Classical worst-case optimization theory neither explains the success of optimization in machine learning, nor does it help with step size selection. In this paper we demonstrate the viability and advantages of replacing the classical'convex function' framework with a'random function' framework. With complexity $\mathcal{O}(n^3d^3)$, where $n$ is the number of steps and $d$ the number of dimensions, Bayesian optimization with gradients has not been viable in large dimension so far.

In Wireless Networked Control Systems (WNCSs), control and communication systems must be co-designed due to their strong interdependence. This paper presents a novel optimization theory-based safe deep reinforcement learning (DRL) framework for ultra-reliable WNCSs, ensuring constraint satisfaction while optimizing performance, for the first time in the literature. The approach minimizes power consumption under key constraints, including Peak Age of Information (PAoI) violation probability, transmit power, and schedulability in the finite blocklength regime. PAoI violation probability is uniquely derived by combining stochastic maximum allowable transfer interval (MATI) and maximum allowable packet delay (MAD) constraints in a multi-sensor network. The framework consists of two stages: optimization theory and safe DRL. The first stage derives optimality conditions to establish mathematical relationships among variables, simplifying and decomposing the problem. The second stage employs a safe DRL model where a teacher-student framework guides the DRL agent (student). The control mechanism (teacher) evaluates compliance with system constraints and suggests the nearest feasible action when needed. Extensive simulations show that the proposed framework outperforms rule-based and other optimization theory based DRL benchmarks, achieving faster convergence, higher rewards, and greater stability.

Classical worst-case optimization theory neither explains the success of optimization in machine learning, nor does it help with step size selection. In this paper we demonstrate the viability and advantages of replacing the classical'convex function' framework with a'random function' framework. With complexity \mathcal{O}(n 3d 3), where n is the number of steps and d the number of dimensions, Bayesian optimization with gradients has not been viable in large dimension so far. Specifically, we use a'stochastic Taylor approximation' to rediscover gradient descent, which is scalable in high dimension due to \mathcal{O}(nd) complexity. This rediscovery yields a specific step size schedule we call Random Function Descent (RFD).

Abstract--Mobile Edge Computing (MEC) offers low-latency and high-bandwidth support for Internet-of-Vehicles (IoV) applications. However, due to high vehicle mobility and finite communication coverage of base stations, it is hard to maintain uninterrupted and high-quality services without proper service migration among MEC servers. Existing solutions commonly rely on prior knowledge and rarely consider efficient resource allocation during the service migration process, making it hard to reach optimal performance in dynamic IoV environments. To address these important challenges, we propose SR-CL, a novel mobility-aware seamless Service migration and Resource allocation framework via Convex-optimization-enabled deep reinforcement Learning in multi-edge IoV systems. First, we decouple the Mixed Integer Nonlinear Programming (MINLP) problem of service migration and resource allocation into two sub-problems. Next, we design a new actor-critic-based asynchronous-update deep reinforcement learning method to handle service migration, where the delayed-update actor makes migration decisions and the one-step-update critic evaluates the decisions to guide the policy update. Notably, we theoretically derive the optimal resource allocation with convex optimization for each MEC server, thereby further improving system performance. Using the real-world datasets of vehicle trajectories and testbed, extensive experiments are conducted to verify the effectiveness of the proposed SR-CL. Compared to benchmark methods, the SR-CL achieves superior convergence and delay performance under various scenarios. However, the real-time demands of IoV applications pose When vehicles offload tasks, MEC servers create dedicated significant challenges for onboard processors with limited service instances via virtualization techniques for the vehicles computational capabilities [2]. Although Cloud Computing and allocate proper resources to them [7].

Lyapunov optimization theory has recently emerged as a powerful mathematical framework for solving complex stochastic optimization problems by transforming long-term objectives into a sequence of real-time short-term decisions while ensuring system stability. This theory is particularly valuable in unmanned aerial vehicle (UAV)-based low-altitude economy (LAE) networking scenarios, where it could effectively address inherent challenges of dynamic network conditions, multiple optimization objectives, and stability requirements. Recently, generative artificial intelligence (GenAI) has garnered significant attention for its unprecedented capability to generate diverse digital content. Extending beyond content generation, in this paper, we propose a framework integrating generative diffusion models with reinforcement learning to address Lyapunov optimization problems in UAV-based LAE networking. We begin by introducing the fundamentals of Lyapunov optimization theory and analyzing the limitations of both conventional methods and traditional AI-enabled approaches. We then examine various GenAI models and comprehensively analyze their potential contributions to Lyapunov optimization. Subsequently, we develop a Lyapunov-guided generative diffusion model-based reinforcement learning framework and validate its effectiveness through a UAV-based LAE networking case study. Finally, we outline several directions for future research.

Traditional wireless network design relies on optimization algorithms derived from domain-specific mathematical models, which are often inefficient and unsuitable for dynamic, real-time applications due to high complexity. Deep learning has emerged as a promising alternative to overcome complexity and adaptability concerns, but it faces challenges such as accuracy issues, delays, and limited interpretability due to its inherent black-box nature. This paper introduces a novel approach that integrates optimization theory with deep learning methodologies to address these issues. The methodology starts by constructing the block diagram of the optimization theory-based solution, identifying key building blocks corresponding to optimality conditions and iterative solutions. Selected building blocks are then replaced with deep neural networks, enhancing the adaptability and interpretability of the system. Extensive simulations show that this hybrid approach not only reduces runtime compared to optimization theory based approaches but also significantly improves accuracy and convergence rates, outperforming pure deep learning models.

Optimization theory serves as a pivotal scientific instrument for achieving optimal system performance, with its origins in economic applications to identify the best investment strategies for maximizing benefits. Over the centuries, from the geometric inquiries of ancient Greece to the calculus contributions by Newton and Leibniz, optimization theory has significantly advanced. The persistent work of scientists like Lagrange, Cauchy, and von Neumann has fortified its progress. The modern era has seen an unprecedented expansion of optimization theory applications, particularly with the growth of computer science, enabling more sophisticated computational practices and widespread utilization across engineering, decision analysis, and operations research. This paper delves into the profound relationship between optimization theory and deep learning, highlighting the omnipresence of optimization problems in the latter. We explore the gradient descent algorithm and its variants, which are the cornerstone of optimizing neural networks. The chapter introduces an enhancement to the SGD optimizer, drawing inspiration from numerical optimization methods, aiming to enhance interpretability and accuracy. Our experiments on diverse deep learning tasks substantiate the improved algorithm's efficacy. The paper concludes by emphasizing the continuous development of optimization theory and its expanding role in solving intricate problems, enhancing computational capabilities, and informing better policy decisions.

The design of Wireless Networked Control System (WNCS) requires addressing critical interactions between control and communication systems with minimal complexity and communication overhead while providing ultra-high reliability. This paper introduces a novel optimization theory based deep reinforcement learning (DRL) framework for the joint design of controller and communication systems. The objective of minimum power consumption is targeted while satisfying the schedulability and rate constraints of the communication system in the finite blocklength regime and stability constraint of the control system. Decision variables include the sampling period in the control system, and blocklength and packet error probability in the communication system. The proposed framework contains two stages: optimization theory and DRL. In the optimization theory stage, following the formulation of the joint optimization problem, optimality conditions are derived to find the mathematical relations between the optimal values of the decision variables. These relations allow the decomposition of the problem into multiple building blocks. In the DRL stage, the blocks that are simplified but not tractable are replaced by DRL. Via extensive simulations, the proposed optimization theory based DRL approach is demonstrated to outperform the optimization theory and pure DRL based approaches, with close to optimal performance and much lower complexity.

The success of deep neural networks is in part due to the use of normalization layers. Normalization layers like Batch Normalization, Layer Normalization and Weight Normalization are ubiquitous in practice, as they improve generalization performance and speed up training significantly. Nonetheless, the vast majority of current deep learning theory and non-convex optimization literature focuses on the un-normalized setting, where the functions under consideration do not exhibit the properties of commonly normalized neural networks. In this paper, we bridge this gap by giving the first global convergence result for two-layer neural networks with ReLU activations trained with a normalization layer, namely Weight Normalization. Our analysis shows how the introduction of normalization layers changes the optimization landscape and can enable faster convergence as compared with un-normalized neural networks.

Federated learning (FL) is a distributed learning approach where a set of end-user devices participate in the learning process by acting on their isolated local data sets. Here, we process local data sets of users where worst-case optimization theory is used to reformulate the FL problem where the impact of local data sets in training phase is considered as an uncertain function bounded in a closed uncertainty region. This representation allows us to compare the performance of FL with its centralized counterpart, and to replace the uncertain function with a concept of protection functions leading to more tractable formulation. The latter supports applying a regularization factor in each user cost function in FL to reach a better performance. We evaluated our model using the MNIST data set versus the protection function parameters, e.g., regularization factors.