  In this paper, we consider a mobile-edge computing system, where an access
point assists a mobile device (MD) to execute an application consisting of
multiple tasks following a general task call graph. The objective is to jointly
determine the offloading decision of each task and the resource allocation
under time-varying wireless fading channels and stochastic edge computing
capability, so that the energy-time cost (ETC) of the MD is minimized. Solving
the problem is particularly hard due to the combinatorial offloading decisions
and the strong coupling among task executions under the general dependency
model. Conventional numerical optimization methods are inefficient to solve
such a problem, especially when the problem size is large. To address the
issue, we propose a deep reinforcement learning (DRL) framework based on the
actor-critic learning structure. In particular, the actor network utilizes a
DNN to learn the optimal mapping from the input states to the binary offloading
decision of each task. Meanwhile, by analyzing the structure of the optimal
solution, we derive a low-complexity algorithm for the critic network to
quickly evaluate the ETC performance of the offloading decisions output by the
actor network. With the low-complexity critic network, we can quickly select
the best offloading action and subsequently store the state-action pair in an
experience replay memory as the training dataset to continuously improve the
action generation DNN. To further reduce the complexity, we show that the
optimal offloading decision exhibits an one-climb structure, which can be
utilized to significantly reduce the search space of action generation.
Numerical results show that for various types of task graphs, the proposed
algorithm achieves up to $99.1\%$ of the optimal performance while
significantly reducing the computational complexity compared to the existing
optimization methods.
