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• Proactive auto-scaling method for container-based edge computing web applications. • Multi-step Federated Learning prediction model using BiGRU model in Edge Computing. • Reducing the data transmission between edge nodes and the cloud server. • Multi-step prediction in proactive auto-scaler approach in Kubernetes. Edge computing has emerged as an attractive alternative to traditional cloud computing by utilizing processing, network, and storage resources close to end devices, such as Internet of Things (IoT) sensors. Edge computing is still in its infancy, and resource provisioning and service scheduling remain research concerns. Kubernetes is a container orchestration tool in distributed environments. Proactive auto-scaling techniques in Kubernetes improve utilization by allocating resources based on future workload prediction. However, prediction models run on central cloud nodes, necessitating data transfer between edge and cloud nodes, which increases latency and response time. We present FedAvg-BiGRU, a proactive auto-scaling method in edge computing based on FedAvg and multi-step prediction by a Bidirectional Gated Recurrent Unit (BiGRU). FedAvg is a technique for training machine learning models in a Federated Learning (FL) model. FL reduces network traffic by exchanging only model updates rather than raw data, relieving learning models of the need to store data on a centralized cloud server. In addition, a technique for determining the number of Kubernetes pods based on the Cool Down Time (CDT) concept has been developed, preventing contradictory scaling actions. To our knowledge, our work is the first to employ FL for proactive auto-scaling in cloud and edge computing. The results demonstrate that the FedAvg-BiGRU method has a slightly higher prediction error than the load centralized processing mode, although this difference is not statistically significant. At the same time, it reduces the amount of data transmission between the edge nodes and the cloud server. [ABSTRACT FROM AUTHOR]