Improving subjective sound quality of enhanced signals is one of the most important missions in speech enhancement. For evaluating the subjective quality, several methods related to perceptually-motivated objective sound quality assessment (OSQA) have been proposed such as PESQ (perceptual evaluation of speech quality). However, direct use of such measures for training deep neural network (DNN) is not allowed in most cases because popular OSQAs are non-differentiable with respect to DNN parameters. Therefore, the previous study has proposed to approximate the score of OSQAs by an auxiliary DNN so that its gradient can be used for training the primary DNN. One problem with this approach is instability of the training caused by the approximation error of the score. To overcome this problem, we propose to use stabilization techniques borrowed from reinforcement learning. The experiments, aimed to increase the score of PESQ as an example, show that the proposed method (i) can stably train a DNN to increase PESQ, (ii) achieved the state-of-the-art PESQ score on a public dataset, and (iii) resulted in better sound quality than conventional methods based on subjective evaluation.