Edge computing enables Mobile Autonomous Systems (MASs) to execute continuous streams of heavy-duty mission-critical processing tasks, such as real-time obstacle detection and navigation. However, in practical applications, erratic patterns in channel quality, network load, and edge server load can interrupt the task flow execution, which necessarily leads to severe disruption of the system's key operations. Existing work has mostly tackled the problem with reactive approaches, which cannot guarantee task-level reliability. Conversely, in this paper we focus on learning-based predictive edge computing to achieve self-resilient task offloading. By conducting a preliminary experimental evaluation, we show that there is no dominant feature that can predict the edge-MAS system reliability, which calls for an ensemble and selection of weaker features. To tackle the complexity of the problem, we propose SeReMAS, a data-driven optimization framework. We first mathematically formulate a Redundant Task Offloading Problem (RTOP), where a MAS may connect to multiple edge servers for redundancy, and needs to select which server(s) to transmit its computing tasks in order to maximize the probability of task execution while minimizing channel and edge resource utilization. We then create a predictor based on Deep Reinforcement Learning (DRL), which produces the optimum task assignment based on application-, network- and telemetry-based features. We prototype SeReMAS on a testbed composed by a drone, mounting a PixHawk flight controller, a Jetson Nano board, and three 802.11n WiFi interfaces. We extensively evaluate SeReMAS by considering an application where one drone offloads high-resolution images for real-time analysis to three edge servers on the ground. Experimental results show that SeReMAS improves task execution probability by $17\%$ with respect to existing reactive-based approaches.