Many adversarial defense methods have been proposed to enhance the adversarial robustness of natural language processing models. However, most of them introduce additional pre-set linguistic knowledge and assume that the synonym candidates used by attackers are accessible, which is an ideal assumption. We delve into adversarial training in the embedding space and propose a Fast Adversarial Training (FAT) method to improve the model robustness in the synonym-unaware scenario from the perspective of single-step perturbation generation and perturbation initialization. Based on the observation that the adversarial perturbations crafted by single-step and multi-step gradient ascent are similar, FAT uses single-step gradient ascent to craft adversarial examples in the embedding space to expedite the training process. Based on the observation that the perturbations generated on the identical training sample in successive epochs are similar, FAT fully utilizes historical information when initializing the perturbation. Extensive experiments demonstrate that FAT significantly boosts the robustness of BERT models in the synonym-unaware scenario, and outperforms the defense baselines under various attacks with character-level and word-level modifications.