In this paper, we propose an efficient beam training technique for millimeter-wave (mmWave) communications. When some mobile users are under high mobility, the beam training should be performed frequently to ensure the accurate acquisition of the channel state information. In order to reduce the resource overhead caused by frequent beam training, we introduce a dedicated beam training strategy which sends the training beams separately to a specific high mobility user (called a target user) without changing the periodicity of the conventional beam training. The dedicated beam training requires small amount of resources since the training beams can be optimized for the target user. In order to satisfy the performance requirement with low training overhead, we propose the optimal training beam selection strategy which finds the best beamforming vectors yielding the lowest channel estimation error based on the target user's probabilistic channel information. Such dedicated beam training is combined with the greedy channel estimation algorithm that accounts for sparse characteristics and temporal dynamics of the target user's channel. Our numerical evaluation demonstrates that the proposed scheme can maintain good channel estimation performance with significantly less training overhead compared to the conventional beam training protocols.