Training speaker-discriminative and robust speaker verification systems without speaker labels is still challenging and worthwhile to explore. In this study, we propose an effective self-supervised learning framework and a novel regularization strategy to facilitate self-supervised speaker representation learning. Different from contrastive learning-based self-supervised learning methods, the proposed self-supervised regularization (SSReg) focuses exclusively on the similarity between the latent representations of positive data pairs. We also explore the effectiveness of alternative online data augmentation strategies on both the time domain and frequency domain. With our strong online data augmentation strategy, the proposed SSReg shows the potential of self-supervised learning without using negative pairs and it can significantly improve the performance of self-supervised speaker representation learning with a simple Siamese network architecture. Comprehensive experiments on the VoxCeleb datasets demonstrate that our proposed self-supervised approach obtains a 23.4% relative improvement by adding the effective self-supervised regularization and outperforms other previous works.