Device-edge co-inference, which partitions a deep neural network between a resource-constrained mobile device and an edge server, recently emerges as a promising paradigm to support intelligent mobile applications. To accelerate the in-ference process, on-device model sparsification and intermediate feature compression are regarded as two prominent techniques. However, as the on-device model sparsity level and intermediate feature compression ratio have direct impacts on computation workload and communication overhead respectively, and both of them affect the inference accuracy, finding the optimal values of these hyper-parameters brings a major challenge due to the large search space. In this paper, we endeavor to develop an efficient algorithm to determine these hyper-parameters. By selecting a suitable model split point and a pair of encoder/decoder for the intermediate feature vector, this problem is casted as a sequential decision problem, for which, a novel automated machine learning (AutoML) framework is proposed based on deep reinforcement learning (DRL). Experiment results on an image classification task demonstrate the effectiveness of the proposed framework in achieving a better communication-computation trade-off and significant inference speedup against various baseline schemes.