta\_\{j\}\+ P2\.16±0\.362\.16\\pm 0\.362\.41±0\.382\.41\\pm 0\.384\.08±0\.414\.08\\pm 0\.41RSPO\+ \(Full\)1\.86±0\.26\\mathbf\{1\.86\}\\pm 0\.262\.02±0\.26\\mathbf\{2\.02\}\\pm 0\.262\.58±0\.40\\mathbf\{2\.58\}\\pm 0\.40

### C\.4PJM Energy Pricing

We use hourly day\-ahead locational marginal pricing \(LMP\) data from the PJM Interconnection, a regional transmission organization spanning 13 U\.S\. states\[[25](https://arxiv.org/html/2610.09091#bib.bib36)\]\. We use the following nodes as clients: DOM, PEPCO, PPL, DUQ, BGE, AECO, ATSI, RECO, PSEG, DPL, DAY, METED, JCPL, APS, DEOK, AEP, COMED, PECO, and PENELEC\. Prices are retrieved for the year 2025 and assigned to clients, with variation arising from localized congestion effects and are in dollars per MWh\[[68](https://arxiv.org/html/2610.09091#bib.bib55)\]\. Energy markets provide a natural setting for decision\-focused federated learning, as operational data are often siloed across entities and subject to regulatory protections such as NERC CIP\-011\[[69](https://arxiv.org/html/2610.09091#bib.bib56)\]\.

#### Data processing\.

The 8 weather covariates are: mean/min/max temperature \(∘C\), mean humidity, mean wind speed, mean/max solar irradiance, and day of week\. The 24 target dimensions arech=LMPh\+congestionhc\_\{h\}=\\mathrm\{LMP\}\_\{h\}\+\\mathrm\{congestion\}\_\{h\}for hoursh=0,…,23h=0,\\ldots,23\. The demand satisfaction constraint for clientkkis∑hwh=demandk\\sum\_\{h\}w\_\{h\}=\\mathrm\{demand\}\_\{k\}withwh∈\[0,1\]w\_\{h\}\\in\[0,1\], solved as a fractional knapsack with equality constraint\.

Each client corresponds to a fixed daily demand customer with demand normalized to a range of 4 to 12 MWh\. This setup captures heterogeneity across participants while maintaining realistic operational constraints\.

#### Optimization problem\.

The downstream task is to minimize daily electricity cost subject to daily demand\. This can be formulated as a capacity\-constrained fractional knapsack problem, as described in[Section4\.1](https://arxiv.org/html/2610.09091#S4.SS1)\. Specifically, the feasible set is defined asS=\{w∈\[0,1\]24:𝟏T​w=Dj\}S=\\\{w\\in\[0,1\]^\{24\}:\\mathbf\{1\}^\{T\}w=D\_\{j\}\\\}, whereDjD\_\{j\}is the demand at clientjj, and the problem is defined asminw∈S⁡cT​w\\min\_\{w\\in S\}c^\{T\}w\. Demand is randomly assigned from\{4,6,8,10,12\}\\\{4,6,8,10,12\\\}to the PJM clients\.

#### Data splits\.

The data is split chronologically with no shuffling: 50% train, 10% validation, 40% test, allocated by date index, with 19 total clients\.

#### Ablation Study\.

For our ablation study, we first evaluate a federated\-only training run with standard MSE loss and evaluate the mean and standard deviation across 5 seeds\. We then replace the MSE loss with the DFL SPO\+ loss and further incorporate RSPO\+\. We use the average of the would\-be per\-client calibrated parameters, or1\|C\|​∑j=1\|C\|λj\\frac\{1\}\{\|C\|\}\\sum\_\{j=1\}^\{\|C\|\}\\lambda\_\{j\}and1\|C\|​∑j=1\|C\|ηj\\frac\{1\}\{\|C\|\}\\sum\_\{j=1\}^\{\|C\|\}\\eta\_\{j\}from[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11), where\|C\|\|C\|is the number of clients\. We setκλ=κLR=0\\kappa^\{\\lambda\}=\\kappa^\{\\mathrm\{LR\}\}=0for the first 2 RSPO\+ ablations\. We then gradually introduce more aspects of the FedRSPO\+ algorithm and evaluate similarly\. To showcase the modularity of the algorithm, we incorporate two popular FL methods: FedProx\[[14](https://arxiv.org/html/2610.09091#bib.bib25)\]and Ditto\[[16](https://arxiv.org/html/2610.09091#bib.bib16)\]\. During the federated stage, FedProx adds a regularization termμ/2​‖θt−θ‖2\\mu/2\\\|\\theta\_\{t\}\-\\theta\\\|^\{2\}to each client’s local objective, whereθt\\theta\_\{t\}is the global model at the start of the round andθ\\thetais the client model\. That means that FedProx only affects the federated training stage, not the personalization stage\. We useμ=1\\mu=1for our experiment, and keep all other parameters the same as in standard FedRSPO\+\. For Ditto, the federated training phase is identical to standard FedRSPO\+\. Similarly to FedProx, each client solves a regularized version of their local objective, adding the termμ/2​‖θt−θ‖2\\mu/2\\\|\\theta\_\{t\}\-\\theta\\\|^\{2\}\. As opposed to FedProx, Ditto is introduced in the personalization phase only\. We useμ=1\\mu=1for the regularization parameter, and keep all other settings the same as before\.

Table 10:PJM hyperparameters\.HyperparameterWarm\-startPersonalizationCommunication rounds1010Local epochs per round31Batch size6464Client fraction0\.41\.0κλ\\kappa^\{\\lambda\}\(RSPO\+ only\)0\.01\.0η0\\eta\_\{0\}\(baseline\)10−310^\{\-3\}10−310^\{\-3\}κLR\\kappa^\{\\mathrm\{LR\}\}\(RSPO\+ only\)0\.00\.1Per\-clientλ/η\\lambda/\\eta\(RSPO\+ only\)✓✓α\\alpha\([Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\)0\.0110−310^\{\-3\}Freeze thresholdτ\\tau—0\.03Validation samples/client—8
#### Leave\-one\-client\-out PJM\.

In the original experiment, each client has roughly six months of training data, and local\-only RSPO\+ already achieves 61\.53 mean regret, compared with 60\.65 for full FedRSPO\+\. The large local datasets leave little estimation error for federation to reduce\. To investigate a case where federation would be very helpful for PJM, we design a leave\-one\-client\-out experiment, where we train 18/19 clients normally in a federation, and leave one client out, who has access to only \[3, 7, 14, 30, 60\] days of data\. The client then compares their local model with the federated model, fine\-tuned on their data\. We repeat this procedure for all 19 client nodes\. In both cases we use RSPO\+ to isolate the federated effect\. The results are shown in[Tables11](https://arxiv.org/html/2610.09091#A3.T11)and[12](https://arxiv.org/html/2610.09091#A3.T12)\.

Table 11:Leave\-one\-client\-out experiment for PJM\. Results are reported in absolute decision regret; lower is better\.Target trainingdaysLocalRSPO\+FederatedonlyFedRSPO\+personalizedGap vs\. local\(Fed\+P\)Relative reductionFed\-onlyFed\+P3168\.48±10\.29168\.48\\pm 10\.2973\.58±1\.8673\.58\\pm 1\.8673\.06±2\.1973\.06\\pm 2\.1995\.43±9\.3095\.43\\pm 9\.3056\.3%56\.3\\%56\.6%56\.6\\%7167\.49±9\.49167\.49\\pm 9\.4973\.58±1\.8673\.58\\pm 1\.8673\.64±2\.1473\.64\\pm 2\.1493\.85±8\.7193\.85\\pm 8\.7156\.1%56\.1\\%56\.0%56\.0\\%14164\.44±10\.91164\.44\\pm 10\.9173\.58±1\.8673\.58\\pm 1\.8673\.31±2\.1573\.31\\pm 2\.1591\.13±10\.4691\.13\\pm 10\.4655\.3%55\.3\\%55\.4%55\.4\\%30164\.37±7\.91164\.37\\pm 7\.9173\.58±1\.8673\.58\\pm 1\.8673\.12±2\.1173\.12\\pm 2\.1191\.26±7\.6291\.26\\pm 7\.6255\.2%55\.2\\%55\.5%55\.5\\%60153\.86±8\.17153\.86\\pm 8\.1773\.58±1\.8673\.58\\pm 1\.8672\.20±1\.8472\.20\\pm 1\.8481\.67±6\.5181\.67\\pm 6\.5152\.2%52\.2\\%53\.1%53\.1\\%Table 12:Client\-level benefit of federated\-only training over the original local RSPO\+baseline with the PJM experiment\.Target trainingdaysFederated\-onlybeats localMedian client reduction\[IQR\]319/1919/1951\.1%​\[37\.2%,66\.3%\]51\.1\\%\\;\[37\.2\\%,\\,66\.3\\%\]719/1919/1947\.9%​\[33\.9%,65\.4%\]47\.9\\%\\;\[33\.9\\%,\\,65\.4\\%\]1419/1919/1948\.6%​\[27\.4%,65\.6%\]48\.6\\%\\;\[27\.4\\%,\\,65\.6\\%\]3019/1919/1945\.1%​\[28\.1%,64\.8%\]45\.1\\%\\;\[28\.1\\%,\\,64\.8\\%\]6018/1918/1941\.0%​\[27\.4%,57\.2%\]41\.0\\%\\;\[27\.4\\%,\\,57\.2\\%\]The federated\-only stays constant since it does not train on the held\-out target’s dataset, so is unaffected by a larger target dataset\. In the data\-poor regimes the federated model provides the largest benefit, with the personalization providing marginal or no benefit \(even 0\.06 worse than federated\-only at 7 days, which is negligible\), due to the small target dataset\. Personalization performance improves noticeably at 60 target days\. In terms of per\-client differences, 19/19 clients prefer the federated\-only model over the local model for target days = 3,7,14, and 30, and 18/19 for target days = 60\. We provide a second table in[Table12](https://arxiv.org/html/2610.09091#A3.T12)with inter\-client median reduction and the interquartile range \(IQR\) for federated only versus local\.

At 60 days, PENELEC is the only client with negative transfer from local to federated\. These values are lower than those in[Table11](https://arxiv.org/html/2610.09091#A3.T11)because the[Table11](https://arxiv.org/html/2610.09091#A3.T11)computes a single ratio of aggregate means, which weights high\-regret clients more heavily, whereas in[Table12](https://arxiv.org/html/2610.09091#A3.T12)each client contributes one percentage equally\. We report only federated\-only gains, since the benefits of personalization do not meaningfully change the results in this experiment\. These IQRs are across the 19 client\-level percentage reductions\. For each client, regret is first averaged over five seeds and then converted to a percentage relative to that client’s local mean\. Each experiment rotates all 19 PJM clients as the held\-out target and uses five training seeds, giving19×5=9519\\times 5=95observations per target\-data budget\. For each seed, we first average the metric over the 19 held\-out targets\.

#### Fixed demand in PJM\.

To isolate the effect of feasible set heterogeneity in the PJM experiment and to show that optimization heterogeneity plays a role and FedRSPO\+ performs better than federated SPO\+ under heterogeneity, we reran the PJM experiment but set each client’s demand to be equal\. We repeat this for all 5 demand values \[4,6,8,10,12\] for 5 seeds\. At each demand level, the gap between FedRSPO\+ and FedSPO\+ decreased on average\. For seedss, letGhet,sG\_\{\\mathrm\{het\},s\}andGd,sG\_\{d,s\}denote the regret gaps under heterogeneous demands and common demanddd, respectively\. Each gap is defined as federated\-only SPO\+regret minus personalized FedRSPO\+regret\. We compute,

Δd,s=100​Gd,s−Ghet,sGhet,s\.\\Delta\_\{d,s\}=100\\,\\frac\{G\_\{d,s\}\-G\_\{\\mathrm\{het\},s\}\}\{G\_\{\\mathrm\{het\},s\}\}\.In[Table13](https://arxiv.org/html/2610.09091#A3.T13)we report the percentage change, where a negative gap corresponds to a narrower gap\. In other words, a negative gap means FedRSPO\+ and federated SPO\+’s performance was closer with homogeneous feasible sets\. The mean percentage change is negative at every demand level\. This is consistent with feasible\-set heterogeneity contributing to the observed performance difference\.

Table 13:Effect of equal client demands on the regret gap between federated\-only SPO\+and personalized FedRSPO\+on the PJM\-derived constructed\-cost benchmark\.Common demandPercentage change in gap4−18\.50%±9\.73%\-18\.50\\%\\pm 9\.73\\%6−8\.37%±10\.94%\-8\.37\\%\\pm 10\.94\\%8−4\.97%±10\.03%\-4\.97\\%\\pm 10\.03\\%10−24\.33%±8\.71%\-24\.33\\%\\pm 8\.71\\%12−20\.13%±8\.16%\-20\.13\\%\\pm 8\.16\\%

### C\.5Centralized and Local\-only Baselines

To contextualize the federated results in[Table3](https://arxiv.org/html/2610.09091#S4.T3)and[Table5](https://arxiv.org/html/2610.09091#S4.T5), we additionally evaluate two non\-federated training regimes\.Centralizedpools all client training data into a single dataset and trains one global model, evaluated on each client’s downstream objective at test time\. It is a centralized proxy for the value of data sharing when no privacy constraint applies\.Local\-onlytrains a separate model per client on that client’s data alone, with no cross\-client communication\. It is a lower\-bound proxy for federation that quantifies how much each client gains from collaboration\. Both regimes use the same architectures, optimizer, and number of gradient updates per client as the federated runs to keep compute comparable, and we report mean±\\pmstd over the same 5 seeds\.

Table 14:Synthetic knapsack: relative regret \(%\) of centralized and local\-only baselines on representative heterogeneity configurations, alongside the federated results from[Table3](https://arxiv.org/html/2610.09091#S4.T3)for reference\. Mean±\\pmstd over 5 seeds\. Lower is better \(↓\\downarrow\)\.\(ηo,ηc,ηd\)\(\\eta\_\{o\},\\eta\_\{c\},\\eta\_\{d\}\)CentralizedLocal\-onlyFedFed \+ PSPO\+\(0,0,0\)\(0,0,0\)0\.32±0\.060\.32\\pm 0\.062\.42±0\.302\.42\\pm 0\.309\.7±2\.19\.7\\pm 2\.15\.6±1\.45\.6\\pm 1\.4\(0,0,0\.5\)\(0,0,0\.5\)3\.49±0\.623\.49\\pm 0\.622\.93±0\.542\.93\\pm 0\.5414\.1±2\.014\.1\\pm 2\.06\.9±1\.36\.9\\pm 1\.3\(0,0,1\)\(0,0,1\)6\.97±0\.896\.97\\pm 0\.892\.47±0\.392\.47\\pm 0\.3917\.7±2\.117\.7\\pm 2\.16\.9±0\.86\.9\\pm 0\.8RSPO\+ \(Ours\)\(0,0,0\)\(0,0,0\)0\.75±0\.080\.75\\pm 0\.089\.74±2\.739\.74\\pm 2\.732\.3±0\.42\.3\\pm 0\.41\.9±0\.31\.9\\pm 0\.3\(0,0,0\.5\)\(0,0,0\.5\)4\.12±0\.404\.12\\pm 0\.406\.93±1\.156\.93\\pm 1\.156\.5±0\.86\.5\\pm 0\.83\.8±0\.23\.8\\pm 0\.2\(0,0,1\)\(0,0,1\)7\.42±0\.727\.42\\pm 0\.724\.15±0\.694\.15\\pm 0\.699\.3±1\.29\.3\\pm 1\.23\.4±0\.23\.4\\pm 0\.2#### Synthetic knapsack\.

Table[14](https://arxiv.org/html/2610.09091#A3.T14)reports relative regret for different heterogeneity configurations\. We can see two patterns\. First, centralized training is the strongest baseline when clients share a data distribution \(rows withηd=0\\eta\_\{d\}=0\): centralized SPO\+ achieves0\.32%0\.32\\%relative regret at\(0,0,0\)\(0,0,0\), far below any federated or local\-only number\. But centralized performance degrades sharply once clients differ in their feature distributions: at\(0,0,1\)\(0,0,1\), centralized SPO\+ rises to6\.97%6\.97\\%, and centralized RSPO\+ to7\.42%7\.42\\%, because pooling distributionally distinct clients creates a misspecified training problem\. Second, local\-only training stays relatively stable under heterogeneity: local\-only SPO\+ averages2\.96±0\.73%2\.96\\pm 0\.73\\%across all 27 heterogeneity configurations and beats centralized SPO\+ in every configuration withηd\>0\\eta\_\{d\}\>0\. Local\-only RSPO\+, by contrast, is much weaker than its federated counterpart \(9\.74±2\.73%9\.74\\pm 2\.73\\%at\(0,0,0\)\(0,0,0\)versus1\.9%1\.9\\%for FedRSPO\+ with personalization in[Table3](https://arxiv.org/html/2610.09091#S4.T3)\), supporting the interpretation that RSPO\+’s advantage lies in federation and mitigating heterogeneity\.

Table 15:PJM energy pricing: mean absolute regret across training regimes\. Mean±\\pmstd over 5 seeds\. Federated rows are reproduced from[Table5](https://arxiv.org/html/2610.09091#S4.T5)for reference\. Lower is better \(↓\\downarrow\)\.RegimeMSESPO\+RSPO\+ \(Ours\)Centralized70\.26±0\.6170\.26\\pm 0\.6169\.37±0\.4469\.37\\pm 0\.4469\.32±1\.1869\.32\\pm 1\.18Local\-only163\.76±10\.12163\.76\\pm 10\.1269\.57±1\.7269\.57\\pm 1\.7261\.53±0\.6661\.53\\pm 0\.66Federated \(naive\)175\.45±7\.26175\.45\\pm 7\.2681\.68±1\.7781\.68\\pm 1\.7772\.89±4\.2372\.89\\pm 4\.23FedRSPO\+ \(Full\)——60\.65±0\.80\\mathbf\{60\.65\\pm 0\.80\}
#### PJM energy pricing\.

[Table15](https://arxiv.org/html/2610.09091#A3.T15)reports the same comparison on the PJM case study, using mean absolute regret to match[Table5](https://arxiv.org/html/2610.09091#S4.T5)\. Three observations stand out\. First, centralized training close the gap between the three methods: MSE, SPO\+, and RSPO\+ all sit between6969and7171, indicating that the PJM clients share enough structure that pooled fitting is a strong strategy regardless of the loss\. Second, local\-only MSE remains poor \(163\.76±10\.12163\.76\\pm 10\.12, close to the federated MSE result of175\.45±7\.26175\.45\\pm 7\.26\), confirming that the improvements in the federated pipeline require decision\-aware training to achieve competitive results\. Third, local\-only RSPO\+ achieves61\.53±0\.6661\.53\\pm 0\.66, nearly matching the full FedRSPO\+ result of60\.65±0\.8060\.65\\pm 0\.80from Table[5](https://arxiv.org/html/2610.09091#S4.T5)\. This reflects the relatively homogeneous downstream structure of PJM \(all clients face the same top\-kkselection problem on overlapping price signals\), where personalization captures most of the available gain\. On synthetic instances with structurally heterogeneous feasible sets, the analogous local\-only RSPO\+ baseline is much weaker, indicating that federation contributes most when clients differ in their optimization structure rather than only in scale\.

#### Discussion\.

Together, these baselines reinforce the interpretation of the main results\. Centralized training is a useful upper\-bound comparator but is not uniformly dominant: it excels when clients are statistically aligned and degrades sharply under data heterogeneity\. Local\-only training is a strong personalization baseline in two specific regimes \(SPO\+ in synthetic data\-heterogeneous settings and RSPO\+ on PJM\) but offers no shared model and is inconsistent across heterogeneity types, due to the addition of the downstream objective\. The main FedRSPO\+ result is a robust middle ground that retains performance across the heterogeneity axes while supporting data privacy\. The PJM observation that local\-only RSPO\+ is competitive with FedRSPO\+ also identifies a regime where the federated machinery is nearly redundant\. We view formalizing this regime, where shared training adds little beyond the surrogate itself, as a useful direction for follow\-up work\.

### C\.6QP–LP decision alignment

To complement the oracle\-bias freeze statisticU~t\\tilde\{U\}\_\{t\}, we directly measure the decision\-space discrepancy between the regularized predicted decision and the LP predicted decision,

ΔQP−LP=𝔼\[‖ΠS\(−c^/λt\)−argminw∈Sc^Tw‖\]\.\\Delta\_\{\\mathrm\{QP\-LP\}\}=\\mathbb\{E\}\\left\[\\left\\\|\\Pi\_\{S\}\(\-\\hat\{c\}/\\lambda\_\{t\}\)\-\\arg\\min\_\{w\\in S\}\\hat\{c\}^\{T\}w\\right\\\|\\right\]\.On synthetic knapsack, personalization consistently reduces this gap across all configurations: the mean gap decreases from3\.7153\.715after the federated warm start to1\.1661\.166after personalization, a69%69\\%reduction, and this decrease occurs in all135/135135/135configurations\. The freeze statistic aligns with this result\. Clients that satisfy the freeze criterion have lower average decision gaps \(1\.1121\.112\) than clients that do not freeze \(1\.3381\.338\), and lower final oracle\-bias valuesU~t\\tilde\{U\}\_\{t\}\(0\.008580\.00858versus0\.017350\.01735\)\. The hardest cases are primarily driven by data heterogeneity, where increasingηd\\eta\_\{d\}from00to11raises the personalized decision gap from1\.0361\.036to1\.3351\.335and reduces the frozen\-client fraction from0\.9170\.917to0\.5780\.578\.

We observe the same qualitative effect in the PJM case study, where personalization reduces the predicted QP–LP gap from1\.9551\.955to1\.1491\.149, a41%41\\%reduction\. This residual gap is not negligible, about42\.6%42\.6\\%of the average predicted LP decision norm, but it confirms that personalization moves the regularized predicted decision substantially closer to the LP decision used at evaluation\. Finally, the final freeze statistic is strongly correlated with the predicted QP–LP decision gap on knapsack at the configuration level \(r=0\.807r=0\.807\), suggesting thatU~t\\tilde\{U\}\_\{t\}is a useful proxy for QP–LP alignment\. The direct decision\-gap diagnostic captures predicted\-decision alignment more explicitly, but adds computational overhead if tracked in situ \(\|Vc\|\|V\_\{c\}\|additional projections per client per round\)\.

Table 16:QP–LP discrepancies after personalization\. Values are mean normalized gaps \(%; lower is better\), averaged over clients and five seeds, and additionally over 27 heterogeneity configurations for knapsack\. The PJM diagnostic uses 10 personalization rounds\.DatasetDecision gap \(%\)Objective gap \(%\)Knapsack21\.443\.26PJM42\.5710\.43We also evaluate the decision and objective gaps for the fractional knapsack and PJM experiments\. For clientjj, letw^j​iQP\\widehat\{w\}^\{\\mathrm\{QP\}\}\_\{ji\}andw^j​iLP\\widehat\{w\}^\{\\mathrm\{LP\}\}\_\{ji\}denote the QP and LP decisions under the same predicted cost vectorc^j​i\\widehat\{c\}\_\{ji\}, with the QP using the client’s final smoothing coefficientλ\\lambda\. Letzj​i⋆=minw∈Sj⁡cj​i⊤​wz^\{\\star\}\_\{ji\}=\\min\_\{w\\in S\_\{j\}\}c\_\{ji\}^\{\\top\}wbe the LP optimum under the true cost vector\. Withϵ=10−8\\epsilon=10^\{\-8\}, the reported values in[Table16](https://arxiv.org/html/2610.09091#A3.T16)use the following normalizations,

Dj\\displaystyle D\_\{j\}=100​1nj​∑i=1nj‖w^j​iQP−w^j​iLP‖21nj​∑i=1nj‖w^j​iLP‖2\+ϵ,\\displaystyle=100\\,\\frac\{\\frac\{1\}\{n\_\{j\}\}\\sum\_\{i=1\}^\{n\_\{j\}\}\\left\\\|\\widehat\{w\}^\{\\mathrm\{QP\}\}\_\{ji\}\-\\widehat\{w\}^\{\\mathrm\{LP\}\}\_\{ji\}\\right\\\|\_\{2\}\}\{\\frac\{1\}\{n\_\{j\}\}\\sum\_\{i=1\}^\{n\_\{j\}\}\\left\\\|\\widehat\{w\}^\{\\mathrm\{LP\}\}\_\{ji\}\\right\\\|\_\{2\}\+\\epsilon\},\(28\)Oj\\displaystyle O\_\{j\}=100nj​∑i=1nj\|cj​i⊤​\(w^j​iQP−w^j​iLP\)\|\|zj​i⋆\|\+ϵ\.\\displaystyle=\\frac\{100\}\{n\_\{j\}\}\\sum\_\{i=1\}^\{n\_\{j\}\}\\frac\{\\left\|c\_\{ji\}^\{\\top\}\\left\(\\widehat\{w\}^\{\\mathrm\{QP\}\}\_\{ji\}\-\\widehat\{w\}^\{\\mathrm\{LP\}\}\_\{ji\}\\right\)\\right\|\}\{\|z^\{\\star\}\_\{ji\}\|\+\\epsilon\}\.\(29\)Both objectives are evaluated using the true linear costs, without the quadratic penalty\.

Table 17:QP–LP decision alignment on synthetic knapsack\. Personalization substantially reduces the predicted decision discrepancy between the regularized RSPO\+ decision and the LP decision used at evaluation\.MetricMeanMedianRangeGlobal QP–LP gap3\.7153\.6543\.080–4\.456Personalized QP–LP gap1\.1661\.1180\.922–1\.521Personalized / global ratio0\.3140\.3160\.258–0\.354Final freeze statistic0\.010670\.009590\.00822–0\.01761Frozen\-client fraction0\.7610\.8000\.300–1\.000
### C\.7Linear predictor results

Table 18:Bias\-free linear predictor results on synthetic knapsack\. Entries report mean client relative regret \(%\) as mean±\\pmsample standard deviation across five seeds\. The heterogeneity tuple represents objective, constraint, and data\-distribution heterogeneity, respectively\. “Fed” denotes the federated endpoint; “\+P” includes subsequent personalization\. Bold indicates the lowest mean regret within each training stage\.MSEPFYLDPOSPO\+RSPO\+HeterogeneityFed\+PFed\+PFed\+PFed\+PFed\+P\(0,0,0\)\(0,0,0\)37\.76±2\.4437\.76\\pm 2\.4434\.89±2\.1634\.89\\pm 2\.1641\.08±3\.1441\.08\\pm 3\.1438\.93±3\.2238\.93\\pm 3\.2244\.99±4\.0344\.99\\pm 4\.0344\.43±4\.3244\.43\\pm 4\.3239\.57±3\.1639\.57\\pm 3\.1636\.79±3\.1436\.79\\pm 3\.1412\.64±0\.91\\mathbf\{12\.64\\pm 0\.91\}11\.04±0\.70\\mathbf\{11\.04\\pm 0\.70\}\(0\.5,0,0\)\(0\.5,0,0\)37\.88±2\.4537\.88\\pm 2\.4534\.97±2\.2034\.97\\pm 2\.2041\.17±3\.1041\.17\\pm 3\.1039\.01±3\.1939\.01\\pm 3\.1945\.02±4\.0045\.02\\pm 4\.0044\.45±4\.2744\.45\\pm 4\.2739\.68±3\.1039\.68\\pm 3\.1036\.90±3\.1636\.90\\pm 3\.1613\.30±1\.04\\mathbf\{13\.30\\pm 1\.04\}11\.54±0\.82\\mathbf\{11\.54\\pm 0\.82\}\(1,0,0\)\(1,0,0\)38\.42±2\.4338\.42\\pm 2\.4335\.42±2\.0435\.42\\pm 2\.0441\.57±2\.9541\.57\\pm 2\.9539\.38±3\.0639\.38\\pm 3\.0645\.08±3\.7845\.08\\pm 3\.7844\.49±4\.0544\.49\\pm 4\.0540\.20±3\.0040\.20\\pm 3\.0037\.35±3\.0237\.35\\pm 3\.0215\.09±1\.14\\mathbf\{15\.09\\pm 1\.14\}13\.00±0\.97\\mathbf\{13\.00\\pm 0\.97\}\(0,0\.5,0\)\(0,0\.5,0\)42\.26±2\.6942\.26\\pm 2\.6939\.38±2\.6839\.38\\pm 2\.6847\.22±2\.9647\.22\\pm 2\.9645\.43±2\.8945\.43\\pm 2\.8949\.87±3\.2749\.87\\pm 3\.2749\.38±3\.3149\.38\\pm 3\.3145\.00±3\.0645\.00\\pm 3\.0642\.46±2\.9542\.46\\pm 2\.9514\.83±1\.19\\mathbf\{14\.83\\pm 1\.19\}12\.98±0\.93\\mathbf\{12\.98\\pm 0\.93\}\(0,1,0\)\(0,1,0\)44\.03±2\.2444\.03\\pm 2\.2441\.08±2\.2241\.08\\pm 2\.2249\.45±2\.8249\.45\\pm 2\.8247\.93±2\.7547\.93\\pm 2\.7551\.97±2\.8751\.97\\pm 2\.8751\.57±2\.9751\.57\\pm 2\.9747\.11±2\.7647\.11\\pm 2\.7644\.68±2\.8744\.68\\pm 2\.8715\.71±0\.58\\mathbf\{15\.71\\pm 0\.58\}13\.73±0\.52\\mathbf\{13\.73\\pm 0\.52\}\(0,0,0\.5\)\(0,0,0\.5\)40\.93±2\.0240\.93\\pm 2\.0238\.34±1\.6738\.34\\pm 1\.6743\.42±2\.5943\.42\\pm 2\.5941\.60±2\.6641\.60\\pm 2\.6646\.07±3\.0446\.07\\pm 3\.0445\.52±3\.3245\.52\\pm 3\.3242\.43±2\.4942\.43\\pm 2\.4940\.01±2\.5440\.01\\pm 2\.5418\.85±1\.67\\mathbf\{18\.85\\pm 1\.67\}16\.62±1\.48\\mathbf\{16\.62\\pm 1\.48\}\(0,0,1\)\(0,0,1\)44\.53±1\.9944\.53\\pm 1\.9941\.90±1\.5141\.90\\pm 1\.5145\.96±2\.6245\.96\\pm 2\.6244\.20±2\.5644\.20\\pm 2\.5647\.39±2\.8147\.39\\pm 2\.8146\.90±2\.8846\.90\\pm 2\.8845\.35±2\.3245\.35\\pm 2\.3243\.05±2\.2643\.05\\pm 2\.2628\.29±2\.31\\mathbf\{28\.29\\pm 2\.31\}25\.19±2\.35\\mathbf\{25\.19\\pm 2\.35\}

To strengthen the connection to[Theorem4](https://arxiv.org/html/2610.09091#Thmtheorem4), we reran all seven synthetic\-knapsack settings using a bias\-free linear predictor, consistent with the predictor class assumed in the convergence analysis\. RSPO\+ achieves the lowest regret in all 14 federated and personalized comparisons\. Averaged across settings, RSPO\+ obtains 16\.96% federated regret, compared with 40\.83% for the strongest baseline, and 14\.87% personalized regret, compared with 38\.00% for the strongest baseline\. This corresponds to approximately2\.4×2\.4\\timesand2\.6×2\.6\\timeslower regret, respectively\. Detailed results are shown in[Table18](https://arxiv.org/html/2610.09091#A3.T18)\.

### C\.8Hyperparameter Selection

Hyperparameters were not selected by automated search\. The RSPO\+\-specific parameters \(λj\\lambda\_\{j\},ηj\\eta\_\{j\}\) were set using the principled per\-client defaults derived in[Eq\.11](https://arxiv.org/html/2610.09091#S3.E11)\. The remaining parameters \(rounds, epochs, batch size, client fraction\) were fixed to standard values from the FL literature\. A sensitivity analysis over personalization parameters for the knapsack experiment is available in the supplementary code and the results are shown in[Fig\.4](https://arxiv.org/html/2610.09091#A3.F4)\.

1%1\\%5%5\\%10%10\\%00%22%44%66%88%3\.6%3\.6\\%3\.2%3\.2\\%2\.4%2\.4\\%Freezing criterionτ\\tauRelative Regret

00\.511\.5200%22%44%66%88%3\.2%3\.2\\%4\.7%4\.7\\%5\.0%5\.0\\%5\.1%5\.1\\%5\.2%5\.2\\%κλ\\kappa\_\{\\lambda\}Relative RegretRSPO\+SPO\+

0\.20\.50\.800%22%44%66%88%4\.7%4\.7\\%4\.2%4\.2\\%3\.8%3\.8\\%κLR\\kappa\_\{\\mathrm\{LR\}\}Relative Regret

Figure 4:\(a\) Relative regret \(%\) of SPO\+ and RSPO\+ across different \(a\) values of the freezing thresholdτ\\tau, \(b\) values ofκλ\\kappa^\{\\lambda\}, and \(c\) values ofκLR\\kappa^\{\\mathrm\{LR\}\}\. Lower is better \(↓\\downarrow\)\.
### C\.9Algorithms

Input:client datasets

\{\(𝒟itr,𝒱i,ni\)\}i=1N\\\{\(\\mathcal\{D\}\_\{i\}^\{\\mathrm\{tr\}\},\\mathcal\{V\}\_\{i\},n\_\{i\}\)\\\}\_\{i=1\}^\{N\}, rounds

Rfed,RperR\_\{\\mathrm\{fed\}\},R\_\{\\mathrm\{per\}\}, local epochs

EE, participation rate

ρ\\rho, freeze threshold

τ\\tau, post\-freeze budget

KK
initialize global model

ϕ0\\phi^\{0\}
for*i=1i=1toNN*do

initialize per\-client schedules

\(λi,0,ηi,0\)\(\\lambda\_\{i,0\},\\,\\eta\_\{i,0\}\)
end for

global freeze flag

g←0g\\leftarrow 0
/\*Phase 1: Federated training \*/

for*r=0r=0toRfed−1R\_\{\\mathrm\{fed\}\}\-1*do

if*g=0g=0*thenadvance schedule counters

λi\(r\),ηi\(r\)\\lambda\_\{i\}^\{\(r\)\},\\eta\_\{i\}^\{\(r\)\}for all

ii
sample

Ir⊆\{1,…,N\}I\_\{r\}\\subseteq\\\{1,\\dots,N\\\}with

\|Ir\|=⌈ρ​N⌉\|I\_\{r\}\|=\\lceil\\rho N\\rceil
forall*i∈Iri\\in I\_\{r\}in parallel*do

ϕ~ir\+1←\\tilde\{\\phi\}\_\{i\}^\{r\+1\}\\leftarrow\{\}LocalTrain\(*ϕr,𝒟itr,λi\(r\),ηi\(r\),E\\phi^\{r\},\\mathcal\{D\}\_\{i\}^\{\\mathrm\{tr\}\},\\lambda\_\{i\}^\{\(r\)\},\\eta\_\{i\}^\{\(r\)\},E*\)

Δir←ϕ~ir\+1−ϕr\\Delta\_\{i\}^\{r\}\\leftarrow\\tilde\{\\phi\}\_\{i\}^\{r\+1\}\-\\phi^\{r\}
end forall

ϕr\+1←ϕr\+∑i∈Irni∑j∈Irnj​Δir\\phi^\{r\+1\}\\leftarrow\\phi^\{r\}\+\\sum\_\{i\\in I\_\{r\}\}\\frac\{n\_\{i\}\}\{\\sum\_\{j\\in I\_\{r\}\}n\_\{j\}\}\\,\\Delta\_\{i\}^\{r\}
if*StopMetric\(*ϕr\+1,\{𝒱i\},\{λi\(r\)\}\\phi^\{r\+1\},\\\{\\mathcal\{V\}\_\{i\}\\\},\\\{\\lambda\_\{i\}^\{\(r\)\}\\\}*\)≤τ\\leq\\tau*then

g←1g\\leftarrow 1//freeze annealing

end for

/\*Phase 2: Personalization \(warm\-started fromϕRfed\\phi^\{R\_\{\\mathrm\{fed\}\}\}\) \*/

forall*i=1,…,Ni=1,\\dots,Nin parallel*do

ϕiper←ϕRfed\\phi\_\{i\}^\{\\mathrm\{per\}\}\\leftarrow\\phi^\{R\_\{\\mathrm\{fed\}\}\}
initialize personal schedules

\(λi,per\(0\),ηi,per\(0\)\)\(\\lambda\_\{i,\\mathrm\{per\}\}^\{\(0\)\},\\,\\eta\_\{i,\\mathrm\{per\}\}^\{\(0\)\}\)
client freeze flag

gi←0g\_\{i\}\\leftarrow 0,

freeze round

ti←\+∞t\_\{i\}\\leftarrow\+\\infty
for*r=0r=0toRper−1R\_\{\\mathrm\{per\}\}\-1*do

if*r\>ti\+Kr\>t\_\{i\}\+K*thenbreak

if*gi=0g\_\{i\}=0*thenadvance personal schedule counters

λi,per\(r\),ηi,per\(r\)\\lambda\_\{i,\\mathrm\{per\}\}^\{\(r\)\},\\eta\_\{i,\\mathrm\{per\}\}^\{\(r\)\}
ϕiper←\\phi\_\{i\}^\{\\mathrm\{per\}\}\\leftarrow\{\}LocalTrain\(*ϕiper,𝒟itr,λi,per\(r\),ηi,per\(r\),E\\phi\_\{i\}^\{\\mathrm\{per\}\},\\mathcal\{D\}\_\{i\}^\{\\mathrm\{tr\}\},\\lambda\_\{i,\\mathrm\{per\}\}^\{\(r\)\},\\eta\_\{i,\\mathrm\{per\}\}^\{\(r\)\},E*\)

if*StopMetric\(*ϕiper,𝒱i,λi,per\(r\)\\phi\_\{i\}^\{\\mathrm\{per\}\},\\mathcal\{V\}\_\{i\},\\lambda\_\{i,\\mathrm\{per\}\}^\{\(r\)\}*\)≤τ\\leq\\tauandgi=0g\_\{i\}=0*then

gi←1g\_\{i\}\\leftarrow 1,

ti←rt\_\{i\}\\leftarrow r
end for

end forall

return

ϕRfed\\phi^\{R\_\{\\mathrm\{fed\}\}\}and

\{ϕi,⋆per\}i=1N\\\{\\phi\_\{i,\\star\}^\{\\mathrm\{per\}\}\\\}\_\{i=1\}^\{N\}

Algorithm 1FedRSPO\+: federated warm start with co\-annealing scheduling and personalization\.

## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The abstract and introduction clearly state our three contributions: the RSPO\+ surrogate with regret/error bounds, FedRSPO\+ with heterogeneity bounds that vanish at homogeneity, and empirical evaluation across three tasks\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: We discuss limitations of our approach in[Section5](https://arxiv.org/html/2610.09091#S5)\.
10. Guidelines: - •The answer\[N/A\]means that the paper has no limitation while the answer\[No\]means that the paper has limitations, but those are not discussed in the paper\. - •The authors are encouraged to create a separate “Limitations” section in their paper\. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions \(e\.g\., independence assumptions, noiseless settings, model well\-specification, asymptotic approximations only holding locally\)\. The authors should reflect on how these assumptions might be violated in practice and what the implications would be\. - •The authors should reflect on the scope of the claims made, e\.g\., if the approach was only tested on a few datasets or with a few runs\. In general, empirical results often depend on implicit assumptions, which should be articulated\. - •The authors should reflect on the factors that influence the performance of the approach\. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting\. Or a speech\-to\-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon\. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size\. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness\. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper\. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community\. Reviewers will be specifically instructed to not penalize honesty concerning limitations\.
11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[Yes\]
14. Justification: All proofs are provided in the Appendix,[AppendixB](https://arxiv.org/html/2610.09091#A2)\.
15. Guidelines: - •The answer\[N/A\]means that the paper does not include theoretical results\. - •All the theorems, formulas, and proofs in the paper should be numbered and cross\-referenced\. - •All assumptions should be clearly stated or referenced in the statement of any theorems\. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition\. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material\. - •Theorems and Lemmas that the proof relies upon should be properly referenced\.
16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: We have implementation details for each experiment, including hyperparameter choices and data splits used, to reproduce all our results\. All data sources are provided and well\-documented\. We also provide the anonymized code for reproducibility\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[Yes\]
24. Justification: Our paper provides the anonymized code necessary to reproduce the experiments, and all data sources are open and well\-documented\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: We explicitly provide all the experimental details and setups for each experiment in the Appendix,[AppendixC](https://arxiv.org/html/2610.09091#A3)\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: We report mean±1​σ\\pm 1\\sigma\(standard deviation\) over 5 random seeds for all experiments and describe error bars and methodology in the Appendix,[AppendixC](https://arxiv.org/html/2610.09091#A3)\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: We provide the computer resources in the implementation details in the Appendix\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: We read carefully and align with the Ethics Guidelines shown\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[Yes\]
49. Justification: We state the potential broader impacts in[Section5](https://arxiv.org/html/2610.09091#S5)\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: Our datasets are widely used ones, such as PJM energy markets and the Warcraft DFL benchmark\. Their safety has been proven by existing work\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: We cite all works and tools used to properly credit their authors\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[Yes\]
64. Justification: We have documented our models and methods used, and release them along with the anonymized code\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: This paper does not involve crowdsourcing or human subjects\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: This paper does not need to do IRB approvals or equivalent since it does not involve human subject research\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: We do not use LLMs for any non\-standard uses\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.