To capture the ubiquitous randomness such as time-varying wireless channel and unpredictable task arrivals in the non-orthogonal multiple access aided multi-access edge computing networks, we formulate a stochastic optimization problem aiming to minimize the time-average cost for Internet of Things devices in this paper. Due to the absence of distribution of random network information, we develop a stochastic gradient descent (SGD) based method to learn the randomness online and minimize the cost asymptotically. The proposed SGD method makes decisions only depending on the observed network information in each time-slot and achieves an [O(ε),O(1/ε)]-tradeoff between the cost-optimality and task queue backlog. To polish this tradeoff, we further propose a momentum-based SGD method by amending SGD iterations with momentum terms, which can efficiently accelerate algorithm convergence while reducing the task queue backlog without loss of cost-optimality. Finally, simulation results confirm the outstanding performance of the proposed methods.