## 1 Introduction  
In recent years, with the rapid development of cloud computing and the Internet of Things (IoT), the combination of cloud and edge has become one of the research hotspots. Traditional cloud computing has the advantage of strong resource service capability but the disadvantage of distant transmission [1], while edge computing has the advantage of low transmission latency but limited resource constraints [2]. Cloud-edge integration refers to a computing mode that combines cloud-computing resources with edge-computing resources. The goal is to make more efficient use of edge-computing resources, while also fully leveraging the advantages of cloud computing [3, 4]. Caching plays a pivotal role in reducing bandwidth traffic and the number of remote server accesses, thereby enhancing the system response time and reliability in the collaborative environment of cloud-edge computing [5]. However, caching in cloud-edge environments poses several challenges. Firstly, limited hardware resources impose constraints on cache capacity [6, 7]. Secondly, restricted network bandwidth may result in prolonged response times for the cache when cache hit rates are low [8]. Cloud-edge caching is typically divided into cloud cache and edge cache, and a caching policy is employed to ensure that popular content is cached as much as possible at the edge. However, existing cloud-edge caching policies face two major problems. Firstly, due to the heterogeneous resources and communication links of the edge base station (BS), such caching strategies can become exceedingly complex. Secondly, the dynamics of user demandslead to real-time changes in content popularity, necessitating time-varying cache content. This requires caching policies capable of adapting in real time to the dynamic caching environment. Therefore, effectively managing caching in real time to improve cache hit rates and reduce response time remains a persistent and crucial challenge. DRL, an AI model that can dynamically generate policies based on the environment, has seen successful applications in various cloud-edge collaborative environments, including wireless access network mode selection and resource management [9], autonomous driving strategy [10], and computation offloading optimization [11]. In the context of cache optimization, DRL methods have been utilized to improve cache efficiency by predicting content popularity [12] and dynamically adjusting cache policies [13; 14; 15]. Despite these achievements, using DRL-based caching strategies in cloud-edge environments still poses two significant challenges: (1) limited hardware resources, which leads to non-negligible energy consumption during DRL training, and (2) the need to jointly schedule cloud and edge storage resources for caching, which raises security and privacy concerns. Federated learning can be conducted at local BSs for learning and inference, and models can be aggregated and distributed in the cloud. Therefore, under federated learning, each BS can take advantage of its local characteristics while also learning from the experiences of other BSs, allowing it to adjust caching policies in real-time while maintaining data privacy. Building on the work of [16] that proposed dynamically adjusting the number of local iterations depending on resource utilization in federated learning, we propose a new federated deep reinforcement learning framework that can effectively reduce energy consumption while ensuring cache performance through the sharing and collaboration of models among multiple nodes. The federated nature of the learning framework maximizes data security and prevents data leaks. This paper's main contributions include the following:  
(1) Creating a system model for the cache environment that combines cloud and edge, including network, cache performance, DRL, and energy-consumption models, and proposing a new objective function of the cache energy efficiency ratio to balance performance and energy consumption.