The prediction of network latency with partial measurements is of importance for ever-increasing personal devices to ensure their Quality of Service (QoS). However, the current matrix-factorization-based efforts, as a promising paradigm, for network latency prediction have failed to intelligently exploit inherent factors hidden in networks to accurately infer the unknown network latency. Furthermore, it is more complicated to execute extensive network measurements on pervasive personal devices due to unstable communication environments. To alleviate these problems, in this paper, a novel accurate network latency prediction (DALP) solution via Deep Reinforcement Learning (DRL) is proposed for personal devices under cost-aware sampling. Specifically, we first alternately implement cost-aware latency measurement based on temporal correlation, and model it as a network latency matrix, in which unmeasured and missing elements need to be inferred. In order to achieve accurate prediction performance, the DRL-based Matrix Factorization with Double Weights (DWMF) is designed to exploit the potential network factors and multiple rules of matrix factorization, which can be alternatively executed, to minimize the prediction errors. Furthermore, an angle-loss-based reward strategy is designed to enhance the quality of model training. Simulation results on real-world datasets illustrate that DALP outperforms the previous approaches with quicker convergence and lower prediction errors.