First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection

# First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection
Source: [https://arxiv.org/html/2607.27632](https://arxiv.org/html/2607.27632)
Kaixuan Jiao Kai Yang Nadjib Aitsaadi Ilhem Fajjari Renwei \(Richard\) Li

###### Abstract

With the rapid advancement of the Internet of Things \(IoT\), massive amounts of data are generated across distributed edge networks\. Training models on full data incurs significant computational overhead and storage bottlenecks, rendering coreset selection a critical paradigm\. Furthermore, given the privacy\-sensitive nature of local data and the escalating demand for model robustness in real\-world deployments, developing an effective distributed optimization framework for robust coreset selection is vital, yet remains largely unexplored\. To this end, this work first characterizes the hierarchical dependencies among coreset selection, robust optimization, and distributed learning, and formulates the distributed robust coreset selection as a trilevel optimization problem with level\-wise constraints\. Furthermore, to effectively solve the trilevel problem in a distributed manner, theFederatedFirst\-orderConstrainedTrilevelOptimization \(F2CTO\) is proposed, which synergistically integrates a hierarchical composite value\-function reformulation and a distributed alternating projected gradient algorithm\. To the best of our knowledge, F2CTO is the first method developed for distributed robust coreset selection, as well as the first distributed optimization approach for trilevel optimization problems with level\-wise constraints\. Additionally, we prove that the proposed method achieves a non\-asymptotic convergence rate of𝒪​\(ϵ−3/2\)\\mathcal\{O\}\(\\epsilon^\{\-3/2\}\)for finding anϵ\\epsilon\-stationary point\. Extensive empirical evaluations on reliable continual learning demonstrate the effectiveness and efficiency of the proposed F2CTO\.

## IIntroduction

Driven by the rapid development of the Internet of Things \(IoT\), massive volumes of data are continuously generated and dispersed across nodes over distributed networks\[[36](https://arxiv.org/html/2607.27632#bib.bib12)\]\. Training machine learning models directly on the entirety of this data incurs prohibitive computational overhead and severe storage bottlenecks\[[37](https://arxiv.org/html/2607.27632#bib.bib31)\]\. To alleviate these training and memory constraints, coreset selection has garnered significant attention\[[25](https://arxiv.org/html/2607.27632#bib.bib25)\]\. This technique involves extracting a representative and information\-dense subset \(i\.e\., the coreset\) from a large\-scale dataset, such that a model trained on this subset can closely approximate the performance of one trained on the entire dataset\. Coreset selection has been widely used across various fields because of its remarkable ability to alleviate computational and storage bottlenecks while preserving the informational integrity of large\-scale datasets\. For instance, to reduce the computational costs associated with training large language models \(LLMs\) on massive corpora, coreset selection has been employed to extract representative subsets of instruction data, thereby significantly improving training efficiency\[[31](https://arxiv.org/html/2607.27632#bib.bib32)\]\. Additionally, to enable models deployed in Internet of Things \(IoT\) environments to acquire new knowledge without suffering from catastrophic forgetting, coreset selection has been leveraged to facilitate continual learning\[[3](https://arxiv.org/html/2607.27632#bib.bib16)\]\. Moreover, in the field of network security, coreset selection has been utilized to reduce memory and storage requirements, ultimately improving the efficiency of Bayesian learning for network intrusion detection\[[43](https://arxiv.org/html/2607.27632#bib.bib15)\]\.

In addition to computational and memory constraints, the highly sensitive nature of local data in distributed networks poses a significant challenge\. Centralizing this data not only risks severe privacy breaches but also incurs massive communication overhead\[[19](https://arxiv.org/html/2607.27632#bib.bib167)\], thereby necessitating the development of distributed optimization methods for coreset selection\. Moreover, as the demand for reliable machine learning models grows, extracting critical data from large\-scale datasets to enhance model robustness, thereby achieving efficient robust optimization, has attracted considerable research attention\[[17](https://arxiv.org/html/2607.27632#bib.bib127)\]\. However, existing coreset selection methods either overlook the robustness of coresets or rely on centralized setups, which limit their applicability to distributed robust coreset selection\. In light of these limitations, the first key question this paper aims to address is:Can we construct an effective distributed robust coreset selection framework?Such a framework is expected to enable individual nodes in a network to identify representative data without sharing raw data, while more effectively leveraging the global information through coordination by a master node\. The motivating applications of studying distributed robust coreset selection are provided in Sec\.[III\-A](https://arxiv.org/html/2607.27632#S3.SS1)\.

Coreset selection is typically formulated as a bilevel optimization problem\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], where the lower\-level optimization trains model parameters based on the selected coreset, and the upper\-level optimization evaluates these parameters to further refine the coreset selection\. Building upon this paradigm and investigating the inherent coupling among coreset selection, robust optimization, and distributed learning, distributed robust coreset selection can be formulated as a constrained trilevel optimization problem\. Compared to trilevel optimization problems without additional constraints, which are the primary focus of existing literature\[[19](https://arxiv.org/html/2607.27632#bib.bib167),[9](https://arxiv.org/html/2607.27632#bib.bib61)\], this constrained variant is harder to solve, leaving the corresponding distributed algorithms largely unexplored\. Motivated by this gap, this work seeks to answer a second key question:Can we design an efficient distributed first\-order algorithm with non\-asymptotic convergence guarantees to solve the constrained trilevel optimization problem?

To this end, we propose aFederatedFirst\-orderConstrainedTrilevelOptimization framework, termed F2CTO, for distributed robust coreset selection\. Within this framework, we first investigate the underlying coupling inherent in robust coreset selection and formulate it as a trilevel optimization problem with level\-wise constraints\. Subsequently, a hierarchical composite value\-function method is proposed to transform the original optimization problem into a formulation amenable to distributed optimization, comprising one inner\-level and two outer\-level value\-functions\. Building upon this, we introduce a distributed first\-order alternating projected gradient algorithm to efficiently address the resulting problem\. To the best of our knowledge, this is the first work to address the distributed robust coreset selection problem, and the first distributed algorithm designed to solve constrained trilevel optimization problems\. Theoretically, we analyze the non\-asymptotic convergence rate of the proposed method to achieve anϵ\\epsilon\-stationary point, explicitly characterizing both its iteration and communication complexities\. The contributions of this work are summarized as follows:

1\.Unlike existing coreset selection works that either focus exclusively on centralized settings or overlook the necessity of robustness, this work takes an initial step toward integrating coreset selection, robust optimization, and distributed learning into a unified trilevel optimization framework to address the distributed robust coreset selection problem\.

2\.Compared to existing trilevel optimization methods, to the best of our knowledge, the proposed F2CTO is the first to address trilevel optimization with level\-wise constraints in a distributed manner\. Theoretically, we demonstrate that the non\-asymptotic convergence rate for the proposed F2CTO to achieve anϵ\\epsilon\-stationary point is upper bounded by𝒪​\(ϵ−3/2\)\\mathcal\{O\}\(\\epsilon^\{\-3/2\}\)\.

3\.Extensive experiments on reliable continual learning demonstrate the superior effectiveness and efficiency of F2CTO in addressing distributed robust coreset selection\.

## IIRelated Work

### II\-ACoreset Selection

Coreset selection aims to extract a representative subset𝒟∗\\mathcal\{D\}^\{\*\}from a large\-scale dataset𝒟\\mathcal\{D\}\(𝒟∗⊂𝒟\\mathcal\{D\}^\{\*\}\\\!\\subset\\\!\\mathcal\{D\}\) to improve training efficiency and reduce memory overhead\. As discussed in\[[25](https://arxiv.org/html/2607.27632#bib.bib25)\], existing coreset selection methods can be broadly categorized into training\-free\[[38](https://arxiv.org/html/2607.27632#bib.bib33)\], scoring\[[37](https://arxiv.org/html/2607.27632#bib.bib31)\], decision boundary\[[39](https://arxiv.org/html/2607.27632#bib.bib30)\], submodularity\[[21](https://arxiv.org/html/2607.27632#bib.bib29)\], gradient matching\[[22](https://arxiv.org/html/2607.27632#bib.bib28)\], and bilevel optimization\[[31](https://arxiv.org/html/2607.27632#bib.bib32)\]approaches\. As an alternative to coreset selection, data distillation aims to synthesize a compact dataset but is highly sensitive to model architecture\[[44](https://arxiv.org/html/2607.27632#bib.bib13)\], making a direct performance comparison with coreset selection inappropriate\[[37](https://arxiv.org/html/2607.27632#bib.bib31)\]\. In addition, recent studies have demonstrated the promise of robust coreset selection in enhancing the robustness of machine learning models\[[5](https://arxiv.org/html/2607.27632#bib.bib19),[32](https://arxiv.org/html/2607.27632#bib.bib18)\]\. Furthermore, in real\-world scenarios, data is frequently generated and distributed across various nodes over networks\. To preserve data privacy while extracting representative subsets, distributed coreset selection methods have received research attention\[[11](https://arxiv.org/html/2607.27632#bib.bib17),[33](https://arxiv.org/html/2607.27632#bib.bib14)\]\. Nevertheless, current methods either neglect the robustness in coreset or remain confined to centralized settings\. How to perform robust coreset selection within distributed networksremains under\-explored\. Notably, existing robust coreset selection methods \(e\.g\.,\[[5](https://arxiv.org/html/2607.27632#bib.bib19),[32](https://arxiv.org/html/2607.27632#bib.bib18)\]\) cannot be effectively applied in distributed robust coreset selection, as it is impractical to centralize distributed data due to data privacy, and performing selection locallyfails tofully exploit global information\. Meanwhile, existing distributed coreset selection methods \(e\.g\.,\[[11](https://arxiv.org/html/2607.27632#bib.bib17),[33](https://arxiv.org/html/2607.27632#bib.bib14)\]\) are also not directly applicable\. This is because robust optimization introduces an additional optimization level into the original optimization problem, fundamentally nesting its structure\. In fact, the distributed robust coreset selection problem admitsno downward polynomial\-time reductionto its non\-robust counterpart unless the polynomial hierarchy collapses \(e\.g\.,ΣrP=Σr\+1P\\Sigma\_\{r\}^\{\\mathrm\{P\}\}=\\Sigma\_\{r\+1\}^\{\\mathrm\{P\}\}\)\[[34](https://arxiv.org/html/2607.27632#bib.bib21)\], rendering it inherently intractable\. To bridge this gap, this work takes apioneering steptowards addressing the distributed robust coreset selection problem\.

TABLE I:Comparison in terms of key properties between the proposed F2CTO and related state\-of\-the\-art approaches\.
### II\-BTrilevel Optimization

Trilevel optimization refers to a class of optimization problems characterized by a three\-tiered nested structure, wherein the three levels are mutually interdependent and dynamically influence one another\. Compared to bilevel optimization\[[28](https://arxiv.org/html/2607.27632#bib.bib221)\], trilevel optimization is significantly more challenging to solve due to its highly complex nested structure\. In fact, merely identifying a feasible solution within a trilevel framework necessitates solving an underlying bilevel optimization problem\. Trilevel optimization has been widely used in machine learning \(e\.g\.,\[[4](https://arxiv.org/html/2607.27632#bib.bib27),[7](https://arxiv.org/html/2607.27632#bib.bib120),[18](https://arxiv.org/html/2607.27632#bib.bib196),[13](https://arxiv.org/html/2607.27632#bib.bib126)\]\) and networking \(e\.g\.,\[[8](https://arxiv.org/html/2607.27632#bib.bib26),[41](https://arxiv.org/html/2607.27632#bib.bib125)\]\)\. Since data is often distributed across multiple nodes, distributed trilevel optimization has garnered attention as an effective approach to solving these problems without transmitting raw data\. In\[[19](https://arxiv.org/html/2607.27632#bib.bib167)\], a federated trilevel optimization framework based on two\-layer polyhedral approximation is introduced\. A zeroth\-order distributed trilevel optimization method is proposed to address the trilevel optimization problems without relying on gradients in\[[15](https://arxiv.org/html/2607.27632#bib.bib111)\]\. However, existing distributed trilevel optimization methods only consider the trilevel optimization without level\-wise constraints\. To the best of our knowledge, this is thefirst workthat addresses the constrained trilevel optimization problems in a distributed manner and provides the non\-asymptotic convergence guarantees\. To explicitly illustrate the superiority of the proposed method, we provide a comprehensive comparison with existing state\-of\-the\-art multilevel optimization methods regarding key properties and convergence rates \(see Tables[I](https://arxiv.org/html/2607.27632#S2.T1)and[II](https://arxiv.org/html/2607.27632#S3.T2)\)\.

## IIIDistributed Robust Coreset Selection

In this part, we first provide the motivating applications for investigating distributed robust coreset selection in Sec\.[III\-A](https://arxiv.org/html/2607.27632#S3.SS1)\. Then, the fundamental interplay between coreset selection, robust optimization, and distributed learning is studied in Sec\.[III\-B](https://arxiv.org/html/2607.27632#S3.SS2), and distributed robust coreset selection is formulated as a trilevel optimization problem with level\-wise constraints\. To effectively solve it, a federated first\-order constrained trilevel optimization method F2CTO is proposed in Sec\.[III\-C](https://arxiv.org/html/2607.27632#S3.SS3), which consists of a hierarchical composite value\-function reformulation and a distributed alternating projected gradient algorithm\. The overview of the proposed framework is shown in Fig\.[1](https://arxiv.org/html/2607.27632#S3.F1)\.

![Refer to caption](https://arxiv.org/html/2607.27632v1/x1.png)Figure 1:Illustration of the proposed distributed robust coreset selection framework\. The framework is formulated as a trilevel optimization problem\. At the first level, the data weights𝜶i\{\\boldsymbol\{\\alpha\}\_\{i\}\}are optimized to select the representative data samples\. The second level serves two coupled purposes: generating the evaluation\-side perturbations𝒒i\{\\boldsymbol\{q\}\_\{i\}\}for the first\-level objective, and training a robust model𝒘\\boldsymbol\{w\}using the optimized data weights together with the training\-side perturbations\. The training\-side perturbations𝒑i\{\\boldsymbol\{p\}\_\{i\}\}are obtained through the third\-level optimization\. During the distributed optimization, each worker locally refines hierarchical composite value\-functions and updates variables, communicating only local parameters to the master without sharing raw data\.### III\-AMotivating Applications and Problem Setup

#### III\-A1Reliable Continual Learning

Continual learning is well suited to IoT applications in which privacy\-sensitive data streams are continuously generated at various edge devices\[[3](https://arxiv.org/html/2607.27632#bib.bib16)\]\. For example, in sensor\-based human activity recognition, wearable devices continuously observe new activity patterns\. Due to limited storage capacity, retaining all historical data is impractical, whereas learning only from new data may cause catastrophic forgetting\[[30](https://arxiv.org/html/2607.27632#bib.bib8)\]\. Furthermore, since adversarial perturbations to sensor data can mislead activity recognition models\[[29](https://arxiv.org/html/2607.27632#bib.bib7)\], ensuring the reliability of continual learning is essential\. This motivates our investigation on distributed robust coreset selection, which identifies robust coresets without sharing privacy\-sensitive data, thereby reducing storage costs while preserving critical knowledge for reliable continual learning\.

#### III\-A2Adversarially Robust Network Intrusion Detection

Network intrusion detection aims to detect attacks and compromised devices in distributed IoT networks\[[27](https://arxiv.org/html/2607.27632#bib.bib10)\]\. In such systems, massive volumes of privacy\-sensitive traffic data are continuously generated and distributed across multiple nodes, training on all the data incurs substantial computation overhead\[[43](https://arxiv.org/html/2607.27632#bib.bib15)\]\. Moreover, it is essential to defend against adversaries who deliberately manipulate network traffic characteristics to evade machine\-learning\-based intrusion detectors\[[10](https://arxiv.org/html/2607.27632#bib.bib4)\]\. To reduce training overhead while maintaining model robustness in network intrusion detection, we are motivated to study the distributed robust coreset selection for identifying representative coresets without transmitting local privacy\-sensitive data\.

Problem Setup\.Following the parameter\-server architecture widely used in federated learning\[[19](https://arxiv.org/html/2607.27632#bib.bib167)\], we consider a distributed network comprising a master node \(central server\) andNNworkers \(clients\)\. The global training dataset𝒟=⋃i=1N𝒟i\\mathcal\{D\}=\\bigcup\_\{i=1\}^\{N\}\\mathcal\{D\}\_\{i\}is partitioned across the workers, where each local dataset𝒟i\\mathcal\{D\}\_\{i\}containsMiM\_\{i\}samples\. Our objective is to construct a representative global coreset𝒟∗=⋃i=1N𝒟i∗\\mathcal\{D\}^\{\*\}=\\bigcup\_\{i=1\}^\{N\}\\mathcal\{D\}\_\{i\}^\{\*\}\(with𝒟i∗⊂𝒟i\\mathcal\{D\}\_\{i\}^\{\*\}\\subset\\mathcal\{D\}\_\{i\}\) that preserves the robust training objective of the full dataset while substantially reducing the sample size, i\.e\.,Ki≪MiK\_\{i\}\\ll M\_\{i\}\.

TABLE II:Comparison in terms of convergence rate between the proposed F2CTO and related state\-of\-the\-art approaches\.Here, the subscriptswwandw/ow/odenote optimization with and without level\-wise constraints, respectively\.

### III\-BOverall Constrained Trilevel Optimization Problem

To mathematically formalize the distributed robust coreset selection, we adopt a multilevel optimization framework\. Coreset selection is typically formulated as a bilevel optimization problem\[[31](https://arxiv.org/html/2607.27632#bib.bib32)\]\. Building upon the continuous regularized bilevel framework proposed in\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], we jointly incorporate min\-max robust optimization and distributed learning\. This allows us to cast the distributed robust coreset selection as the following trilevel optimization problem with level\-wise constraints:

min⁡1N​∑i=1N∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k,yi,k\)−λ​∑b=1Ki𝔼𝒛i​\(𝜶i\+𝒛i\)\[b\]s\.t​\. 0≤αi,k≤1,‖𝜶i‖=1,∀i,k,𝒒i=arg⁡min𝒒i′:‖qi,k′‖∞≤c1,∀k−∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k′,yi,k\),∀i,𝒘=arg⁡min𝒘′⁡1N​∑i=1N∑k=1Miαi,k​ℓi,k​\(𝒘′;xi,k\+pi,k′,yi,k\)s\.t\.‖𝒘′‖2≤c2,𝒑i=arg⁡min𝒑i′−∑k=1Miαi,k​ℓi,k​\(𝒘′;xi,k\+pi,k′,yi,k\),∀i,s\.t\.‖pi,k′‖∞≤c3,∀kvar\.\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\}\.\\begin\{array\}\[\]\{l\}\\min\\frac\{1\}\{N\}\\sum\\limits\_\{i=1\}^\{N\}\{\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}\},\{y\_\{i,k\}\}\)\}\\\!\-\\\!\\lambda\\sum\\limits\_\{b=1\}^\{\{K\_\{i\}\}\}\\\!\{\{\\mathbb\{E\}\_\{\\boldsymbol\{z\}\_\{i\}\}\}\}\{\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\!\+\\\!\\boldsymbol\{z\}\_\{i\}\)\}\_\{\[b\]\}\}\}\\\\ \{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;0\\leq\{\\alpha\_\{i,k\}\}\\leq 1,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1,\\forall i,k,\\\\ \\quad\{\\boldsymbol\{q\}\_\{i\}\}=\\mathop\{\\arg\\min\}\\limits\_\{\{\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\}:\|\|\{q\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{1\},\\forall k\}\}\-\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\),\\forall i,\\\\ \\quad\\boldsymbol\{w\}=\\arg\{\\min\_\{\\boldsymbol\{w\}^\{\\prime\}\}\}\\frac\{1\}\{N\}\\sum\\limits\_\{i=1\}^\{N\}\{\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\alpha\_\{i,k\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\}^\{\\prime\};\{x\_\{i,k\}\}\\\!\+\\\!\{p\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\)\}\\\\ \\qquad\{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\|\|\{\\boldsymbol\{w\}^\{\\prime\}\}\|\{\|\_\{2\}\}\\leq c\_\{2\},\\\\ \\qquad\{\\boldsymbol\{p\}\_\{i\}\}=\\arg\{\\min\_\{\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\}\}\-\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\\\!\{\{\\alpha\_\{i,k\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\}^\{\\prime\};\{x\_\{i,k\}\}\\\!\+\\\!\{p\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\),\\forall i,\\\\ \\qquad\\quad\{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\|\|\{p\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall k\\\\ \{\\rm\{var\}\}\.\\qquad\\qquad\\\{\\boldsymbol\{\\alpha\}\_\{i\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}\}\\\},\\boldsymbol\{w\},\\\{\\boldsymbol\{p\}\_\{i\}\\\}\.\\end\{array\}\(1\)whereNNrepresents the total number of workers\. For each workerii,MiM\_\{i\}andKiK\_\{i\}denote the sizes of its local dataset and robust coreset, respectively\. The pair\(xi,k,yi,k\)\(\{x\_\{i,k\}\},\{y\_\{i,k\}\}\)denotes thekthk^\{\\rm\{th\}\}data sample and label, withℓi,k​\(⋅\)\{\\ell\_\{i,k\}\}\(\\cdot\)being the corresponding loss function\.𝜶i∈ℝMi\\boldsymbol\{\\alpha\}\_\{i\}\\in\\mathbb\{R\}^\{M\_\{i\}\}is the local data weight vector \(whereαi,k\{\\alpha\_\{i,k\}\}is itskthk^\{\\rm\{th\}\}element\), while𝒒i=\[qi,1,⋯,qi,Mi\]\\boldsymbol\{q\}\_\{i\}=\[q\_\{i,1\},\\cdots,q\_\{i,M\_\{i\}\}\]and𝒑i=\[pi,1,⋯,pi,Mi\]\\boldsymbol\{p\}\_\{i\}=\[p\_\{i,1\},\\cdots,p\_\{i,M\_\{i\}\}\]represent the evaluation\-side and training\-side adversarial perturbations, respectively\.𝒘∈ℝd\\boldsymbol\{w\}\\in\\mathbb\{R\}^\{d\}denotes the model parameters, and𝒛i∼𝒩​\(𝟎,δ2​𝐈\)\\boldsymbol\{z\}\_\{i\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\delta^\{2\}\\mathbf\{I\}\)is a Gaussian noise vector, where\(𝜶i\+𝒛i\)\[b\]\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\}\+\\boldsymbol\{z\}\_\{i\}\)\}\_\{\[b\]\}identifies thebthb^\{\\rm\{th\}\}largest component of the resulting noisy vector\. Moreover,αi,k≤1,‖𝜶i‖=1,‖qi,k′‖∞≤c1,‖𝒘′‖2≤c2,‖pi,k′‖∞≤c3,∀i,k\{\\alpha\_\{i,k\}\}\\leq 1,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1,\|\|\{q\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{1\},\|\|\{\\boldsymbol\{w\}^\{\\prime\}\}\|\{\|\_\{2\}\}\\leq c\_\{2\},\|\|\{p\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall i,kare the level\-wise constraints, wherec1\>0,c2\>0,c3\>0c\_\{1\}\>0,c\_\{2\}\>0,c\_\{3\}\>0are constants\. To clearly articulate the overall trilevel optimization problem, we systematically analyze each level as follows:

① First\-level optimization:The first\-level local objectivef1,i​\(𝜶i,𝒒i,𝒘,𝒑i\)\{\{f\_\{1,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}on workeriiconsists of two parts\. The first term∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k,yi,k\)\\sum\\nolimits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\+\{q\_\{i,k\}\},\{y\_\{i,k\}\}\)\}evaluates the robust empirical loss of this trained model𝒘\\boldsymbol\{w\}on the full dataset under worst\-case evaluation\-side perturbations𝒒i\{\\boldsymbol\{q\}\_\{i\}\}\. Although𝜶i\\boldsymbol\{\\alpha\}\_\{i\}does not appear explicitly in this term, it affects this term indirectly through the trained model𝒘\\boldsymbol\{w\}\. The second term−λ​∑b=1Ki𝔼𝒛i​\(𝜶i\+𝒛i\)\[b\]\-\\lambda\\sum\\nolimits\_\{b=1\}^\{\{K\_\{i\}\}\}\\\!\{\{\\mathbb\{E\}\_\{\\boldsymbol\{z\}\_\{i\}\}\}\}\{\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\!\+\\\!\\boldsymbol\{z\}\_\{i\}\)\}\_\{\[b\]\}\}acts as a smoothed top\-KiK\_\{i\}regularization\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\],λ\>0\\lambda\>0is the regularization coefficient\. Minimizing it encourages the largestKiK\_\{i\}entries of𝜶i\\boldsymbol\{\\alpha\}\_\{i\}to carry more weight, thereby promoting a robust coreset of sizeKiK\_\{i\}\.

② Second\-level optimization:The second level consists of two optimization problems with different roles\. The first one is associated with the evaluation\-side perturbation𝒒i\\boldsymbol\{q\}\_\{i\}used in the first\-level robust assessment, which isf2,i\(1\)​\(𝜶i,𝒒i,𝒘\)=−∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k,yi,k\)\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\}\)\}=\-\\sum\\nolimits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}\},\{y\_\{i,k\}\}\)\. This objective generates the worst\-case evaluation perturbations for the first\-level objective\. The second one corresponds to the weighted robust training objectivef2,i\(2\)​\(𝜶i,𝒘,𝒑i\)=∑k=1Miαi,k​ℓi,k​\(𝒘;xi,k\+pi,k,yi,k\)\{\{f\_\{2,i\}^\{\(2\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}=\\sum\\nolimits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\alpha\_\{i,k\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\+\{p\_\{i,k\}\},\{y\_\{i,k\}\}\)\. This objective uses the data weights𝜶i\\boldsymbol\{\\alpha\}\_\{i\}and the adversarial perturbations𝒑i\\boldsymbol\{p\}\_\{i\}determined by the first and third level to train the robust model\.

③ Third\-level optimization:The third level aims to generate the training\-side perturbations for the second\-level robust optimization\. Given fixed data weights𝜶i\\boldsymbol\{\\alpha\}\_\{i\}and candidate model parameters𝒘′\\boldsymbol\{w\}^\{\\prime\}, the local third\-level objective on workeriiis defined asf3,i​\(𝜶i,𝒘′,𝒑i′\)=−∑k=1Miαi,k​ℓi,k​\(𝒘′;xi,k\+pi,k′,yi,k\)\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\)=\-\\sum\\nolimits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\alpha\_\{i,k\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\}^\{\\prime\};\{x\_\{i,k\}\}\+\{p\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\)\. Minimizingf3,if\_\{3,i\}can effectively identify the worst\-case perturbations for the second\-level optimization\. Since the loss is weighted by𝜶i\\boldsymbol\{\\alpha\}\_\{i\}, data assigned larger weights exert a stronger influence on the generated perturbations\.

Challenges in Solving the Problem \([1](https://arxiv.org/html/2607.27632#S3.E1)\):In trilevel optimization, even identifying a feasible solution is NP\-hard\[[14](https://arxiv.org/html/2607.27632#bib.bib34)\]\. From a polynomial hierarchy perspective, the decision versions of bilevel and trilevel linear optimization areΣ1P\\Sigma\_\{1\}^\{\\mathrm\{P\}\}\-complete andΣ2P\\Sigma\_\{2\}^\{\\mathrm\{P\}\}\-complete, respectively\[[34](https://arxiv.org/html/2607.27632#bib.bib21)\]\. Unless polynomial hierarchy collapses \(e\.g\.,ΣrP=Σr\+1P\\Sigma\_\{r\}^\{\\mathrm\{P\}\}\\\!=\\\!\\Sigma\_\{r\+1\}^\{\\mathrm\{P\}\}\), higher\-level structures represent harder problem classes that admit no downward polynomial\-time reduction\. Moreover, unlike the distributed trilevel optimization studied in previous works, Eq\. \([1](https://arxiv.org/html/2607.27632#S3.E1)\) presentstwo additional challenges:1\) it incorporates auxiliary constraints at each level, and 2\) it involves two distinct optimization problems at the second level\. The complex interplay between these level\-wise constraints and the multiple second\-level subproblems significantly complicates the geometry in this optimization problem,making existing trilevel optimization methods fail to solve it, which motivates us to develop an efficient distributed constrained trilevel optimization method\.

### III\-CFederated First\-order Constrained Trilevel Optimization

To effectively solve the trilevel problem \([1](https://arxiv.org/html/2607.27632#S3.E1)\), a federated first\-order constrained trilevel optimization F2CTO is proposed\. Firstly, given that this highly nested structure is intractable to solve directly, Sec\.[III\-C1](https://arxiv.org/html/2607.27632#S3.SS3.SSS1)introduces a hierarchical composite value\-function method, which transforms the problem into a single\-level structure, making it amenable to first\-order optimization algorithms\. Building upon this, Sec\.[III\-C2](https://arxiv.org/html/2607.27632#S3.SS3.SSS2)presents a distributed first\-order alternating projected gradient algorithm to efficiently address this problem\.

#### III\-C1Hierarchical Composite Value\-function Reformulation

Value\-function methods are widely used in bilevel optimization to reduce the optimization problems into tractable structures\[[40](https://arxiv.org/html/2607.27632#bib.bib224)\]\. Motivated by this, this work proposes a hierarchical composite value\-function reformulation tailored for constrained trilevel optimization, which consists of an inner\-layer value\-function and two outer\-layer value\-functions\. Specifically, the third\-level problem, i\.e\.,𝒑i=arg⁡min𝒑i′⁡f3,i​\(𝜶i,𝒘′,𝒑i′\)​s\.t\.‖pi,k′‖∞≤c3,∀k\{\\boldsymbol\{p\}\_\{i\}\}=\\arg\\min\_\{\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\}\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\)\\,\{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\,\|\|\{p\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall k, operates as a constraint on the second\-level optimization, which can be equivalently expressed by the value\-function constraint:

f3,i​\(𝜶i,𝒘′,𝒑i\)−V3,i​\(𝜶i,𝒘′\)≤0,∀i,V3,i​\(𝜶i,𝒘′\)=min𝒑i′:‖pi,k′‖∞≤c3,∀k⁡f3,i​\(𝜶i,𝒘′,𝒑i′\),\\begin\{array\}\[\]\{l\}\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}\}\)\-\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\}\)\\leq 0,\\forall i,\\\\ \{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\}\)=\{\\min\_\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}:\|\|\{p\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall k\}\}\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}\}^\{\\prime\}\),\\end\{array\}\\vskip\-3\.55658pt\(2\)whereV3,i​\(⋅\)V\_\{3,i\}\(\\cdot\)is the inner\-layer value\-function\. Thus, by utilizing the inner\-layer value\-function constraint in Eq\. \([2](https://arxiv.org/html/2607.27632#S3.E2)\), the original optimization problem can be reformulated as a constrained bilevel optimization problem as follows:

min⁡1N​∑i=1N∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k,yi,k\)−λ​∑b=1Ki𝔼𝒛i​\(𝜶i\+𝒛i\)\[b\]s\.t​\. 0≤αi,k≤1,‖𝜶i‖=1,∀i,k,𝒒i=arg⁡min𝒒i′:‖qi,k′‖∞≤c1,∀k−∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k′,yi,k\),∀i,𝒘=arg⁡min𝒘′⁡1N​∑i=1N∑k=1Miαi,k​ℓi,k​\(𝒘′;xi,k\+pi,k′,yi,k\)s\.t\.‖𝒘′‖2≤c2,f3,i​\(𝜶i,𝒘′,𝒑i\)−V3,i​\(𝜶i,𝒘′\)≤0,∀i,var\.\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\}\.\\begin\{array\}\[\]\{l\}\\min\\frac\{1\}\{N\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\{\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}\},\{y\_\{i,k\}\}\)\}\\\!\-\\\!\\lambda\\sum\\limits\_\{b=1\}^\{\{K\_\{i\}\}\}\\\!\{\{\\mathbb\{E\}\_\{\\boldsymbol\{z\}\_\{i\}\}\}\}\{\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\!\+\\\!\\boldsymbol\{z\}\_\{i\}\)\}\_\{\[b\]\}\}\}\\\\ \{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;0\\leq\{\\alpha\_\{i,k\}\}\\leq 1,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1,\\forall i,k,\\\\ \\quad\{\\boldsymbol\{q\}\_\{i\}\}=\\mathop\{\\arg\\min\}\\limits\_\{\{\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\}:\|\|\{q\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{1\},\\forall k\}\}\-\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\),\\forall i,\\\\ \\quad\\boldsymbol\{w\}=\\arg\{\\min\_\{\\boldsymbol\{w\}^\{\\prime\}\}\}\\frac\{1\}\{N\}\\sum\\limits\_\{i=1\}^\{N\}\{\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\{\{\\alpha\_\{i,k\}\}\}\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\}^\{\\prime\};\{x\_\{i,k\}\}\\\!\+\\\!\{p\_\{i,k\}^\{\\prime\}\},\{y\_\{i,k\}\}\)\}\\\\ \\qquad\{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\|\|\{\\boldsymbol\{w\}^\{\\prime\}\}\|\{\|\_\{2\}\}\\\!\\leq\\\!c\_\{2\},\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}\}\)\\\!\-\\\!\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\}\)\\\!\\leq\\\!0,\\forall i,\\\\ \{\\rm\{var\}\}\.\\qquad\\qquad\\\{\\boldsymbol\{\\alpha\}\_\{i\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}\}\\\},\\boldsymbol\{w\},\\\{\\boldsymbol\{p\}\_\{i\}\\\}\.\\end\{array\}\(3\)
Likewise, in Eq\. \([3](https://arxiv.org/html/2607.27632#S3.E3)\), the lower\-level optimization also acts as a constraint on the upper\-level problem\. Specifically, among the two lower\-level subproblems, the first can be equivalently expressed via the following value\-function constraint:

f2,i\(1\)​\(𝜶i,𝒒i,𝒘\)−V2,i\(1\)​\(𝜶i,𝒘\)≤0,‖qi,k‖∞≤c1,∀i,k,V2,i\(1\)​\(𝜶i,𝒘\)=min𝒒i′:‖qi,k′‖∞≤c1,∀kf2,i\(1\)​\(𝜶i,𝒒i′,𝒘\),\\\!\\\!\\begin\{array\}\[\]\{l\}\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\}\)\}\\\!\-\\\!V\_\{2,i\}^\{\(1\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}\)\\\!\\leq\\\!0,\|\|\{q\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\\!\\leq\\\!c\_\{1\},\\forall i,k,\\\\ V\_\{2,i\}^\{\(1\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}\)=\\mathop\{\\min\}\\nolimits\_\{\{\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\}:\|\|\{q\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{1\},\\forall k\}\}\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\},\\boldsymbol\{w\}\)\},\\end\{array\}\\vskip\-3\.55658pt\(4\)whereV2,i\(1\)​\(⋅\)V\_\{2,i\}^\{\(1\)\}\(\\cdot\)represents the first outer\-layer value\-function\. Analogously, the second lower\-level subproblem admits a similar reformulation, expressed as:

1N​∑i=1Nf2,i\(2\)​\(𝜶i,𝒘,𝒑i\)−V2\(2\)​\(\{𝜶i\}\)≤0,V2\(2\)​\(\{𝜶i\}\)=min𝒘′,𝒑i′​∑i=1Nf2,i\(2\)​\(𝜶i,𝒘′,𝒑i′\)s\.t\.‖𝒘′‖2≤c2,‖pi,k′‖∞≤c3,∀i,k,f3,i​\(𝜶i,𝒘′,𝒑i′\)−V3,i​\(𝜶i,𝒘′\)≤0,∀i,\\begin\{array\}\[\]\{l\}\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\{f\_\{2,i\}^\{\(2\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\-\{V\_\{2\}^\{\(2\)\}\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\}\)\\leq 0,\\\\ \{V\_\{2\}^\{\(2\)\}\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\}\)=\{\\min\_\{\\boldsymbol\{w\}^\{\\prime\},\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\{f\_\{2,i\}^\{\(2\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\)\}\\\\ \{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\|\|\{\\boldsymbol\{w\}^\{\\prime\}\}\|\{\|\_\{2\}\}\\leq c\_\{2\},\|\|\{p\_\{i,k\}^\{\\prime\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall i,k,\\\\ \{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\},\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\)\-\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}^\{\\prime\}\)\\leq 0,\\forall i,\\end\{array\}\\vskip\-3\.55658pt\(5\)whereV2\(2\)​\(⋅\)\{V\_\{2\}^\{\(2\)\}\}\(\\cdot\)denotes the second outer\-layer value\-function\. Consequently, by incorporating these two outer\-layer constraints in Eqs\. \([4](https://arxiv.org/html/2607.27632#S3.E4)\) and \([5](https://arxiv.org/html/2607.27632#S3.E5)\), the resulting hierarchical composite optimization problem can be formulated as follows:

min⁡1N​∑i=1N∑k=1Miℓi,k​\(𝒘;xi,k\+qi,k,yi,k\)−λ​∑b=1Ki𝔼𝒛i​\(𝜶i\+𝒛i\)\[b\]s\.t​\. 0≤αi,k≤1,‖𝜶i‖=1,∀i,k,‖𝒘‖2≤c2,‖qi,k‖∞≤c1,‖pi,k‖∞≤c3,∀i,k,f2,i\(1\)​\(𝜶i,𝒒i,𝒘\)−V2,i\(1\)​\(𝜶i,𝒘\)≤0,∀i,1N​∑i=1Nf2,i\(2\)​\(𝜶i,𝒘,𝒑i\)−V2\(2\)​\(\{𝜶i\}\)≤0,f3,i​\(𝜶i,𝒘,𝒑i\)−V3,i​\(𝜶i,𝒘\)≤0,∀i,var\.\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\}\.\\begin\{array\}\[\]\{l\}\\min\\frac\{1\}\{N\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\{\\sum\\limits\_\{k=1\}^\{\{M\_\{i\}\}\}\\\!\{\{\\ell\_\{i,k\}\}\(\\boldsymbol\{w\};\{x\_\{i,k\}\}\\\!\+\\\!\{q\_\{i,k\}\},\{y\_\{i,k\}\}\)\}\\\!\-\\\!\\lambda\\sum\\limits\_\{b=1\}^\{\{K\_\{i\}\}\}\\\!\{\{\\mathbb\{E\}\_\{\\boldsymbol\{z\}\_\{i\}\}\}\}\{\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\!\+\\\!\\boldsymbol\{z\}\_\{i\}\)\}\_\{\[b\]\}\}\}\\\\ \{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\\;\\;\\;0\\leq\{\\alpha\_\{i,k\}\}\\leq 1,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1,\\forall i,k,\|\|\{\\boldsymbol\{w\}\}\|\{\|\_\{2\}\}\\\!\\leq\\\!c\_\{2\},\\\\ \\quad\\;\\;\\;\\;\\,\|\|\{q\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\\!\\leq\\\!c\_\{1\},\|\|\{p\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\\!\\leq\\\!c\_\{3\},\\forall i,k,\\\\ \\quad\\;\\;\\;\\;\\,\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\}\)\}\-V\_\{2,i\}^\{\(1\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}\)\\leq 0,\\forall i,\\\\ \\quad\\;\\;\\;\\;\\,\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\{f\_\{2,i\}^\{\(2\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\-\{V\_\{2\}^\{\(2\)\}\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\}\)\\leq 0,\\\\ \\quad\\;\\;\\;\\;\\,\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\-\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}\)\\leq 0,\\forall i,\\\\ \{\\rm\{var\}\}\.\\qquad\\qquad\\\{\\boldsymbol\{\\alpha\}\_\{i\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}\}\\\},\\boldsymbol\{w\},\\\{\\boldsymbol\{p\}\_\{i\}\\\}\.\\end\{array\}\(6\)

#### III\-C2Distributed Alternating Projected Gradient Algorithm

In this section, we propose a first\-order distributed algorithm to effectively solve the resulting problem\. The proposed method consists of two key steps: refining the hierarchical composite value\-functions and updating the variables alternately\. Specifically, in the\(t\+1\)th\(t\+1\)^\{\\text\{th\}\}iteration, inspired by the fact that 1\) lower\-level optimization often serves as asoft constraintto the upper\-level ones in multilevel optimization\[[14](https://arxiv.org/html/2607.27632#bib.bib34)\], which can be approximated to a certain extent without resulting in meaningless solutions, and 2\) it is common practice to employ multiple gradient descent steps to approximate the optimal lower\-level solution\[[40](https://arxiv.org/html/2607.27632#bib.bib224)\]\. Each workeriiperformsRRsteps of projected gradient descent to update value\-functionsV3,i​\(𝜶it,𝒘t\)=f3,i​\(𝜶it,𝒘t,𝒑i∗\),𝒑i∗=arg⁡min𝒑i′∈𝑷⁡f3,i​\(𝜶it,𝒘t,𝒑i′\)\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{w\}^\{t\}\}\)=\{f\_\{3,i\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\{\\boldsymbol\{w\}^\{t\}\},\{\\boldsymbol\{p\}\_\{i\}^\{\*\}\}\),\{\\boldsymbol\{p\}\_\{i\}^\{\*\}\}=\\arg\{\\min\_\{\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\\in\\boldsymbol\{P\}\}\}\{f\_\{3,i\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\{\\boldsymbol\{w\}^\{t\}\},\{\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\}\)andV2,i\(1\)​\(𝜶it,𝒘t\)=f2,i\(1\)​\(𝜶it,𝒒i∗,𝒘t\)V\_\{2,i\}^\{\(1\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\\boldsymbol\{w\}^\{t\}\)=\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{q\}\_\{i\}^\{\*\}\},\\boldsymbol\{w\}^\{t\}\)\},𝒒i∗=arg⁡min𝒒i′∈𝑸f2,i\(1\)​\(𝜶it,𝒒i′,𝒘t\)\\boldsymbol\{q\}\_\{i\}^\{\*\}=\\mathop\{\\arg\\min\}\\nolimits\_\{\{\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\}\\in\\boldsymbol\{Q\}\}\}\{\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{q\}\_\{i\}^\{\\prime\}\},\\boldsymbol\{w\}^\{t\}\)\}as:

𝒑ir\+1=𝒫𝑷​\(𝒑ir−η𝒑​∇𝒑f3,i​\(𝜶it,𝒘t,𝒑ir\)\),𝒑i∗≈𝒑iR,\\boldsymbol\{p\}\_\{i\}^\{r\+1\}=\{\\mathcal\{P\}\_\{\\boldsymbol\{P\}\}\}\(\\boldsymbol\{p\}\_\{i\}^\{r\}\-\\eta\_\{\\boldsymbol\{p\}\}\\nabla\_\{\\boldsymbol\{p\}\}\{f\_\{3,i\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\{\\boldsymbol\{w\}^\{t\}\},\{\\boldsymbol\{p\}\_\{i\}^\{r\}\}\)\),\{\\boldsymbol\{p\}\_\{i\}^\{\*\}\}\\approx\\boldsymbol\{p\}\_\{i\}^\{R\},\(7\)𝒒ir\+1=𝒫𝑸\(𝒒ir−η𝒒∇𝒒f2,i\(1\)\(𝜶it,𝒒ir,𝒘t\),𝒒i∗≈𝒒iR,\\boldsymbol\{q\}\_\{i\}^\{r\+1\}=\{\\mathcal\{P\}\_\{\\boldsymbol\{Q\}\}\}\(\\boldsymbol\{q\}\_\{i\}^\{r\}\-\\eta\_\{\\boldsymbol\{q\}\}\\nabla\_\{\\boldsymbol\{q\}\}\{f\_\{2,i\}^\{\(1\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{q\}\_\{i\}^\{r\}\},\\boldsymbol\{w\}^\{t\}\),\{\\boldsymbol\{q\}\_\{i\}^\{\*\}\}\\approx\\boldsymbol\{q\}\_\{i\}^\{R\},\\vskip\-3\.55658pt\(8\)where𝒫𝑷\{\\mathcal\{P\}\_\{\\boldsymbol\{P\}\}\}and𝒫𝑸\{\\mathcal\{P\}\_\{\\boldsymbol\{Q\}\}\}are the projection operator onto the feasible sets𝑸=\{𝒒i:‖qi,k‖∞≤c1,∀k\}\{\\boldsymbol\{Q\}\}=\\\{\{\\boldsymbol\{q\}\_\{i\}\}:\|\|\{q\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{1\},\\forall k\\\}and𝑷=\{𝒑i:‖pi,k‖∞≤c3,∀k\}\{\\boldsymbol\{P\}\}=\\\{\{\\boldsymbol\{p\}\_\{i\}\}:\|\|\{p\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\leq c\_\{3\},\\forall k\\\}, respectively\.η𝒑\\eta\_\{\\boldsymbol\{p\}\}andη𝒒\\eta\_\{\\boldsymbol\{q\}\}are step\-sizes\. Likewise, for the second outer\-layer value\-functionV2\(2\)​\(\{𝜶it\}\)=∑if2,i​\(𝜶it,𝒘∗,𝒑i∗\)V\_\{2\}^\{\(2\)\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\\\}\)=\\sum\_\{i\}f\_\{2,i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{w\}^\{\*\},\\boldsymbol\{p\}\_\{i\}^\{\*\}\), where𝒘∗,𝒑i∗=arg⁡min𝒘′∈𝑾,𝒑i′∈𝑷​∑if2,i​\(𝜶it,𝒘′,𝒑i′\)\+ϕN​\(∑if3,i​\(𝜶it,𝒘′,𝒑i′\)−V3,i​\(𝜶it,𝒘′\)\)\\boldsymbol\{w\}^\{\*\},\\boldsymbol\{p\}\_\{i\}^\{\*\}=\\arg\\min\_\{\\boldsymbol\{w\}^\{\\prime\}\\in\\boldsymbol\{W\},\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\\in\\boldsymbol\{P\}\}\\sum\_\{i\}f\_\{2,i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{w\}^\{\\prime\},\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\)\+\\frac\{\\phi\}\{N\}\\big\(\\sum\_\{i\}f\_\{3,i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{w\}^\{\\prime\},\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\)\-V\_\{3,i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{w\}^\{\\prime\}\)\\big\), we approximate the optimal𝒘∗\\boldsymbol\{w\}^\{\*\}and𝒑i∗\\boldsymbol\{p\}\_\{i\}^\{\*\}using the results afterR^\\hat\{R\}steps of alternating projected gradient descent\. Specifically, definingg2,i​\(𝜶i,𝒘,𝒑i\)=f2,i\(2\)​\(𝜶i,𝒘,𝒑i\)\+ϕ​\(f3,i​\(𝜶i,𝒘,𝒑i\)−V3,i​\(𝜶i,𝒘\)\)g\_\{2,i\}\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{w\},\\boldsymbol\{p\}\_\{i\}\)=f\_\{2,i\}^\{\(2\)\}\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{w\},\\boldsymbol\{p\}\_\{i\}\)\+\\phi\\big\(f\_\{3,i\}\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{w\},\\boldsymbol\{p\}\_\{i\}\)\-V\_\{3,i\}\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{w\}\)\\big\), whereϕ\\phiis a penalty parameter, for eachr=0,⋯,R^−1r=0,\\cdots,\\hat\{R\}\-1, workeriicomputes:

𝒑ir\+1=𝒫𝑷​\(𝒑ir−η𝒑​∇𝒑g2,i​\(𝜶it,𝒘ir,𝒑ir\)\),𝒘ir\+1=𝒫𝑾​\(𝒘ir−η𝒘​∇𝒘g2,i​\(𝜶it,𝒘ir,𝒑ir\+1\)\),\\begin\{array\}\[\]\{l\}\\boldsymbol\{p\}\_\{i\}^\{r\+1\}=\{\{\\mathcal\{P\}\_\{\\boldsymbol\{P\}\}\}\}\(\\boldsymbol\{p\}\_\{i\}^\{r\}\-\{\\eta\_\{\\boldsymbol\{p\}\}\}\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{g\_\{2,i\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\{\\boldsymbol\{w\}\_\{i\}^\{r\}\},\\boldsymbol\{p\}\_\{i\}^\{r\}\)\),\\\\ \\boldsymbol\{w\}\_\{i\}^\{r\+1\}=\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{r\}\}\-\{\\eta\_\{\\boldsymbol\{w\}\}\}\{\\nabla\_\{\\boldsymbol\{w\}\}\}\{g\_\{2,i\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\{\\boldsymbol\{w\}\_\{i\}^\{r\}\},\\boldsymbol\{p\}\_\{i\}^\{r\+1\}\)\),\\end\{array\}\\vskip\-3\.55658pt\(9\)where𝒫𝑾\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}is the projection operator onto the feasible set𝑾=\{𝒘:‖𝒘‖2≤c2\}\{\\boldsymbol\{W\}\}=\\\{\{\\boldsymbol\{w\}\}:\|\|\{\\boldsymbol\{w\}\}\|\{\|\_\{2\}\}\\leq c\_\{2\}\\\}, andη𝒘\{\\eta\_\{\\boldsymbol\{w\}\}\}is the step\-size\. Then, the local parameters𝒘iR^\\boldsymbol\{w\}\_\{i\}^\{\\hat\{R\}\}are transmitted to the master, which computes their average as𝒘R^=1N​∑i=1N𝒘iR^\\boldsymbol\{w\}^\{\\hat\{R\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\boldsymbol\{w\}\_\{i\}^\{\\hat\{R\}\}\. Thus, the approximations can be obtained:𝒘∗≈𝒘R^\\boldsymbol\{w\}^\{\*\}\\approx\\boldsymbol\{w\}^\{\\hat\{R\}\}and𝒑i∗≈𝒑iR^\\boldsymbol\{p\}\_\{i\}^\{\*\}\\approx\\boldsymbol\{p\}\_\{i\}^\{\\hat\{R\}\}\.

Building upon the refined hierarchical composite value\-functions and incorporating the penalty method commonly used in bilevel optimization\[[26](https://arxiv.org/html/2607.27632#bib.bib222)\], we formulate the penalty function for the problem in Eq\. \([6](https://arxiv.org/html/2607.27632#S3.E6)\) as follows:

min⁡ℒ​\(\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\}\)=1N​∑i=1NLi​\(𝜶i,𝒒i,𝒘,𝒑i\)=1N​∑i=1Nf1,i​\(𝜶i,𝒒i,𝒘,𝒑i\)\+ρ1N​∑i=1Nf2,i\(1\)​\(𝜶i,𝒒i,𝒘\)−V2,i\(1\)​\(𝜶i,𝒘\)\+ρ2N​\(∑i=1Nf2,i\(2\)​\(𝜶i,𝒘,𝒑i\)−V2\(2\)​\(\{𝜶i\}\)\)\+ρ3N​∑i=1Nf3,i​\(𝜶i,𝒘,𝒑i\)−V3,i​\(𝜶i,𝒘\)s\.t​\. 0≤αi,k≤1,‖𝜶i‖=1,∀i,k,‖𝒘‖2≤c2,‖qi,k‖∞≤c1,‖pi,k‖∞≤c3,∀i,k,var\.\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\},\\begin\{array\}\[\]\{l\}\\min\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}\}\\\},\\boldsymbol\{w\},\\\{\{\\boldsymbol\{p\}\_\{i\}\}\\\}\)=\\frac\{1\}\{N\}\\\!\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\{L\_\{i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\\\\ \\\!=\\\!\\\!\\frac\{1\}\{N\}\\\!\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\{\{f\_\{1,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\\\!\+\\\!\\\!\\frac\{\{\{\\rho\_\{1\}\}\}\}\{N\}\\\!\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\{\{f\_\{2,i\}^\{\(1\)\}\}\\\!\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\}\)\}\\\!\-\\\!V\_\{2,i\}^\{\(1\)\}\\\!\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\\!\\boldsymbol\{w\}\)\\\\ \\;\\;\+\\frac\{\{\{\\rho\_\{2\}\}\}\}\{N\}\(\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\{f\_\{2,i\}^\{\(2\)\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\\\!\-\\\!\{V\_\{2\}^\{\(2\)\}\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}\\\}\)\)\\\\ \\;\\;\+\\frac\{\{\{\\rho\_\{3\}\}\}\}\{N\}\\\!\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\{f\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\\\!\-\\\!\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\\boldsymbol\{w\}\)\}\\\\ \{\\rm\{s\}\}\.\{\\rm\{t\}\}\.\\;\\;0\\leq\{\\alpha\_\{i,k\}\}\\leq 1,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1,\\forall i,k,\|\|\{\\boldsymbol\{w\}\}\|\{\|\_\{2\}\}\\\!\\leq\\\!c\_\{2\},\\\\ \\qquad\|\|\{q\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\\!\\leq\\\!c\_\{1\},\|\|\{p\_\{i,k\}\}\|\{\|\_\{\\infty\}\}\\\!\\leq\\\!c\_\{3\},\\forall i,k,\\\\ \{\\rm\{var\}\}\.\\qquad\\qquad\\\{\\boldsymbol\{\\alpha\}\_\{i\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}\}\\\},\\boldsymbol\{w\},\\\{\\boldsymbol\{p\}\_\{i\}\\\},\\end\{array\}\(10\)whereρ1,ρ2,ρ3\\rho\_\{1\},\\rho\_\{2\},\\rho\_\{3\}are the penalty parameters\. To optimize the penalized objective in Eq\. \([10](https://arxiv.org/html/2607.27632#S3.E10)\), and owing to the variable dependencies across the trilevel architecture, the local variables on workeriiare updated in aGauss\-Seidelmanner as follows:

𝜶it\+1=𝒫𝑨​\(𝜶it−η𝜶​∇𝜶ℒi​\(𝜶it,𝒒it,𝒘t,𝒑it\)\),\\begin\{array\}\[\]\{l\}\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}=\{\\mathcal\{P\}\_\{\\boldsymbol\{A\}\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\-\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\{\\nabla\_\{\\boldsymbol\{\\alpha\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{q\}\_\{i\}^\{t\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\),\\end\{array\}\\vskip\-5\.69054pt\(11\)𝒒it\+1=𝒫𝑸​\(𝒒it−η𝒒​∇𝒒ℒi​\(𝜶it\+1,𝒒it,𝒘t,𝒑it\)\),\\begin\{array\}\[\]\{l\}\\boldsymbol\{q\}\_\{i\}^\{t\+1\}=\{\\mathcal\{P\}\_\{\\boldsymbol\{Q\}\}\}\(\{\\boldsymbol\{q\}\_\{i\}^\{t\}\}\-\{\\eta\_\{\\boldsymbol\{q\}\}\}\{\\nabla\_\{\\boldsymbol\{q\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\),\\end\{array\}\\vskip\-5\.69054pt\(12\)𝒘it\+1=𝒘t−η𝒘​∇𝒘ℒi​\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\),\\begin\{array\}\[\]\{l\}\\boldsymbol\{w\}\_\{i\}^\{t\+1\}=\{\{\\boldsymbol\{w\}^\{t\}\}\}\-\{\\eta\_\{\\boldsymbol\{w\}\}\}\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\),\\end\{array\}\\vskip\-5\.69054pt\(13\)𝒑it\+1=𝒫𝑷​\(𝒑it−η𝒑​∇𝒑ℒi​\(𝜶it\+1,𝒒it\+1,𝒫𝑾​\(𝒘it\+1\),𝒑it\)\),\\begin\{array\}\[\]\{l\}\\boldsymbol\{p\}\_\{i\}^\{t\+1\}=\{\{\\mathcal\{P\}\_\{\\boldsymbol\{P\}\}\}\}\(\\boldsymbol\{p\}\_\{i\}^\{t\}\-\{\\eta\_\{\\boldsymbol\{p\}\}\}\{\\nabla\_\{\\boldsymbol\{p\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\}\_\{i\}^\{t\+1\}\),\\boldsymbol\{p\}\_\{i\}^\{t\}\)\),\\end\{array\}\(14\)where𝒫𝑨\{\\mathcal\{P\}\_\{\\boldsymbol\{A\}\}\}represents the projection operator onto the feasible set𝑨=\{𝜶i:0≤αi,k≤1,∀k,‖𝜶i‖=1\}\{\\boldsymbol\{A\}\}=\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}\}:0\\leq\{\\alpha\_\{i,k\}\}\\leq 1,\\forall k,\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}\}\|\|=1\\\}, andη𝜶\\eta\_\{\\boldsymbol\{\\alpha\}\}is the step\-size\. Subsequently, the updated local model parameters𝒘it\+1\\boldsymbol\{w\}\_\{i\}^\{t\+1\}are transmitted to the master, which then updates and broadcasts the global model parameters as follows:

𝒘t\+1=𝒫𝑾​\(1N​∑i=1N𝒘it\+1\)\.\\begin\{array\}\[\]\{l\}\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}=\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\\left\(\{\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\}\\right\)\.\\end\{array\}\(15\)
It can be seen from Eq\. \([7](https://arxiv.org/html/2607.27632#S3.E7)\) to Eq\. \([15](https://arxiv.org/html/2607.27632#S3.E15)\) that the projection operation at each step iscomputationally efficientdue to the highly structured nature of the closed convex sets𝑨,𝑸,𝑾,𝑷\{\\boldsymbol\{A\}\},\{\\boldsymbol\{Q\}\},\{\\boldsymbol\{W\}\},\{\\boldsymbol\{P\}\}\[[1](https://arxiv.org/html/2607.27632#bib.bib11)\]\. Furthermore, we emphasize that the proposed method is a single\-loopfirst\-orderdistributed algorithm that avoids the computation of hyper\-gradients\. The complete procedure of the proposed method is summarized in Algorithm[1](https://arxiv.org/html/2607.27632#alg1)\.

## IVTheoretical Analyses

###### Definition 1

\(Stationarity gap\)Following\[[23](https://arxiv.org/html/2607.27632#bib.bib24),[19](https://arxiv.org/html/2607.27632#bib.bib167),[16](https://arxiv.org/html/2607.27632#bib.bib252)\], the projected gradient mappings at thettht^\{\\rm\{th\}\}iteration are defined asg𝛂,it=1η𝛂​N​\(𝛂it−𝛂it\+1\)g\_\{\\boldsymbol\{\\alpha\},i\}^\{t\}=\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}N\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\-\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\),g𝐪,it=1η𝐪​N​\(𝐪it−𝐪it\+1\)g\_\{\\boldsymbol\{q\},i\}^\{t\}=\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{q\}\}\}\}N\}\(\\boldsymbol\{q\}\_\{i\}^\{t\}\-\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\),g𝐰t=1η𝐰​\(𝐰t−𝐰t\+1\)g\_\{\\boldsymbol\{w\}\}^\{t\}=\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\(\{\{\\boldsymbol\{w\}^\{t\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\), andg𝐩,it=1η𝐩​N​\(𝐩it−𝐩it\+1\)g\_\{\\boldsymbol\{p\},i\}^\{t\}=\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}N\}\(\\boldsymbol\{p\}\_\{i\}^\{t\}\-\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\)fori=1,⋯,Ni\\\!=\\\!1,\\\!\\cdots\\\!,N\. Consequently, the stationarity gap of the studied problem at thettht^\{\\rm\{th\}\}iteration can be expressed as:

Gt=\[1η𝜶​N​\(𝜶it−𝜶it\+1\),∀i1η𝒒​N​\(𝒒it−𝒒it\+1\),∀i1η𝒘​\(𝒘t−𝒘t\+1\)1η𝒑​N​\(𝒑it−𝒑it\+1\),∀i\]\.\{G^\{t\}\}=\\left\[\\begin\{array\}\[\]\{l\}\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}N\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\-\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\),\\forall i\\\\ \\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{q\}\}\}\}N\}\(\\boldsymbol\{q\}\_\{i\}^\{t\}\-\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\),\\forall i\\\\ \\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\(\{\{\\boldsymbol\{w\}^\{t\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\)\\\\ \\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}N\}\(\\boldsymbol\{p\}\_\{i\}^\{t\}\-\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\),\\forall i\\end\{array\}\\right\]\.\\vskip\-2\.84526pt\(16\)
Thus, we can also obtain:‖Gt‖2=∑i=1N‖g𝛂,it‖2\+∑i=1N‖g𝐪,it‖2\+‖g𝐰t‖2\+∑i=1N‖g𝐩,it‖2\.\|\|\{G^\{t\}\}\|\{\|^\{2\}\}=\\sum\\nolimits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{\\alpha\},i\}^\{t\}\|\{\|^\{2\}\}\}\+\\sum\\nolimits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{q\},i\}^\{t\}\|\{\|^\{2\}\}\}\+\|\|g\_\{\\boldsymbol\{w\}\}^\{t\}\|\{\|^\{2\}\}\+\\sum\\nolimits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{p\},i\}^\{t\}\|\{\|^\{2\}\}\}\.

###### Definition 2

\(ϵ\\epsilon\-stationary point\)\(\{𝛂it\},\{𝐪it\},𝐰t,\{𝐩it\}\)\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\\\},\\\{\{\\boldsymbol\{q\}\_\{i\}^\{t\}\}\\\},\\boldsymbol\{w\}^\{t\},\\\{\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\\\}\)is defined as anϵ\\epsilon\-stationary point when‖Gt‖2≤ϵ\|\|G^\{t\}\|\|^\{2\}\\leq\\epsilon, andT​\(ϵ\)T\(\\epsilon\)is defined as the first iteration to achieve theϵ\\epsilon\-stationary point, i\.e\.,T​\(ϵ\)=min⁡\{t∣‖Gt‖2≤ϵ\}T\(\\epsilon\)=\\min\\\{t\|\\;\|\|G^\{t\}\|\|^\{2\}\\leq\\epsilon\\\}\.

Algorithm 1F2CTO:FederatedFirst\-orderConstrainedTrilevelOptimizationInitialization:iteration

t=0t=0, variables

\{𝜶i0\}\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\}\\\},

\{𝒒i0\}\\\{\{\\boldsymbol\{q\}\_\{i\}^\{0\}\}\\\},

𝒘0\\boldsymbol\{w\}^\{0\},

\{𝒑i0\}\\\{\{\\boldsymbol\{p\}\_\{i\}^\{0\}\}\\\},

i=1,⋯,Ni=1,\\cdots,N\.

repeat

Refinement of Hierarchical Composite Value\-functions:

for*local workerii*do

computes

𝒑iR\\boldsymbol\{p\}\_\{i\}^\{R\}to update value\-function

V3,i​\(𝜶it,𝒘t\)\{V\_\{3,i\}\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{w\}^\{t\}\}\);

computes

𝒒iR\\boldsymbol\{q\}\_\{i\}^\{R\}to update value\-function

V2,i\(1\)​\(𝜶it,𝒘t\)V\_\{2,i\}^\{\(1\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\\boldsymbol\{w\}^\{t\}\);

computes

𝒘iR^,𝒑iR^\{\\boldsymbol\{w\}\}\_\{i\}^\{\\hat\{R\}\},\\boldsymbol\{p\}\_\{i\}^\{\\hat\{R\}\}and transmits

𝒘iR^\{\\boldsymbol\{w\}\}\_\{i\}^\{\\hat\{R\}\}to the master\.

endfor

for*master*do

aggregates local parameters as

𝒘R^=1N​∑i=1N𝒘iR^\{\\boldsymbol\{w\}\}^\{\\hat\{R\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\{\\boldsymbol\{w\}\}\_\{i\}^\{\\hat\{R\}\};

broadcasts

𝒘R^\{\\boldsymbol\{w\}\}^\{\\hat\{R\}\}to update value\-function

V2\(2\)​\(𝜶it\)\{V\_\{2\}^\{\(2\)\}\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\)\.

endfor

Update of Optimization Variables:

for*local workerii*do

updates local variables

𝜶it\+1,𝒒it\+1,𝒘it\+1,𝒑it\+1\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\\boldsymbol\{w\}\_\{i\}^\{t\+1\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}in a Gauss\-Seidel manner according to Eq\. \([11](https://arxiv.org/html/2607.27632#S3.E11)\)\-Eq\. \([14](https://arxiv.org/html/2607.27632#S3.E14)\):

transmits the updated

𝒘it\+1\\boldsymbol\{w\}\_\{i\}^\{t\+1\}to the master\.

endfor

for*master*do

aggregates local model parameters to get the global model parameters

𝒘t\+1=𝒫𝑾​\(1N​∑i=1N𝒘it\+1\)\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}=\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\}\);

broadcasts the obtained

𝒘t\+1\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}to the workers\.

endfor

t=t\+1t=t\+1;

untiltermination\.

###### Assumption 1

\(LL\-smoothness\)Following previous work\[[12](https://arxiv.org/html/2607.27632#bib.bib22),[19](https://arxiv.org/html/2607.27632#bib.bib167),[24](https://arxiv.org/html/2607.27632#bib.bib260)\], we assume functionℒi​\(⋅\)\\mathcal\{L\}\_\{i\}\(\\cdot\)has anLL\-Lipschitz continuous gradient\. That is, there exists a constant0<L<∞0<L<\\inftysuch that for any𝐮=\(𝛂i,𝐪i,𝐰,𝐩i\)\\boldsymbol\{u\}=\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{q\}\_\{i\},\\boldsymbol\{w\},\\boldsymbol\{p\}\_\{i\}\)and𝐮′=\(𝛂i′,𝐪i′,𝐰′,𝐩i′\)\\boldsymbol\{u\}^\{\\prime\}=\(\\boldsymbol\{\\alpha\}\_\{i\}^\{\\prime\},\\boldsymbol\{q\}\_\{i\}^\{\\prime\},\\boldsymbol\{w\}^\{\\prime\},\\boldsymbol\{p\}\_\{i\}^\{\\prime\}\), we have:

‖∇ℒi​\(𝒖\)−∇ℒi​\(𝒖′\)‖≤L​‖𝒖−𝒖′‖\.\|\|\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{u\}\)\-\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{u\}^\{\\prime\}\)\|\|\\leq L\|\|\\boldsymbol\{u\}\-\\boldsymbol\{u\}^\{\\prime\}\|\|\.\(17\)
TABLE III:Comparison of the proposed F2CTO with state\-of\-the\-art methods in terms of average robustness \(%\\%\) against various adversarial attacks across multiple datasets, all experiments were repeated five times\.By combining Eq\. \([17](https://arxiv.org/html/2607.27632#S4.E17)\) with the Triangle inequality, we can derive that functionℒ​\(⋅\)\\mathcal\{L\}\(\\cdot\)also satisfies theLL\-smoothness:

‖∇ℒ​\(𝒖\)−∇ℒ​\(𝒖′\)‖=‖1N​∑i=1N∇ℒi​\(𝒖\)−∇ℒi​\(𝒖′\)‖≤L​‖𝒖−𝒖′‖\.\\begin\{array\}\[\]\{l\}\|\|\\nabla\\mathcal\{L\}\(\\boldsymbol\{u\}\)\\\!\-\\\!\\nabla\\mathcal\{L\}\(\\boldsymbol\{u\}^\{\\prime\}\)\|\|\\\!=\\\!\|\|\\frac\{1\}\{N\}\\\!\\sum\_\{i=1\}^\{N\}\\\!\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{u\}\)\\\!\-\\\!\\nabla\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{u\}^\{\\prime\}\)\|\|\\\\ \\qquad\\qquad\\qquad\\qquad\\;\\leq L\|\|\\boldsymbol\{u\}\-\\boldsymbol\{u\}^\{\\prime\}\|\|\.\\end\{array\}\(18\)

###### Lemma 1

\(Local\-Global Projection Gap\)Under Assumption 1, the discrepancy between the local projected update𝒫𝐖​\(𝐰it\+1\)\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\)and the aggregated global update𝐰t\+1\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}is uniformly bounded\. Specifically, for any workerii, we have:

‖𝒫𝑾​\(𝒘it\+1\)−𝒘t\+1‖≤2​η𝒘​B𝒘,\|\|\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\)\-\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\|\|\\leq 2\{\\eta\_\{\\boldsymbol\{w\}\}\}\{B\_\{\\boldsymbol\{w\}\}\},\\vskip\-3\.55658pt\(19\)whereB𝐰=maxi⁡max\(𝛂i,𝐪i,𝐰,𝐩i\)​‖∇𝐰ℒi​\(𝛂i,𝐪i,𝐰,𝐩i\)‖\{B\_\{\\boldsymbol\{w\}\}\}=\\max\_\{i\}\\max\_\{\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\}\|\|\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}\},\{\\boldsymbol\{q\}\_\{i\}\},\\boldsymbol\{w\},\{\\boldsymbol\{p\}\_\{i\}\}\)\|\|is a finite constant due to the smoothness ofℒi\\mathcal\{L\}\_\{i\}over the compact feasible domains, andη𝐰\{\\eta\_\{\\boldsymbol\{w\}\}\}is the step\-size\.

###### Lemma 2

\(One\-Step Descent Inequality\)Under Assumption 1, one complete iteration of the proposed algorithm decreases the objective function up to a controllable error\. Specifically, the following inequality holds:

ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\+1\}\)−ℒ​\(\{𝜶it\},\{𝒒it\},𝒘t,\{𝒑it\}\)≤−1N​∑i=1N\(1η𝜶−L2\)​‖𝜶it\+1−𝜶it‖2−\(1η𝒘−L2\)​‖𝒘t\+1−𝒘t‖2\+2​L2​η𝒘2​B𝒘21η𝒑−L2−12​N​∑i=1N\(\(1η𝒑−L2\)​‖𝒑it\+1−𝒑it‖2\+\(1η𝒒−L2\)​‖𝒒it\+1−𝒒it‖2\)\.\\begin\{array\}\[\]\{l\}\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\!\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\!\\\{\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\}\)\\\!\-\\\!\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\\\},\\\!\\\{\\boldsymbol\{q\}\_\{i\}^\{t\}\\\},\{\\boldsymbol\{w\}^\{t\}\},\\\!\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\\ \\leq\-\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\-\\frac\{L\}\{2\}\}\)\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\-\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\|\{\|^\{2\}\}\\\\ \-\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\-\\frac\{L\}\{2\}\}\)\|\|\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\}\}\}\|\{\|^\{2\}\}\+\\frac\{\{2\{L^\{2\}\}\{\\eta\_\{\\boldsymbol\{w\}\}^\{2\}\}\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\}\{\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\-\\frac\{L\}\{2\}\}\}\\\\ \-\\frac\{1\}\{2N\}\\\!\\sum\_\{i=1\}^\{N\}\\\!\(\(\\frac\{1\}\{\\eta\_\{\\boldsymbol\{p\}\}\}\\\!\-\\\!\\frac\{L\}\{2\}\)\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\|\|^\{2\}\\\!\+\\\!\(\\frac\{1\}\{\\eta\_\{\\boldsymbol\{q\}\}\}\\\!\-\\\!\\frac\{L\}\{2\}\)\|\|\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{q\}\_\{i\}^\{t\}\|\|^\{2\}\)\.\\end\{array\}\(20\)

###### Theorem 1

\(Iteration Complexity\)Under Assumption[1](https://arxiv.org/html/2607.27632#Thmassumption1), setting step\-sizesη𝛂=η𝐪=η𝐰=η𝐩=η=T−1/3\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}=\{\\eta\_\{\\boldsymbol\{q\}\}\}=\{\\eta\_\{\\boldsymbol\{w\}\}\}=\{\\eta\_\{\\boldsymbol\{p\}\}\}=\\eta=\{T^\{\-1/3\}\}, whenT≥L3T\\geq L^\{3\}, the iteration complexity to achieveϵ\\epsilon\-stationary point is upper bounded by

T​\(ϵ\)∼max⁡\{L3,\(8​\(ℒ​\(\{𝜶i0\},\{𝒒i0\},𝒘0,\{𝒑i0\}\)−ℒ∗\)\+32​L2​B𝒘2\)3/2ϵ3/2\},\\begin\{array\}\[\]\{l\}T\(\\epsilon\)\\\!\\sim\\\!\\max\\left\\\{\{\{L^\{3\}\},\\frac\{\{\{\{\(\{8\(\{\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{0\}\\\},\{\\boldsymbol\{w\}^\{0\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{0\}\\\}\)\-\\mathcal\{L\}^\{\*\}\}\)\+32\{L^\{2\}\}\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\)\}^\{3/2\}\}\}\}\{\{\{\\epsilon^\{3/2\}\}\}\}\}\\right\\\}\\\!,\\end\{array\}\(21\)

whereℒ∗=min⁡ℒ​\(\{𝜶i\},\{𝒒i\},𝒘,\{𝒑i\}\)\\mathcal\{L\}^\{\*\}=\\min\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}\\\},\{\\boldsymbol\{w\}\},\\\{\\boldsymbol\{p\}\_\{i\}\\\}\)is a constant due to the continuity ofℒ\\mathcal\{L\}over the compact sets, andB𝒘\{B\_\{\\boldsymbol\{w\}\}\}is also a constant\. The detailed proofs are shown in Sec\.[VII](https://arxiv.org/html/2607.27632#S7)\.

###### Theorem 2

\(Communication Complexity\)Following\[[19](https://arxiv.org/html/2607.27632#bib.bib167)\], communication complexity is defined as the total volume of information transmitted until the algorithm converges\. The communication complexity for the proposed method to achieve theϵ\\epsilon\-stationary point isCcomm​\(ϵ\)=𝒪​\(dϵ3/2\)C\_\{\\mathrm\{comm\}\}\(\\epsilon\)=\\mathcal\{O\}\(\\frac\{d\}\{\{\\epsilon^\{3/2\}\}\}\), whereddis a constant representing the dimension of the model parameters\. Detailed proofs are provided in Sec\.[VII](https://arxiv.org/html/2607.27632#S7)\.

## VExperiment

Building upon the experimental settings of prior coreset selection work\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], the proposed F2CTO is evaluated on rehearsal\-based continual learning tasks, with the evaluation further extended to distributed scenarios\. We compare F2CTO with a range of state\-of\-the\-art methods, including the distributed trilevel optimization methods AFTO\[[19](https://arxiv.org/html/2607.27632#bib.bib167)\]and DTZO\[[15](https://arxiv.org/html/2607.27632#bib.bib111)\], and the distributed coreset selection methods FedCS\[[11](https://arxiv.org/html/2607.27632#bib.bib17)\]and GCFL\[[33](https://arxiv.org/html/2607.27632#bib.bib14)\]\. Additionally, we adapt several centralized coreset selection methods to the distributed setting and include them as additional baselines, including ACS\[[5](https://arxiv.org/html/2607.27632#bib.bib19)\], BCSR\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], and Greedy Coreset\[[2](https://arxiv.org/html/2607.27632#bib.bib3)\]\. The experiments are conducted on a server equipped with two NVIDIA GeForce RTX 5090 GPUs\. All models are implemented in PyTorch, and the detailed experimental settings are summarized in Table[IV](https://arxiv.org/html/2607.27632#S5.T4)\. In our experiments, we aim to answer the following questions:\(Q1\)Can the proposed F2CTO achieve superior performance in distributed robust coreset selection?\(Q2\)Is the optimization at each level of the proposed framework effective?\(Q3\)Can the proposed framework be applied to large\-scale settings?\(Q4\)Is the proposed first\-order algorithm more efficient than traditional algorithms?\(Q5\)Can periodic communication be effectively integrated into the proposed framework to further improve efficiency?

![Refer to caption](https://arxiv.org/html/2607.27632v1/x2.png)Figure 2:The comparisons between F2CTO and its two bilevel variants\.![Refer to caption](https://arxiv.org/html/2607.27632v1/x3.png)\(a\)Robustness against FGSM attack
![Refer to caption](https://arxiv.org/html/2607.27632v1/x4.png)\(b\)Robustness against PGD attack

Figure 3:Results on large\-scale IoT setting\.### V\-AMain Results \(Answering Q1\)

Following the experimental setup in\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], we conduct experiments on Split CIFAR\-100, Permuted MNIST, and Tiny\-ImageNet\. Two separate models are employed for robust coreset selection and model training, respectively\. Consistent with\[[12](https://arxiv.org/html/2607.27632#bib.bib22)\], we use the MLP for Permuted MNIST and ResNet\-18 for the other datasets\. Moreover, we use the average adversarial robustness across all tasks as an evaluation metric, and we assess adversarial robustness against three representative attacks: FGSM, PGD\-10, and AutoAttack, following\[[42](https://arxiv.org/html/2607.27632#bib.bib6)\]\. The comparisons between the proposed F2CTO and the state\-of\-the\-art methods across various datasets are presented in Table[III](https://arxiv.org/html/2607.27632#S4.T3), it is seen that the F2CTO outperforms all compared methods\. This improvement can be attributed to two main factors: \(1\) In contrast to the state\-of\-the\-art coreset selection methods \(FedCS, GCFL, ACS, BCSR, and Greedy Coreset\), this is the first work to unify robust optimization, coreset selection, and distributed learning\. The resulting trilevel optimization framework effectively addresses robust coreset selection in a distributed manner, yielding improved performance\. \(2\) Compared with the state\-of\-the\-art distributed trilevel optimization methods AFTO and DTZO, which are designed for unconstrained trilevel optimization, the proposed F2CTO is the first distributed method tailored to constrained trilevel optimization and can effectively handle level\-wise constraints\.

### V\-BAblation Study \(Answering Q2\)

This work introduces a trilevel optimization framework for distributed robust coreset selection\. Within this framework, the upper two levels and the lower two levels each constitute a bilevel optimization problem\. To assess the necessity of each optimization level, we conduct an ablation study with two bilevel variants\. The upper\-bilevel variant \(UBV\) retains the upper two levels while removing the third\-level optimization, whereas the lower\-bilevel variant \(LBV\) retains the lower two levels while removing the first\-level optimization\. Since LBV does not update data weights during optimization, a greedy\-based strategy is adopted for coreset selection\. It is worth noting that the second level cannot be removed independently, as doing so would disrupt the nested optimization structure\. As shown in Fig\.[2](https://arxiv.org/html/2607.27632#S5.F2), the proposed F2CTO consistently outperforms both UBV and LBV across all datasets, demonstrating the effectiveness of the complete trilevel optimization framework\.

### V\-CScalability \(Answering Q3\)

To evaluate the scalability of the proposed method in large\-scale IoT scenarios, we conduct experiments on the Edge\-IIoTset dataset\[[6](https://arxiv.org/html/2607.27632#bib.bib5)\]in a distributed network involving 200 workers\. As shown in Fig\.[3](https://arxiv.org/html/2607.27632#S5.F3), the proposed F2CTO maintains stable and superior performance compared with the baseline methods as the network size increases\. These results demonstrate the scalability of F2CTO and suggest that the proposed framework is applicable not only to cross\-silo scenarios but also to large\-scale cross\-device environments\.

![Refer to caption](https://arxiv.org/html/2607.27632v1/x5.png)Figure 4:Comparisons between the proposed F2CTO and the hypergradient\-based method in terms of running time \(100 iterations\) and memory usage\.
### V\-DRunning Time and Memory Usage \(Answering Q4\)

Compared with traditional hypergradient\-based approaches to trilevel optimization\[[9](https://arxiv.org/html/2607.27632#bib.bib61)\], the proposed F2CTO is afirst\-orderdistributed algorithm that avoids computing hypergradients at each iteration, thereby reducing computational overhead\. To empirically evaluate its computational efficiency, we compare the running time and memory usage of F2CTO with the hypergradient\-based trilevel method\. As shown in Fig\.[4](https://arxiv.org/html/2607.27632#S5.F4), F2CTO consistently achieves shorter running time and lower memory usage, demonstrating its superior efficiency\.

![Refer to caption](https://arxiv.org/html/2607.27632v1/x6.png)\(a\)Permuted MNIST
![Refer to caption](https://arxiv.org/html/2607.27632v1/x7.png)\(b\)Split CIFAR\-100
![Refer to caption](https://arxiv.org/html/2607.27632v1/x8.png)\(c\)Tiny\-ImageNet

Figure 5:F2CTO with periodic communication strategy \(I=5I=5\)\.
### V\-EPeriodic Communication \(Answering Q5\)

To further improve the communication efficiency of F2CTO, we incorporate a periodic communication strategy, yielding F2CTO\+P\. Instead of transmitting local model parameters to the master after every execution of the alternating projected gradient updates in Eqs\. \([11](https://arxiv.org/html/2607.27632#S3.E11)\)\-\([14](https://arxiv.org/html/2607.27632#S3.E14)\), each worker performsIIrounds of local updates between two consecutive communication rounds, whereI≥1I\\geq 1is a pre\-specified constant\. As shown in Fig\.[5](https://arxiv.org/html/2607.27632#S5.F5), F2CTO\+P achieves convergence with fewer communication rounds, indicating that periodic communication can be effectively integrated into the proposed framework\.

TABLE IV:Detailed experimental settings\.Here,NNis the number of workers in distributed systems,η𝜶\\eta\_\{\\boldsymbol\{\\alpha\}\},η𝒒\\eta\_\{\\boldsymbol\{q\}\},η𝒘\\eta\_\{\\boldsymbol\{w\}\},η𝒑\\eta\_\{\\boldsymbol\{p\}\}are step\-sizes\.εF\\varepsilon\_\{\\rm\{F\}\},εP\\varepsilon\_\{\\rm\{P\}\}, andεA\\varepsilon\_\{\\rm\{A\}\}are perturbation budgets for FGSM, PGD\-10, and AutoAttack\.R\{R\}andR^\\hat\{R\}are the parameters for value\-function refinement\.

## VIConclusion

In this work, we explore the fundamental coupling among coreset selection, robust optimization, and distributed learning, formulating the distributed robust coreset selection as a trilevel optimization problem with level\-wise constraints\. To effectively address this, we propose F2CTO, a federated first\-order constrained trilevel optimization framework\. Specifically, a hierarchical composite value\-function reformulation is first introduced in F2CTO to handle the trilevel structure, followed by a distributed first\-order alternating projected gradient descent algorithm to solve the resulting problem\. Theoretically, we provide the non\-asymptotic convergence guarantee for the proposed F2CTO, demonstrating its iteration and communication complexities to achieve theϵ\\epsilon\-stationary point\. Extensive experimental results on reliable continual learning have demonstrated the superior performance of F2CTO\.

## VIIAppendix: Proofs of Theorem[1](https://arxiv.org/html/2607.27632#Thmtheorem1)and[2](https://arxiv.org/html/2607.27632#Thmtheorem2)

In this section, we present thekey stepsin the proofs of Theorems[1](https://arxiv.org/html/2607.27632#Thmtheorem1)and[2](https://arxiv.org/html/2607.27632#Thmtheorem2)\. To facilitate this, we first establish Lemmas[1](https://arxiv.org/html/2607.27632#Thmlemma1)and[2](https://arxiv.org/html/2607.27632#Thmlemma2)\. By combining the definition of𝒫𝑾\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}, the update rule in Eq\. \([15](https://arxiv.org/html/2607.27632#S3.E15)\), and the Triangle inequality, we can obtain:

‖𝒫𝑾​\(𝒘it\+1\)−𝒘t\+1‖=\|\|𝒫𝑾\(𝒘t−η𝒘∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)\)−𝒫𝑾\(𝒘t−1N∑i=1Nη𝒘∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)\)\|\|≤\(a\)η𝒘\|\|∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)−1N∑i=1N∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)\|\|≤\(b\)η𝒘\(\|\|∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)\|\|\+1N∑i=1N\|\|∇𝒘ℒi\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)\|\|\)≤\(c\)​2​η𝒘​B𝒘,\\begin\{array\}\[\]\{l\}\|\|\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\)\-\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\|\|\\\\ =\|\|\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\{\\boldsymbol\{w\}^\{t\}\}\}\-\{\\eta\_\{\\boldsymbol\{w\}\}\}\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\)\\\\ \-\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\{\\boldsymbol\{w\}^\{t\}\}\}\\\!\-\\\!\\frac\{1\}\{N\}\\\!\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\}\)\|\|\\\\ \\overset\{\(a\)\}\{\\leq\}\{\\eta\_\{\\boldsymbol\{w\}\}\}\|\|\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\\\\ \\quad\-\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\}\|\|\\\\ \\overset\{\(b\)\}\{\\leq\}\{\\eta\_\{\\boldsymbol\{w\}\}\}\(\|\|\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\|\|\\\\ \\quad\+\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\|\|\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)\|\|\}\)\\\\ \\overset\{\(c\)\}\{\\leq\}2\{\\eta\_\{\\boldsymbol\{w\}\}\}\{B\_\{\\boldsymbol\{w\}\}\},\\end\{array\}\\vskip\-2\.84526pt\(22\)where inequality \(a\) follows from the non\-expansiveness property of the projection operator𝒫𝑾\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}, inequality\(b\)\(b\)is obtained by applying the Triangle inequality, and inequality\(c\)\(c\)holds based on Assumption[1](https://arxiv.org/html/2607.27632#Thmassumption1)\. This completes the proof of Lemma[1](https://arxiv.org/html/2607.27632#Thmlemma1)\. By the optimality condition of the projection operator in Eq\. \([11](https://arxiv.org/html/2607.27632#S3.E11)\), it follows that

⟨∇𝜶ℒi​\(𝜶it,𝒒it,𝒘t,𝒑it\),𝜶it\+1−𝜶i⟩≤⟨𝜶it\+1−𝜶it,𝜶i−𝜶it\+1⟩η𝜶\.\\begin\{array\}\[\]\{l\}\\langle\{\\nabla\_\{\\boldsymbol\{\\alpha\}\}\}\\mathcal\{L\}\_\{i\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{q\}\_\{i\}^\{t\}\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\),\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\-\\boldsymbol\{\\alpha\}\_\{i\}\\rangle\\\!\\leq\\\!\\frac\{\\langle\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\-\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\\boldsymbol\{\\alpha\}\_\{i\}\-\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\rangle\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\.\\end\{array\}\(23\)
By substituting𝜶i=𝜶it\\boldsymbol\{\\alpha\}\_\{i\}=\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}into Eq\. \([23](https://arxiv.org/html/2607.27632#S7.E23)\), we can obtain:

⟨∇𝜶ℒi​\(𝜶it,𝒒it,𝒘t,𝒑it\),𝜶it\+1−𝜶it⟩≤−‖𝜶it\+1−𝜶it‖2η𝜶\.\\begin\{array\}\[\]\{l\}\\langle\{\\nabla\_\{\\boldsymbol\{\\alpha\}\}\}\\mathcal\{L\}\_\{i\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\},\{\\boldsymbol\{q\}\_\{i\}^\{t\}\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\),\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\!\-\\\!\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\\rangle\\\!\\leq\\\!\-\\frac\{\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\!\-\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\|\{\|^\{2\}\}\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\.\\end\{array\}\(24\)
In addition, according to theLL\-smoothness in Assumption[1](https://arxiv.org/html/2607.27632#Thmassumption1)and the fact thatℒ​\(\{𝜶it\+1\},\{𝒒it\},𝒘t,\{𝒑it\}\)=1N​∑i=1Nℒi​\(𝜶it\+1,𝒒it,𝒘t,𝒑it\)\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathcal\{L\}\_\{i\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\},\{\\boldsymbol\{q\}\_\{i\}^\{t\}\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\), it follows that

ℒ​\(\{𝜶it\+1\},\{𝒒it\},𝒘t,\{𝒑it\}\)≤ℒ​\(\{𝜶it\},\{𝒒it\},𝒘t,\{𝒑it\}\)−1N​∑i=1N\(1η𝜶−L2\)​‖𝜶it\+1−𝜶it‖2\.\\begin\{array\}\[\]\{l\}\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\},\\\!\\\{\\boldsymbol\{q\}\_\{i\}^\{t\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\!\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\!\\leq\\\!\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\\\},\\\!\\\{\\boldsymbol\{q\}\_\{i\}^\{t\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\\ \-\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\-\\frac\{L\}\{2\}\}\)\|\|\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\-\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\|\{\|^\{2\}\}\.\\end\{array\}\(25\)
For the variable𝒒i\\boldsymbol\{q\}\_\{i\}, by combining the optimality condition of the projection operator in Eq\. \([12](https://arxiv.org/html/2607.27632#S3.E12)\) withLL\-smoothness in Assumption[1](https://arxiv.org/html/2607.27632#Thmassumption1), we can obtain:

ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t,\{𝒑it\}\)≤ℒ​\(\{𝜶it\+1\},\{𝒒it\},𝒘t,\{𝒑it\}\)−1N​∑i=1N\(1η𝒒−L2\)​‖𝒒it\+1−𝒒it‖2\.\\begin\{array\}\[\]\{l\}\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\!\\leq\\\!\\mathcal\{L\}\(\\\{\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\\ \-\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{q\}\}\}\}\}\-\\frac\{L\}\{2\}\}\)\|\|\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\}\-\{\\boldsymbol\{q\}\_\{i\}^\{t\}\}\|\{\|^\{2\}\}\.\\end\{array\}\(26\)
For the variable𝒘\\boldsymbol\{w\}, the projection happens after aggregating local parameters in Eq\. \([15](https://arxiv.org/html/2607.27632#S3.E15)\)\. According to the optimality condition of the projection operator and by substituting𝒘=𝒘t\\boldsymbol\{w\}=\{\\boldsymbol\{w\}^\{t\}\}:

⟨∑i=1N∇𝒘ℒi​\(𝜶it\+1,𝒒it\+1,𝒘t,𝒑it\)N,𝒘t\+1−𝒘t⟩≤−‖𝒘t\+1−𝒘t‖2η𝒘\.\\begin\{array\}\[\]\{l\}\\\!\\left\\langle\{\\frac\{\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\\nabla\_\{\\boldsymbol\{w\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\)\}\{N\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\}\}\}\}\\right\\rangle\\\!\\leq\\\!\-\\frac\{\|\|\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\}\}\}\|\{\|^\{2\}\}\}\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\.\\end\{array\}\(27\)
Combining Eq\. \([27](https://arxiv.org/html/2607.27632#S7.E27)\) with Eq\. \([18](https://arxiv.org/html/2607.27632#S4.E18)\), we have:

ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\}\)≤−\(1η𝒘−L2\)​‖𝒘t\+1−𝒘t‖2\+ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t,\{𝒑it\}\)\.\\begin\{array\}\[\]\{l\}\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\\\}\)\\\!\\leq\\\!\-\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\\\!\-\\\!\\frac\{L\}\{2\}\}\)\|\|\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\-\{\{\\boldsymbol\{w\}^\{t\}\}\}\|\{\|^\{2\}\}\\\\ \+\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\}\}\},\\\{\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\\\}\)\.\\end\{array\}\(28\)
Likewise, for variable𝒑i\\boldsymbol\{p\}\_\{i\}, according to the optimality condition of the projection operator in Eq\. \([14](https://arxiv.org/html/2607.27632#S3.E14)\), we can get:

⟨∇𝒑ℒi​\(𝜶it\+1,𝒒it\+1,𝒫𝑾​\(𝒘it\+1\),𝒑it\),𝒑it\+1−𝒑it⟩≤−1η𝒑​‖𝒑it\+1−𝒑it‖2\.\\begin\{array\}\[\]\{l\}\\left\\langle\{\{\{\\nabla\_\{\\boldsymbol\{p\}\}\}\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\),\{\\boldsymbol\{p\}\_\{i\}^\{t\}\}\),\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\}\\right\\rangle\\\\ \\leq\-\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\.\\end\{array\}\(29\)
For simplicity, letℒit\+1=ℒi​\(𝜶it\+1,𝒒it\+1,𝒘t\+1,𝒑it\)\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}=\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\}\)andℒ^it\+1=ℒi​\(𝜶it\+1,𝒒it\+1,𝒫𝑾​\(𝒘it\+1\),𝒑it\)\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\}=\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\}\),\\boldsymbol\{p\}\_\{i\}^\{t\}\), we can obtain:

⟨∇𝒑ℒit\+1,𝒑it\+1−𝒑it⟩=⟨∇𝒑ℒ^it\+1,𝒑it\+1−𝒑it⟩\+⟨∇𝒑ℒit\+1−∇𝒑ℒ^it\+1,𝒑it\+1−𝒑it⟩≤−1η𝒑​‖𝒑it\+1−𝒑it‖2\+⟨∇𝒑ℒit\+1−∇𝒑ℒ^it\+1,𝒑it\+1−𝒑it⟩\.\\begin\{array\}\[\]\{l\}\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\\\\ =\\\!\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\+\\\!\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\\\!\-\\\!\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\\\\ \\\!\\leq\\\!\-\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\\\!\+\\\!\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\\\!\-\\\!\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\.\\end\{array\}\(30\)
By combining Eq\. \([30](https://arxiv.org/html/2607.27632#S7.E30)\) with theLL\-smoothness, i\.e\.,ℒi​\(𝜶it\+1,𝒒it\+1,𝒘t\+1,𝒑it\+1\)≤ℒit\+1\+⟨∇𝒑ℒit\+1,𝒑it\+1−𝒑it⟩\+L2​‖𝒑it\+1−𝒑it‖2\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\)\\leq\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\+\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\+\\frac\{L\}\{2\}\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}andℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\+1\}\)=1N​∑i=1Nℒi​\(𝜶it\+1,𝒒it\+1,𝒘t\+1,𝒑it\+1\)\{\\mathcal\{L\}\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\}\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\mathcal\{L\}\_\{i\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\},\\boldsymbol\{q\}\_\{i\}^\{t\+1\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\), we can obtain:

ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\+1\}\)≤−\(1η𝒑−L2\)​∑i=1N‖𝒑it\+1−𝒑it‖2N\+ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\}\)\+1N​∑i=1N⟨∇𝒑ℒit\+1−∇𝒑ℒ^it\+1,𝒑it\+1−𝒑it⟩⏟E1\.\\begin\{array\}\[\]\{l\}\\\!\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\}\)\\\!\\leq\\\!\-\(\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\\\!\-\\\!\\frac\{L\}\{2\}\)\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\\frac\{\{\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\}\}\{N\}\\\\ \+\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\\\\ \+\\underbrace\{\\begin\{array\}\[\]\{l\}\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\{\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\-\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\}\\end\{array\}\}\_\{\{E\_\{1\}\}\}\.\\end\{array\}\(31\)
For the termE1E\_\{1\}in Eq\. \([31](https://arxiv.org/html/2607.27632#S7.E31)\), we can derive:

E1=1N​∑i=1N⟨∇𝒑ℒit\+1−∇𝒑ℒ^it\+1,𝒑it\+1−𝒑it⟩≤\(a\)​12​β​N​∑i=1N‖∇𝒑ℒit\+1−∇𝒑ℒ^it\+1‖2\+β2​N​∑i=1N‖𝒑it\+1−𝒑it‖2≤\(b\)​L22​β​N​∑i=1N‖𝒘t\+1−𝒫𝑾​\(𝒘it\+1\)‖2\+β2​N​∑i=1N‖𝒑it\+1−𝒑it‖2≤\(c\)​2​L2​η𝒘2​B𝒘21η𝒑−L2\+12​N​\(1η𝒑−L2\)​∑i=1N‖𝒑it\+1−𝒑it‖2,\\begin\{array\}\[\]\{l\}\{E\_\{1\}\}=\\frac\{1\}\{N\}\\sum\\nolimits\_\{i=1\}^\{N\}\{\\langle\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\-\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\},\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\\rangle\}\\\\ \\overset\{\(a\)\}\{\\leq\}\\\!\\frac\{1\}\{\{2\\beta N\}\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\{\|\|\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\mathcal\{L\}\_\{i\}^\{t\+1\}\}\\\!\-\\\!\{\\nabla\_\{\\boldsymbol\{p\}\}\}\{\\hat\{\\mathcal\{L\}\}\_\{i\}^\{t\+1\}\}\|\{\|^\{2\}\}\}\\\!\+\\\!\\frac\{\\beta\}\{2N\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\\\\ \\overset\{\(b\)\}\{\\leq\}\\\!\\frac\{\{\{L^\{2\}\}\}\}\{\{2\\beta N\}\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\{\|\|\{\{\\boldsymbol\{w\}^\{t\+1\}\}\}\\\!\-\\\!\{\{\\mathcal\{P\}\_\{\\boldsymbol\{W\}\}\}\}\(\{\\boldsymbol\{w\}\_\{i\}^\{t\+1\}\}\)\|\{\|^\{2\}\}\}\\\!\+\\\!\\frac\{\\beta\}\{2N\}\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\!\-\\\!\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\\\\ \\overset\{\(c\)\}\{\\leq\}\\\!\\frac\{\{2\{L^\{2\}\}\{\\eta\_\{\\boldsymbol\{w\}\}^\{2\}\}\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\}\{\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\-\\frac\{L\}\{2\}\}\}\+\\frac\{1\}\{2N\}\(\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\-\\frac\{L\}\{2\}\)\\sum\\nolimits\_\{i=1\}^\{N\}\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\},\\end\{array\}\\vskip\-2\.84526pt\(32\)where inequality \(a\) is based on the Young’s inequality, inequality \(b\) applies theLL\-smoothness, and inequality \(c\) holds due to Eq\. \([22](https://arxiv.org/html/2607.27632#S7.E22)\)\. Substituting the upper bound ofE1E\_\{1\}from Eq\. \([32](https://arxiv.org/html/2607.27632#S7.E32)\) into Eq\. \([31](https://arxiv.org/html/2607.27632#S7.E31)\) yields:

ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\+1\}\)≤2​L2​η𝒘2​B𝒘21/η𝒑−L/2−\(1η𝒑−L2\)​∑i=1N‖𝒑it\+1−𝒑it‖22​N\+ℒ​\(\{𝜶it\+1\},\{𝒒it\+1\},𝒘t\+1,\{𝒑it\}\)\.\\begin\{array\}\[\]\{l\}\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\\\}\)\\leq\\frac\{\{2\{L^\{2\}\}\{\\eta\_\{\\boldsymbol\{w\}\}^\{2\}\}\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\}\{\{\{\{\{1/\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\-L/2\}\}\\\\ \-\(\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\\\!\-\\\!\\frac\{L\}\{2\}\)\\\!\\sum\\limits\_\{i=1\}^\{N\}\\\!\\frac\{\{\|\|\\boldsymbol\{p\}\_\{i\}^\{t\+1\}\-\\boldsymbol\{p\}\_\{i\}^\{t\}\|\{\|^\{2\}\}\}\}\{2N\}\\\!\+\\\!\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\+1\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{t\+1\}\\\},\{\{\\boldsymbol\{w\}^\{t\+1\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{t\}\\\}\)\.\\end\{array\}\(33\)
Consequently, combining Eqs\. \([25](https://arxiv.org/html/2607.27632#S7.E25)\), \([26](https://arxiv.org/html/2607.27632#S7.E26)\), \([28](https://arxiv.org/html/2607.27632#S7.E28)\), and \([33](https://arxiv.org/html/2607.27632#S7.E33)\) completes the proof of Lemma[2](https://arxiv.org/html/2607.27632#Thmlemma2)\. Summing both sides of Eq\. \([20](https://arxiv.org/html/2607.27632#S4.E20)\) fromt=0t=0tot=T−1t=T\-1, and using Definition[1](https://arxiv.org/html/2607.27632#Thmdefinition1), we get:

∑t=0T−1\(η𝜶2\(1−L​η𝜶2\)∑i=1N\|\|g𝜶,it\|\|2\+η𝒒2\(1−L​η𝒒2\)∑i=1N\|\|g𝒒,it\|\|2\+η𝒘2\(1−L​η𝒘2\)\|\|g𝒘t\|\|\+2η𝒑2\(1−L​η𝒑2\)∑i=1N\|\|g𝒑,it\|\|2\)−L2​η𝒘2​B𝒘2​T2​\(1η𝒑−L2\)≤ℒ​\(\{𝜶i0\},\{𝒒i0\},𝒘0,\{𝒑i0\}\)−ℒ​\(\{𝜶iT\},\{𝒒iT\},𝒘T,\{𝒑iT\}\)\.\\begin\{array\}\[\]\{l\}\\sum\\limits\_\{t=0\}^\{T\-1\}\(\\frac\{\{\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\{2\}\(\{1\-\\frac\{\{L\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}\}\}\{2\}\}\)\\sum\\limits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{\\alpha\},i\}^\{t\}\|\{\|^\{2\}\}\}\+\\frac\{\{\{\\eta\_\{\\boldsymbol\{q\}\}\}\}\}\{2\}\(\{1\-\\frac\{\{L\{\\eta\_\{\\boldsymbol\{q\}\}\}\}\}\{2\}\}\)\\sum\\limits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{q\},i\}^\{t\}\|\{\|^\{2\}\}\}\\\\ \+\\\!\\frac\{\{\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\{2\}\(\{1\\\!\-\\\!\\frac\{\{L\{\\eta\_\{\\boldsymbol\{w\}\}\}\}\}\{2\}\}\)\|\|g\_\{\\boldsymbol\{w\}\}^\{t\}\|\|\{\{\}^\{2\}\}\\\!\+\\\!\\frac\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\{2\}\(\{1\\\!\-\\\!\\frac\{\{L\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\{2\}\}\)\\sum\\limits\_\{i=1\}^\{N\}\{\|\|g\_\{\\boldsymbol\{p\},i\}^\{t\}\|\{\|^\{2\}\}\}\)\-\\frac\{\{\{L^\{2\}\}\\eta\_\{\\boldsymbol\{w\}\}^\{2\}B\_\{\\boldsymbol\{w\}\}^\{2\}T\}\}\{\{2\(\{\\frac\{1\}\{\{\{\\eta\_\{\\boldsymbol\{p\}\}\}\}\}\-\\frac\{L\}\{2\}\}\)\}\}\\\\ \\leq\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{0\}\\\},\{\\boldsymbol\{w\}^\{0\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{0\}\\\}\)\-\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{T\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{T\}\\\},\{\{\\boldsymbol\{w\}^\{T\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{T\}\\\}\)\.\\end\{array\}\(34\)
By the choice of the step\-sizes, i\.e\.,η𝜶=η𝒒=η𝒘=η𝒑=η=T−1/3\{\\eta\_\{\\boldsymbol\{\\alpha\}\}\}=\{\\eta\_\{\\boldsymbol\{q\}\}\}=\{\\eta\_\{\\boldsymbol\{w\}\}\}=\{\\eta\_\{\\boldsymbol\{p\}\}\}=\\eta=\{T^\{\-1/3\}\}, whenT≥L3T\\geq L^\{3\}, we can getη2​\(1−L​η2\)≥η4​\(1−L​η2\)≥η8,1η−L2≥12​η\\frac\{\\eta\}\{2\}\(\{1\-\\frac\{\{L\\eta\}\}\{2\}\}\)\\geq\\frac\{\\eta\}\{4\}\(\{1\-\\frac\{\{L\\eta\}\}\{2\}\}\)\\geq\\frac\{\\eta\}\{8\},\\frac\{1\}\{\\eta\}\-\\frac\{L\}\{2\}\\geq\\frac\{1\}\{\{2\\eta\}\}\. Thus, we can obtain:

1T∑t=0T−1\|\|Gt\|\|2=1T∑t=0T−1\(∑i=1N\(\|\|g𝜶,it\|\|\+2\|\|g𝒒,it\|\|\+2\|\|g𝒑,it\|\|\)2\+\|\|g𝒘t\|\|\)2≤ℒ​\(\{𝜶i0\},\{𝒒i0\},𝒘0,\{𝒑i0\}\)−ℒ​\(\{𝜶iT\},\{𝒒iT\},𝒘T,\{𝒑iT\}\)η4​\(1−L​η2\)​T\+2​L2​η2​B𝒘2η4​\(1−L​η2\)​\(1η−L2\)≤8​\(ℒ​\(\{𝜶i0\},\{𝒒i0\},𝒘0,\{𝒑i0\}\)−ℒ​\(\{𝜶iT\},\{𝒒iT\},𝒘T,\{𝒑iT\}\)\)\+32​L2​η3​T​B𝒘2η​T≤8​\(ℒ​\(\{𝜶i0\},\{𝒒i0\},𝒘0,\{𝒑i0\}\)−ℒ​\(\{𝜶i∗\},\{𝒒i∗\},𝒘∗,\{𝒑i∗\}\)\)\+32​L2​B𝒘2T2/3\.\\begin\{array\}\[\]\{l\}\\frac\{1\}\{T\}\\sum\\nolimits\_\{t=0\}^\{T\-1\}\{\|\|\{G^\{t\}\}\|\|\{\{\}^\{2\}\}\}\\\\ \\\!=\\frac\{1\}\{T\}\\\!\\sum\\nolimits\_\{t=0\}^\{T\-1\}\\left\(\\sum\\nolimits\_\{i=1\}^\{N\}\\\!\\left\(\|\{\}\|\{\}g\_\{\\boldsymbol\{\\alpha\},i\}^\{t\}\|\{\}\|\{\}^\{2\}\+\|\{\}\|\{\}g\_\{\\boldsymbol\{q\},i\}^\{t\}\|\{\}\|\{\}^\{2\}\\\!\+\\\!\|\{\}\|\{\}g\_\{\\boldsymbol\{p\},i\}^\{t\}\|\{\}\|\{\}^\{2\}\\right\)\+\|\{\}\|\{\}g\_\{\\boldsymbol\{w\}\}^\{t\}\|\{\}\|\{\}^\{2\}\\right\)\\\\ \\\!\\leq\\\!\\frac\{\{\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{0\}\\\},\{\\boldsymbol\{w\}^\{0\}\}\\\!,\\\{\\boldsymbol\{p\}\_\{i\}^\{0\}\\\}\)\-\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{T\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{T\}\\\},\{\{\\boldsymbol\{w\}^\{T\}\}\}\\\!,\\\{\\boldsymbol\{p\}\_\{i\}^\{T\}\\\}\)\}\}\{\{\\frac\{\\eta\}\{4\}\(\{1\-\\frac\{\{L\\eta\}\}\{2\}\}\)\}T\}\\\!\+\\\!\\frac\{\{2\{L^\{2\}\}\{\\eta^\{2\}\}B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\{\{\\frac\{\\eta\}\{4\}\(\{1\\\!\-\\\!\\frac\{\{L\\eta\}\}\{2\}\}\)\(\{\\frac\{1\}\{\\eta\}\\\!\-\\\!\\frac\{L\}\{2\}\}\)\}\}\\\\ \\\!\\leq\\\!\\frac\{\{8\(\{\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{0\}\\\},\{\\boldsymbol\{w\}^\{0\}\}\\\!,\\\{\\boldsymbol\{p\}\_\{i\}^\{0\}\\\}\)\-\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{T\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{T\}\\\},\{\{\\boldsymbol\{w\}^\{T\}\}\}\\\!,\\\{\\boldsymbol\{p\}\_\{i\}^\{T\}\\\}\)\}\)\}\+32\{L^\{2\}\}\{\\eta^\{3\}\}T\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\{\\eta T\}\\\\ \\\!\\leq\\\!\\frac\{\{8\(\{\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{0\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{0\}\\\},\{\\boldsymbol\{w\}^\{0\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{0\}\\\}\)\-\\mathcal\{L\}\(\\\{\\boldsymbol\{\\alpha\}\_\{i\}^\{\*\}\\\},\\\{\\boldsymbol\{q\}\_\{i\}^\{\*\}\\\},\{\{\\boldsymbol\{w\}^\{\*\}\}\},\\\{\\boldsymbol\{p\}\_\{i\}^\{\*\}\\\}\)\}\)\+32\{L^\{2\}\}\{B\_\{\\boldsymbol\{w\}\}^\{2\}\}\}\}\{\{\{T^\{2/3\}\}\}\}\.\\end\{array\}\(35\)
Consequently, combining Eq\. \([35](https://arxiv.org/html/2607.27632#S7.E35)\) with the definition of anϵ\\epsilon\-stationary point in Definition[2](https://arxiv.org/html/2607.27632#Thmdefinition2)yields the iteration complexity result stated in Theorem[1](https://arxiv.org/html/2607.27632#Thmtheorem1)\.

The communication complexity of the proposed method comprises the costs of value\-function refinement and variable updates\. In\(t\+1\)th\(t\+1\)^\{\\rm\{th\}\}iteration, each workeriifirst locally updates the value\-functionsV3,i​\(𝜶i,𝒘\)V\_\{3,i\}\(\\boldsymbol\{\\alpha\}\_\{i\},\\boldsymbol\{w\}\)andV2,i\(1\)​\(𝜶it,𝒘t\)V\_\{2,i\}^\{\(1\)\}\(\\boldsymbol\{\\alpha\}\_\{i\}^\{t\},\\boldsymbol\{w\}^\{t\}\)completely on\-device, incurring zero communication cost\. Then, to refineV2\(2\)​\(𝜶it\)V\_\{2\}^\{\(2\)\}\(\{\\boldsymbol\{\\alpha\}\_\{i\}^\{t\}\}\), each workeriitransmits𝒘​iQ∈ℝd\\boldsymbol\{w\}i^\{Q\}\\in\\mathbb\{R\}^\{d\}to the master, which aggregates them and broadcasts the global average𝒘Q=1N​∑i=1N​𝒘iQ∈ℝd\\boldsymbol\{w\}^\{Q\}=\\frac\{1\}\{N\}\\sum\{i=1\}^\{N\}\\boldsymbol\{w\}\_\{i\}^\{Q\}\\in\\mathbb\{R\}^\{d\}back to the workers\.

Following this, the local variables are updated alternately, and each worker uploads𝒘​it\+1∈ℝd\\boldsymbol\{w\}i^\{t\+1\}\\in\\mathbb\{R\}^\{d\}to the master\. The master then broadcasts the projected aggregate as the new global parameter,𝒘t\+1=𝒫​𝑾​\(1N​∑i=1N𝒘​it\+1\)∈ℝd\\boldsymbol\{w\}^\{t\+1\}=\\mathcal\{P\}\{\\boldsymbol\{W\}\}\(\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\boldsymbol\{w\}i^\{t\+1\}\)\\in\\mathbb\{R\}^\{d\}\. Multiplying the per\-iteration communication cost by the iteration complexityT​\(ϵ\)T\(\\epsilon\)in Theorem[1](https://arxiv.org/html/2607.27632#Thmtheorem1), the overall communication complexity satisfiesC​comm​\(ϵ\)≤128​d⋅T​\(ϵ\)=𝒪​\(dϵ3/2\)C\{\\mathrm\{comm\}\}\(\\epsilon\)\\leq 128d\\cdot T\(\\epsilon\)=\\mathcal\{O\}\(\\frac\{d\}\{\{\\epsilon^\{3/2\}\}\}\),completing the proof of Theorem[2](https://arxiv.org/html/2607.27632#Thmtheorem2)\.

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