Rare events, despite their infrequency, often carry critical information and require immediate attentions in mission-critical applications such as autonomous driving, healthcare, and industrial automation. The data-intensive nature of these tasks and their need for prompt responses, combined with designing edge AI (or edge inference), pose significant challenges in systems and techniques. Existing edge inference approaches often suffer from communication bottlenecks due to high-dimensional data transmission and fail to provide timely responses to rare events, limiting their effectiveness for mission-critical applications in the sixth-generation (6G) mobile networks. To overcome these challenges, we propose a channel-adaptive, event-triggered edge-inference framework that prioritizes efficient rare-event processing. Central to this framework is a dual-threshold, multi-exit architecture, which enables early local inference for rare events detected locally while offloading more complex rare events to edge servers for detailed classification. To further enhance the system's performance, we developed a channel-adaptive offloading policy paired with an online algorithm to dynamically determine the optimal confidence thresholds for controlling offloading decisions. The associated optimization problem is solved by reformulating the original non-convex function into an equivalent strongly convex one. Using deep neural network classifiers and real medical datasets, our experiments demonstrate that the proposed framework not only achieves superior rare-event classification accuracy, but also effectively reduces communication overhead, as opposed to existing edge-inference approaches.

Federated Dropout is an efficient technique to overcome both communication and computation bottlenecks for deploying federated learning at the network edge. In each training round, an edge device only needs to update and transmit a sub-model, which is generated by the typical method of dropout in deep learning, and thus effectively reduces the per-round latency. \textcolor{blue}{However, the theoretical convergence analysis for Federated Dropout is still lacking in the literature, particularly regarding the quantitative influence of dropout rate on convergence}. To address this issue, by using the Taylor expansion method, we mathematically show that the gradient variance increases with a scaling factor of $\gamma/(1-\gamma)$, with $\gamma \in [0, \theta)$ denoting the dropout rate and $\theta$ being the maximum dropout rate ensuring the loss function reduction. Based on the above approximation, we provide the convergence analysis for Federated Dropout. Specifically, it is shown that a larger dropout rate of each device leads to a slower convergence rate. This provides a theoretical foundation for reducing the convergence latency by making a tradeoff between the per-round latency and the overall rounds till convergence. Moreover, a low-complexity algorithm is proposed to jointly optimize the dropout rate and the bandwidth allocation for minimizing the loss function in all rounds under a given per-round latency and limited network resources. Finally, numerical results are provided to verify the effectiveness of the proposed algorithm.