Recent advancements in Deep Neural Networks (DNNs) have catalyzed the development of numerous intelligent mobile applications and services. However, they also introduce significant computational challenges for resource-constrained mobile devices. To address this, collaborative edge inference has been proposed. This method involves partitioning a DNN inference task into several subtasks and distributing these across multiple network nodes. Despite its potential, most current approaches presume known network parameters -- like node processing speeds and link transmission rates -- or rely on a fixed sequence of nodes for processing the DNN subtasks. In this paper, we tackle a more complex scenario where network parameters are unknown and must be learned, and multiple network paths are available for distributing inference tasks. Specifically, we explore the learning problem of selecting the optimal network path and assigning DNN layers to nodes along this path, considering potential security threats and the costs of switching paths. We begin by deriving structural insights from the DNN layer assignment with complete network information, which narrows down the decision space and provides crucial understanding of optimal assignments. We then cast the learning problem with incomplete network information as a novel adversarial group linear bandits problem with switching costs, featuring rewards generation through a combined stochastic and adversarial process. We introduce a new bandit algorithm, B-EXPUCB, which combines elements of the classical blocked EXP3 and LinUCB algorithms, and demonstrate its sublinear regret. Extensive simulations confirm B-EXPUCB's superior performance in learning for collaborative edge inference over existing algorithms.