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arXiv:2406.17470v1 [cs.LG] 25 Jun 2024
arXiv:2406.17470v1 [cs.LG] 2024 年 6 月 25 日

Dynamic Scheduling for Vehicle-to-Vehicle Communications Enhanced Federated Learning
车辆到车辆通信增强联邦学习的动态调度

Jintao Yan,  Tan Chen, Yuxuan Sun, ,
Zhaojun Nan,  Sheng Zhou,  and Zhisheng Niu, 
J. Yan, T. Chen, Z. Nan, S. Zhou (Corresponding Author) and Z. Niu are with the Beijing National Research Center for Information Science and Technology, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. (email: {yanjt22, chent21}@mails.tsinghua.edu.cn, nzj660624@mail.tsinghua.edu.cn, {sheng.zhou, niuzhs}@tsinghua.edu.cn). Y. Sun is with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China. (e-mail: yxsun@bjtu.edu.cn).
Abstract 摘要

Leveraging the computing and sensing capabilities of vehicles, vehicular federated learning (VFL) has been applied to edge training for connected vehicles. The dynamic and interconnected nature of vehicular networks presents unique opportunities to harness direct vehicle-to-vehicle (V2V) communications, enhancing VFL training efficiency. In this paper, we formulate a stochastic optimization problem to optimize the VFL training performance, considering the energy constraints and mobility of vehicles, and propose a V2V-enhanced dynamic scheduling (VEDS) algorithm to solve it. The model aggregation requirements of VFL and the limited transmission time due to mobility result in a stepwise objective function, which presents challenges in solving the problem. We thus propose a derivative-based drift-plus-penalty method to convert the long-term stochastic optimization problem to an online mixed integer nonlinear programming (MINLP) problem, and provide a theoretical analysis to bound the performance gap between the online solution and the offline optimal solution. Further analysis of the scheduling priority reduces the original problem into a set of convex optimization problems, which are efficiently solved using the interior-point method. Experimental results demonstrate that compared with the state-of-the-art benchmarks, the proposed algorithm enhances the image classification accuracy on the CIFAR-10 dataset by 3.18%percent3.183.18\%3.18 % and reduces the average displacement errors on the Argoverse trajectory prediction dataset by 10.21%percent10.2110.21\%10.21 %.
利用车辆的计算和感知能力,车联网联邦学习(VFL)已被应用于联网车辆的边缘训练。车联网的动态和互联特性为利用直接车对车(V2V)通信提供了独特的机会,从而提高了 VFL 训练效率。本文针对车辆的能量约束和移动性,将 VFL 训练性能优化问题转化为随机优化问题,并提出了一种 V2V 增强动态调度(VEDS)算法来解决该问题。VFL 的模型聚合需求以及移动性带来的有限传输时间导致了分段目标函数,这给问题的求解带来了挑战。因此,我们提出了一种基于导数的漂移加惩罚方法,将长期随机优化问题转化为在线混合整数非线性规划(MINLP)问题,并提供理论分析来界定在线解与离线最优解之间的性能差距。 对调度优先级的进一步分析将原始问题简化为一组凸优化问题,这些问题可以使用内点法有效地解决。实验结果表明,与最先进的基准相比,所提出的算法将 CIFAR-10 数据集上的图像分类精度提高了 3.18%3.18\%3.18 % ,并将 Argoverse 轨迹预测数据集上的平均位移误差降低了 10.21%10.21\%10.21 %

I Introduction
简介

The rapid advancement of vehicular networks has enabled various new applications, including vehicular cooperative perception, trajectory prediction, and route planning. These applications produce vast amounts of data and require timely training of machine learning (ML) models to adapt to changing road conditions [1]. In conventional ML frameworks, data is transmitted to a central server for model training, which poses privacy risks and incurs significant delays. As more and more vehicles are equipped with powerful computing capabilities and can collect data via on-board sensors, the ML training process can shift from centralized servers to the vehicles themselves. Therefore, vehicular federated learning (VFL) is a promising framework for timely training and privacy conservation [2].
车辆网络的快速发展催生了各种新应用,包括车辆协同感知、轨迹预测和路线规划。这些应用产生了海量数据,需要及时训练机器学习 (ML) 模型以适应不断变化的路况 [1]。在传统的 ML 框架中,数据被传输到中央服务器进行模型训练,这会带来隐私风险并造成重大延迟。随着越来越多的车辆配备强大的计算能力,并能够通过车载传感器收集数据,ML 训练过程可以从集中式服务器转移到车辆本身。因此,车联网联邦学习 (VFL) 是一个很有前景的框架,可以实现及时训练和隐私保护 [2]

VFL is a distributed ML framework, where an ML model is trained over multiple vehicles. Vehicles with local data and computing capabilities are called source vehicles (SOVs). Each SOV trains an ML model based on the local dataset and uploads the model parameters to the roadside unit (RSU). The RSU aggregates the received parameters to obtain a global model and then broadcasts the new models to vehicles to start a new round. Implemented in vehicular networks, VFL takes advantage of the distributed data and processing capabilities while maintaining data privacy [3, 4].
VFL 是一种分布式机器学习框架,其中机器学习模型在多辆车之间进行训练。拥有本地数据和计算能力的车辆被称为源车辆 (SOV)。每个 SOV 基于本地数据集训练一个机器学习模型,并将模型参数上传到路边单元 (RSU)。RSU 聚合接收到的参数以获得全局模型,然后将新模型广播到车辆以开始新一轮。VFL 在车联网中实现,利用了分布式数据和处理能力,同时维护数据隐私[3, 4]

The distinguished characteristic of VFL is the high mobility of vehicles [4], bringing about challenges and opportunities. On the one hand, mobility leads to many challenges. Firstly, the channel conditions of vehicular networks change rapidly due to the high mobility, which complicates the channel estimation and leads to unreliable data transmissions [5]. Secondly, the connections between vehicle-to-infrastructure (V2I) are intermittent. One vehicle may leave the coverage of an RSU before uploading all of the local model parameters [6], which imposes stringent latency requirements for model aggregation in VFL. The current solution to this problem is to increase the processor frequency and transmission power to reduce the computation and communication latency [7]. However, this may greatly increase the energy consumption of SOVs.
VFL 的显著特点是车辆的高度移动性 [4],带来了挑战和机遇。一方面,移动性带来了许多挑战。首先,由于高移动性,车联网的信道条件变化迅速,这使得信道估计变得复杂,并导致数据传输不可靠 [5]。其次,车辆到基础设施 (V2I) 之间的连接是间歇性的。一辆车可能在上传所有本地模型参数之前就离开了 RSU 的覆盖范围 [6],这对 VFL 中的模型聚合提出了严格的延迟要求。目前解决这个问题的方法是提高处理器频率和传输功率,以降低计算和通信延迟 [7]。然而,这可能会大大增加 SOV 的能耗。

On the other hand, mobility also brings about communication opportunities [8, 9]. Recent advancements in vehicle-to-vehicle (V2V) communications via sidelinks enable vehicles to communicate directly with each other, enhancing transmission rates and reliability in vehicular networks [10, 11]. Many vehicles that are not scheduled for training can also be involved in VFL by relaying the model uploads, which are namely opportunistic vehicles (OPVs). Utilizing the sidelinks, SOVs can upload their model parameters to the RSUs with the help of OPVs. Mobility increases the likelihood of scheduled vehicles encountering OPVs at closer ranges, under better channel conditions, or with line-of-sight paths. Leveraging these OPVs may increase the success rate of model uploading and therefore enhance the learning performance.
另一方面,移动性也带来了通信机会[8, 9]。近年来,车联网(V2V)通信技术的进步,使得车辆能够通过侧链直接相互通信,从而提高了车联网的传输速率和可靠性[10, 11]。许多未安排培训的车辆也可以通过中继模型上传参与 VFL,这些车辆被称为机会车辆(OPVs)。利用侧链,SOVs 可以在 OPVs 的帮助下将模型参数上传到 RSUs。移动性增加了计划车辆在更近的范围内、更好的信道条件下或视线路径下遇到 OPVs 的可能性。利用这些 OPVs 可以提高模型上传的成功率,从而提高学习性能。

Currently, many studies have leveraged V2V sidelinks to support various applications in vehicular networks, such as vehicular task offloading [12, 13, 14], vehicular edge caching [15, 16] and cooperative perception [17, 18, 19]. However, few works utilize V2V sidelinks to improve the performance of VFL. Different from other applications [12, 13, 14, 15, 16, 17, 18, 19], VFL operates on a longer time scale with model aggregation requirements. Therefore, a dynamic scheduling algorithm is needed to adapt to the changing environment throughout the VFL training.
目前,许多研究利用 V2V 侧链来支持车联网中的各种应用,例如车辆任务卸载 [12, 13, 14]、车辆边缘缓存 [15, 16] 和协同感知 [17, 18, 19]。然而,很少有工作利用 V2V 侧链来提高 VFL 的性能。与其他应用 [12, 13, 14, 15, 16, 17, 18, 19] 不同,VFL 在更长的时间尺度上运行,并具有模型聚合需求。因此,需要一种动态调度算法来适应 VFL 训练过程中的不断变化的环境。

In this work, we consider a VFL system that utilizes the V2V communication resources and employs the OPVs to assist SOVs in model uploading, enhancing the VFL performance. The main contributions are summarized as follows:
在这项工作中,我们考虑了一种利用 V2V 通信资源并使用 OPV 来协助 SOV 上传模型的 VFL 系统,从而提高 VFL 性能。主要贡献总结如下:

  • We characterize the convergence bound of the VFL system, and formulate a stochastic optimization problem to minimize the global loss function, considering the energy constraints and the channel uncertainty caused by vehicle mobility. A V2V-enhanced dynamic scheduling (VEDS) algorithm is proposed to solve it.


    我们刻画了 VFL 系统的收敛界,并考虑到车辆移动带来的能量约束和信道不确定性,将全局损失函数最小化问题转化为随机优化问题。提出了一种 V2V 增强动态调度(VEDS)算法来解决该问题。
  • The model aggregation requirements and the limited transmission time in VFL result in a stepwise objective function, which is non-convex and hard to solve. We propose a derivative-based drift-plus-penalty method to convert the long-term stochastic optimization problem to an online mixed integer nonlinear programming (MINLP) problem. We provide a theoretical performance guarantee for the proposed transformation by bounding the performance gap between the online and offline solutions. Our analysis further shows the impact of approximation parameters on the performance bound.


    • VFL 中的模型聚合需求和有限传输时间导致了分步目标函数,该函数是非凸的且难以求解。我们提出了一种基于导数的漂移加惩罚方法,将长期随机优化问题转换为在线混合整数非线性规划 (MINLP) 问题。我们通过对在线和离线解决方案之间的性能差距进行界定,为提出的转换提供了理论性能保证。我们的分析进一步表明了近似参数对性能界限的影响。
  • Through the analysis of the MINLP problem, we identify the priority in the OPV scheduling and reduce the original problem to a set of convex optimization problems, which are solved using the interior-point method.


    通过对 MINLP 问题的分析,我们确定了 OPV 调度中的优先级,并将原始问题简化为一组凸优化问题,这些问题使用内点法求解。
  • Experimental results show that, compared with the state-of-the-art benchmarks, the test accuracy is increased by 3.18%percent3.183.18\%3.18 % for image classification on the CIFAR-10 dataset, and the average displacement error (ADE) is reduced by 10.21%percent10.2110.21\%10.21 % for trajectory prediction on the Argoverse dataset.


    • 实验结果表明,与最先进的基准相比,在 CIFAR-10 数据集上的图像分类测试精度提高了 3.18%3.18\%3.18 % ,在 Argoverse 数据集上的轨迹预测平均位移误差 (ADE) 降低了 10.21%10.21\%10.21 %

The rest of this paper is organized as follows. The related papers are reviewed in Section II. Section III introduces the system model, including the FL, computation, and communication models. The convergence analysis and problem formulation are provided in Section IV, and the VEDS algorithm is proposed in Section V. Experimental results are shown in Section VI, and conclusions are drawn in Section VII.
本文的其余部分组织如下:第二部分回顾了相关文献;第三部分介绍了系统模型,包括联邦学习、计算和通信模型;第四部分提供了收敛分析和问题公式化,第五部分提出了 VEDS 算法;第六部分展示了实验结果,第七部分得出结论。

Refer to caption
Figure 1: The VFL framework.
图 1: VFL 框架。

II Related Works
II 相关工作

Many studies have explored the application of federated learning (FL) in wireless networks [20], addressing critical issues such as wireless resource management[21, 22, 23, 24, 25, 26, 27], compression and sparsification[28, 29, 30, 31, 32], and training algorithm design [33, 34, 35]. However, these studies rarely consider the unique characteristics of vehicular networks, such as high mobility and rapidly changing channel conditions.
许多研究探索了联邦学习 (FL) 在无线网络中的应用[20],解决无线资源管理[21, 22, 23, 24, 25, 26, 27]、压缩和稀疏化[28, 29, 30, 31, 32]以及训练算法设计[33, 34, 35]等关键问题。然而,这些研究很少考虑车联网的独特特性,例如高移动性和快速变化的信道条件。

More recent studies have begun to investigate FL in vehicular networks. These studies recognize the challenges posed by the high mobility of vehicles and the dynamic nature of vehicular environments [36, 5, 7, 37]. In [36], the impact of vehicle mobility on data quality, such as noise, motion blur, and distortion, is considered, and a resource optimization and vehicle selection scheme is proposed in the context of VFL. The proposed scheme dynamically schedules vehicles with higher image quality, increasing the convergence rate and reducing the time and energy consumption in FL training. In [5], the short-lived connections between vehicles and RSUs are considered, and a mobility-aware optimization algorithm is proposed. The proposed algorithm enhances the convergence performance of VFL by optimizing the duration of each training round and the number of local iterations. In [7, 37], the impact of rapidly time-varying channels resulting from vehicle mobility is considered. Specifically, a mobility and channel dynamic aware FL (MADCA-FL) scheme is proposed in [7], which optimizes the success probability of vehicle selection and model parameter updating based on the analysis of vehicle mobility and channel dynamics. In [37], a more realistic scenario is explored within a 5G new radio framework, and a joint VFL and radio access technology parameter optimization scheme is proposed under the constraints of delay, energy, and cost, aiming to maximize the successful transmission rate of locally trained models. However, most existing studies focus on V2I aggregation, overlooking the potential of harnessing V2V sidelinks to enhance the VFL training efficiency.
最近的研究开始调查车辆网络中的联邦学习。这些研究认识到车辆高移动性和车辆环境动态性带来的挑战[36, 5, 7, 37]。在[36]中,考虑了车辆移动性对数据质量的影响,例如噪声、运动模糊和失真,并在 VFL 的背景下提出了一种资源优化和车辆选择方案。该方案动态地调度具有更高图像质量的车辆,从而提高了收敛速度,并减少了 FL 训练中的时间和能量消耗。在[5]中,考虑了车辆与 RSU 之间短暂的连接,并提出了一种移动感知优化算法。该算法通过优化每个训练轮次的持续时间和本地迭代次数来提高 VFL 的收敛性能。在[7, 37]中,考虑了车辆移动性导致的快速时变信道的影响。 具体来说,[7]中提出了一种移动性和信道动态感知的联邦学习(MADCA-FL)方案,该方案基于对车辆移动性和信道动态的分析,优化了车辆选择和模型参数更新的成功概率。在[37]中,在 5G 新无线电框架内探索了更现实的场景,并提出了一种联合 VFL 和无线接入技术参数优化方案,该方案在延迟、能量和成本约束下,旨在最大化本地训练模型的成功传输率。然而,大多数现有研究都集中在 V2I 聚合上,忽略了利用 V2V 侧链来提高 VFL 训练效率的潜力。

Enhancements in V2V communications through sidelinks, as introduced in the recent updates by the Third Generation Partnership Project (3GPP) [10, 11], enable vehicles to communicate with each other directly. This advancement supports a variety of vehicular applications, including vehicular task offloading [12, 13, 14], vehicular edge caching [15, 16] and cooperative perception [17, 18, 19]. In [12, 13], vehicular task offloading strategies are proposed based on V2V communications, where tasks from one vehicle are offloaded to another to reduce the computational load on the original vehicle and enhance the task execution performance. Further investigations [14] have explored the integration of V2I and V2V communications, utilizing vehicles within the network as relays to improve the efficiency of task offloading processes. In terms of vehicular edge caching, the V2V sidelinks are utilized to enhance the caching hit rate and reduce the content access latency [15, 16]. In [17, 18, 19], the scenario of vehicular cooperative perception is explored, where V2V assistance expands the sensing range and enhances the accuracy of vehicle perception.
第三代合作伙伴计划 (3GPP) 最近的更新中引入了侧链,增强了 V2V 通信[10, 11],使车辆能够直接相互通信。这一进步支持各种车辆应用,包括车辆任务卸载[12, 13, 14]、车辆边缘缓存[15, 16]和协同感知[17, 18, 19]。在[12, 13]中,提出了基于 V2V 通信的车辆任务卸载策略,其中一辆车的任务被卸载到另一辆车,以减少原始车的计算负荷并提高任务执行性能。进一步的研究[14]探索了 V2I 和 V2V 通信的集成,利用网络中的车辆作为中继来提高任务卸载过程的效率。 在车辆边缘缓存方面,V2V 侧链被用来提高缓存命中率并降低内容访问延迟[15, 16]。在[17, 18, 19]中,探索了车辆协同感知的场景,其中 V2V 辅助扩展了感知范围并提高了车辆感知的准确性。

In the context of VFL, V2V communication resources have great potential for optimizing training efficiency. By appropriately utilizing these resources, the convergence speed of FL can be significantly improved, and the energy consumption of vehicles can be balanced.
在 VFL 的背景下,V2V 通信资源在优化训练效率方面具有巨大潜力。通过合理利用这些资源,可以显著提高 FL 的收敛速度,并平衡车辆的能耗。

III System Model
III 系统模型

III-A VFL Model
III-A VFL 模型

We consider a VFL system as shown in Fig. 1, where an RSU (indexed by r𝑟ritalic_r in the following) orchestrates the training of a neural network model 𝒘𝒘\boldsymbol{w}bold_italic_w with the assistance of vehicles that enter its coverage area. During the kthsuperscript𝑘thk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT training round, the vehicles that possess local datasets and are willing to participate in the collaborative training of the neural network model are referred to as SOVs, denoted by 𝒮ksubscript𝒮𝑘\mathcal{S}_{k}caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. The vehicles that do not participate in model training, but have communication capabilities and can help SOVs upload the models are referred to as OPVs, denoted by 𝒰ksubscript𝒰𝑘\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.
我们考虑一个如图 1 所示的 VFL 系统,其中一个 RSU(在以下内容中用 rritalic_r 索引)在进入其覆盖区域的车辆的帮助下协调神经网络模型 𝒘\boldsymbol{w}bold_italic_w 的训练。在 kthk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT 训练轮次中,拥有本地数据集并愿意参与神经网络模型协同训练的车辆被称为 SOV,用 𝒮k\mathcal{S}_{k}caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 表示。不参与模型训练,但具有通信能力,可以帮助 SOV 上传模型的车辆被称为 OPV,用 𝒰k\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 表示。

Each SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT holds a local dataset with an associated distribution 𝒟msubscript𝒟𝑚\mathcal{D}_{m}caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT over the space of samples 𝒳msubscript𝒳𝑚\mathcal{X}_{m}caligraphic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. For each data sample 𝒙𝒳m𝒙subscript𝒳𝑚\boldsymbol{x}\in\mathcal{X}_{m}bold_italic_x ∈ caligraphic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, a loss function f(𝒘;𝒙)𝑓𝒘𝒙f(\boldsymbol{w};\boldsymbol{x})italic_f ( bold_italic_w ; bold_italic_x ) is used to measure the fitting performance of the model vector 𝒘𝒘\boldsymbol{w}bold_italic_w. The local loss function of vehicle m𝑚mitalic_m is defined as the average loss over the distribution 𝒟msubscript𝒟𝑚\mathcal{D}_{m}caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, i.e., fm(𝒘)𝔼𝒙𝒟m[f(𝒘;𝒙)].subscript𝑓𝑚𝒘similar-to𝒙subscript𝒟𝑚𝔼delimited-[]𝑓𝒘𝒙f_{m}(\boldsymbol{w})\triangleq\underset{\boldsymbol{x}\sim\mathcal{D}_{m}}{% \operatorname*{\mathbb{E}}}[f(\boldsymbol{w};\boldsymbol{x})].italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) ≜ start_UNDERACCENT bold_italic_x ∼ caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_UNDERACCENT start_ARG blackboard_E end_ARG [ italic_f ( bold_italic_w ; bold_italic_x ) ] .
每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 都包含一个本地数据集,该数据集与样本空间上的关联分布 𝒟m\mathcal{D}_{m}caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT 相关联。对于每个数据样本 𝒙𝒳m\boldsymbol{x}\in\mathcal{X}_{m}bold_italic_x ∈ caligraphic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ,使用损失函数 f(𝒘;𝒙)f(\boldsymbol{w};\boldsymbol{x})italic_f ( bold_italic_w ; bold_italic_x ) 来衡量模型向量 𝒘\boldsymbol{w}bold_italic_w 的拟合性能。车辆 mmitalic_m 的局部损失函数定义为分布 𝒟m\mathcal{D}_{m}caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT 上的平均损失,即 fm(𝒘)𝔼𝒙𝒟m[f(𝒘;𝒙)].f_{m}(\boldsymbol{w})\triangleq\underset{\boldsymbol{x}\sim\mathcal{D}_{m}}{% \operatorname*{\mathbb{E}}}[f(\boldsymbol{w};\boldsymbol{x})].italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) ≜ start_UNDERACCENT bold_italic_x ∼ caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_UNDERACCENT start_ARG blackboard_E end_ARG [ italic_f ( bold_italic_w ; bold_italic_x ) ] .

Different from traditional FL, where the set of clients participating in model training is fixed, the set of vehicles participating in VFL training varies in each round due to mobility. We assume that the vehicles are drawn from a given distribution 𝒫𝒫\mathcal{P}caligraphic_P, and the global loss function is defined as the average local loss function over the distribution 𝒫𝒫\mathcal{P}caligraphic_P, i.e.,
与传统的联邦学习 (FL) 不同,传统的 FL 中参与模型训练的客户端集合是固定的,而 VFL 训练中参与的车辆集合由于移动性而在每一轮中都会发生变化。我们假设车辆是从给定分布 𝒫\mathcal{P}caligraphic_P 中抽取的,全局损失函数定义为分布 𝒫\mathcal{P}caligraphic_P 上的平均局部损失函数,即:

F(𝒘)𝔼m𝒫[fm(𝒘)].𝐹𝒘similar-to𝑚𝒫𝔼delimited-[]subscript𝑓𝑚𝒘F(\boldsymbol{w})\triangleq\underset{m\sim\mathcal{P}}{\operatorname*{\mathbb{% E}}}[f_{m}(\boldsymbol{w})].italic_F ( bold_italic_w ) ≜ start_UNDERACCENT italic_m ∼ caligraphic_P end_UNDERACCENT start_ARG blackboard_E end_ARG [ italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) ] . (1)

The goal is to minimize the global loss function by optimizing the global parameter 𝒘𝒘\boldsymbol{w}bold_italic_w through K𝐾Kitalic_K rounds of training. 𝒦={1,2,,K}𝒦12𝐾\mathcal{K}=\{1,2,...,K\}caligraphic_K = { 1 , 2 , … , italic_K } denotes the index of training rounds.
目标是通过 KKitalic_K 轮训练,优化全局参数 𝒘\boldsymbol{w}bold_italic_w 来最小化全局损失函数。 𝒦={1,2,,K}\mathcal{K}=\{1,2,...,K\}caligraphic_K = { 1 , 2 , … , italic_K } 表示训练轮次的索引。

The VFL training process in each round includes three stages: local updates, model uploading and model aggregation.
每轮 VFL 训练过程包括三个阶段:本地更新、模型上传和模型聚合。

III-A1 Local Updates
III-A1 本地更新

At the start of kthsuperscript𝑘thk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT round, the RSU r𝑟ritalic_r broadcasts its model parameters 𝒘k1subscript𝒘𝑘1\boldsymbol{w}_{k-1}bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT to the SOVs. After receiving the global model 𝒘k1subscript𝒘𝑘1\boldsymbol{w}_{k-1}bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT, every SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT uses stochastic gradient descent (SGD) algorithm to update the local model:
kthk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT 回合开始时,RSU rritalic_r 将它的模型参数 𝒘k1\boldsymbol{w}_{k-1}bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT 广播给 SOVs。在收到全局模型 𝒘k1\boldsymbol{w}_{k-1}bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT 后,每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 使用随机梯度下降 (SGD) 算法来更新本地模型:

𝒘m,k=𝒘k1ηkBk𝒙m,kf(𝒘k1;𝒙),subscript𝒘𝑚𝑘subscript𝒘𝑘1subscript𝜂𝑘subscript𝐵𝑘subscript𝒙subscript𝑚𝑘𝑓subscript𝒘𝑘1𝒙\boldsymbol{w}_{m,k}=\boldsymbol{w}_{k-1}-\frac{\eta_{k}}{B_{k}}\sum_{% \boldsymbol{x}\in\mathcal{B}_{m,k}}\nabla f\left(\boldsymbol{w}_{k-1};% \boldsymbol{x}\right),bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT = bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT - divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT bold_italic_x ∈ caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∇ italic_f ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ; bold_italic_x ) , (2)

where ηksubscript𝜂𝑘\eta_{k}italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the learning rate, m,ksubscript𝑚𝑘\mathcal{B}_{m,k}caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT is a subset randomly sampled from the sample space 𝒳msubscript𝒳𝑚\mathcal{X}_{m}caligraphic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. We assume that the batch size of all SOVs is the same, and denote it by Bk=|m,k|subscript𝐵𝑘subscript𝑚𝑘B_{k}=|\mathcal{B}_{m,k}|italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = | caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT |.
其中 ηk\eta_{k}italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 是学习率, m,k\mathcal{B}_{m,k}caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT 是从样本空间 𝒳m\mathcal{X}_{m}caligraphic_X start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT 中随机抽取的子集。我们假设所有 SOV 的批次大小相同,并用 Bk=|m,k|B_{k}=|\mathcal{B}_{m,k}|italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = | caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | 表示。

III-A2 Model Uploading
III-A2 模型上传

After an SOV completes the local updates, it uploads its model parameters to the RSU for model aggregation. SOVs can upload their model either via a direct V2I link or with the help of the OPVs via a V2V sidelink. The set of SOVs that successfully upload their model to the RSU is denoted by 𝒮^k𝒮ksubscript^𝒮𝑘subscript𝒮𝑘\hat{\mathcal{S}}_{k}\in\mathcal{S}_{k}over^ start_ARG caligraphic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. The detailed communication model for model uploading is described in Section III-C.
在 SOV 完成本地更新后,它会将其模型参数上传到 RSU 进行模型聚合。SOV 可以通过直接的 V2I 链接或借助 OPV 通过 V2V 侧链上传其模型。成功将模型上传到 RSU 的 SOV 集合用 𝒮^k𝒮k\hat{\mathcal{S}}_{k}\in\mathcal{S}_{k}over^ start_ARG caligraphic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 表示。模型上传的详细通信模型在第 III-C 节中描述。

III-A3 Model Aggregation
III-A3 模型聚合

At the end of the kthsuperscript𝑘thk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT round, the RSU aggregates the received model parameters:
kthk^{\text{th}}italic_k start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT 轮结束时,RSU 会聚合接收到的模型参数:

𝒘k=m𝒮^k|𝒟m|𝒘m,km𝒮^k|𝒟m|,subscript𝒘𝑘subscript𝑚subscript^𝒮𝑘subscript𝒟𝑚subscript𝒘𝑚𝑘subscript𝑚subscript^𝒮𝑘subscript𝒟𝑚\boldsymbol{w}_{k}=\frac{\sum_{m\in\hat{\mathcal{S}}_{k}}|\mathcal{D}_{m}|% \boldsymbol{w}_{m,k}}{\sum_{m\in\hat{\mathcal{S}}_{k}}|\mathcal{D}_{m}|},bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ over^ start_ARG caligraphic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ over^ start_ARG caligraphic_S end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG , (3)

and then starts a new round.
然后开始新一轮。

III-B Computation Model
III-B 计算模型

We adopt a standard computation model [38] [39] for local updates. The total workload for computing local updates for each vehicle is NflopBksubscript𝑁flopsubscript𝐵𝑘N_{\text{flop}}B_{k}italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, where Nflopsubscript𝑁flopN_{\text{flop}}italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT is the number of floating point operations (FLOPs) needed for processing each sample. Further, we define lm,ksubscript𝑙𝑚𝑘l_{m,k}italic_l start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT (in cycle/s) as the clock frequency of the vehicular processor in round k𝑘kitalic_k. Hence, the computation latency for updating the local model is determined as follows:
我们采用标准计算模型[38][39]进行本地更新。每辆车进行本地更新的总工作量为 NflopBkN_{\text{flop}}B_{k}italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,其中 NflopN_{\text{flop}}italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT 表示处理每个样本所需的浮点运算次数(FLOPs)。此外,我们将 lm,kl_{m,k}italic_l start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT (以周期/秒为单位)定义为第 kkitalic_k 轮中车辆处理器的时钟频率。因此,更新本地模型的计算延迟确定如下:

tm,kcp=NflopBklm,k,subscriptsuperscript𝑡cp𝑚𝑘subscript𝑁flopsubscript𝐵𝑘subscript𝑙𝑚𝑘t^{\text{cp}}_{m,k}=\frac{N_{\text{flop}}B_{k}}{l_{m,k}},italic_t start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT = divide start_ARG italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_l start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG ,

and the computation energy usage is
以及计算能耗是

em,kcp=ρlm,k2NflopBk,subscriptsuperscript𝑒cp𝑚𝑘𝜌superscriptsubscript𝑙𝑚𝑘2subscript𝑁flopsubscript𝐵𝑘e^{\text{cp}}_{m,k}=\rho l_{m,k}^{2}N_{\text{flop}}B_{k},italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT = italic_ρ italic_l start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT flop end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,

where ρ𝜌\rhoitalic_ρ is the energy consumption coefficient that depends on the chip architecture of the processor.
其中 ρ\rhoitalic_ρ 是能耗系数,它取决于处理器的芯片架构。

III-C Communication Model
III-C 沟通模型

We assume that the vehicular network operates in a discrete time-slotted manner. The slots in round k𝑘kitalic_k are denoted by 𝒯k={1,2,3,,Tk}subscript𝒯𝑘123subscript𝑇𝑘\mathcal{T}_{k}=\{1,2,3,...,T_{k}\}caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = { 1 , 2 , 3 , … , italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }, where Tksubscript𝑇𝑘T_{k}italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is the number of slots in round k𝑘kitalic_k and the slot length is denoted by κ𝜅\kappaitalic_κ. The round duration κTk𝜅subscript𝑇𝑘\kappa T_{k}italic_κ italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is set to be the average sojourn time of vehicles in the RSU coverage. We assume that, based on historical information, the average sojourn time of vehicles within the RSU coverage area can be estimated, but the specific sojourn time of each vehicle cannot be known in advance. The timeline of the proposed system is shown in Fig. 2.
我们假设车辆网络以离散时隙的方式运行。回合 kkitalic_k 中的时隙用 𝒯k={1,2,3,,Tk}\mathcal{T}_{k}=\{1,2,3,...,T_{k}\}caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = { 1 , 2 , 3 , … , italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } 表示,其中 TkT_{k}italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 是回合 kkitalic_k 中的时隙数,时隙长度用 κ\kappaitalic_κ 表示。回合持续时间 κTk\kappa T_{k}italic_κ italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 设置为车辆在 RSU 覆盖范围内的平均停留时间。我们假设,根据历史信息,可以估计车辆在 RSU 覆盖区域内的平均停留时间,但无法提前知道每辆车的具体停留时间。所提议系统的时序图如图 2 所示。

In every slot, one SOV is scheduled to upload its model parameters to the RSU either via a direct V2I link, called direct transmission (DT), or with the help of the OPVs, called cooperative transmission (COT). We use 𝒔(t)=[s1(t),,s|𝒮k|(t)]𝒔𝑡subscript𝑠1𝑡subscript𝑠subscript𝒮𝑘𝑡\boldsymbol{s}(t)=[s_{1}(t),...,s_{|\mathcal{S}_{k}|}(t)]bold_italic_s ( italic_t ) = [ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_s start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] to denote the SOV scheduling decision. sm(t)=1subscript𝑠𝑚𝑡1s_{m}(t)=1italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 1 if the SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is scheduled for model uploading in slot t𝑡titalic_t. Otherwise, sm(t)=0subscript𝑠𝑚𝑡0s_{m}(t)=0italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0. Note that since t𝒯k𝑡subscript𝒯𝑘t\in\mathcal{T}_{k}italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, the subscript k𝑘kitalic_k of sm(t)subscript𝑠𝑚𝑡s_{m}(t)italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) is omitted for simplicity, and the same applies in the following text. sm(t)subscript𝑠𝑚𝑡s_{m}(t)italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) has the following constraints:
在每个时隙中,一个 SOV 被安排将它的模型参数上传到 RSU,可以通过直接的 V2I 链接,称为直接传输 (DT),或者借助 OPV,称为协作传输 (COT)。我们使用 𝒔(t)=[s1(t),,s|𝒮k|(t)]\boldsymbol{s}(t)=[s_{1}(t),...,s_{|\mathcal{S}_{k}|}(t)]bold_italic_s ( italic_t ) = [ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_s start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] 来表示 SOV 调度决策。 sm(t)=1s_{m}(t)=1italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 1 如果 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 被安排在时隙 ttitalic_t 中上传模型。否则, sm(t)=0s_{m}(t)=0italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 。注意,由于 t𝒯kt\in\mathcal{T}_{k}italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPTsm(t)s_{m}(t)italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 的下标 kkitalic_k 为了简便省略了,以下文本中也是如此。 sm(t)s_{m}(t)italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 有以下约束:

sm(t){0,1},m𝒮k,t𝒯k,formulae-sequencesubscript𝑠𝑚𝑡01formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑡subscript𝒯𝑘s_{m}(t)\in\{0,1\},\quad\forall m\in\mathcal{S}_{k},\forall t\in\mathcal{T}_{k% },\\ italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ∈ { 0 , 1 } , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (4)
m𝒮ksm(t)1,t𝒯k.formulae-sequencesubscript𝑚subscript𝒮𝑘subscript𝑠𝑚𝑡1for-all𝑡subscript𝒯𝑘\sum_{m\in\mathcal{S}_{k}}s_{m}(t)\leq 1,\quad\forall t\in\mathcal{T}_{k}.∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≤ 1 , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (5)

We use a binary variable c(t)𝑐𝑡c(t)italic_c ( italic_t ) to denote the transmission mode. c(t)=0𝑐𝑡0c(t)=0italic_c ( italic_t ) = 0 if the SOV transmits its model to the RSU via DT. c(t)=1𝑐𝑡1c(t)=1italic_c ( italic_t ) = 1 if the SOV transmits its model to the RSU via COT. c(t)𝑐𝑡c(t)italic_c ( italic_t ) has the binary constraint:
我们使用一个二元变量 c(t)c(t)italic_c ( italic_t ) 来表示传输模式。 c(t)=0c(t)=0italic_c ( italic_t ) = 0 如果 SOV 通过 DT 将其模型传输到 RSU。 c(t)=1c(t)=1italic_c ( italic_t ) = 1 如果 SOV 通过 COT 将其模型传输到 RSU。 c(t)c(t)italic_c ( italic_t ) 具有二元约束:

c(t){0,1},t𝒯k.formulae-sequence𝑐𝑡01for-all𝑡subscript𝒯𝑘c(t)\in\{0,1\},\quad\forall t\in\mathcal{T}_{k}.\\ italic_c ( italic_t ) ∈ { 0 , 1 } , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (6)
Refer to caption
Figure 2: The timeline of the proposed VFL system.
图 2: 拟议的 VFL 系统时间线。

For DT, the scheduled SOV uploads its model parameters to the RSU directly using the whole bandwidth β𝛽\betaitalic_β. The transmission rate (bit/s) for the SOV m𝑚mitalic_m is
对于 DT,计划的 SOV 使用整个带宽 β\betaitalic_β 将其模型参数直接上传到 RSU。SOV mmitalic_m 的传输速率(bit/s)为

RmDT(t)=βlog2(1+pm(t)|hm,r(t)|2βN0),superscriptsubscript𝑅𝑚DT𝑡𝛽subscript21subscript𝑝𝑚𝑡superscriptsubscript𝑚𝑟𝑡2𝛽subscript𝑁0R_{m}^{\text{DT}}(t)=\beta\log_{2}\left(1+\frac{p_{m}(t)|h_{m,r}(t)|^{2}}{% \beta N_{0}}\right),italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT DT end_POSTSUPERSCRIPT ( italic_t ) = italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) ,

where hm,r(t)subscript𝑚𝑟𝑡h_{m,r}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) is the channel coefficient between vehicle m𝑚mitalic_m and the RSU. Due to the high mobility of vehicles, the channel coefficient varies in different slots. If vehicle m𝑚mitalic_m leaves the RSU coverage, hm,r(t)=0subscript𝑚𝑟𝑡0h_{m,r}(t)=0italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) = 0. pm(t)subscript𝑝𝑚𝑡p_{m}(t)italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) is the transmission power of vehicle m𝑚mitalic_m, and N0subscript𝑁0N_{0}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the noise power spectrum density.
其中 hm,r(t)h_{m,r}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) 表示车辆 mmitalic_m 与 RSU 之间的信道系数。由于车辆的高机动性,信道系数在不同的时隙中会发生变化。如果车辆 mmitalic_m 离开 RSU 覆盖范围, hm,r(t)=0h_{m,r}(t)=0italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) = 0pm(t)p_{m}(t)italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 是车辆 mmitalic_m 的传输功率, N0N_{0}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT 是噪声功率谱密度。

For COT, the scheduled SOV uses the first half of the slot to transmit its model parameters to the OPVs, and the OPVs use distributed space-time code (DSTC) [40] to relay the model parameters to the RSU in the second half of the slot, as shown in Fig. 2. We use 𝒖(t)=[u1(t),,u|𝒰k|(t)]𝒖𝑡subscript𝑢1𝑡subscript𝑢subscript𝒰𝑘𝑡\boldsymbol{u}(t)=[u_{1}(t),...,u_{|\mathcal{U}_{k}|}(t)]bold_italic_u ( italic_t ) = [ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_u start_POSTSUBSCRIPT | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] to denote the OPV scheduling decision in slot t𝑡titalic_t, where un(t)=1subscript𝑢𝑛𝑡1u_{n}(t)=1italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 if the OPV n𝒰k𝑛subscript𝒰𝑘n\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is scheduled for COT, and un(t)=0subscript𝑢𝑛𝑡0u_{n}(t)=0italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 0 otherwise. un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) has the binary constraint:
对于 COT,计划的 SOV 使用时隙的前半部分将模型参数传输到 OPV,而 OPV 使用分布式时空码 (DSTC) [40] 在时隙的后半部分将模型参数中继到 RSU,如图 2 所示。我们使用 𝒖(t)=[u1(t),,u|𝒰k|(t)]\boldsymbol{u}(t)=[u_{1}(t),...,u_{|\mathcal{U}_{k}|}(t)]bold_italic_u ( italic_t ) = [ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_u start_POSTSUBSCRIPT | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] 来表示时隙 ttitalic_t 中的 OPV 调度决策,其中 un(t)=1u_{n}(t)=1italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 如果 OPV n𝒰kn\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 被调度用于 COT,否则为 un(t)=0u_{n}(t)=0italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 0un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) 具有二进制约束:

un(t){0,1},n𝒰k,t𝒯k.formulae-sequencesubscript𝑢𝑛𝑡01formulae-sequencefor-all𝑛subscript𝒰𝑘for-all𝑡subscript𝒯𝑘u_{n}(t)\in\{0,1\},\quad\forall n\in\mathcal{U}_{k},\forall t\in\mathcal{T}_{k% }.\\ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ∈ { 0 , 1 } , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (7)

The transmission rate of SOV m𝑚mitalic_m using COT is [40, 41, 42]
使用 COT 的 SOV mmitalic_m 传输速率为 [40, 41, 42]

RmCOT(t)=βlog2(1+pm(t)|hm,r(t)|2βN0\displaystyle R_{m}^{\text{COT}}(t)=\beta\log_{2}\bigg{(}1+\frac{p_{m}(t)|h_{m% ,r}(t)|^{2}}{\beta N_{0}}italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) = italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG
+n𝒰kun(t)pn(t)|hn,r(t)|2βN0).\displaystyle+\sum_{n\in\mathcal{U}_{k}}\frac{u_{n}(t)p_{n}(t)|h_{n,r}(t)|^{2}% }{\beta N_{0}}\bigg{)}.+ ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_n , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) .

The V2V transmission rate between SOV m𝑚mitalic_m and OPV n𝑛nitalic_n is
SOV mmitalic_m 和 OPV nnitalic_n 之间的 V2V 传输速率为

Rm,nCOT-V(t)=βlog2(1+pm(t)|hm,n(t)|2βN0),superscriptsubscript𝑅𝑚𝑛COT-V𝑡𝛽subscript21subscript𝑝𝑚𝑡superscriptsubscript𝑚𝑛𝑡2𝛽subscript𝑁0\displaystyle R_{m,n}^{\text{COT-V}}(t)=\beta\log_{2}\bigg{(}1+\frac{p_{m}(t)|% h_{m,n}(t)|^{2}}{\beta N_{0}}\bigg{)},italic_R start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT-V end_POSTSUPERSCRIPT ( italic_t ) = italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) ,

where hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) is the channel coefficient between SOV m𝑚mitalic_m and OPV n𝑛nitalic_n. To ensure that the scheduled OPVs can reliably decode the signal before it begins to transmit, we have the following constraint:
其中 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 表示 SOV mmitalic_m 和 OPV nnitalic_n 之间的信道系数。为了确保计划的 OPV 能够在开始传输之前可靠地解码信号,我们有以下约束:

sm(t)c(t)un(t)RmCOT(t)un(t)Rm,nCOT-V(t),subscript𝑠𝑚𝑡𝑐𝑡subscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚COT𝑡subscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚𝑛COT-V𝑡\displaystyle s_{m}(t)c(t)u_{n}(t)R_{m}^{\text{COT}}(t)\leq u_{n}(t)R_{m,n}^{% \text{COT-V}}(t),italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) italic_c ( italic_t ) italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) ≤ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT-V end_POSTSUPERSCRIPT ( italic_t ) , (8)
m𝒮k,n𝒰k,t𝒯k.formulae-sequencefor-all𝑚subscript𝒮𝑘formulae-sequencefor-all𝑛subscript𝒰𝑘for-all𝑡subscript𝒯𝑘\displaystyle\forall m\in\mathcal{S}_{k},\forall n\in\mathcal{U}_{k},\forall t% \in\mathcal{T}_{k}.∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .

We use 𝒑(t)=[p1(t),,p(|𝒮k|+|𝒰k|)(t)]𝒑𝑡subscript𝑝1𝑡subscript𝑝subscript𝒮𝑘subscript𝒰𝑘𝑡\boldsymbol{p}(t)=[p_{1}(t),...,p_{\left(|\mathcal{S}_{k}|+|\mathcal{U}_{k}|% \right)}(t)]bold_italic_p ( italic_t ) = [ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_p start_POSTSUBSCRIPT ( | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | + | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ) end_POSTSUBSCRIPT ( italic_t ) ] to denote the transmission power allocation in slot t𝑡titalic_t. There is a power constraint for SOVs:
我们使用 𝒑(t)=[p1(t),,p(|𝒮k|+|𝒰k|)(t)]\boldsymbol{p}(t)=[p_{1}(t),...,p_{\left(|\mathcal{S}_{k}|+|\mathcal{U}_{k}|% \right)}(t)]bold_italic_p ( italic_t ) = [ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_p start_POSTSUBSCRIPT ( | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | + | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ) end_POSTSUBSCRIPT ( italic_t ) ] 来表示时隙 ttitalic_t 中的传输功率分配。SOV 存在功率约束:

0pm(t)pmmax,m𝒮k,t𝒯k,formulae-sequence0subscript𝑝𝑚𝑡subscriptsuperscript𝑝max𝑚formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑡subscript𝒯𝑘0\leq p_{m}(t)\leq p^{\text{max}}_{m},\quad\forall m\in\mathcal{S}_{k},\forall t% \in\mathcal{T}_{k},0 ≤ italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≤ italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (9)

and for OPVs: 以及用于 OPV:

0pn(t)pnmax,n𝒰k,t𝒯k.formulae-sequence0subscript𝑝𝑛𝑡subscriptsuperscript𝑝max𝑛formulae-sequencefor-all𝑛subscript𝒰𝑘for-all𝑡subscript𝒯𝑘0\leq p_{n}(t)\leq p^{\text{max}}_{n},\quad\forall n\in\mathcal{U}_{k},\forall t% \in\mathcal{T}_{k}.0 ≤ italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ≤ italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (10)

In every slot, the communication energy consumption for each SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is
在每个时隙中,每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 的通信能耗为

emcm(t)=κpm(t)[sm(t)(1c(t))+12sm(t)c(t)],superscriptsubscript𝑒𝑚cm𝑡𝜅subscript𝑝𝑚𝑡delimited-[]subscript𝑠𝑚𝑡1𝑐𝑡12subscript𝑠𝑚𝑡𝑐𝑡e_{m}^{\text{cm}}(t)=\kappa p_{m}(t)\left[s_{m}(t)(1-c(t))+\frac{1}{2}s_{m}(t)% c(t)\right],italic_e start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT ( italic_t ) = italic_κ italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) [ italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ( 1 - italic_c ( italic_t ) ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) italic_c ( italic_t ) ] ,

and for each OPV n𝒰k𝑛subscript𝒰𝑘n\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, it is
并且对于每个 OPV n𝒰kn\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,它是

encm(t)=12κpn(t)un(t)c(t).superscriptsubscript𝑒𝑛cm𝑡12𝜅subscript𝑝𝑛𝑡subscript𝑢𝑛𝑡𝑐𝑡e_{n}^{\text{cm}}(t)=\frac{1}{2}\kappa p_{n}(t)u_{n}(t)c(t).italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT ( italic_t ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_c ( italic_t ) .

The data transmitted for each SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is
每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 传输的数据是

zm(t)=κ[sm(t)(1c(t))RmDT(t)+12sm(t)c(t)RmCOT(t)].subscript𝑧𝑚𝑡𝜅delimited-[]subscript𝑠𝑚𝑡1𝑐𝑡superscriptsubscript𝑅𝑚DT𝑡12subscript𝑠𝑚𝑡𝑐𝑡superscriptsubscript𝑅𝑚COT𝑡z_{m}(t)=\kappa\left[s_{m}(t)(1-c(t))R_{m}^{\text{DT}}(t)+\frac{1}{2}s_{m}(t)c% (t)R_{m}^{\text{COT}}(t)\right].italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_κ [ italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ( 1 - italic_c ( italic_t ) ) italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT DT end_POSTSUPERSCRIPT ( italic_t ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) italic_c ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) ] .

The SOV m𝑚mitalic_m has successfully transmitted its model to the RSU if the amount of transmitted model parameters in all slots is greater than or equal to the model size, i.e., t𝒯kzm(t)Qsubscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q, where Q𝑄Qitalic_Q denotes the model size. We use an indicator function 𝕀(t𝒯kzm(t)Q)𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) to denote whether the vehicle m𝑚mitalic_m has successfully transmitted its model, where 𝕀(a)=1𝕀𝑎1\mathbb{I}(a)=1blackboard_I ( italic_a ) = 1 if condition a𝑎aitalic_a is true, and 𝕀(a)=0𝕀𝑎0\mathbb{I}(a)=0blackboard_I ( italic_a ) = 0 otherwise. Using this notation, the aggregation rule (3) can be rewritten as
如果所有时隙中传输的模型参数量大于或等于模型大小,即 t𝒯kzm(t)Q\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ,其中 QQitalic_Q 表示模型大小,则 SOV mmitalic_m 已成功将模型传输到 RSU。我们使用指示函数 𝕀(t𝒯kzm(t)Q)\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) 来表示车辆 mmitalic_m 是否已成功传输其模型,其中 𝕀(a)=1\mathbb{I}(a)=1blackboard_I ( italic_a ) = 1 如果条件 aaitalic_a 为真,否则为 𝕀(a)=0\mathbb{I}(a)=0blackboard_I ( italic_a ) = 0 。使用此符号,聚合规则 (3) 可以改写为

𝒘k=m𝒮k𝕀(t𝒯kzm(t)Q)|𝒟m|𝒘m,km𝒮k𝕀(t𝒯kzm(t)Q)|𝒟m|.subscript𝒘𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄subscript𝒟𝑚subscript𝒘𝑚𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄subscript𝒟𝑚\boldsymbol{w}_{k}=\frac{\sum_{m\in{\mathcal{S}}_{k}}\mathbb{I}\left(\sum_{t% \in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)|\mathcal{D}_{m}|\boldsymbol{w}_{m,k}}% {\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)% \geq Q\right)|\mathcal{D}_{m}|}.bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG . (11)

IV Problem Formulation
IV 问题陈述

IV-A Convergence Analysis
IV-A 收敛分析

The goal of the VFL is to minimize the global loss function (1). However, this objective function is implicit due to the deep and diverse neural network architectures of ML. Therefore, convergence analysis is performed for an explicit objective function. Following the state-of-the-art literature [23, 24, 27, 28, 29], we make the following assumptions:
VFL 的目标是最小化全局损失函数 (1)。然而,由于 ML 的深度和多样化的神经网络架构,该目标函数是隐式的。因此,对显式目标函数进行收敛分析。遵循最先进的文献 [23, 24, 27, 28, 29],我们做出以下假设:

Assumption 1: The local loss function fm(𝒘)subscript𝑓𝑚𝒘f_{m}(\boldsymbol{w})italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) is L𝐿Litalic_L-smooth for each SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in each round k𝒦𝑘𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K, i.e.,
假设 1: 局部损失函数 fm(𝒘)f_{m}(\boldsymbol{w})italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) 对于每个回合 k𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K 中的每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 都是 LLitalic_L -光滑的,即,

fm(𝒘k\displaystyle f_{m}(\boldsymbol{w}_{k}italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )fm(𝒘k1)\displaystyle)-f_{m}(\boldsymbol{w}_{k-1})) - italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT )
fm(𝒘k1),𝒘k𝒘k1+L2𝒘k𝒘k12.absentsubscript𝑓𝑚subscript𝒘𝑘1subscript𝒘𝑘subscript𝒘𝑘1𝐿2superscriptnormsubscript𝒘𝑘subscript𝒘𝑘12\displaystyle\leq\left<{\nabla f_{m}(\boldsymbol{w}_{k-1}),\boldsymbol{w}_{k}-% \boldsymbol{w}_{k-1}}\right>+\frac{L}{2}\left\|{\boldsymbol{w}_{k}-\boldsymbol% {w}_{k-1}}\right\|^{2}.≤ ⟨ ∇ italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ⟩ + divide start_ARG italic_L end_ARG start_ARG 2 end_ARG ∥ bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Assumption 2: The local loss function fm(𝒘)subscript𝑓𝑚𝒘f_{m}(\boldsymbol{w})italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) is μ𝜇\muitalic_μ-strongly convex for each SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in each round k𝒦𝑘𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K, i.e.,
假设 2: 局部损失函数 fm(𝒘)f_{m}(\boldsymbol{w})italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) 对于每一轮 k𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K 中的每个 SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 都是 μ\muitalic_μ -强凸的,即,

fm(𝒘k\displaystyle f_{m}(\boldsymbol{w}_{k}italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )fm(𝒘k1)\displaystyle)-f_{m}(\boldsymbol{w}_{k-1})) - italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT )
fm(𝒘k1),𝒘k𝒘k1+μ2𝒘k𝒘k12.absentsubscript𝑓𝑚subscript𝒘𝑘1subscript𝒘𝑘subscript𝒘𝑘1𝜇2superscriptnormsubscript𝒘𝑘subscript𝒘𝑘12\displaystyle\geq\left<{\nabla f_{m}(\boldsymbol{w}_{k-1}),\boldsymbol{w}_{k}-% \boldsymbol{w}_{k-1}}\right>+\frac{\mu}{2}\left\|{\boldsymbol{w}_{k}-% \boldsymbol{w}_{k-1}}\right\|^{2}.≥ ⟨ ∇ italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ⟩ + divide start_ARG italic_μ end_ARG start_ARG 2 end_ARG ∥ bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Assumption 3: The stochastic gradient is unbiased and variance-bounded, i.e.,
假设 3: 随机梯度是无偏的且方差有界的,即,

𝔼𝒙𝒟m[f(𝒘;𝒙)]=𝔼m𝒫m[fm(𝒘)]=F(𝒘),similar-to𝒙subscript𝒟𝑚𝔼delimited-[]𝑓𝒘𝒙similar-to𝑚subscript𝒫𝑚𝔼delimited-[]subscript𝑓𝑚𝒘𝐹𝒘\underset{\boldsymbol{x}\sim\mathcal{D}_{m}}{\operatorname*{\mathbb{E}}}\left[% \nabla f(\boldsymbol{w};\boldsymbol{x})\right]=\underset{m\sim\mathcal{P}_{m}}% {\operatorname*{\mathbb{E}}}\left[\nabla f_{m}(\boldsymbol{w})\right]=\nabla F% (\boldsymbol{w}),start_UNDERACCENT bold_italic_x ∼ caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_UNDERACCENT start_ARG blackboard_E end_ARG [ ∇ italic_f ( bold_italic_w ; bold_italic_x ) ] = start_UNDERACCENT italic_m ∼ caligraphic_P start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_UNDERACCENT start_ARG blackboard_E end_ARG [ ∇ italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( bold_italic_w ) ] = ∇ italic_F ( bold_italic_w ) ,
𝔼𝒙𝒟m[f(𝒘;𝒙)F(𝒘)2]G2.similar-to𝒙subscript𝒟𝑚𝔼delimited-[]superscriptnorm𝑓𝒘𝒙𝐹𝒘2superscript𝐺2\underset{\boldsymbol{x}\sim\mathcal{D}_{m}}{\operatorname*{\mathbb{E}}}\left[% \left\|\nabla f(\boldsymbol{w};\boldsymbol{x})-\nabla F(\boldsymbol{w})\right% \|^{2}\right]\leq G^{2}.start_UNDERACCENT bold_italic_x ∼ caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_UNDERACCENT start_ARG blackboard_E end_ARG [ ∥ ∇ italic_f ( bold_italic_w ; bold_italic_x ) - ∇ italic_F ( bold_italic_w ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ≤ italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Then, the following Lemma is derived:
然后,推导出以下引理:

Lemma 1. Based on the given assumptions and the aggregation rule (11), the expected loss decreases after one round is upper bounded by
引理 1.基于给定的假设和聚合规则 (11),一轮后预期损失的下降上限为

𝔼[F(𝒘k)]𝔼[F(𝒘k1)]ηk(Lηk21)F(𝒘k1)2𝔼delimited-[]𝐹subscript𝒘𝑘𝔼delimited-[]𝐹subscript𝒘𝑘1subscript𝜂𝑘𝐿subscript𝜂𝑘21superscriptnorm𝐹subscript𝒘𝑘12\displaystyle\mathbb{E}[F(\boldsymbol{w}_{k})]-\mathbb{E}[F(\boldsymbol{w}_{k-% 1})]\leq\eta_{k}\left(\frac{L\eta_{k}}{2}-1\right)\left\|{\nabla F(\boldsymbol% {w}_{k-1})}\right\|^{2}blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] - blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ] ≤ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG - 1 ) ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
+Lηk22G2Bkm𝒮k𝕀(t𝒯kzm(t)Q).𝐿superscriptsubscript𝜂𝑘22superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\displaystyle+\frac{L\eta_{k}^{2}}{2}\frac{G^{2}}{B_{k}\sum_{m\in\mathcal{S}_{% k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)}.+ divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG . (12)

where the expectation is taken over the randomness of SGD.
其中期望值是在 SGD 的随机性上取的。

Proof: See Appendix A. \square
证明:参见附录 A\square□

Based on Lemma 1, the convergence performance of the proposed VFL after K𝐾Kitalic_K rounds of training is given by:
基于引理 1,所提出的 VFL 在 KKitalic_K 轮训练后的收敛性能由下式给出:

Theorem 1. After K𝐾Kitalic_K round of training, the difference between F(𝐰K)𝐹subscript𝐰𝐾F({\boldsymbol{w}}_{K})italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) and the optimal global loss function F(𝐰)𝐹superscript𝐰F(\boldsymbol{w}^{*})italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) is upper bounded by
定理 1.经过 KKitalic_K 轮训练后,F(𝐰K)F({\boldsymbol{w}}_{K})italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) 与最优全局损失函数 F(𝐰)F(\boldsymbol{w}^{*})italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) 之间的差值的上界为

𝔼[F(𝒘K)]F(𝒘)𝔼delimited-[]𝐹subscript𝒘𝐾𝐹superscript𝒘\displaystyle\mathbb{E}[F({\boldsymbol{w}}_{K})]-F(\boldsymbol{w}^{*})blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )
(𝔼[F(𝒘0)]F(𝒘))k=1K(1μηk)absent𝔼delimited-[]𝐹subscript𝒘0𝐹superscript𝒘superscriptsubscriptproduct𝑘1𝐾1𝜇subscript𝜂𝑘\displaystyle\leq(\mathbb{E}[F({\boldsymbol{w}}_{0})]-F(\boldsymbol{w}^{*}))% \prod_{k=1}^{K}(1-\mu\eta_{k})≤ ( blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) ∏ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ( 1 - italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )
+k=1K1ηk2G2Bkm𝒮k𝕀(t𝒯kzm(t)Q)j=k+1K(1μηk)superscriptsubscript𝑘1𝐾1subscript𝜂𝑘2superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄superscriptsubscriptproduct𝑗𝑘1𝐾1𝜇subscript𝜂𝑘\displaystyle+\sum_{k=1}^{K-1}\frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in% \mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right% )}\prod_{j=k+1}^{K}(1-\mu\eta_{k})+ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K - 1 end_POSTSUPERSCRIPT divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG ∏ start_POSTSUBSCRIPT italic_j = italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ( 1 - italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )
+ηK2G2BKm𝒮K𝕀(t𝒯Kzm(t)Q).subscript𝜂𝐾2superscript𝐺2subscript𝐵𝐾subscript𝑚subscript𝒮𝐾𝕀subscript𝑡subscript𝒯𝐾subscript𝑧𝑚𝑡𝑄\displaystyle+\frac{\eta_{K}}{2}\frac{G^{2}}{B_{K}\sum_{m\in\mathcal{S}_{K}}% \mathbb{I}\left(\sum_{t\in\mathcal{T}_{K}}z_{m}(t)\geq Q\right)}.+ divide start_ARG italic_η start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG . (13)

Proof: See Appendix B. \square
证明:参见附录 B\square□

IV-B Problem Formulation
IV-B 问题公式化

Based on Theorem 1, we alternatively minimize the upper bound of 𝔼[F(𝒘K)]F(𝒘)𝔼delimited-[]𝐹subscript𝒘𝐾𝐹superscript𝒘\mathbb{E}[F({\boldsymbol{w}}_{K})]-F(\boldsymbol{w}^{*})blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) in (13), which is equivalent to minimizing ηk2G2Bkm𝒮k𝕀(t𝒯kzm(t)Q)subscript𝜂𝑘2superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(% \sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)}divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG in each round k𝒦𝑘𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K. The optimization problem is formulated as
基于定理 1,我们交替最小化 𝔼[F(𝒘K)]F(𝒘)\mathbb{E}[F({\boldsymbol{w}}_{K})]-F(\boldsymbol{w}^{*})blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) 在 (13) 中的上界,这等价于在每一轮 k𝒦k\in\mathcal{K}italic_k ∈ caligraphic_K 中最小化 ηk2G2Bkm𝒮k𝕀(t𝒯kzm(t)Q)\frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(% \sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)}divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG 。优化问题被表述为

P0::𝑃0absent\displaystyle P0:italic_P 0 : min𝑺k,𝒄k,𝑼k,𝑷kηk2G2Bkm𝒮k𝕀(t𝒯kzm(t)Q)subscript𝑺𝑘subscript𝒄𝑘subscript𝑼𝑘subscript𝑷𝑘subscript𝜂𝑘2superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\displaystyle\underset{\boldsymbol{S}_{k},\boldsymbol{c}_{k},\boldsymbol{U}_{k% },\boldsymbol{P}_{k}}{\min}\ \frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in% \mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)}start_UNDERACCENT bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG roman_min end_ARG divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG (14a)
s.t. t𝒯kemcm(t)+em,kcpEmcons,m𝒮k,formulae-sequencesubscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝑒cp𝑚𝑘subscriptsuperscript𝐸cons𝑚for-all𝑚subscript𝒮𝑘\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{m}(t)+e^{\text{cp}}_{m,k% }\leq E^{\text{cons}}_{m},\quad\forall m\in\mathcal{S}_{k},∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) + italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (14b)
t𝒯kencm(t)Encons,n𝒰k,formulae-sequencesubscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑛𝑡subscriptsuperscript𝐸cons𝑛for-all𝑛subscript𝒰𝑘\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{n}(t)\leq E^{\text{cons}% }_{n},\quad\forall n\in\mathcal{U}_{k},∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (14c) (14 世纪)
𝕀(tm,kcp(t1)κ)sm(t)=0,m𝒮k,t𝒯k,formulae-sequence𝕀subscriptsuperscript𝑡cp𝑚𝑘𝑡1𝜅subscript𝑠𝑚𝑡0formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑡subscript𝒯𝑘\displaystyle\mathbb{I}\left(t^{\text{cp}}_{m,k}\geq(t-1)\kappa\right)s_{m}(t)% =0,\ \forall m\in\mathcal{S}_{k},\forall t\in\mathcal{T}_{k},blackboard_I ( italic_t start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≥ ( italic_t - 1 ) italic_κ ) italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (14d) (14 天)
constraints (4)--(10),
约束 (4) -- (10),

where 𝑺k=[𝒔(1),,𝒔(Tk)]subscript𝑺𝑘𝒔1𝒔subscript𝑇𝑘\boldsymbol{S}_{k}=[\boldsymbol{s}(1),...,\boldsymbol{s}(T_{k})]bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_s ( 1 ) , … , bold_italic_s ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] denotes the SOV scheduling, 𝒄k=[c(1),,c(Tk)]subscript𝒄𝑘𝑐1𝑐subscript𝑇𝑘\boldsymbol{c}_{k}=[c(1),...,c(T_{k})]bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ italic_c ( 1 ) , … , italic_c ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] denotes the transmission mode, 𝑼k=[𝒖(1),,𝒖(Tk)]subscript𝑼𝑘𝒖1𝒖subscript𝑇𝑘\boldsymbol{U}_{k}=[\boldsymbol{u}(1),...,\boldsymbol{u}(T_{k})]bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_u ( 1 ) , … , bold_italic_u ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] is the OPV scheduling, 𝑷k=[𝒑(1),,𝒑(Tk)]subscript𝑷𝑘𝒑1𝒑subscript𝑇𝑘\boldsymbol{P}_{k}=[\boldsymbol{p}(1),...,\boldsymbol{p}(T_{k})]bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_p ( 1 ) , … , bold_italic_p ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] is the power allocation throughout round k𝑘kitalic_k. The constraints (14b) and (14c) indicate that for each vehicle, the total energy consumption cannot exceed the given energy budget. The constraint (14d) ensures that the vehicles begin to transmit after they finish local updates. The constraints (4)--(10) limit the range of optimization variables.
其中 𝑺k=[𝒔(1),,𝒔(Tk)]\boldsymbol{S}_{k}=[\boldsymbol{s}(1),...,\boldsymbol{s}(T_{k})]bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_s ( 1 ) , … , bold_italic_s ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] 表示 SOV 调度, 𝒄k=[c(1),,c(Tk)]\boldsymbol{c}_{k}=[c(1),...,c(T_{k})]bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ italic_c ( 1 ) , … , italic_c ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] 表示传输模式, 𝑼k=[𝒖(1),,𝒖(Tk)]\boldsymbol{U}_{k}=[\boldsymbol{u}(1),...,\boldsymbol{u}(T_{k})]bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_u ( 1 ) , … , bold_italic_u ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] 表示 OPV 调度, 𝑷k=[𝒑(1),,𝒑(Tk)]\boldsymbol{P}_{k}=[\boldsymbol{p}(1),...,\boldsymbol{p}(T_{k})]bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = [ bold_italic_p ( 1 ) , … , bold_italic_p ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] 表示第 kkitalic_k 轮的功率分配。约束 (14b) 和 (14c) 表明,对于每辆车,总能耗不能超过给定的能量预算。约束 (14d) 确保车辆在完成本地更新后开始传输。约束 (4) -- (10) 限制了优化变量的范围。

Since ηksubscript𝜂𝑘\eta_{k}italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, G𝐺Gitalic_G and Bksubscript𝐵𝑘B_{k}italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT are all constants in round k𝑘kitalic_k, minimizing (14a) is equivalent to minimizing 1m𝒮k𝕀(t𝒯kzm(t)Q)1subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\frac{1}{\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z% _{m}(t)\geq Q\right)}divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG. Also, since 1m𝒮k𝕀(t𝒯kzm(t)Q)>01subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄0\frac{1}{\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z% _{m}(t)\geq Q\right)}>0divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG > 0, there is
由于 ηk\eta_{k}italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPTGGitalic_GBkB_{k}italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 在第 kkitalic_k 轮中都是常数,因此最小化 (14a) 等同于最小化 1m𝒮k𝕀(t𝒯kzm(t)Q)\frac{1}{\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z% _{m}(t)\geq Q\right)}divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG 。此外,由于 1m𝒮k𝕀(t𝒯kzm(t)Q)>0\frac{1}{\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z% _{m}(t)\geq Q\right)}>0divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG > 0 ,因此存在

argmin𝑺k,𝒄k,𝑼k,𝑷k1m𝒮k𝕀(t𝒯kzm(t)Q)subscriptsubscript𝑺𝑘subscript𝒄𝑘subscript𝑼𝑘subscript𝑷𝑘1subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\displaystyle\mathop{\arg\min}\limits_{\boldsymbol{S}_{k},\boldsymbol{c}_{k},% \boldsymbol{U}_{k},\boldsymbol{P}_{k}}\frac{1}{\sum_{m\in\mathcal{S}_{k}}% \mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)}start_BIGOP roman_arg roman_min end_BIGOP start_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) end_ARG
=argmax𝑺k,𝒄k,𝑼k,𝑷km𝒮k𝕀(t𝒯kzm(t)Q).absentsubscriptsubscript𝑺𝑘subscript𝒄𝑘subscript𝑼𝑘subscript𝑷𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\displaystyle\quad\quad=\mathop{\arg\max}\limits_{\boldsymbol{S}_{k},% \boldsymbol{c}_{k},\boldsymbol{U}_{k},\boldsymbol{P}_{k}}\sum_{m\in\mathcal{S}% _{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right).= start_BIGOP roman_arg roman_max end_BIGOP start_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) .

Therefore, we transform the objective of P0𝑃0P0italic_P 0 from (14a) to maxm𝒮k𝕀(t𝒯kzm(t)Q)subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\max\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(% t)\geq Q\right)roman_max ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ), and reformulate P0𝑃0P0italic_P 0 as
因此,我们将 P0P0italic_P 0 的目标从(14a)转换为 maxm𝒮k𝕀(t𝒯kzm(t)Q)\max\sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(% t)\geq Q\right)roman_max ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) ,并将 P0P0italic_P 0 重新表述为

P1::𝑃1absent\displaystyle P1:italic_P 1 : max𝑺k,𝒄k,𝑼k,𝑷km𝒮k𝕀(t𝒯kzm(t)Q)subscript𝑺𝑘subscript𝒄𝑘subscript𝑼𝑘subscript𝑷𝑘subscript𝑚subscript𝒮𝑘𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\displaystyle\underset{\boldsymbol{S}_{k},\boldsymbol{c}_{k},\boldsymbol{U}_{k% },\boldsymbol{P}_{k}}{\max}\ \sum_{m\in\mathcal{S}_{k}}\mathbb{I}\left(\sum_{t% \in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)start_UNDERACCENT bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG roman_max end_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) (15)
s.t. constraints (4)(10), (14b)(14d).constraints (4)(10), (14b)(14d)\displaystyle\text{constraints (\ref{trans1})$-$(\ref{power1}), (\ref{energy1}% )$-$(\ref{long1})}.constraints ( ) - ( ), ( ) - ( ) .

V V2V-Enhanced Dynamic Scheduling Algorithm
V V2V 增强动态调度算法

In this section, we propose the VEDS algorithm that solves P1𝑃1P1italic_P 1 in an online fashion. Firstly, we propose a derivative-based drift-plus-penalty method to convert the long-term stochastic optimization problem into an online MINLP problem. The converted MINLP problem is then decoupled into a DT problem and a COT problem. The DT problem is convex and is directly solved using the Karush-Kuhn-Tucker (KKT) conditions. Analysis of the OPV scheduling priority reduces the COT problem to a set of convex problems, which are solved using the interior-point method.
在本节中,我们提出了 VEDS 算法,该算法以在线方式解决 P1P1italic_P 1 。首先,我们提出了一种基于导数的漂移加惩罚方法,将长期随机优化问题转化为在线 MINLP 问题。然后将转换后的 MINLP 问题解耦为 DT 问题和 COT 问题。DT 问题是凸的,可以直接使用 Karush-Kuhn-Tucker(KKT)条件求解。对 OPV 调度优先级的分析将 COT 问题简化为一组凸问题,这些问题使用内点法求解。

V-A Transformation of the stochastic optimization problem
V-A 随机优化问题的转化

P1𝑃1P1italic_P 1 is a stochastic optimization problem. The greatest challenge to solving this problem lies in the uncertainty of channel state information. In vehicular networks, this results from the rapid changes in channels due to the high mobility of vehicles. In reality, future channel information is often difficult to predict, and even if we could acquire future channel information, addressing this problem remains highly complex due to the integer optimization variables and the non-convex objective function.
P1P1italic_P 1 是一个 随机优化问题。解决这个问题的最大挑战在于信道状态信息的不可预测性。在车联网中,这是由于车辆的高机动性导致信道快速变化造成的。实际上,未来的信道信息往往难以预测,即使我们能够获取未来的信道信息,由于整数优化变量和非凸目标函数,解决这个问题仍然非常复杂。

One effective way to tackle this kind of problem is the drift-plus-penalty method in Lyapunov optimization [43][44]. By constructing virtual queues, the long-term stochastic optimization problem is transformed into an online problem and online decision-making algorithms can be designed to solve it. However, the model aggregation requirements and the limited transmission time of VFL result in a stepwise objective function (15), which cannot be handled by the typical drift-plus-penalty method. Therefore, we propose a derivative-based drift-plus-penalty method to address this challenge. Firstly, we use the shifted sigmoid function to approximate it and transform P1𝑃1P1italic_P 1 into P2𝑃2P2italic_P 2.
解决这类问题的有效方法之一是 Lyapunov 优化中的漂移加惩罚方法 [43][44]。通过构建虚拟队列,将长期随机优化问题转化为在线问题,并可以设计在线决策算法来解决它。然而,VFL 的模型聚合需求和有限传输时间导致了阶梯式目标函数 (15),这无法通过典型的漂移加惩罚方法处理。因此,我们提出了一种基于导数的漂移加惩罚方法来应对这一挑战。首先,我们使用移位 sigmoid 函数来近似它,并将 P1P1italic_P 1 转换为 P2P2italic_P 2

P2::𝑃2absent\displaystyle P2:italic_P 2 : max𝑺k,𝒄k,𝑼k,𝑷km𝒮kσ(t𝒯kzm(t))subscript𝑺𝑘subscript𝒄𝑘subscript𝑼𝑘subscript𝑷𝑘subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡\displaystyle\underset{\boldsymbol{S}_{k},\boldsymbol{c}_{k},\boldsymbol{U}_{k% },\boldsymbol{P}_{k}}{\max}\ \sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t\in% \mathcal{T}_{k}}z_{m}(t)\right)start_UNDERACCENT bold_italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_italic_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG roman_max end_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) (16a)
s.t. 𝕀(ζm(t)=Q)sm(t)=0,m𝒮k,t𝒯k,formulae-sequence𝕀subscript𝜁𝑚𝑡𝑄subscript𝑠𝑚𝑡0formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑡subscript𝒯𝑘\displaystyle\mathbb{I}\left(\zeta_{m}(t)=Q\right)s_{m}(t)=0,\ \forall m\in% \mathcal{S}_{k},\forall t\in\mathcal{T}_{k},blackboard_I ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_Q ) italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (16b)
constraints (4)--(10), (14b)--(14d),
约束 (4) -- (10), (14b) -- (14d),

where σ(t𝒯kzm(t))𝜎subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\right)italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) is a shifted sigmoid function, defined as σ(t𝒯kzm(t))[1+exp(αt𝒯kzm(t)QQ)]1,𝜎subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡superscriptdelimited-[]1𝛼subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄𝑄1\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\right)\triangleq\left[1+\exp% \left(-\alpha\frac{\sum_{t\in\mathcal{T}_{k}}z_{m}(t)-Q}{Q}\right)\right]^{-1},italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) ≜ [ 1 + roman_exp ( - italic_α divide start_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - italic_Q end_ARG start_ARG italic_Q end_ARG ) ] start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , and α𝛼\alphaitalic_α is an approximation parameter. As α𝛼\alphaitalic_α increases, the function σ()𝜎\sigma(\cdot)italic_σ ( ⋅ ) converges towards the indicator function 𝕀()𝕀\mathbb{I}(\cdot)blackboard_I ( ⋅ ), becoming a more precise approximation. Constraint (16b) ensures that a vehicle will not be scheduled after it finishes transmitting its model.
其中 σ(t𝒯kzm(t))\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\right)italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) 是一个移位的 sigmoid 函数,定义为 σ(t𝒯kzm(t))[1+exp(αt𝒯kzm(t)QQ)]1,\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\right)\triangleq\left[1+\exp% \left(-\alpha\frac{\sum_{t\in\mathcal{T}_{k}}z_{m}(t)-Q}{Q}\right)\right]^{-1},italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) ≜ [ 1 + roman_exp ( - italic_α divide start_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - italic_Q end_ARG start_ARG italic_Q end_ARG ) ] start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , 并且 α\alphaitalic_α 是一个近似参数。随着 α\alphaitalic_α 的增加,函数 σ()\sigma(\cdot)italic_σ ( ⋅ ) 收敛到指标函数 𝕀()\mathbb{I}(\cdot)blackboard_I ( ⋅ ) ,成为一个更精确的近似。约束 (16b) 确保车辆在完成模型传输后不会被调度。

We define 𝜻(t)=[ζ1(t),,ζ|𝒮k|(t)]𝜻𝑡subscript𝜁1𝑡subscript𝜁subscript𝒮𝑘𝑡\boldsymbol{\zeta}(t)=[\zeta_{1}(t),...,\zeta_{|\mathcal{S}_{k}|}(t)]bold_italic_ζ ( italic_t ) = [ italic_ζ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_ζ start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] as the amount of model parameters that has been transmitted, where
我们定义 𝜻(t)=[ζ1(t),,ζ|𝒮k|(t)]\boldsymbol{\zeta}(t)=[\zeta_{1}(t),...,\zeta_{|\mathcal{S}_{k}|}(t)]bold_italic_ζ ( italic_t ) = [ italic_ζ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , … , italic_ζ start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT ( italic_t ) ] 为模型参数的数量, 已被传输,在

ζm(t)={min(τ=1t1zm(τ),Q),for t>1,0,for t=1,subscript𝜁𝑚𝑡casessuperscriptsubscript𝜏1𝑡1subscript𝑧𝑚𝜏𝑄for t>10for t=1\zeta_{m}(t)=\begin{cases}\min\left(\sum_{\tau=1}^{t-1}z_{m}(\tau),Q\right),&% \text{for $t>1$},\\ 0,&\text{for $t=1$},\end{cases}italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = { start_ROW start_CELL roman_min ( ∑ start_POSTSUBSCRIPT italic_τ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_τ ) , italic_Q ) , end_CELL start_CELL for italic_t > 1 , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL for italic_t = 1 , end_CELL end_ROW (17)

The derivative of σ(ζm(t))𝜎subscript𝜁𝑚𝑡\sigma(\zeta_{m}(t))italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) with respect to ζm(t)subscript𝜁𝑚𝑡\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) is
σ(ζm(t))\sigma(\zeta_{m}(t))italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) )ζm(t)\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 的导数是

dσ(ζm(t))dζm(t)=α(1σ(ζm(t)))σ(ζm(t))Q.𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝛼1𝜎subscript𝜁𝑚𝑡𝜎subscript𝜁𝑚𝑡𝑄\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}=\frac{\alpha(1-\sigma(\zeta_{m}(t)% ))\cdot\sigma(\zeta_{m}(t))}{Q}.divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG = divide start_ARG italic_α ( 1 - italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) ) ⋅ italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_Q end_ARG .

According to (17), ζm(t)[0,Q]subscript𝜁𝑚𝑡0𝑄\zeta_{m}(t)\in[0,Q]italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , italic_Q ]. As ζm(t)subscript𝜁𝑚𝑡\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) increases from 00 to Q𝑄Qitalic_Q, σ(ζm(t))𝜎subscript𝜁𝑚𝑡\sigma(\zeta_{m}(t))italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) increases from 00 to 0.50.50.50.5. Therefore, dσ(ζm(t))dζm(t)𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG is an increasing function with respect to ζm(t)subscript𝜁𝑚𝑡\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ), reaching its minimum when ζm(t)=0subscript𝜁𝑚𝑡0\zeta_{m}(t)=0italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0, and reaching its maximum when ζm(t)=Qsubscript𝜁𝑚𝑡𝑄\zeta_{m}(t)=Qitalic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_Q. We define
根据(17), ζm(t)[0,Q]\zeta_{m}(t)\in[0,Q]italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , italic_Q ] 。当 ζm(t)\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )0 增加到 QQitalic_Q 时, σ(ζm(t))\sigma(\zeta_{m}(t))italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) )0 增加到 0.50.50.5 。因此, dσ(ζm(t))dζm(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG 是关于 ζm(t)\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 的增函数,当 ζm(t)=0\zeta_{m}(t)=0italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 时达到最小值,当 ζm(t)=Q\zeta_{m}(t)=Qitalic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_Q 时达到最大值。我们定义

ψ(α)σ(0)ζm(t)/σ(Q)ζm(t).𝜓𝛼𝜎0subscript𝜁𝑚𝑡𝜎𝑄subscript𝜁𝑚𝑡\psi(\alpha)\triangleq\frac{\partial\sigma(0)}{\partial\zeta_{m}(t)}\bigg{/}% \frac{\partial\sigma(Q)}{\partial\zeta_{m}(t)}.italic_ψ ( italic_α ) ≜ divide start_ARG ∂ italic_σ ( 0 ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG / divide start_ARG ∂ italic_σ ( italic_Q ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG .

ψ(α)𝜓𝛼\psi(\alpha)italic_ψ ( italic_α ) is a decreasing function with respect to α𝛼\alphaitalic_α. Since ζm(t)[0,Q]subscript𝜁𝑚𝑡0𝑄\zeta_{m}(t)\in[0,Q]italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , italic_Q ], there is
ψ(α)\psi(\alpha)italic_ψ ( italic_α ) 是关于 α\alphaitalic_α 的递减函数。由于 ζm(t)[0,Q]\zeta_{m}(t)\in[0,Q]italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ∈ [ 0 , italic_Q ] ,所以有

dσ(ζm(t))dζm(t)ψ(α)σ(Q)ζm(t).𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝜓𝛼𝜎𝑄subscript𝜁𝑚𝑡\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\geq\psi(\alpha)\frac{\partial% \sigma(Q)}{\partial\zeta_{m}(t)}.divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG ≥ italic_ψ ( italic_α ) divide start_ARG ∂ italic_σ ( italic_Q ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG . (18)

We convert the long-term stochastic optimization problem into an online optimization problem as follows. For the SOVs, virtual queues 𝒒SOV(t)=[q1SOV(t),,q|𝒮k|SOV(t)]superscript𝒒SOV𝑡superscriptsubscript𝑞1SOV𝑡superscriptsubscript𝑞subscript𝒮𝑘SOV𝑡\boldsymbol{q}^{\text{SOV}}(t)=[q_{1}^{\text{SOV}}(t),...,q_{|\mathcal{S}_{k}|% }^{\text{SOV}}(t)]bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) = [ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) , … , italic_q start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ] are created to represent the difference between the cumulative energy consumption up to slot t𝑡titalic_t and the budget, evolving as follows:
我们将长期随机优化问题转化为在线优化问题,如下所示。对于 SOVs,创建虚拟队列 𝒒SOV(t)=[q1SOV(t),,q|𝒮k|SOV(t)]\boldsymbol{q}^{\text{SOV}}(t)=[q_{1}^{\text{SOV}}(t),...,q_{|\mathcal{S}_{k}|% }^{\text{SOV}}(t)]bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) = [ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) , … , italic_q start_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ] 来表示直到时隙 ttitalic_t 的累积能耗与预算之间的差值,其演变方式如下:

qmSOV(t+1)=max{qmSOV(t)+emcm(t)Emconsem,kcpT,0}.superscriptsubscript𝑞𝑚SOV𝑡1superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝑒𝑚cm𝑡superscriptsubscript𝐸𝑚conssuperscriptsubscript𝑒𝑚𝑘cp𝑇0q_{m}^{\text{SOV}}(t+1)=\max\left\{q_{m}^{\text{SOV}}(t)+e_{m}^{\text{cm}}(t)-% \frac{E_{m}^{\text{cons}}-e_{m,k}^{\text{cp}}}{T},0\right\}.italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t + 1 ) = roman_max { italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) + italic_e start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT - italic_e start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT end_ARG start_ARG italic_T end_ARG , 0 } . (19)

Likewise, virtual queues 𝒒OPV(t)=[q1OPV(t),,q|𝒰k|OPV(t)]superscript𝒒OPV𝑡superscriptsubscript𝑞1OPV𝑡superscriptsubscript𝑞subscript𝒰𝑘OPV𝑡\boldsymbol{q}^{\text{OPV}}(t)=[q_{1}^{\text{OPV}}(t),...,q_{|\mathcal{U}_{k}|% }^{\text{OPV}}(t)]bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) = [ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) , … , italic_q start_POSTSUBSCRIPT | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ] are created for the OPVs, evolving as follows:
同样,OPV 的虚拟队列 𝒒OPV(t)=[q1OPV(t),,q|𝒰k|OPV(t)]\boldsymbol{q}^{\text{OPV}}(t)=[q_{1}^{\text{OPV}}(t),...,q_{|\mathcal{U}_{k}|% }^{\text{OPV}}(t)]bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) = [ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) , … , italic_q start_POSTSUBSCRIPT | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ] 按以下方式演变:

qnOPV(t+1)=max{qnOPV(t)+encm(t)EnconsT,0}.superscriptsubscript𝑞𝑛OPV𝑡1superscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝑒𝑛cm𝑡superscriptsubscript𝐸𝑛cons𝑇0q_{n}^{\text{OPV}}(t+1)=\max\left\{q_{n}^{\text{OPV}}(t)+e_{n}^{\text{cm}}(t)-% \frac{E_{n}^{\text{cons}}}{T},0\right\}.italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t + 1 ) = roman_max { italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) + italic_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT end_ARG start_ARG italic_T end_ARG , 0 } . (20)

All virtual queues are initialized to 0, i.e., 𝒒SOV(t)=𝟎superscript𝒒SOV𝑡0\boldsymbol{q}^{\text{SOV}}(t)=\boldsymbol{0}bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) = bold_0, and 𝒒OPV(t)=𝟎superscript𝒒OPV𝑡0\boldsymbol{q}^{\text{OPV}}(t)=\boldsymbol{0}bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) = bold_0. Then, problem P2𝑃2P2italic_P 2 can be transformed to P3𝑃3P3italic_P 3:
所有虚拟队列都初始化为 0,即 𝒒SOV(t)=𝟎\boldsymbol{q}^{\text{SOV}}(t)=\boldsymbol{0}bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) = bold_0𝒒OPV(t)=𝟎\boldsymbol{q}^{\text{OPV}}(t)=\boldsymbol{0}bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) = bold_0 。然后,问题 P2P2italic_P 2 可以转换为 P3P3italic_P 3

P3:max𝒔(t),c(t),𝒖(t),𝒑(t)Vm𝒮kzm(t)dσ(ζm(t))dζm(t):𝑃3𝒔𝑡𝑐𝑡𝒖𝑡𝒑𝑡𝑉subscript𝑚subscript𝒮𝑘subscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle P3:\underset{\boldsymbol{s}(t),c(t),\boldsymbol{u}(t),% \boldsymbol{p}(t)}{\max}V\sum_{m\in\mathcal{S}_{k}}z_{m}(t)\frac{d\sigma(\zeta% _{m}(t))}{d\zeta_{m}(t)}italic_P 3 : start_UNDERACCENT bold_italic_s ( italic_t ) , italic_c ( italic_t ) , bold_italic_u ( italic_t ) , bold_italic_p ( italic_t ) end_UNDERACCENT start_ARG roman_max end_ARG italic_V ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
m𝒮kqmSOVemcm(t)n𝒰kqnOPVencm(t)subscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOVsubscriptsuperscript𝑒cm𝑚𝑡subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPVsubscriptsuperscript𝑒cm𝑛𝑡\displaystyle\quad\quad\quad-\sum_{m\in\mathcal{S}_{k}}q_{m}^{\text{SOV}}e^{% \text{cm}}_{m}(t)-\sum_{n\in\mathcal{U}_{k}}q_{n}^{\text{OPV}}e^{\text{cm}}_{n% }(t)- ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) (21a)
s.t. sm(t),c(t),un(t){0,1},m𝒮k,n𝒰k,formulae-sequencesubscript𝑠𝑚𝑡𝑐𝑡subscript𝑢𝑛𝑡01formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑛subscript𝒰𝑘\displaystyle s_{m}(t),c(t),u_{n}(t)\in\{0,1\},\ \forall m\in\mathcal{S}_{k},% \forall n\in\mathcal{U}_{k},italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) , italic_c ( italic_t ) , italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ∈ { 0 , 1 } , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21b)
m𝒮ksm(t)1,subscript𝑚subscript𝒮𝑘subscript𝑠𝑚𝑡1\displaystyle\sum_{m\in\mathcal{S}_{k}}s_{m}(t)\leq 1,∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≤ 1 , (21c) (21 世纪)
0pm(t)pmmax,m𝒮k,formulae-sequence0subscript𝑝𝑚𝑡subscriptsuperscript𝑝max𝑚for-all𝑚subscript𝒮𝑘\displaystyle 0\leq p_{m}(t)\leq p^{\text{max}}_{m},\quad\forall m\in\mathcal{% S}_{k},0 ≤ italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≤ italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21d) (21 天)
0pn(t)pnmax,n𝒰k,formulae-sequence0subscript𝑝𝑛𝑡subscriptsuperscript𝑝max𝑛for-all𝑛subscript𝒰𝑘\displaystyle 0\leq p_{n}(t)\leq p^{\text{max}}_{n},\quad\forall n\in\mathcal{% U}_{k},0 ≤ italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ≤ italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21e)
sm(t)c(t)un(t)RmCOT(t)un(t)Rm,nCOT-V(t),subscript𝑠𝑚𝑡𝑐𝑡subscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚COT𝑡subscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚𝑛COT-V𝑡\displaystyle s_{m}(t)c(t)u_{n}(t)R_{m}^{\text{COT}}(t)\leq u_{n}(t)R_{m,n}^{% \text{COT-V}}(t),italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) italic_c ( italic_t ) italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) ≤ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT-V end_POSTSUPERSCRIPT ( italic_t ) ,
m𝒮k,n𝒰k,formulae-sequencefor-all𝑚subscript𝒮𝑘for-all𝑛subscript𝒰𝑘\displaystyle\forall m\in\mathcal{S}_{k},\forall n\in\mathcal{U}_{k},∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21f)
𝕀(tm,kcp(t1)κ)sm(t)=0,m𝒮k,formulae-sequence𝕀subscriptsuperscript𝑡cp𝑚𝑘𝑡1𝜅subscript𝑠𝑚𝑡0for-all𝑚subscript𝒮𝑘\displaystyle\mathbb{I}\left(t^{\text{cp}}_{m,k}\geq(t-1)\kappa\right)s_{m}(t)% =0,\ \forall m\in\mathcal{S}_{k},blackboard_I ( italic_t start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≥ ( italic_t - 1 ) italic_κ ) italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (21g) (21 克)
𝕀(ζm(t)=Q)sm(t)=0,m𝒮k.formulae-sequence𝕀subscript𝜁𝑚𝑡𝑄subscript𝑠𝑚𝑡0for-all𝑚subscript𝒮𝑘\displaystyle\mathbb{I}\left(\zeta_{m}(t)=Q\right)s_{m}(t)=0,\ \forall m\in% \mathcal{S}_{k}.blackboard_I ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = italic_Q ) italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 0 , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (21h) (21 小时)

We derive the following theorem to guarantee the performance of the proposed transformation. Superscript is used to denote the solution to P3𝑃3P3italic_P 3, and is used to denote the optimal offline solution to P2𝑃2P2italic_P 2.
我们推导出以下定理来保证所提变换的性能。上标 用于表示 P3P3italic_P 3 的解,而 用于表示 P2P2italic_P 2 的最佳离线解。

Theorem 2. Suppose all queues are initialized to 0, the difference between the optimal value of solving P2𝑃2P2italic_P 2 and the counterpart of solving P3𝑃3P3italic_P 3 is bounded by:
定理 2.假设所有队列都初始化为 0,则求解 P2P2italic_P 2 的最优值与求解 P3P3italic_P 3 的对应值的差值受以下限制:

m𝒮kσ(t𝒯kzm(t))m𝒮kσ(t𝒯kzm(t))Tk2ΦVψ(α).subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡superscriptsubscript𝑇𝑘2Φ𝑉𝜓𝛼\sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}^{*}(t)% \right)-\sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}^% {\dagger}(t)\right)\leq\frac{T_{k}^{2}\Phi}{V\psi(\alpha)}.∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) - ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) ≤ divide start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG italic_V italic_ψ ( italic_α ) end_ARG . (22)

The energy consumption of the SOV m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is bounded by
SOV m𝒮km\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 的能耗受以下因素限制:

t𝒯kemcm(t)+em,kcpsubscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝑒cp𝑚𝑘\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{m}(t)+e^{\text{cp}}_{m,k}∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) + italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT (23)
Emcons+2Tk2Φ2Vt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t),absentsubscriptsuperscript𝐸cons𝑚2superscriptsubscript𝑇𝑘2Φ2𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq E^{\text{cons}}_{m}+\sqrt{2T_{k}^{2}\Phi-2V\sum_{t\in% \mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(% t))}{d\zeta_{m}(t)}},≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + square-root start_ARG 2 italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - 2 italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG end_ARG ,

and that of the OPV n𝒰k𝑛subscript𝒰𝑘n\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is bounded by
以及 OPV n𝒰kn\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 的边界是

t𝒯kencm(t)subscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑛𝑡\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{n}(t)∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) (24)
Encons+2Tk2Φ2Vt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t),absentsubscriptsuperscript𝐸cons𝑛2superscriptsubscript𝑇𝑘2Φ2𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq E^{\text{cons}}_{n}+\sqrt{2T_{k}^{2}\Phi-2V\sum_{t\in% \mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(% t))}{d\zeta_{m}(t)}},≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + square-root start_ARG 2 italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - 2 italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG end_ARG ,

where δmSOV(t)emcm(t)Emconsem,kcpTksuperscriptsubscript𝛿𝑚SOV𝑡subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝐸cons𝑚subscriptsuperscript𝑒cp𝑚𝑘subscript𝑇𝑘\delta_{m}^{\text{SOV}}(t)\triangleq e^{\text{cm}}_{m}(t)-\frac{E^{\text{cons}% }_{m}-e^{\text{cp}}_{m,k}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG, δnOPV(t)encm(t)EnconsTksuperscriptsubscript𝛿𝑛OPV𝑡subscriptsuperscript𝑒cm𝑛𝑡subscriptsuperscript𝐸cons𝑛subscript𝑇𝑘\delta_{n}^{\text{OPV}}(t)\triangleq e^{\text{cm}}_{n}(t)-\frac{E^{\text{cons}% }_{n}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG, ϕmSOVmaxt{|δm(t)|}superscriptsubscriptitalic-ϕ𝑚SOVsubscript𝑡subscript𝛿𝑚𝑡\phi_{m}^{\text{SOV}}\triangleq\max_{t}\{|\delta_{m}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | }, ϕnOPVmaxt{|δn(t)|}superscriptsubscriptitalic-ϕ𝑛OPVsubscript𝑡subscript𝛿𝑛𝑡\phi_{n}^{\text{OPV}}\triangleq\max_{t}\{|\delta_{n}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | }, and Φm𝒮k(ϕmSOV)2+n𝒰k(ϕnOPV)2Φsubscript𝑚subscript𝒮𝑘superscriptsuperscriptsubscriptitalic-ϕ𝑚SOV2subscript𝑛subscript𝒰𝑘superscriptsuperscriptsubscriptitalic-ϕ𝑛OPV2\Phi\triangleq\sum_{m\in\mathcal{S}_{k}}(\phi_{m}^{\text{SOV}})^{2}+\sum_{n\in% \mathcal{U}_{k}}(\phi_{n}^{\text{OPV}})^{2}roman_Φ ≜ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.
其中 δmSOV(t)emcm(t)Emconsem,kcpTk\delta_{m}^{\text{SOV}}(t)\triangleq e^{\text{cm}}_{m}(t)-\frac{E^{\text{cons}% }_{m}-e^{\text{cp}}_{m,k}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARGδnOPV(t)encm(t)EnconsTk\delta_{n}^{\text{OPV}}(t)\triangleq e^{\text{cm}}_{n}(t)-\frac{E^{\text{cons}% }_{n}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARGϕmSOVmaxt{|δm(t)|}\phi_{m}^{\text{SOV}}\triangleq\max_{t}\{|\delta_{m}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | }ϕnOPVmaxt{|δn(t)|}\phi_{n}^{\text{OPV}}\triangleq\max_{t}\{|\delta_{n}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | }Φm𝒮k(ϕmSOV)2+n𝒰k(ϕnOPV)2\Phi\triangleq\sum_{m\in\mathcal{S}_{k}}(\phi_{m}^{\text{SOV}})^{2}+\sum_{n\in% \mathcal{U}_{k}}(\phi_{n}^{\text{OPV}})^{2}roman_Φ ≜ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

Proof: See Appendix C. \square
证明:参见附录 C\square□

Theorem 2 shows that, instead of solving the long-term stochastic optimization problem P2𝑃2P2italic_P 2, we alternatively solve the online problem P3𝑃3P3italic_P 3. The performance is bounded with respect to the optimal offline solution to P2𝑃2P2italic_P 2, and the energy consumption for each vehicle is also bounded. The trade-off between the objective function (16a) and the energy consumption is balanced by the weight parameter V𝑉Vitalic_V. The worst-case performance can be improved by increasing the parameter ψ(α)𝜓𝛼\psi(\alpha)italic_ψ ( italic_α ), equivalent to reducing the approximation parameter α𝛼\alphaitalic_α. However, choosing overly small α𝛼\alphaitalic_α values compromises the precision of approximating the indicator function 𝕀()𝕀\mathbb{I}(\cdot)blackboard_I ( ⋅ ) with the sigmoid function σ()𝜎\sigma(\cdot)italic_σ ( ⋅ ). Therefore, in practice, it is crucial to carefully choose the values of V𝑉Vitalic_V and α𝛼\alphaitalic_α to ensure optimal approximation performance under the energy constraints.
定理 2 表明,我们不是直接求解长期随机优化问题 P2P2italic_P 2 ,而是通过交替求解在线问题 P3P3italic_P 3 。性能受限于 P2P2italic_P 2 的最优离线解,每辆车的能耗也受到限制。目标函数 (16a) 和能耗之间的权衡由权重参数 VVitalic_V 平衡。通过增加参数 ψ(α)\psi(\alpha)italic_ψ ( italic_α ) 可以改善最坏情况下的性能,这相当于减少近似参数 α\alphaitalic_α 。然而,选择过小的 α\alphaitalic_α 值会影响用 sigmoid 函数 σ()\sigma(\cdot)italic_σ ( ⋅ ) 近似指示函数 𝕀()\mathbb{I}(\cdot)blackboard_I ( ⋅ ) 的精度。因此,在实践中,仔细选择 VVitalic_Vα\alphaitalic_α 的值以确保在能量约束下获得最佳近似性能至关重要。

P3𝑃3P3italic_P 3 is an MINLP with binary variables 𝒔(t),c(t),𝒖(t)𝒔𝑡𝑐𝑡𝒖𝑡\boldsymbol{s}(t),c(t),\boldsymbol{u}(t)bold_italic_s ( italic_t ) , italic_c ( italic_t ) , bold_italic_u ( italic_t ) and continuous variables 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t ), which exhibits high computational complexity for direct solution. However, due to the existence of constraint (21c), enumerating 𝒔(t)𝒔𝑡\boldsymbol{s}(t)bold_italic_s ( italic_t ) and c(t)𝑐𝑡c(t)italic_c ( italic_t ) only introduces a linear increase in computational complexity. Therefore, we fix the SOV scheduling decision 𝒔(t)𝒔𝑡\boldsymbol{s}(t)bold_italic_s ( italic_t ) and transmission mode c(t)𝑐𝑡c(t)italic_c ( italic_t ), and focus on solving 𝒖(t)𝒖𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) and 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t ). Specifically, when the SOV scheduling 𝒔(t)𝒔𝑡\boldsymbol{s}(t)bold_italic_s ( italic_t ) and transmission mode c(t)𝑐𝑡c(t)italic_c ( italic_t ) are decided, P3𝑃3P3italic_P 3 is reduced to the following sub-problems.
P3P3italic_P 3 是一个包含二元变量 𝒔(t),c(t),𝒖(t)\boldsymbol{s}(t),c(t),\boldsymbol{u}(t)bold_italic_s ( italic_t ) , italic_c ( italic_t ) , bold_italic_u ( italic_t ) 和连续变量 𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t ) 的混合整数非线性规划 (MINLP) 问题,直接求解具有很高的计算复杂度。然而,由于存在约束 (21c),枚举 𝒔(t)\boldsymbol{s}(t)bold_italic_s ( italic_t )c(t)c(t)italic_c ( italic_t ) 仅会线性地增加计算复杂度。因此,我们固定 SOV 调度决策 𝒔(t)\boldsymbol{s}(t)bold_italic_s ( italic_t ) 和传输模式 c(t)c(t)italic_c ( italic_t ) ,并专注于求解 𝒖(t)\boldsymbol{u}(t)bold_italic_u ( italic_t )𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t ) 。具体来说,当 SOV 调度 𝒔(t)\boldsymbol{s}(t)bold_italic_s ( italic_t ) 和传输模式 c(t)c(t)italic_c ( italic_t ) 被确定后, P3P3italic_P 3 被简化为以下子问题。

V-B Direct Transmission Problem
V-B 直接传输问题

When SOV m𝑚mitalic_m is scheduled for transmission and DT mode is selected (c(t)=0𝑐𝑡0c(t)=0italic_c ( italic_t ) = 0), P3𝑃3P3italic_P 3 is reduced to
当 SOV mmitalic_m 计划传输且选择 DT 模式( c(t)=0c(t)=0italic_c ( italic_t ) = 0 )时, P3P3italic_P 3 将减少到

P3.1::𝑃3.1absent\displaystyle P3.1:italic_P 3.1 : maxpm(t)Vdσ(ζm(t))dζm(t)κRmDT(t)κqmSOV(t)pm(t)subscript𝑝𝑚𝑡𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝜅superscriptsubscript𝑅𝑚DT𝑡𝜅superscriptsubscript𝑞𝑚SOV𝑡subscript𝑝𝑚𝑡\displaystyle\underset{p_{m}(t)}{\max}\ V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{% m}(t)}\kappa R_{m}^{\text{DT}}(t)-\kappa q_{m}^{\text{SOV}}(t)p_{m}(t)start_UNDERACCENT italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_UNDERACCENT start_ARG roman_max end_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_κ italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT DT end_POSTSUPERSCRIPT ( italic_t ) - italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) (25a)
s.t. 0pm(t)pmmax.0subscript𝑝𝑚𝑡subscriptsuperscript𝑝max𝑚\displaystyle 0\leq p_{m}(t)\leq p^{\text{max}}_{m}.0 ≤ italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≤ italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT . (25b)

P3.1𝑃3.1P3.1italic_P 3.1 is a convex problem. The optimal solution is derived using the KKT conditions, in Proposition 1:
P3.1P3.1italic_P 3.1 是一个凸问题。命题 1 中使用 KKT 条件推导出最优解:

Proposition 1. Given the SOV scheduling decision, the optimal power allocation strategy for DT is given by
命题 1.在 SOV 调度决策下,DT 的最优功率分配策略为

pm(t)=[Vdσ(ζm(t))dζm(t)βqmSOV(t)βN0|hm,r(t)|2]0pmmax,superscriptsubscript𝑝𝑚𝑡subscriptsuperscriptdelimited-[]𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝛽superscriptsubscript𝑞𝑚SOV𝑡𝛽subscript𝑁0superscriptsubscript𝑚𝑟𝑡2subscriptsuperscript𝑝max𝑚0p_{m}^{*}(t)=\left[\frac{V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\beta}{q_% {m}^{\text{SOV}}(t)}-\frac{\beta N_{0}}{|h_{m,r}(t)|^{2}}\right]^{p^{\text{max% }}_{m}}_{0},italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = [ divide start_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_β end_ARG start_ARG italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) end_ARG - divide start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , (26)

where [a]0pmmaxsubscriptsuperscriptdelimited-[]𝑎subscriptsuperscript𝑝max𝑚0[a]^{p^{\text{max}}_{m}}_{0}[ italic_a ] start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is defined as min(max(a,0),pmmax)𝑎0subscriptsuperscript𝑝max𝑚\min(\max(a,0),p^{\text{max}}_{m})roman_min ( roman_max ( italic_a , 0 ) , italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ).

Proof: See Appendix D. \square
证明:参见附录 D\square□

Algorithm 1 The procedure of solving P3𝑃3P3italic_P 3
算法 1 解决 P3P3italic_P 3 的过程
  Input: (t)𝑡\mathcal{H}(t)caligraphic_H ( italic_t ), 𝒒SOV(t)superscript𝒒SOV𝑡\boldsymbol{q}^{\text{SOV}}(t)bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ), 𝒒OPV(t)superscript𝒒OPV𝑡\boldsymbol{q}^{\text{OPV}}(t)bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) and 𝜻(t)𝜻𝑡\boldsymbol{\zeta}(t)bold_italic_ζ ( italic_t );
输入: (t)\mathcal{H}(t)caligraphic_H ( italic_t )𝒒SOV(t)\boldsymbol{q}^{\text{SOV}}(t)bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t )𝒒OPV(t)\boldsymbol{q}^{\text{OPV}}(t)bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t )𝜻(t)\boldsymbol{\zeta}(t)bold_italic_ζ ( italic_t )
  Output: The solution 𝒔(t),c(t),𝒖(t),𝒑(t)superscript𝒔𝑡superscript𝑐𝑡superscript𝒖𝑡superscript𝒑𝑡\boldsymbol{s}^{*}(t),c^{*}(t),\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) to P3𝑃3P3italic_P 3; 重试    错误原因
  Initialize 𝒔(t)=𝟎superscript𝒔𝑡0\boldsymbol{s}^{*}(t)=\boldsymbol{0}bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0, c(t)=0superscript𝑐𝑡0c^{*}(t)=0italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = 0, 𝒖(t)=𝟎superscript𝒖𝑡0\boldsymbol{u}^{*}(t)=\boldsymbol{0}bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0 and 𝒑(t)=𝟎superscript𝒑𝑡0\boldsymbol{p}^{*}(t)=\boldsymbol{0}bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0.
初始化 𝒔(t)=𝟎\boldsymbol{s}^{*}(t)=\boldsymbol{0}bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0c(t)=0c^{*}(t)=0italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = 0𝒖(t)=𝟎\boldsymbol{u}^{*}(t)=\boldsymbol{0}bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0𝒑(t)=𝟎\boldsymbol{p}^{*}(t)=\boldsymbol{0}bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_0
  for m𝑚mitalic_m in 𝒮ksubscript𝒮𝑘\mathcal{S}_{k}caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT do
     Set 𝒔(t)=𝟎𝒔𝑡0\boldsymbol{s}(t)=\boldsymbol{0}bold_italic_s ( italic_t ) = bold_0;  设置 𝒔(t)=𝟎\boldsymbol{s}(t)=\boldsymbol{0}bold_italic_s ( italic_t ) = bold_0
     Set sm(t)=1subscript𝑠𝑚𝑡1s_{m}(t)=1italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 1;  设置 sm(t)=1s_{m}(t)=1italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) = 1
     if tm,kcp(t1)κsubscriptsuperscript𝑡cp𝑚𝑘𝑡1𝜅t^{\text{cp}}_{m,k}\leq(t-1)\kappaitalic_t start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≤ ( italic_t - 1 ) italic_κ and ζm(t)Qsubscript𝜁𝑚𝑡𝑄\zeta_{m}(t)\neq Qitalic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≠ italic_Q then
如果 tm,kcp(t1)κt^{\text{cp}}_{m,k}\leq(t-1)\kappaitalic_t start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≤ ( italic_t - 1 ) italic_κ 并且 ζm(t)Q\zeta_{m}(t)\neq Qitalic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≠ italic_Q 那么
        Set c(t)=0𝑐𝑡0c(t)=0italic_c ( italic_t ) = 0, 𝒖(t)=𝟎𝒖𝑡0\boldsymbol{u}(t)=\boldsymbol{0}bold_italic_u ( italic_t ) = bold_0 and 𝒑(t)=𝟎𝒑𝑡0\boldsymbol{p}(t)=\boldsymbol{0}bold_italic_p ( italic_t ) = bold_0;
设置 c(t)=0c(t)=0italic_c ( italic_t ) = 0𝒖(t)=𝟎\boldsymbol{u}(t)=\boldsymbol{0}bold_italic_u ( italic_t ) = bold_0𝒑(t)=𝟎\boldsymbol{p}(t)=\boldsymbol{0}bold_italic_p ( italic_t ) = bold_0
        Solve P3.1𝑃3.1P3.1italic_P 3.1 to obtain 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t );
P3.1P3.1italic_P 3.1 以获得 𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t ) ;
        Take 𝒔(t)𝒔𝑡\boldsymbol{s}(t)bold_italic_s ( italic_t ), c(t)𝑐𝑡c(t)italic_c ( italic_t ), 𝒖(t)𝒖𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) and 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t ) to (21a) to get y(t)𝑦𝑡y(t)italic_y ( italic_t );
𝒔(t)\boldsymbol{s}(t)bold_italic_s ( italic_t )c(t)c(t)italic_c ( italic_t )𝒖(t)\boldsymbol{u}(t)bold_italic_u ( italic_t )𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t ) 带到 (21a) 以获得 y(t)y(t)italic_y ( italic_t )
        if y(t)y(t)𝑦𝑡superscript𝑦𝑡y(t)\geq y^{*}(t)italic_y ( italic_t ) ≥ italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) then
如果 y(t)y(t)y(t)\geq y^{*}(t)italic_y ( italic_t ) ≥ italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) 那么
           Set 𝒔(t)=𝒔(t)superscript𝒔𝑡𝒔𝑡\boldsymbol{s}^{*}(t)=\boldsymbol{s}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_s ( italic_t ), c(t)=c(t)superscript𝑐𝑡𝑐𝑡c^{*}(t)=c(t)italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = italic_c ( italic_t ), 𝒖(t)=𝒖(t)superscript𝒖𝑡𝒖𝑡\boldsymbol{u}^{*}(t)=\boldsymbol{u}(t)bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_u ( italic_t ), and 𝒑(t)=𝒑(t)superscript𝒑𝑡𝒑𝑡\boldsymbol{p}^{*}(t)=\boldsymbol{p}(t)bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_p ( italic_t );
设置 𝒔(t)=𝒔(t)\boldsymbol{s}^{*}(t)=\boldsymbol{s}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_s ( italic_t )c(t)=c(t)c^{*}(t)=c(t)italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = italic_c ( italic_t )𝒖(t)=𝒖(t)\boldsymbol{u}^{*}(t)=\boldsymbol{u}(t)bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_u ( italic_t )𝒑(t)=𝒑(t)\boldsymbol{p}^{*}(t)=\boldsymbol{p}(t)bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_p ( italic_t )
        end if
结束如果
        Set c(t)=1𝑐𝑡1c(t)=1italic_c ( italic_t ) = 1;  设置 c(t)=1c(t)=1italic_c ( italic_t ) = 1
        Sort the elements of 𝒰ksubscript𝒰𝑘\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in descending order based on the values of hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t );
根据 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 的值,将 𝒰k\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 中的元素按降序排列;
        for i=1𝑖1i=1italic_i = 1 to |𝒰k|subscript𝒰𝑘|\mathcal{U}_{k}|| caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | do
i=1i=1italic_i = 1 |𝒰k||\mathcal{U}_{k}|| caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |
           For the top i𝑖iitalic_i elements in 𝒰ksubscript𝒰𝑘\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, set the corresponding un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )to 1111. For all other elements, set un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) to 00;
对于前 iiitalic_i𝒰k\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT 元素,将相应的 un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) 设置为 111 。对于所有其他元素,将 un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) 设置为 0
           Solving P4𝑃4P4italic_P 4 based to obtain 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t );
解决 P4P4italic_P 4 基于获得 𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t )
           Taking 𝒔(t)𝒔𝑡\boldsymbol{s}(t)bold_italic_s ( italic_t ), c(t)𝑐𝑡c(t)italic_c ( italic_t ), 𝒖(t)𝒖𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) and 𝒑(t)𝒑𝑡\boldsymbol{p}(t)bold_italic_p ( italic_t ) back to (21a) to obtain y(t)𝑦𝑡y(t)italic_y ( italic_t );
𝒔(t)\boldsymbol{s}(t)bold_italic_s ( italic_t )c(t)c(t)italic_c ( italic_t )𝒖(t)\boldsymbol{u}(t)bold_italic_u ( italic_t )𝒑(t)\boldsymbol{p}(t)bold_italic_p ( italic_t ) 带回 (21a) 以获得 y(t)y(t)italic_y ( italic_t )
           if y(t)y(t)𝑦𝑡superscript𝑦𝑡y(t)\geq y^{*}(t)italic_y ( italic_t ) ≥ italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) then
如果 y(t)y(t)y(t)\geq y^{*}(t)italic_y ( italic_t ) ≥ italic_y start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) 那么
              Set 𝒔(t)=𝒔(t)superscript𝒔𝑡𝒔𝑡\boldsymbol{s}^{*}(t)=\boldsymbol{s}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_s ( italic_t ), c(t)=c(t)superscript𝑐𝑡𝑐𝑡c^{*}(t)=c(t)italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = italic_c ( italic_t ), 𝒖(t)=𝒖(t)superscript𝒖𝑡𝒖𝑡\boldsymbol{u}^{*}(t)=\boldsymbol{u}(t)bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_u ( italic_t ), and 𝒑(t)=𝒑(t)superscript𝒑𝑡𝒑𝑡\boldsymbol{p}^{*}(t)=\boldsymbol{p}(t)bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_p ( italic_t ).
设置 𝒔(t)=𝒔(t)\boldsymbol{s}^{*}(t)=\boldsymbol{s}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_s ( italic_t )c(t)=c(t)c^{*}(t)=c(t)italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = italic_c ( italic_t )𝒖(t)=𝒖(t)\boldsymbol{u}^{*}(t)=\boldsymbol{u}(t)bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_u ( italic_t )𝒑(t)=𝒑(t)\boldsymbol{p}^{*}(t)=\boldsymbol{p}(t)bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = bold_italic_p ( italic_t )
           end if
结束如果
        end for
结束
     end if
结束如果
  end for
结束

V-C Cooperative Transmission Problem
V-C 协作传输问题

When SOV m𝑚mitalic_m is scheduled for transmission and COT mode is selected (c(t)=1𝑐𝑡1c(t)=1italic_c ( italic_t ) = 1), P3𝑃3P3italic_P 3 is reduced to
当 SOV mmitalic_m 计划传输且选择 COT 模式( c(t)=1c(t)=1italic_c ( italic_t ) = 1 )时, P3P3italic_P 3 将减少到

P3.2::𝑃3.2absent\displaystyle P3.2:italic_P 3.2 : max𝒖(t),𝒑(t)Vdσ(ζm(t))dζm(t)12κRmCOT(t)12κqmSOV(t)pm(t)𝒖𝑡𝒑𝑡𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡12𝜅superscriptsubscript𝑅𝑚COT𝑡12𝜅superscriptsubscript𝑞𝑚SOV𝑡subscript𝑝𝑚𝑡\displaystyle\underset{\boldsymbol{u}(t),\boldsymbol{p}(t)}{\max}V\frac{d% \sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\frac{1}{2}\kappa R_{m}^{\text{COT}}(t)-% \frac{1}{2}\kappa q_{m}^{\text{SOV}}(t)p_{m}(t)start_UNDERACCENT bold_italic_u ( italic_t ) , bold_italic_p ( italic_t ) end_UNDERACCENT start_ARG roman_max end_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )
n𝒰k12κun(t)qnOPV(t)pn(t)subscript𝑛subscript𝒰𝑘12𝜅subscript𝑢𝑛𝑡superscriptsubscript𝑞𝑛OPV𝑡subscript𝑝𝑛𝑡\displaystyle-\sum_{n\in\mathcal{U}_{k}}\frac{1}{2}\kappa u_{n}(t)q_{n}^{\text% {OPV}}(t)p_{n}(t)- ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) (27a)
s.t. un(t){0,1},n𝒰k,formulae-sequencesubscript𝑢𝑛𝑡01for-all𝑛subscript𝒰𝑘\displaystyle u_{n}(t)\in\{0,1\},\quad\forall n\in\mathcal{U}_{k},italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ∈ { 0 , 1 } , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (27b)
un(t)RmCOT(t)un(t)Rm,nCOT-V(t),n𝒰k,formulae-sequencesubscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚COT𝑡subscript𝑢𝑛𝑡superscriptsubscript𝑅𝑚𝑛COT-V𝑡for-all𝑛subscript𝒰𝑘\displaystyle u_{n}(t)R_{m}^{\text{COT}}(t)\leq u_{n}(t)R_{m,n}^{\text{COT-V}}% (t),\quad\forall n\in\mathcal{U}_{k},italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT end_POSTSUPERSCRIPT ( italic_t ) ≤ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_R start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT COT-V end_POSTSUPERSCRIPT ( italic_t ) , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (27c)
constraint (21d), (21e).constraint (21d), (21e)\displaystyle\text{constraint (\ref{powercons1}), (\ref{powercons2})}.constraint ( ), ( ) .

P3.2𝑃3.2P3.2italic_P 3.2 is still an MINLP problem, and directly enumerating the binary variable 𝒖(t)𝒖𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) introduces exponential complexity. We further analyze the OPV scheduling priority and prove the following proposition.
P3.2P3.2italic_P 3.2 仍然是一个 MINLP 问题,直接枚举二元变量 𝒖(t)\boldsymbol{u}(t)bold_italic_u ( italic_t ) 会引入指数复杂度。我们进一步分析了 OPV 调度优先级并证明了以下命题。

Proposition 2. Suppose P3.2𝑃3.2P3.2italic_P 3.2 is solvable, then there must exist an optimal set of 𝐮(t)𝐮𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) that adheres to a specific structure: the un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) variables are arranged according to the descending order of hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) values, and the optimal solution involves selecting the top i𝑖iitalic_i un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) based on this ordering.
命题 2.假设 P3.2P3.2italic_P 3. 2 可解,则必须存在一组最优的 𝐮(t)\boldsymbol{u}(t)bold_italic_u ( italic_t ) 符合特定结构:un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) 变量按 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 值的降序排列,最优解涉及选择前 iiitalic_iun(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )。 基于此排序的 STSUBSCRIPT *n* end_POSTSUBSCRIPT (*t*)。 请提供要翻译的文本。

Proof: This proposition is proved by contradiction. Assume that all optimal solutions {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) } do not adhere to the proposed structure, i.e., they do not select the top i𝑖iitalic_i un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) based on the highest hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) values.
证明: 该命题通过反证法证明。假设所有最优解 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) } 不符合所提出的结构,即它们没有根据最高的 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 值选择前 iiitalic_i un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )

Consider one of the optimal solutions {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{\prime}(t),\boldsymbol{p}^{\prime}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) }, that includes some un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) with lower hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) values set to 1, while at least one un(t)subscript𝑢𝑛𝑡u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) with a higher hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) value (within the top i𝑖iitalic_i) is set to 0. Consider another OPV scheduling strategy 𝒖(t)superscript𝒖𝑡\boldsymbol{u}^{\dagger}(t)bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ), where all un(t)superscriptsubscript𝑢𝑛𝑡u_{n}^{\dagger}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) within the top i𝑖iitalic_i highest hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) values are set to 1. For all n𝒰k𝑛subscript𝒰𝑘n\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, we consider the power allocation strategy:
考虑以下最佳解决方案之一 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{\prime}(t),\boldsymbol{p}^{\prime}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) } ,其中包含一些 un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ,其较低的 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 值设置为 1,而至少一个 un(t)u_{n}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) 具有较高的 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 值(在最高 iiitalic_i 中)设置为 0。 考虑另一种 OPV 调度策略 𝒖(t)\boldsymbol{u}^{\dagger}(t)bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ,其中所有位于前 iiitalic_i 个最高 hm,n(t)h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) 值内的 un(t)u_{n}^{\dagger}(t)italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) 都设置为 1。对于所有 n𝒰kn\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,我们考虑以下功率分配策略:

pn(t)={pn(t),if un(t)=1,0,otherwise,superscriptsubscript𝑝𝑛𝑡casessubscriptsuperscript𝑝𝑛𝑡if un(t)=10otherwisep_{n}^{\dagger}(t)=\begin{cases}p^{\prime}_{n}(t),\quad&\text{if $u^{\prime}_{% n}(t)=1$},\\ 0,\quad&\text{otherwise},\end{cases}italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) = { start_ROW start_CELL italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) , end_CELL start_CELL if italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 , end_CELL end_ROW start_ROW start_CELL 0 , end_CELL start_CELL otherwise , end_CELL end_ROW

and set pm(t)=pm(t)superscriptsubscript𝑝𝑚𝑡subscriptsuperscript𝑝𝑚𝑡p_{m}^{\dagger}(t)=p^{\prime}_{m}(t)italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) = italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ). The solution set {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } is a feasible solution, since all constraints of P4𝑃4P4italic_P 4 are satisfied, and the objective function is
并设置 pm(t)=pm(t)p_{m}^{\dagger}(t)=p^{\prime}_{m}(t)italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) = italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) 。解集 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } 是一个可行解,因为 P4P4italic_P 4 的所有约束条件都满足,目标函数是

Vdσ(ζm(t))dζm(t)12κβlog2(1+pm(t)|hm,r(t)|2βN0\displaystyle V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\frac{1}{2}\kappa% \beta\log_{2}\bigg{(}1+\frac{p^{\dagger}_{m}(t)|h_{m,r}(t)|^{2}}{\beta N_{0}}italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG
+n𝒰kun(t)pn(t)|hn,r(t)|2βN0)12κqmSOV(t)pm(t)\displaystyle+\sum_{n\in\mathcal{U}_{k}}\frac{u^{\dagger}_{n}(t)p^{\dagger}_{n% }(t)|h_{n,r}(t)|^{2}}{\beta N_{0}}\bigg{)}-\frac{1}{2}\kappa q_{m}^{\text{SOV}% }(t)p^{\dagger}_{m}(t)+ ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_n , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )
n𝒰k12κun(t)qnOPV(t)pn(t)subscript𝑛subscript𝒰𝑘12𝜅subscriptsuperscript𝑢𝑛𝑡superscriptsubscript𝑞𝑛OPV𝑡subscriptsuperscript𝑝𝑛𝑡\displaystyle-\sum_{n\in\mathcal{U}_{k}}\frac{1}{2}\kappa u^{\dagger}_{n}(t)q_% {n}^{\text{OPV}}(t)p^{\dagger}_{n}(t)- ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )
=Vdσ(ζm(t))dζm(t)12κβlog2(1+pm(t)|hm,r(t)|2βN0\displaystyle=V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\frac{1}{2}\kappa% \beta\log_{2}\bigg{(}1+\frac{p^{\prime}_{m}(t)|h_{m,r}(t)|^{2}}{\beta N_{0}}= italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG
+n𝒰kun(t)pn(t)|hn,r(t)|2βN0)12κqmSOV(t)pm(t)\displaystyle+\sum_{n\in\mathcal{U}_{k}}\frac{u^{\prime}_{n}(t)p^{\prime}_{n}(% t)|h_{n,r}(t)|^{2}}{\beta N_{0}}\bigg{)}-\frac{1}{2}\kappa q_{m}^{\text{SOV}}(% t)p^{\prime}_{m}(t)+ ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_n , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )
n𝒰k12κun(t)qnOPV(t)pn(t).subscript𝑛subscript𝒰𝑘12𝜅subscriptsuperscript𝑢𝑛𝑡superscriptsubscript𝑞𝑛OPV𝑡subscriptsuperscript𝑝𝑛𝑡\displaystyle-\sum_{n\in\mathcal{U}_{k}}\frac{1}{2}\kappa u^{\prime}_{n}(t)q_{% n}^{\text{OPV}}(t)p^{\prime}_{n}(t).- ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) .

Since the objective function of the solution set {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } is equal to that of the optimal solution set {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{\prime}(t),\boldsymbol{p}^{\prime}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) }, {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } is also an optimal solution. This contradicts the assumption that none of the optimal solutions {𝒖(t),𝒑(t)}superscript𝒖𝑡superscript𝒑𝑡\{\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) } adhere to the proposed structure. Proposition 2 is proved. \square
由于解集 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } 的目标函数等于最优解集 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{\prime}(t),\boldsymbol{p}^{\prime}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_t ) } 的目标函数,因此 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{\dagger}(t),\boldsymbol{p}^{\dagger}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) } 也是最优解。这与假设没有最优解 {𝒖(t),𝒑(t)}\{\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)\}{ bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) } 符合所提出的结构相矛盾。命题 2 得证。 \square□

Algorithm 2 The procedure of the VEDS algorithm
算法 2 VEDS 算法的过程
  Initialization Set 𝒒SOV(1)superscript𝒒SOV1\boldsymbol{q}^{\text{SOV}}(1)bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( 1 ), 𝒒OPV(1)superscript𝒒OPV1\boldsymbol{q}^{\text{OPV}}(1)bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( 1 ) and 𝜻(1)𝜻1\boldsymbol{\zeta}(1)bold_italic_ζ ( 1 ) to 𝟎0\boldsymbol{0}bold_0;
初始化 设置 𝒒SOV(1)\boldsymbol{q}^{\text{SOV}}(1)bold_italic_q start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( 1 )𝒒OPV(1)\boldsymbol{q}^{\text{OPV}}(1)bold_italic_q start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( 1 )𝜻(1)\boldsymbol{\zeta}(1)bold_italic_ζ ( 1 )𝟎\boldsymbol{0}bold_0
  for t𝑡titalic_t in 𝒯ksubscript𝒯𝑘\mathcal{T}_{k}caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT do
     Update the amount of transmitted model parameters 𝜻(t)𝜻𝑡\boldsymbol{\zeta}(t)bold_italic_ζ ( italic_t ) according to (17);
根据 (17) 更新传输模型参数的数量 𝜻(t)\boldsymbol{\zeta}(t)bold_italic_ζ ( italic_t )
     Observe the current channel state 𝒉(t)𝒉𝑡\boldsymbol{h}(t)bold_italic_h ( italic_t );
观察当前频道状态 𝒉(t)\boldsymbol{h}(t)bold_italic_h ( italic_t )
     Solve P3𝑃3P3italic_P 3 to get 𝒔(t),c(t),𝒖(t),𝒑(t)superscript𝒔𝑡superscript𝑐𝑡superscript𝒖𝑡superscript𝒑𝑡\boldsymbol{s}^{*}(t),c^{*}(t),\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) based on Algorithm 1, and allocate communication resources;
根据算法解决 P3P3italic_P 3 以获得 𝒔(t),c(t),𝒖(t),𝒑(t)\boldsymbol{s}^{*}(t),c^{*}(t),\boldsymbol{u}^{*}(t),\boldsymbol{p}^{*}(t)bold_italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , italic_c start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_u start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) , bold_italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ,并分配通信资源;
     Update the virtual queue 𝒒(t)𝒒𝑡\boldsymbol{q}(t)bold_italic_q ( italic_t ) according to (19) and (20);
更新虚拟队列 𝒒(t)\boldsymbol{q}(t)bold_italic_q ( italic_t ) 根据 (19) 和 (20);
  end for
结束

Based on Proposition 2, we can sort the elements of 𝒰ksubscript𝒰𝑘\mathcal{U}_{k}caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in descending order based on the values of hm,n(t)subscript𝑚𝑛𝑡h_{m,n}(t)italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ), and schedule the first i𝑖iitalic_i OPVs for COT, i.e., set un(t)=1subscript𝑢𝑛𝑡1u_{n}(t)=1italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 for them, and set un(t)=0subscript𝑢𝑛𝑡0u_{n}(t)=0italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 0 for all other vehicles. 重试    错误原因

When 𝒖(t)𝒖𝑡\boldsymbol{u}(t)bold_italic_u ( italic_t ) is given, the constraint (27c) becomes 重试    错误原因

1+pm(t)|hm,r(t)|2βN0+n(t)pn(t)|hn,r(t)|2βN01subscript𝑝𝑚𝑡superscriptsubscript𝑚𝑟𝑡2𝛽subscript𝑁0subscript𝑛𝑡subscript𝑝𝑛𝑡superscriptsubscript𝑛𝑟𝑡2𝛽subscript𝑁0\displaystyle 1+\frac{p_{m}(t)|h_{m,r}(t)|^{2}}{\beta N_{0}}+\sum_{n\in% \mathcal{R}(t)}\frac{p_{n}(t)|h_{n,r}(t)|^{2}}{\beta N_{0}}1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_R ( italic_t ) end_POSTSUBSCRIPT divide start_ARG italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_n , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG (28)
1+pm(t)|hm,n(t)|2βN0,n(t),formulae-sequenceabsent1subscript𝑝𝑚𝑡superscriptsubscript𝑚𝑛𝑡2𝛽subscript𝑁0for-all𝑛𝑡\displaystyle\leq 1+\frac{p_{m}(t)|h_{m,n}(t)|^{2}}{\beta N_{0}},\quad\forall n% \in\mathcal{R}(t),≤ 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG , ∀ italic_n ∈ caligraphic_R ( italic_t ) ,

where (t)={nun(t)=1,n𝒰k}𝑡conditional-set𝑛formulae-sequencesubscript𝑢𝑛𝑡1𝑛subscript𝒰𝑘\mathcal{R}(t)=\{n\mid u_{n}(t)=1,n\in\mathcal{U}_{k}\}caligraphic_R ( italic_t ) = { italic_n ∣ italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) = 1 , italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. P3.2𝑃3.2P3.2italic_P 3.2 is reduced to 重试    错误原因

P4::𝑃4absent\displaystyle P4:italic_P 4 : max𝒑(t)Vdσ(ζm(t))dζm(t)12κβlog2(1+pm(t)|hm,r(t)|2βN0\displaystyle\underset{\boldsymbol{p}(t)}{\max}\ V\frac{d\sigma(\zeta_{m}(t))}% {d\zeta_{m}(t)}\frac{1}{2}\kappa\beta\log_{2}\bigg{(}1+\frac{p_{m}(t)|h_{m,r}(% t)|^{2}}{\beta N_{0}}start_UNDERACCENT bold_italic_p ( italic_t ) end_UNDERACCENT start_ARG roman_max end_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG
+n(t)pn(t)|hn,r(t)|2βN0)12κqmSOV(t)pm(t)\displaystyle+\sum_{n\in\mathcal{R}(t)}\frac{p_{n}(t)|h_{n,r}(t)|^{2}}{\beta N% _{0}}\bigg{)}-\frac{1}{2}\kappa q_{m}^{\text{SOV}}(t)p_{m}(t)+ ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_R ( italic_t ) end_POSTSUBSCRIPT divide start_ARG italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_n , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t )
n(t)12κqnOPV(t)pn(t)subscript𝑛𝑡12𝜅superscriptsubscript𝑞𝑛OPV𝑡subscript𝑝𝑛𝑡\displaystyle-\sum_{n\in\mathcal{R}(t)}\frac{1}{2}\kappa q_{n}^{\text{OPV}}(t)% p_{n}(t)- ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_R ( italic_t ) end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_κ italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) (29)
s.t. constraints (21e),(25b),(28).constraints (21e)(25b)(28)\displaystyle\text{constraints (\ref{powercons2})},\text{(\ref{powercons3})},% \text{(\ref{relay3})}.constraints ( ) , ( ) , ( ) .

P4𝑃4P4italic_P 4 is a convex optimization problem since the objective (29) is to maximize a concave function and all constraints (21e), (25b) and (28) are convex, which can be solved by optimization tools, such as CVX [45], based on the interior-point method. All transformations of P3𝑃3P3italic_P 3 are equivalent transformations, and the procedure of solving P3𝑃3P3italic_P 3 is summarized in Algorithm 1, where (t)={hm,n(t)|m𝒮k,n𝒰kr}𝑡conditional-setsubscript𝑚𝑛𝑡formulae-sequence𝑚subscript𝒮𝑘𝑛subscript𝒰𝑘𝑟\mathcal{H}(t)=\{h_{m,n}(t)\ |m\in\mathcal{S}_{k},n\in\mathcal{U}_{k}\cup r\}caligraphic_H ( italic_t ) = { italic_h start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT ( italic_t ) | italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∪ italic_r }, and y(t)𝑦𝑡y(t)italic_y ( italic_t ) denotes the value of (21a), i.e., the objective function of P3𝑃3P3italic_P 3. 重试    错误原因

V-D The Complete Algorithm 重试    错误原因

The whole procedure of the proposed VEDS algorithm is summarized in Algorithm 2. At the start of each round, the RSU broadcasts the global model to the SOVs, and the SOVs perform local updates based on their local dataset. In each slot, the RSU solves P3𝑃3P3italic_P 3 based on the current channel state 𝒉(t)𝒉𝑡\boldsymbol{h}(t)bold_italic_h ( italic_t ) and the amount of transmitted model parameters 𝜻(t)𝜻𝑡\boldsymbol{\zeta}(t)bold_italic_ζ ( italic_t ). Based on the solution to P3𝑃3P3italic_P 3, the resources are allocated, and the virtual queues are updated. This process is iterated until the end of the round. 重试    错误原因

Refer to caption
Figure 3: The SUMO road network. 重试    错误原因

V-E Complexity Analysis 重试    错误原因

The complexity of Algorithm 2 is 𝒪(Tk|𝒮k|(Cd+|𝒰k|Cc)),𝒪subscript𝑇𝑘subscript𝒮𝑘subscript𝐶𝑑subscript𝒰𝑘subscript𝐶𝑐\mathcal{O}(T_{k}|\mathcal{S}_{k}|(C_{d}+|\mathcal{U}_{k}|C_{c})),caligraphic_O ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | ( italic_C start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT + | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_C start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) ) , where 𝒪(Cd)𝒪subscript𝐶𝑑\mathcal{O}(C_{d})caligraphic_O ( italic_C start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) and 𝒪(Cc)𝒪subscript𝐶𝑐\mathcal{O}(C_{c})caligraphic_O ( italic_C start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) denote the complexity of solving P3.1𝑃3.1P3.1italic_P 3.1 and P4𝑃4P4italic_P 4, respectively. P3.1𝑃3.1P3.1italic_P 3.1 can be solved in constant time 𝒪(1)𝒪1\mathcal{O}(1)caligraphic_O ( 1 ) according to Proposition 1. P4𝑃4P4italic_P 4 is a convex optimization problem with a convex objective and up to 2|𝒰k|+12subscript𝒰𝑘12|\mathcal{U}_{k}|+12 | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | + 1 linear constraints, involves an optimization variable of dimension |𝒰k|+1subscript𝒰𝑘1|\mathcal{U}_{k}|+1| caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | + 1. Utilizing the interior-point method, P4𝑃4P4italic_P 4 can be addressed with a complexity of 𝒪(|𝒰k|4.5ln1ϵ)𝒪superscriptsubscript𝒰𝑘4.51italic-ϵ\mathcal{O}(|\mathcal{U}_{k}|^{4.5}\ln{\frac{1}{\epsilon}})caligraphic_O ( | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 4.5 end_POSTSUPERSCRIPT roman_ln divide start_ARG 1 end_ARG start_ARG italic_ϵ end_ARG ) for a given precision ϵitalic-ϵ\epsilonitalic_ϵ of the solution. Ignoring lower-order terms, the overall computation complexity of Algorithm 2 is 𝒪(Tk|𝒮k||𝒰k|5.5ln1ϵ).𝒪subscript𝑇𝑘subscript𝒮𝑘superscriptsubscript𝒰𝑘5.51italic-ϵ\mathcal{O}(T_{k}|\mathcal{S}_{k}||\mathcal{U}_{k}|^{5.5}\ln{\frac{1}{\epsilon% }}).caligraphic_O ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | | caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 5.5 end_POSTSUPERSCRIPT roman_ln divide start_ARG 1 end_ARG start_ARG italic_ϵ end_ARG ) . 重试    错误原因

VI Experiments 重试    错误原因

In this section, we evaluate the performance of the proposed VEDS algorithm. Firstly, it is compared with the benchmarks under different vehicle speeds v𝑣vitalic_v, approximation parameters α𝛼\alphaitalic_α, and weight V𝑉Vitalic_V. Then, the proposed VEDS algorithm is evaluated for the CIFAR-10 image classification task [46]. Finally, the VEDS algorithm is applied to a real-world trajectory prediction dataset Argoverse [47] to showcase the value in practical vehicular applications. 重试    错误原因

TABLE I: Simulation Parameters. 重试    错误原因
Simulation Parameters 重试    错误原因 Values 重试    错误原因
System Bandwidth 重试    错误原因 20MHz 
Carrier Frequency  5.9GHz 
Maximum Transmission Power  0.3W
Noise Power Spectrum Density  -174dBm/Hz 
Shadowing Fading Std. Dev.  3dB (LOS, NLOSv), 4dB (NLOS)   
Vehicle Blockage Loss    max{0,𝒩(5,4)}0𝒩54\max\{0,\mathcal{N}(5,4)\}roman_max { 0 , caligraphic_N ( 5 , 4 ) } dB   
Energy Consumption Coefficient    1028superscript102810^{-28}10 start_POSTSUPERSCRIPT - 28 end_POSTSUPERSCRIPT
Energy Constraints    Random selected from 0.050.050.050.05J to 0.10.10.10.1J   

VI-A Simulation setups   

A road network is built based on SUMO [48], as shown in Fig. 3. An RSU is placed at the center of the road network. The vehicles move according to the Manhattan mobility model with a maximum speed of v𝑣vitalic_v m/s, where v𝑣vitalic_v is a variable for the experiments. For wireless communications, we adopt the V2X channel models in 3GPP TR 37.885 [11]. In the urban environment, the pathloss of the LOS and the NLOSv channels are given by PLLOS=38.77+16.7log10d+18.2log10γ,𝑃subscript𝐿LOS38.7716.7subscript10𝑑18.2subscript10𝛾PL_{\text{LOS}}=38.77+16.7\log_{10}{d}+18.2\log_{10}{\gamma},italic_P italic_L start_POSTSUBSCRIPT LOS end_POSTSUBSCRIPT = 38.77 + 16.7 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_d + 18.2 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_γ , where d𝑑ditalic_d is the distance between two devices, γ𝛾\gammaitalic_γ is the carrier frequency. The pathloss of the NLOS channel is specified by PLNLOS=36.85+30log10d+18.9log10γ.𝑃subscript𝐿NLOS36.8530subscript10𝑑18.9subscript10𝛾PL_{\text{NLOS}}=36.85+30\log_{10}{d}+18.9\log_{10}{\gamma}.italic_P italic_L start_POSTSUBSCRIPT NLOS end_POSTSUBSCRIPT = 36.85 + 30 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_d + 18.9 roman_log start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT italic_γ . The simulation parameters are summarized in Table I.

Refer to caption
Figure 4: Performance of the VEDS algorithm and the benchmarks under different vehicle speeds.
Refer to caption
Figure 5: Performance of the VEDS algorithm and the benchmarks under different parameters α𝛼\alphaitalic_α.
Refer to caption
Figure 6: The graph for the sigmoid function.
Refer to caption
Figure 7: The graph for the derivative of the sigmoid function.
Refer to caption
Figure 8: Performance of the VEDS algorithm and the benchmarks under different weight V𝑉Vitalic_V.
Refer to caption
Figure 9: Total energy consumption of all vehicles per round under different weight parameter V𝑉Vitalic_V.

For comparison, the following benchmarks are considered:

VI-A1 Optimal Benchmark

All the SOVs within the RSU coverage can successfully upload their model parameters.

VI-A2 Dynamic algorithm with V2I-only communications (V2I-only)

This framework adjusts transmission strategies dynamically in every time slot, considering vehicle mobility. However, it solely uses V2I communications, meaning that the OPVs are not included. This is a special case of our proposed algorithm.

VI-A3 Mobility and channel dynamic-aware FL (MADCA-FL)

This is a state-of-the-art VFL framework that considers the rapidly changing channel and vehicle mobility [7].

VI-A4 Static resource allocation and device scheduling algorithm (SA)

This framework does not consider the rapidly time-varying channel and vehicle mobility. It schedules vehicles based on their initial channel states and positions, which is a modified version of the state-of-the-art device scheduling and resource allocation scheme [26].

VI-B Performance of VEDS under different parameters

VI-B1 Impact of vehicle speed v𝑣vitalic_v

Firstly, we validate the performance of our algorithm under different vehicle speeds. We use the objective function of P1𝑃1P1italic_P 1, i.e., the number of successful aggregations, as the performance metrics. As illustrated in Fig. 9, the number of successful aggregations of our framework initially increases and then decreases as the vehicle speed is adjusted from 0 (a stationary scenario) to 25252525m/s, achieving 81.03%percent81.0381.03\%81.03 % of the optimal benchmark performance when v=5𝑣5v=5italic_v = 5m/s. This performance increase at low speeds can be attributed to the mobility of vehicles allowing OPVs to enter the coverage of the RSU, while the SOVs largely remain within the RSU coverage area. If the vehicles move at high speed, the departure of some SOVs from RSU coverage results in deteriorated channel conditions. However, with the assistance of OPVs, these SOVs can still transmit the model back to the RSU. In comparison, the V2I-only framework and the MADCA-FL also consider vehicle mobility and exhibit certain robustness to changes in mobility. The SA framework, which employs static device scheduling, shows a significant performance decline in high-speed scenarios.

VI-B2 Impact of α𝛼\alphaitalic_α

We evaluate our proposed algorithm for different values of α𝛼\alphaitalic_α, as shown in Fig. 9. It is illustrated that as α𝛼\alphaitalic_α increases from 102superscript10210^{-2}10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT to 102superscript10210^{2}10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the number of successful aggregations first increases and then decreases, reaching a maximum when α𝛼\alphaitalic_α is approximately equal to 2222. This is because as Theorem 2 suggests, when the parameter α𝛼\alphaitalic_α is too small, the sigmoid function becomes overly smooth, leading to a suboptimal approximation of the indicator function (as shown in Fig. 9). On the other hand, when α𝛼\alphaitalic_α is too large, the term ψ(α)𝜓𝛼\psi(\alpha)italic_ψ ( italic_α ) diminishes, resulting in a loose bound in (22). Both scenarios adversely affect the overall performance of the algorithm. We also explain this phenomenon from a more intuitive perspective. As illustrated in Fig. 9, when α𝛼\alphaitalic_α is small, the weight dσ(ζm(t))dζm(t)𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG increases slowly with respect to ζm(t)subscript𝜁𝑚𝑡\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ), the amount of transmitted model parameters. In this case, the algorithm tends to schedule vehicles evenly to balance their energy consumption. Consequently, it is possible that many vehicles have transmitted most of their model parameters but have not completed the upload. In the FL context, such a scenario is considered a transmission failure. When α𝛼\alphaitalic_α is large, dσ(ζm(t))dζm(t)𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG also increases slowly when ζm(t)subscript𝜁𝑚𝑡\zeta_{m}(t)italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) is small, and thus, the aforementioned phenomenon persists, leading to suboptimal performance.

VI-B3 Impact of V𝑉Vitalic_V

Then, we evaluate our proposed VEDS algorithm for different weight parameters V𝑉Vitalic_V. The number of successful aggregations and the energy consumption of all vehicles are shown in Fig. 9 and Fig. 9, respectively. It is illustrated that as V𝑉Vitalic_V increases from 102superscript10210^{-2}10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT to 102superscript10210^{2}10 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, vehicles tend to consume more energy, which results in higher energy usage and a greater number of successful aggregations. When V𝑉Vitalic_V exceeds a threshold (around V=1𝑉1V=1italic_V = 1), most vehicles use their maximum transmission power to upload their model, and the energy constraints are violated. Therefore, in practical systems, it is crucial to carefully choose the value of V𝑉Vitalic_V to ensure optimal training performance under energy constraints.

VI-C Evaluation on the CIFAR-10 dataset

Then, we evaluate the proposed VEDS algorithm on the CIFAR-10 dataset[46], which comprises 50000500005000050000 training images and 10000100001000010000 test images across ten categories. We consider both the independent and identically distributed (i.i.d.) and non-independent and identically distributed (non-i.i.d) settings. For the i.i.d. setting, the dataset is evenly divided into 40404040 subsets, each containing samples from all 10 categories. For the non-i.i.d. setting, data samples are organized by category, and each vehicle holds a disjoint subset of data with samples from 2222 categories. Using the dataset, we train a convolutional neural network (CNN) with six convolutional layers. The learning rate is 0.10.10.10.1, and the batch size is set to 32323232.

The test accuracy of the VEDS algorithm compared with the benchmarks is illustrated in Fig. 12 (i.i.d.) and Fig. 12 (non-i.i.d.), where α=2,V=0.2formulae-sequence𝛼2𝑉0.2\alpha=2,V=0.2italic_α = 2 , italic_V = 0.2, and vehicle speed v=10𝑣10v=10italic_v = 10 m/s. In the i.i.d. scenario, both VEDS and the benchmarks achieve high test accuracy. The VFL convergence speed of the VEDS algorithm closely approaches that of the optimal benchmark and is significantly higher than other benchmarks. Under the non-i.i.d. scenario, the convergence speed and the highest test accuracy of the VEDS algorithm are close to the optimal benchmark and exceed the other three benchmarks. After 1000 seconds of training, VEDS achieves a test accuracy of 62.45%percent62.4562.45\%62.45 %, exceeding V2I-only, MADCA-FL and SA over 13.86%percent13.8613.86\%13.86 %, 18.28%percent18.2818.28\%18.28 % and 41.50%percent41.5041.50\%41.50 %. After 10000 seconds of training, the highest achievable accuracies are 79.41%percent79.4179.41\%79.41 % for the optimal benchmark, 78.80%percent78.8078.80\%78.80 % for VEDS, 76.33%percent76.3376.33\%76.33 % for V2I-only, 76.37%percent76.3776.37\%76.37 % for MADCA-FL, and 73.43%percent73.4373.43\%73.43 % for SA.

Refer to caption
Figure 10: Test accuracy on the CIFAR-10 dataset (i.i.d. setting).
Refer to caption
Figure 11: Test accuracy on the CIFAR-10 dataset (non-i.i.d. setting).
Refer to caption
Figure 12: ADE for the trajectory prediction task on the Argoverse dataset.

VI-D Evaluation on Argoverse trajectory prediction dataset

Finally, we evaluate the proposed VEDS algorithm on the real-world trajectory prediction dataset Argoverse [47]. Argoverse encompasses more than 300000300000300000300000 sequences gathered from Pittsburgh and Miami. Each sequence is captured from a moving vehicle at a sampling frequency of 10101010 Hz. The task is to predict the position of the vehicle for the next 3 seconds. The dataset is organized into training, validation, and test sets, containing 205942205942205942205942, 39472394723947239472, and 78143781437814378143 sequences, respectively. The sequences are uniformly partitioned into 40 subsets.

Based on the dataset, the VFL system collaboratively trains a lane graph convolutional neural network (LaneGCN) [49]. The LaneGCN includes three sub neural networks: an ActorNet, a MapNet, and a FusionNet. The ActorNet contains a 1D CNN and a Feature Pyramid Network (FPN) to extract features of vehicle trajectories. The MapNet is a graph convolutional neural network that represents and extracts the map features. The FusionNet is used to fuse the vehicle trajectory features and the map features to output the final trajectory prediction results. We employ ADE as the metric for trajectory prediction, which is the average l2subscript𝑙2l_{2}italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT distance between the actual and predicted vehicle positions on the trajectory.

The performance of the proposed framework compared with the benchmarks is illustrated in Fig. 12. It is shown that the proposed VEDS algorithm outperforms the benchmarks both in terms of ADE. Specifically, VEDS achieves an ADE of 1.721.721.721.72 after 2000200020002000 rounds of training, which is 7.98%percent7.987.98\%7.98 %, 10.21%percent10.2110.21\%10.21 %, 22.00%percent22.0022.00\%22.00 % lower than V2I-only, MADCA-FL and SA, respectively. These results validate the strong performance of our proposed VEDS algorithm when applied to real-world autonomous driving datasets.

VII Conclusions

In this paper, we have considered a VFL system, where the SOVs and OPVs in a vehicular network collaborate to train an ML model under the orchestration of the RSU. A VEDS algorithm has been proposed to optimize the VFL training performance under energy constraints and channel uncertainty of vehicles. Convergence analysis has been performed to transform the implicit FL loss function into the number of successful aggregations. Then, a derivative-based drift-plus-penalty method has been proposed to convert the long-term stochastic optimization problem into an online MINLP problem, and a theoretical performance guarantee has been provided for the proposed transformation by bounding the performance gap between the online and offline solutions. Based on the analysis of the scheduling priority, the MINLP problem has been further reduced to a set of convex optimization problems, which can be efficiently solved using the interior-point method. Experimental results have illustrated that our proposed framework is robust under different vehicle speeds. The test accuracy is increased by 3.18%percent3.183.18\%3.18 % for the CIFAR-10 dataset, and the ADE is reduced by 10.21%percent10.2110.21\%10.21 % for the Argoverse dataset.

Appendix A Proof of Lemma 1

For simplicity, we use Im,ksubscript𝐼𝑚𝑘I_{m,k}italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT to denote 𝕀(t𝒯kzm(t)Q)𝕀subscript𝑡subscript𝒯𝑘subscript𝑧𝑚𝑡𝑄\mathbb{I}\left(\sum_{t\in\mathcal{T}_{k}}z_{m}(t)\geq Q\right)blackboard_I ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ≥ italic_Q ) in the appendix. According to Assumption 1 and definition (1), the global loss function is also L𝐿Litalic_L-smooth and μ𝜇\muitalic_μ-strongly convex. There is:

F(𝒘k)𝐹subscript𝒘𝑘\displaystyle F(\boldsymbol{w}_{k})italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) F(𝒘k1)𝐹subscript𝒘𝑘1\displaystyle-F(\boldsymbol{w}_{k-1})- italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) (30)
F(𝒘k1),𝒘k𝒘k1+L2𝒘k𝒘k12.absent𝐹subscript𝒘𝑘1subscript𝒘𝑘subscript𝒘𝑘1𝐿2superscriptnormsubscript𝒘𝑘subscript𝒘𝑘12\displaystyle\leq\left<{\nabla F(\boldsymbol{w}_{k-1}),\boldsymbol{w}_{k}-% \boldsymbol{w}_{k-1}}\right>+\frac{L}{2}\left\|{\boldsymbol{w}_{k}-\boldsymbol% {w}_{k-1}}\right\|^{2}.≤ ⟨ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ⟩ + divide start_ARG italic_L end_ARG start_ARG 2 end_ARG ∥ bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

According to Assumption 3, there is

𝔼[m𝒮kIm,k|𝒟m|(𝒘m,k𝒘k1)m𝒮kIm,k|𝒟m|]𝔼delimited-[]subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚subscript𝒘𝑚𝑘subscript𝒘𝑘1subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚\displaystyle\mathbb{E}\left[\frac{\sum_{m\in{\mathcal{S}}_{k}}I_{m,k}|% \mathcal{D}_{m}|(\boldsymbol{w}_{m,k}-\boldsymbol{w}_{k-1})}{\sum_{m\in% \mathcal{S}_{k}}I_{m,k}|\mathcal{D}_{m}|}\right]blackboard_E [ divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ( bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG ]
=\displaystyle== 𝔼[m𝒮k𝒙m,kIm,k|𝒟m|ηkf(𝒘k1;𝒙)Bkm𝒮kIm,k|𝒟m|]𝔼delimited-[]subscript𝑚subscript𝒮𝑘subscript𝒙subscript𝑚𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚subscript𝜂𝑘𝑓subscript𝒘𝑘1𝒙subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚\displaystyle\mathbb{E}\left[\frac{\sum_{m\in{\mathcal{S}}_{k}}\sum_{% \boldsymbol{x}\in\mathcal{B}_{m,k}}I_{m,k}|\mathcal{D}_{m}|\eta_{k}\nabla f% \left(\boldsymbol{w}_{k-1};\boldsymbol{x}\right)}{B_{k}\sum_{m\in\mathcal{S}_{% k}}I_{m,k}|\mathcal{D}_{m}|}\right]blackboard_E [ divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT bold_italic_x ∈ caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_f ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ; bold_italic_x ) end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG ]
=\displaystyle== ηkF(𝒘k1).subscript𝜂𝑘𝐹subscript𝒘𝑘1\displaystyle\eta_{k}\nabla F(\boldsymbol{w}_{k-1}).italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) .

For the term F(𝒘k1),𝒘k+1𝒘k𝐹subscript𝒘𝑘1subscript𝒘𝑘1subscript𝒘𝑘\left<{\nabla F(\boldsymbol{w}_{k-1}),\boldsymbol{w}_{k+1}-\boldsymbol{w}_{k}}\right>⟨ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , bold_italic_w start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩, we have

F(𝒘k1),𝒘k𝒘k1𝐹subscript𝒘𝑘1subscript𝒘𝑘subscript𝒘𝑘1\displaystyle\left<{\nabla F(\boldsymbol{w}_{k-1}),\boldsymbol{w}_{k}-% \boldsymbol{w}_{k-1}}\right>⟨ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ⟩ (31)
=<F(𝒘k1),m𝒮kIm,k|𝒟m|(𝒘m,k𝒘k1)m𝒮kIm,k|𝒟m|>\displaystyle=\big{<}{\nabla F(\boldsymbol{w}_{k-1}),\frac{\sum_{m\in{\mathcal% {S}}_{k}}I_{m,k}|\mathcal{D}_{m}|(\boldsymbol{w}_{m,k}-\boldsymbol{w}_{k-1})}{% \sum_{m\in\mathcal{S}_{k}}I_{m,k}|\mathcal{D}_{m}|}}\big{>}= < ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) , divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ( bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG >
=ηkF(𝒘k1)2.absentsubscript𝜂𝑘superscriptnorm𝐹subscript𝒘𝑘12\displaystyle=\eta_{k}\nabla\left\|{F(\boldsymbol{w}_{k-1})}\right\|^{2}.= italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ ∥ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

For the term 𝒘k𝒘k12superscriptnormsubscript𝒘𝑘subscript𝒘𝑘12\left\|{\boldsymbol{w}_{k}-\boldsymbol{w}_{k-1}}\right\|^{2}∥ bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, we have

𝔼[𝒘k𝒘k12]𝔼delimited-[]superscriptnormsubscript𝒘𝑘subscript𝒘𝑘12\displaystyle\mathbb{E}\left[\left\|{\boldsymbol{w}_{k}-\boldsymbol{w}_{k-1}}% \right\|^{2}\right]blackboard_E [ ∥ bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]
=𝔼[m𝒮kIm,k|𝒟m|(𝒘m,k𝒘k1)m𝒮kIm,k|𝒟m|2]absent𝔼delimited-[]superscriptnormsubscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚subscript𝒘𝑚𝑘subscript𝒘𝑘1subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚2\displaystyle=\mathbb{E}\left[\left\|{\frac{\sum_{m\in{\mathcal{S}}_{k}}I_{m,k% }|\mathcal{D}_{m}|(\boldsymbol{w}_{m,k}-\boldsymbol{w}_{k-1})}{\sum_{m\in% \mathcal{S}_{k}}I_{m,k}|\mathcal{D}_{m}|}}\right\|^{2}\right]= blackboard_E [ ∥ divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | ( bold_italic_w start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT - bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]
=𝔼[m𝒮k𝒙m,kIm,k|𝒟m|ηkf(𝒘k1;𝒙)Bkm𝒮kIm,k|𝒟m|2]absent𝔼delimited-[]superscriptnormsubscript𝑚subscript𝒮𝑘subscript𝒙subscript𝑚𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚subscript𝜂𝑘𝑓subscript𝒘𝑘1𝒙subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘subscript𝒟𝑚2\displaystyle=\mathbb{E}\left[\left\|{\frac{\sum_{m\in{\mathcal{S}}_{k}}\sum_{% \boldsymbol{x}\in\mathcal{B}_{m,k}}I_{m,k}|\mathcal{D}_{m}|\eta_{k}\nabla f% \left(\boldsymbol{w}_{k-1};\boldsymbol{x}\right)}{B_{k}\sum_{m\in\mathcal{S}_{% k}}I_{m,k}|\mathcal{D}_{m}|}}\right\|^{2}\right]= blackboard_E [ ∥ divide start_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT bold_italic_x ∈ caligraphic_B start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ italic_f ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ; bold_italic_x ) end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT | caligraphic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]
F(𝒘k1)2+G2Bkm𝒮kIm,k.absentsuperscriptnorm𝐹subscript𝒘𝑘12superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘\displaystyle\leq\left\|{\nabla F(\boldsymbol{w}_{k-1})}\right\|^{2}+\frac{G^{% 2}}{B_{k}\sum_{m\in\mathcal{S}_{k}}I_{m,k}}.≤ ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG . (32)

Taking the expectation over stochastic data sampling on both sides of (30) and plugging (31) and (32), we have

𝔼[F(𝒘k)]𝔼[F(𝒘k1)]ηkF(𝒘k1)2𝔼delimited-[]𝐹subscript𝒘𝑘𝔼delimited-[]𝐹subscript𝒘𝑘1subscript𝜂𝑘superscriptnorm𝐹subscript𝒘𝑘12\displaystyle\mathbb{E}[F(\boldsymbol{w}_{k})]-\mathbb{E}[F(\boldsymbol{w}_{k-% 1})]\leq-\eta_{k}\left\|{\nabla F(\boldsymbol{w}_{k-1})}\right\|^{2}blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] - blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ] ≤ - italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
+Lηk22(F(𝒘k1)2+G2Bkm𝒮kIm,k)𝐿superscriptsubscript𝜂𝑘22superscriptnorm𝐹subscript𝒘𝑘12superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘\displaystyle+\frac{L\eta_{k}^{2}}{2}\left(\left\|{\nabla F(\boldsymbol{w}_{k-% 1})}\right\|^{2}+\frac{G^{2}}{B_{k}\sum_{m\in\mathcal{S}_{k}}I_{m,k}}\right)+ divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG )
=ηk(Lηk21)F(𝒘k1)2+Lηk22G2Bkm𝒮kIm,k.absentsubscript𝜂𝑘𝐿subscript𝜂𝑘21superscriptnorm𝐹subscript𝒘𝑘12𝐿superscriptsubscript𝜂𝑘22superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘\displaystyle=\eta_{k}\left(\frac{L\eta_{k}}{2}-1\right)\left\|{\nabla F(% \boldsymbol{w}_{k-1})}\right\|^{2}+\frac{L\eta_{k}^{2}}{2}\frac{G^{2}}{B_{k}% \sum_{m\in\mathcal{S}_{k}}I_{m,k}}.= italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG - 1 ) ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG .

Lemma 1 is proved.

Appendix B Proof of Theorem 1

According to Assumption 2 (μ𝜇\muitalic_μ-strong convexity of the loss functions), the Polyak-Lojasiewicz inequality holds

F(𝒘k1)22μ(F(𝒘k1)F(𝒘)).superscriptdelimited-∥∥𝐹subscript𝒘𝑘122𝜇𝐹subscript𝒘𝑘1𝐹superscript𝒘\lVert\nabla F(\boldsymbol{w}_{k-1})\rVert^{2}\geq 2\mu(F({\boldsymbol{w}}_{k-% 1})-F(\boldsymbol{w}^{*})).∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 italic_μ ( italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) . (33)

Substituting (33) into (12), and set ηk1Lsubscript𝜂𝑘1𝐿\eta_{k}\leq\frac{1}{L}italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_L end_ARG, there is

𝔼[F(𝒘k)]𝔼[F(𝒘k1)]𝔼delimited-[]𝐹subscript𝒘𝑘𝔼delimited-[]𝐹subscript𝒘𝑘1\displaystyle\mathbb{E}[F(\boldsymbol{w}_{k})]-\mathbb{E}[F(\boldsymbol{w}_{k-% 1})]blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] - blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ]
ηk(Lηk21)F(𝒘k1)2+Lηk22G2Bkm𝒮kIm,kabsentsubscript𝜂𝑘𝐿subscript𝜂𝑘21superscriptnorm𝐹subscript𝒘𝑘12𝐿superscriptsubscript𝜂𝑘22superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘\displaystyle\leq\eta_{k}\left(\frac{L\eta_{k}}{2}-1\right)\left\|{\nabla F(% \boldsymbol{w}_{k-1})}\right\|^{2}+\frac{L\eta_{k}^{2}}{2}\frac{G^{2}}{B_{k}% \sum_{m\in\mathcal{S}_{k}}I_{m,k}}≤ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG - 1 ) ∥ ∇ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_L italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG
ηtμ(𝔼[F(𝒘k1)]F(𝒘))+ηk2G2Bkm𝒮kIm,k.absentsubscript𝜂𝑡𝜇𝔼delimited-[]𝐹subscript𝒘𝑘1𝐹superscript𝒘subscript𝜂𝑘2superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘\displaystyle\leq-\eta_{t}\mu(\mathbb{E}[F({\boldsymbol{w}}_{k-1})]-F(% \boldsymbol{w}^{*}))+\frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in\mathcal{S}% _{k}}I_{m,k}}.≤ - italic_η start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_μ ( blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) + divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG .

With recursion, there is

𝔼[F(𝒘K)]F(𝒘)𝔼delimited-[]𝐹subscript𝒘𝐾𝐹superscript𝒘\displaystyle\mathbb{E}[F({\boldsymbol{w}}_{K})]-F(\boldsymbol{w}^{*})blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )
(1μηK)(𝔼[F(𝒘K1)]F(𝒘))+ηK2G2BKm𝒮KIm,Kabsent1𝜇subscript𝜂𝐾𝔼delimited-[]𝐹subscript𝒘𝐾1𝐹superscript𝒘subscript𝜂𝐾2superscript𝐺2subscript𝐵𝐾subscript𝑚subscript𝒮𝐾subscript𝐼𝑚𝐾\displaystyle\leq(1-\mu\eta_{K})(\mathbb{E}[F({\boldsymbol{w}}_{K-1})]-F(% \boldsymbol{w}^{*}))+\frac{\eta_{K}}{2}\frac{G^{2}}{B_{K}\sum_{m\in\mathcal{S}% _{K}}I_{m,K}}≤ ( 1 - italic_μ italic_η start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ) ( blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT italic_K - 1 end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) + divide start_ARG italic_η start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_K end_POSTSUBSCRIPT end_ARG
(𝔼[F(𝒘0)]F(𝒘))k=1K(1μηk)absent𝔼delimited-[]𝐹subscript𝒘0𝐹superscript𝒘superscriptsubscriptproduct𝑘1𝐾1𝜇subscript𝜂𝑘\displaystyle\leq\cdots\leq(\mathbb{E}[F({\boldsymbol{w}}_{0})]-F(\boldsymbol{% w}^{*}))\prod_{k=1}^{K}(1-\mu\eta_{k})≤ ⋯ ≤ ( blackboard_E [ italic_F ( bold_italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ] - italic_F ( bold_italic_w start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) ) ∏ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ( 1 - italic_μ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )
+k=1K1ηk2G2Bkm𝒮kIm,kj=k+1K(1μηj)superscriptsubscript𝑘1𝐾1subscript𝜂𝑘2superscript𝐺2subscript𝐵𝑘subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝑘superscriptsubscriptproduct𝑗𝑘1𝐾1𝜇subscript𝜂𝑗\displaystyle+\,\sum_{k=1}^{K-1}\frac{\eta_{k}}{2}\frac{G^{2}}{B_{k}\sum_{m\in% \mathcal{S}_{k}}I_{m,k}}\prod_{j=k+1}^{K}(1-\mu\eta_{j})+ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K - 1 end_POSTSUPERSCRIPT divide start_ARG italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG ∏ start_POSTSUBSCRIPT italic_j = italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ( 1 - italic_μ italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )
+ηK2G2BKm𝒮kIm,K.subscript𝜂𝐾2superscript𝐺2subscript𝐵𝐾subscript𝑚subscript𝒮𝑘subscript𝐼𝑚𝐾\displaystyle+\frac{\eta_{K}}{2}\frac{G^{2}}{B_{K}\sum_{m\in\mathcal{S}_{k}}I_% {m,K}}.+ divide start_ARG italic_η start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG divide start_ARG italic_G start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_m , italic_K end_POSTSUBSCRIPT end_ARG .

Theorem 1 is proved.

Appendix C Proof of Theorem 2

We define a quadratic Lyapunov function as

L(t)12m𝒮kqmSOV(t)2+12n𝒰kqnOPV(t)2.𝐿𝑡12subscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOVsuperscript𝑡212subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPVsuperscript𝑡2L(t)\triangleq\frac{1}{2}\sum_{m\in\mathcal{S}_{k}}q_{m}^{\text{SOV}}(t)^{2}+% \frac{1}{2}\sum_{n\in\mathcal{U}_{k}}q_{n}^{\text{OPV}}(t)^{2}.italic_L ( italic_t ) ≜ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

We define δmSOV(t)emcm(t)Emconsem,kcpTksuperscriptsubscript𝛿𝑚SOV𝑡subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝐸cons𝑚subscriptsuperscript𝑒cp𝑚𝑘subscript𝑇𝑘\delta_{m}^{\text{SOV}}(t)\triangleq e^{\text{cm}}_{m}(t)-\frac{E^{\text{cons}% }_{m}-e^{\text{cp}}_{m,k}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG, δnOPV(t)encm(t)EnconsTksuperscriptsubscript𝛿𝑛OPV𝑡subscriptsuperscript𝑒cm𝑛𝑡subscriptsuperscript𝐸cons𝑛subscript𝑇𝑘\delta_{n}^{\text{OPV}}(t)\triangleq e^{\text{cm}}_{n}(t)-\frac{E^{\text{cons}% }_{n}}{T_{k}}italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ≜ italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG, ϕmSOVmaxt{|δmSOV(t)|}superscriptsubscriptitalic-ϕ𝑚SOVsubscript𝑡superscriptsubscript𝛿𝑚SOV𝑡\phi_{m}^{\text{SOV}}\triangleq\max_{t}\{|\delta_{m}^{\text{SOV}}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) | }, ϕnOPVmaxt{|δnOPV(t)|}superscriptsubscriptitalic-ϕ𝑛OPVsubscript𝑡superscriptsubscript𝛿𝑛OPV𝑡\phi_{n}^{\text{OPV}}\triangleq\max_{t}\{|\delta_{n}^{\text{OPV}}(t)|\}italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ≜ roman_max start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT { | italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) | }, and Φm𝒮k(ϕmSOV)2+n𝒰k(ϕnOPV)2Φsubscript𝑚subscript𝒮𝑘superscriptsuperscriptsubscriptitalic-ϕ𝑚SOV2subscript𝑛subscript𝒰𝑘superscriptsuperscriptsubscriptitalic-ϕ𝑛OPV2\Phi\triangleq\sum_{m\in\mathcal{S}_{k}}(\phi_{m}^{\text{SOV}})^{2}+\sum_{n\in% \mathcal{U}_{k}}(\phi_{n}^{\text{OPV}})^{2}roman_Φ ≜ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Then, the Lyapunov drift of a single round is defined as

Δ(t)Δ𝑡\displaystyle\Delta(t)roman_Δ ( italic_t ) L(t+1)L(t)absent𝐿𝑡1𝐿𝑡\displaystyle\triangleq L(t+1)-L(t)≜ italic_L ( italic_t + 1 ) - italic_L ( italic_t )
=12m𝒮k(qmSOV(t+1)2qmSOV(t)2)absent12subscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOVsuperscript𝑡12superscriptsubscript𝑞𝑚SOVsuperscript𝑡2\displaystyle=\frac{1}{2}\sum_{m\in\mathcal{S}_{k}}\left(q_{m}^{\text{SOV}}(t+% 1)^{2}-q_{m}^{\text{SOV}}(t)^{2}\right)= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
+12n𝒰k(qnOPV(t+1)2qnOPV(t)2)12subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPVsuperscript𝑡12superscriptsubscript𝑞𝑛OPVsuperscript𝑡2\displaystyle+\frac{1}{2}\sum_{n\in\mathcal{U}_{k}}\left(q_{n}^{\text{OPV}}(t+% 1)^{2}-q_{n}^{\text{OPV}}(t)^{2}\right)+ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
12m𝒮k((qmSOV(t)+δmSOV(t))2qmSOV(t)2)absent12subscript𝑚subscript𝒮𝑘superscriptsuperscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡2superscriptsubscript𝑞𝑚SOVsuperscript𝑡2\displaystyle\leq\frac{1}{2}\sum_{m\in\mathcal{S}_{k}}\left(\left(q_{m}^{\text% {SOV}}(t)+\delta_{m}^{\text{SOV}}(t)\right)^{2}-q_{m}^{\text{SOV}}(t)^{2}% \right)\ ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ( italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) (34)
+12n𝒰k((qnOPV(t)+δnOPV(t))2qnOPV(t)2)12subscript𝑛subscript𝒰𝑘superscriptsuperscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡2superscriptsubscript𝑞𝑛OPVsuperscript𝑡2\displaystyle+\frac{1}{2}\sum_{n\in\mathcal{U}_{k}}\left(\left(q_{n}^{\text{% OPV}}(t)+\delta_{n}^{\text{OPV}}(t)\right)^{2}-q_{n}^{\text{OPV}}(t)^{2}\right)+ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ( italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) + italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
Φ+m𝒮kqmSOV(t)δmSOV(t)+n𝒰kqnOPV(t)δnOPV(t),absentΦsubscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡\displaystyle\leq\Phi+\sum_{m\in\mathcal{S}_{k}}q_{m}^{\text{SOV}}(t)\delta_{m% }^{\text{SOV}}(t)+\sum_{n\in\mathcal{U}_{k}}q_{n}^{\text{OPV}}(t)\delta_{n}^{% \text{OPV}}(t),≤ roman_Φ + ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) ,

By adding Vm𝒮kzm(t)dσ(ζm(t))dζm(t)𝑉subscript𝑚subscript𝒮𝑘subscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡-V\sum_{m\in\mathcal{S}_{k}}z_{m}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}- italic_V ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG on both sides of (34), the upper bound on the derivative-based drift-plus-penalty function is

Δ(t)Vm𝒮kzm(t)dσ(ζm(t))dζm(t)Φ+m𝒮kqmSOV(t)δmSOV(t)Δ𝑡𝑉subscript𝑚subscript𝒮𝑘subscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡Φsubscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡\displaystyle\Delta(t)-V\sum_{m\in\mathcal{S}_{k}}z_{m}(t)\frac{d\sigma(\zeta_% {m}(t))}{d\zeta_{m}(t)}\leq\Phi+\sum_{m\in\mathcal{S}_{k}}q_{m}^{\text{SOV}}(t% )\delta_{m}^{\text{SOV}}(t)roman_Δ ( italic_t ) - italic_V ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG ≤ roman_Φ + ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t )
+n𝒰kqnOPV(t)δnOPV(t)Vm𝒮kzm(t)dσ(ζm(t))dζm(t).subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡𝑉subscript𝑚subscript𝒮𝑘subscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle+\sum_{n\in\mathcal{U}_{k}}q_{n}^{\text{OPV}}(t)\delta_{n}^{\text% {OPV}}(t)-V\sum_{m\in\mathcal{S}_{k}}z_{m}(t)\frac{d\sigma(\zeta_{m}(t))}{d% \zeta_{m}(t)}.+ ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) - italic_V ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG .

We define the Tksubscript𝑇𝑘T_{k}italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-round drift as

ΔTksubscriptΔsubscript𝑇𝑘\displaystyle\Delta_{T_{k}}roman_Δ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT Δ(Tk+1)Δ(1)absentΔsubscript𝑇𝑘1Δ1\displaystyle\triangleq\Delta(T_{k}+1)-\Delta(1)≜ roman_Δ ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + 1 ) - roman_Δ ( 1 )
=m𝒮k12qmSOV(Tk+1)2+n𝒰k12qnOPV(Tk+1)2.absentsubscript𝑚subscript𝒮𝑘12superscriptsubscript𝑞𝑚SOVsuperscriptsubscript𝑇𝑘12subscript𝑛subscript𝒰𝑘12superscriptsubscript𝑞𝑛OPVsuperscriptsubscript𝑇𝑘12\displaystyle=\sum_{m\in\mathcal{S}_{k}}\frac{1}{2}q_{m}^{\text{SOV}}(T_{k}+1)% ^{2}+\sum_{n\in\mathcal{U}_{k}}\frac{1}{2}q_{n}^{\text{OPV}}(T_{k}+1)^{2}.= ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .

Then, the Tksubscript𝑇𝑘T_{k}italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-round drift-plus-penalty function is bounded by:

ΔTkVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)subscriptΔsubscript𝑇𝑘𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\Delta_{T_{k}}-V\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{% k}}z_{m}^{\dagger}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}roman_Δ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
TkΦVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)absentsubscript𝑇𝑘Φ𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq T_{k}\Phi-V\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{% k}}z_{m}^{\dagger}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}≤ italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Φ - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
+t𝒯k(m𝒮kqmSOV(t)δmSOV(t)+n𝒰kqnOPV(t)δnOPV(t))subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡\displaystyle+\sum_{t\in\mathcal{T}_{k}}\left(\sum_{m\in\mathcal{S}_{k}}q_{m}^% {\text{SOV}}(t)\delta_{m}^{\text{SOV}\dagger}(t)+\sum_{n\in\mathcal{U}_{k}}q_{% n}^{\text{OPV}}(t)\delta_{n}^{\text{OPV}\dagger}(t)\right)+ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV † end_POSTSUPERSCRIPT ( italic_t ) + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV † end_POSTSUPERSCRIPT ( italic_t ) )
(a)TkΦVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)𝑎subscript𝑇𝑘Φ𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\overset{(a)}{\leq}T_{k}\Phi-V\sum_{t\in\mathcal{T}_{k}}\sum_{m% \in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}start_OVERACCENT ( italic_a ) end_OVERACCENT start_ARG ≤ end_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Φ - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG (35)
+t𝒯k(m𝒮kqmSOV(t)δmSOV(t)+n𝒰kqnOPV(t)δnOPV(t)),subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡subscript𝑛subscript𝒰𝑘superscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡\displaystyle+\sum_{t\in\mathcal{T}_{k}}\left(\sum_{m\in\mathcal{S}_{k}}q_{m}^% {\text{SOV}}(t)\delta_{m}^{\text{SOV}*}(t)+\sum_{n\in\mathcal{U}_{k}}q_{n}^{% \text{OPV}}(t)\delta_{n}^{\text{OPV}*}(t)\right),+ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV ∗ end_POSTSUPERSCRIPT ( italic_t ) + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV ∗ end_POSTSUPERSCRIPT ( italic_t ) ) ,

where inequality (a)𝑎(a)( italic_a ) holds because solving P3𝑃3P3italic_P 3 yields a minimum value of (21a).

Based on the definition of qmSOV(t)superscriptsubscript𝑞𝑚SOV𝑡q_{m}^{\text{SOV}}(t)italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ), we have qmSOV(t+1)qmSOV(t)ϕmSOV(t),m𝒮k,t𝒯kformulae-sequencesuperscriptsubscript𝑞𝑚SOV𝑡1superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscriptitalic-ϕ𝑚SOV𝑡formulae-sequencefor-all𝑚subscript𝒮𝑘𝑡subscript𝒯𝑘q_{m}^{\text{SOV}}(t+1)-q_{m}^{\text{SOV}}(t)\leq\phi_{m}^{\text{SOV}}(t),% \forall m\in\mathcal{S}_{k},t\in\mathcal{T}_{k}italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t + 1 ) - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ≤ italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, and therefore

qmSOV(t)=qmSOV(t)qmSOV(1)superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝑞𝑚SOV1\displaystyle q_{m}^{\text{SOV}}(t)=q_{m}^{\text{SOV}}(t)-q_{m}^{\text{SOV}}(1)italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) = italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( 1 )
=τ=1t1(qmSOV(t+1)qmSOV(t))(t1)ϕmSOV,m𝒮k,formulae-sequenceabsentsuperscriptsubscript𝜏1𝑡1superscriptsubscript𝑞𝑚SOV𝑡1superscriptsubscript𝑞𝑚SOV𝑡𝑡1superscriptsubscriptitalic-ϕ𝑚SOVfor-all𝑚subscript𝒮𝑘\displaystyle=\sum_{\tau=1}^{t-1}\left(q_{m}^{\text{SOV}}(t+1)-q_{m}^{\text{% SOV}}(t)\right)\leq(t-1)\phi_{m}^{\text{SOV}},\quad\forall m\in\mathcal{S}_{k},= ∑ start_POSTSUBSCRIPT italic_τ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t - 1 end_POSTSUPERSCRIPT ( italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t + 1 ) - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) ) ≤ ( italic_t - 1 ) italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,

and

qmSOV(t)δmSOV(t)(t1)(ϕmSOV)2,m𝒮k.formulae-sequencesuperscriptsubscript𝑞𝑚SOV𝑡superscriptsubscript𝛿𝑚SOV𝑡𝑡1superscriptsuperscriptsubscriptitalic-ϕ𝑚SOV2for-all𝑚subscript𝒮𝑘q_{m}^{\text{SOV}}(t)\delta_{m}^{\text{SOV}*}(t)\leq(t-1)(\phi_{m}^{\text{SOV}% })^{2},\quad\forall m\in\mathcal{S}_{k}.italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV ∗ end_POSTSUPERSCRIPT ( italic_t ) ≤ ( italic_t - 1 ) ( italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (36)

Similarly, there is

qnOPV(t)δnOPV(t)(t1)(ϕnOPV)2,n𝒰k.formulae-sequencesuperscriptsubscript𝑞𝑛OPV𝑡superscriptsubscript𝛿𝑛OPV𝑡𝑡1superscriptsuperscriptsubscriptitalic-ϕ𝑛OPV2for-all𝑛subscript𝒰𝑘q_{n}^{\text{OPV}}(t)\delta_{n}^{\text{OPV}*}(t)\leq(t-1)(\phi_{n}^{\text{OPV}% })^{2},\quad\forall n\in\mathcal{U}_{k}.italic_q start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ( italic_t ) italic_δ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV ∗ end_POSTSUPERSCRIPT ( italic_t ) ≤ ( italic_t - 1 ) ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (37)

Substituting (36) and (37) into (35), we have

ΔTkVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)subscriptΔsubscript𝑇𝑘𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\Delta_{T_{k}}-V\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{% k}}z_{m}^{\dagger}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}roman_Δ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
TkΦVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)absentsubscript𝑇𝑘Φ𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq T_{k}\Phi-V\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{% k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}≤ italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Φ - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
+t𝒯k(m𝒮k(t1)(ϕmSOV)2+n𝒰k(t1)(ϕnOPV)2)subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘𝑡1superscriptsuperscriptsubscriptitalic-ϕ𝑚SOV2subscript𝑛subscript𝒰𝑘𝑡1superscriptsuperscriptsubscriptitalic-ϕ𝑛OPV2\displaystyle+\sum_{t\in\mathcal{T}_{k}}\left(\sum_{m\in\mathcal{S}_{k}}(t-1)(% \phi_{m}^{\text{SOV}})^{2}+\sum_{n\in\mathcal{U}_{k}}(t-1)(\phi_{n}^{\text{OPV% }})^{2}\right)+ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t - 1 ) ( italic_ϕ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_t - 1 ) ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT OPV end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )
=Tk2ΦVt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t).absentsuperscriptsubscript𝑇𝑘2Φ𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle=T_{k}^{2}\Phi-V\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{% k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}.= italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG .

Since ΔTk0subscriptΔsubscript𝑇𝑘0\Delta_{T_{k}}\geq 0roman_Δ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ 0, we have

t𝒯km𝒮ksubscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘\displaystyle\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT zm(t)dσ(ζm(t))dζm(t)superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle z_{m}^{\dagger}(t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
t𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t)Tk2ΦV.absentsubscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡superscriptsubscript𝑇𝑘2Φ𝑉\displaystyle\geq\sum_{t\in\mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}% (t)\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}-\frac{T_{k}^{2}\Phi}{V}.≥ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG - divide start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG italic_V end_ARG .

Since the function σ()𝜎\sigma(\cdot)italic_σ ( ⋅ ) is continuous and derivable, there exist a point ξm(t𝒯kzm(t),t𝒯kzm(t))subscript𝜉𝑚subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡\xi_{m}\in(\sum_{t\in\mathcal{T}_{k}}z_{m}^{\dagger}(t),\sum_{t\in\mathcal{T}_% {k}}z_{m}^{*}(t))italic_ξ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) , ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) such that

σ(ξm)ζm(t)=σ(t𝒯kzm(t))σ(t𝒯kzm(t))t𝒯kzm(t)t𝒯kzm(t).𝜎subscript𝜉𝑚subscript𝜁𝑚𝑡𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡\frac{\partial\sigma(\xi_{m})}{\partial\zeta_{m}(t)}=\frac{\sigma\left(\sum_{t% \in\mathcal{T}_{k}}z_{m}^{*}(t)\right)-\sigma\left(\sum_{t\in\mathcal{T}_{k}}z% _{m}^{\dagger}(t)\right)}{\sum_{t\in\mathcal{T}_{k}}z_{m}^{*}(t)-\sum_{t\in% \mathcal{T}_{k}}z_{m}^{\dagger}(t)}.divide start_ARG ∂ italic_σ ( italic_ξ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG = divide start_ARG italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) - italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) - ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) end_ARG .

Based on (18), we have

Tk2ΦVt𝒯km𝒮k(zm(t)zm(t))dσ(ζm(t))dζm(t)superscriptsubscript𝑇𝑘2Φ𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\frac{T_{k}^{2}\Phi}{V}\geq\sum_{t\in\mathcal{T}_{k}}\sum_{m\in% \mathcal{S}_{k}}\left(z_{m}^{*}(t)-z_{m}^{\dagger}(t)\right)\frac{d\sigma(% \zeta_{m}(t))}{d\zeta_{m}(t)}divide start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG italic_V end_ARG ≥ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) - italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
m𝒮k(t𝒯kzm(t)t𝒯kzm(t))ψ(α)σ(Q)ζm(t)absentsubscript𝑚subscript𝒮𝑘subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡𝜓𝛼𝜎𝑄subscript𝜁𝑚𝑡\displaystyle\geq\sum_{m\in\mathcal{S}_{k}}\left(\sum_{t\in\mathcal{T}_{k}}z_{% m}^{*}(t)-\sum_{t\in\mathcal{T}_{k}}z_{m}^{\dagger}(t)\right)\psi(\alpha)\frac% {\partial\sigma(Q)}{\partial\zeta_{m}(t)}≥ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) - ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) italic_ψ ( italic_α ) divide start_ARG ∂ italic_σ ( italic_Q ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
m𝒮k(t𝒯kzm(t)t𝒯kzm(t))ψ(α)σ(ξm)ζm(t)absentsubscript𝑚subscript𝒮𝑘subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡𝜓𝛼𝜎subscript𝜉𝑚subscript𝜁𝑚𝑡\displaystyle\geq\sum_{m\in\mathcal{S}_{k}}\left(\sum_{t\in\mathcal{T}_{k}}z_{% m}^{*}(t)-\sum_{t\in\mathcal{T}_{k}}z_{m}^{\dagger}(t)\right)\psi(\alpha)\frac% {\partial\sigma(\xi_{m})}{\partial\zeta_{m}(t)}≥ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) - ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) italic_ψ ( italic_α ) divide start_ARG ∂ italic_σ ( italic_ξ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) end_ARG start_ARG ∂ italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG
=ψ(α)[m𝒮kσ(t𝒯kzm(t))m𝒮kσ(t𝒯kzm(t))].absent𝜓𝛼delimited-[]subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡\displaystyle=\psi(\alpha)\left[\sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t% \in\mathcal{T}_{k}}z_{m}^{*}(t)\right)-\sum_{m\in\mathcal{S}_{k}}\sigma\left(% \sum_{t\in\mathcal{T}_{k}}z_{m}^{\dagger}(t)\right)\right].= italic_ψ ( italic_α ) [ ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) - ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) ] .

Finally, there is

m𝒮kσ(t𝒯kzm(t))m𝒮kσ(t𝒯kzm(t))Tk2ΦVψ(α).subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡subscript𝑚subscript𝒮𝑘𝜎subscript𝑡subscript𝒯𝑘superscriptsubscript𝑧𝑚𝑡superscriptsubscript𝑇𝑘2Φ𝑉𝜓𝛼\sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}^{*}(t)% \right)-\sum_{m\in\mathcal{S}_{k}}\sigma\left(\sum_{t\in\mathcal{T}_{k}}z_{m}^% {\dagger}(t)\right)\leq\frac{T_{k}^{2}\Phi}{V\psi(\alpha)}.∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) - ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ ( ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_t ) ) ≤ divide start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ end_ARG start_ARG italic_V italic_ψ ( italic_α ) end_ARG .

For energy consumption, we have

t𝒯k(emcm(t)EmconsTk)+em,kcpt𝒯kqmSOV(t+1)qmSOV(t)subscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝐸cons𝑚subscript𝑇𝑘subscriptsuperscript𝑒cp𝑚𝑘subscript𝑡subscript𝒯𝑘superscriptsubscript𝑞𝑚SOV𝑡1superscriptsubscript𝑞𝑚SOV𝑡\displaystyle\sum_{t\in\mathcal{T}_{k}}\left(e^{\text{cm}}_{m}(t)-\frac{E^{% \text{cons}}_{m}}{T_{k}}\right)+e^{\text{cp}}_{m,k}\leq\sum_{t\in\mathcal{T}_{% k}}q_{m}^{\text{SOV}}(t+1)-q_{m}^{\text{SOV}}(t)∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - divide start_ARG italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG start_ARG italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ) + italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT ≤ ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t + 1 ) - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t )
2ΔTk2Tk2Φ2Vt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t).absent2subscriptΔsubscript𝑇𝑘2superscriptsubscript𝑇𝑘2Φ2𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq\sqrt{2\Delta_{T_{k}}}\leq\sqrt{2T_{k}^{2}\Phi-2V\sum_{t\in% \mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(% t))}{d\zeta_{m}(t)}}.≤ square-root start_ARG 2 roman_Δ start_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ≤ square-root start_ARG 2 italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - 2 italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG end_ARG .

Therefore, the energy consumption of m𝒮k𝑚subscript𝒮𝑘m\in\mathcal{S}_{k}italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is bounded by

t𝒯kemcm(t)+em,kcpsubscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑚𝑡subscriptsuperscript𝑒cp𝑚𝑘\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{m}(t)+e^{\text{cp}}_{m,k}∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) + italic_e start_POSTSUPERSCRIPT cp end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m , italic_k end_POSTSUBSCRIPT
Emcons+2Tk2Φ2Vt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t).absentsubscriptsuperscript𝐸cons𝑚2superscriptsubscript𝑇𝑘2Φ2𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq E^{\text{cons}}_{m}+\sqrt{2T_{k}^{2}\Phi-2V\sum_{t\in% \mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(% t))}{d\zeta_{m}(t)}}.≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + square-root start_ARG 2 italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - 2 italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG end_ARG .

Likewise, the energy consumption of n𝒰k𝑛subscript𝒰𝑘n\in\mathcal{U}_{k}italic_n ∈ caligraphic_U start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is bounded by

t𝒯kencm(t)subscript𝑡subscript𝒯𝑘subscriptsuperscript𝑒cm𝑛𝑡\displaystyle\sum_{t\in\mathcal{T}_{k}}e^{\text{cm}}_{n}(t)∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT cm end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t )
Encons+2Tk2Φ2Vt𝒯km𝒮kzm(t)dσ(ζm(t))dζm(t).absentsubscriptsuperscript𝐸cons𝑛2superscriptsubscript𝑇𝑘2Φ2𝑉subscript𝑡subscript𝒯𝑘subscript𝑚subscript𝒮𝑘superscriptsubscript𝑧𝑚𝑡𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡\displaystyle\leq E^{\text{cons}}_{n}+\sqrt{2T_{k}^{2}\Phi-2V\sum_{t\in% \mathcal{T}_{k}}\sum_{m\in\mathcal{S}_{k}}z_{m}^{*}(t)\frac{d\sigma(\zeta_{m}(% t))}{d\zeta_{m}(t)}}.≤ italic_E start_POSTSUPERSCRIPT cons end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + square-root start_ARG 2 italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ - 2 italic_V ∑ start_POSTSUBSCRIPT italic_t ∈ caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ∈ caligraphic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG end_ARG .

Theorem 2 is proved.

Appendix D Proof of Proposition 1

The Lagrangian of P3.1𝑃3.1P3.1italic_P 3.1 is given by:

=Vdσ(ζm(t))dζm(t)κβlog2(1+pm(t)|hm,r(t)|2βN0)𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝜅𝛽subscript21subscript𝑝𝑚𝑡superscriptsubscript𝑚𝑟𝑡2𝛽subscript𝑁0\displaystyle\mathcal{L}=V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\kappa% \beta\log_{2}\left(1+\frac{p_{m}(t)|h_{m,r}(t)|^{2}}{\beta N_{0}}\right)caligraphic_L = italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_κ italic_β roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG )
κqmSOV(t)pm(t)λmpm(t)+νm(pmmaxpm(t)).𝜅superscriptsubscript𝑞𝑚SOV𝑡subscript𝑝𝑚𝑡subscript𝜆𝑚subscript𝑝𝑚𝑡subscript𝜈𝑚subscriptsuperscript𝑝max𝑚subscript𝑝𝑚𝑡\displaystyle-\kappa q_{m}^{\text{SOV}}(t)p_{m}(t)-\lambda_{m}p_{m}(t)+\nu_{m}% (p^{\text{max}}_{m}-p_{m}(t)).- italic_κ italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) - italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) + italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) .

Then the KKT condition is given by:

λmpm(t)=0,superscriptsubscript𝜆𝑚superscriptsubscript𝑝𝑚𝑡0\displaystyle\lambda_{m}^{*}p_{m}^{*}(t)=0,italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = 0 ,
νm(pmmaxpm(t))=0,superscriptsubscript𝜈𝑚subscriptsuperscript𝑝max𝑚superscriptsubscript𝑝𝑚𝑡0\displaystyle\nu_{m}^{*}(p^{\text{max}}_{m}-p_{m}^{*}(t))=0,italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) ) = 0 ,
λm,νm0,superscriptsubscript𝜆𝑚superscriptsubscript𝜈𝑚0\displaystyle\lambda_{m}^{*},\nu_{m}^{*}\geq 0,italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≥ 0 ,
Vdσ(ζm(t))dζm(t)κβ|hm,r(t)|2βN01+pm(t)|hm,r(t)|2βN0qmSOV(t)κλm+νm=0.𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝜅𝛽superscriptsubscript𝑚𝑟𝑡2𝛽subscript𝑁01superscriptsubscript𝑝𝑚𝑡superscriptsubscript𝑚𝑟𝑡2𝛽subscript𝑁0superscriptsubscript𝑞𝑚SOV𝑡𝜅superscriptsubscript𝜆𝑚superscriptsubscript𝜈𝑚0\displaystyle\frac{V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\kappa\beta% \frac{|h_{m,r}(t)|^{2}}{\beta N_{0}}}{1+\frac{p_{m}^{*}(t)|h_{m,r}(t)|^{2}}{% \beta N_{0}}}-q_{m}^{\text{SOV}}(t)\kappa-\lambda_{m}^{*}+\nu_{m}^{*}=0.divide start_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_κ italic_β divide start_ARG | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG start_ARG 1 + divide start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG - italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) italic_κ - italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT + italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0 .

If neither λmsuperscriptsubscript𝜆𝑚\lambda_{m}^{*}italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT nor νmsuperscriptsubscript𝜈𝑚\nu_{m}^{*}italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is zero, there is no solution to these equations. Therefore, three cases are considered:
1) If λm=0superscriptsubscript𝜆𝑚0\lambda_{m}^{*}=0italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0, νm=0superscriptsubscript𝜈𝑚0\nu_{m}^{*}=0italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0, then

pm(t)=Vdσ(ζm(t))dζm(t)βqmSOV(t)βN0|hm,r(t)|2.superscriptsubscript𝑝𝑚𝑡𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝛽superscriptsubscript𝑞𝑚SOV𝑡𝛽subscript𝑁0superscriptsubscript𝑚𝑟𝑡2p_{m}^{*}(t)=\frac{V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\beta}{q_{m}^{% \text{SOV}}(t)}-\frac{\beta N_{0}}{|h_{m,r}(t)|^{2}}.italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = divide start_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_β end_ARG start_ARG italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) end_ARG - divide start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .

2) If λm=0superscriptsubscript𝜆𝑚0\lambda_{m}^{*}=0italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0, νm0superscriptsubscript𝜈𝑚0\nu_{m}^{*}\neq 0italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≠ 0, then pm(t)=pmmaxsubscript𝑝𝑚superscript𝑡subscriptsuperscript𝑝max𝑚p_{m}(t)^{*}=p^{\text{max}}_{m}italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT.
3) If λm0superscriptsubscript𝜆𝑚0\lambda_{m}^{*}\neq 0italic_λ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ≠ 0, νm=0superscriptsubscript𝜈𝑚0\nu_{m}^{*}=0italic_ν start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0, then pm(t)=0subscript𝑝𝑚superscript𝑡0p_{m}(t)^{*}=0italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 0. We get:

pm(t)=[Vdσ(ζm(t))dζm(t)βqmSOV(t)βN0|hm,r(t)|2]0pmmax,superscriptsubscript𝑝𝑚𝑡subscriptsuperscriptdelimited-[]𝑉𝑑𝜎subscript𝜁𝑚𝑡𝑑subscript𝜁𝑚𝑡𝛽superscriptsubscript𝑞𝑚SOV𝑡𝛽subscript𝑁0superscriptsubscript𝑚𝑟𝑡2subscriptsuperscript𝑝max𝑚0p_{m}^{*}(t)=\left[\frac{V\frac{d\sigma(\zeta_{m}(t))}{d\zeta_{m}(t)}\beta}{q_% {m}^{\text{SOV}}(t)}-\frac{\beta N_{0}}{|h_{m,r}(t)|^{2}}\right]^{p^{\text{max% }}_{m}}_{0},italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_t ) = [ divide start_ARG italic_V divide start_ARG italic_d italic_σ ( italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) ) end_ARG start_ARG italic_d italic_ζ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_t ) end_ARG italic_β end_ARG start_ARG italic_q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT SOV end_POSTSUPERSCRIPT ( italic_t ) end_ARG - divide start_ARG italic_β italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG | italic_h start_POSTSUBSCRIPT italic_m , italic_r end_POSTSUBSCRIPT ( italic_t ) | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,

where [a]0pmmaxsubscriptsuperscriptdelimited-[]𝑎subscriptsuperscript𝑝max𝑚0[a]^{p^{\text{max}}_{m}}_{0}[ italic_a ] start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT denotes min(max(a,0),pmmax)𝑎0subscriptsuperscript𝑝max𝑚\min(\max(a,0),p^{\text{max}}_{m})roman_min ( roman_max ( italic_a , 0 ) , italic_p start_POSTSUPERSCRIPT max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ).

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