In self-supervised learning for speaker recognition, pseudo labels are useful as the supervision signals. It is a known fact that a speaker recognition model doesn't always benefit from pseudo labels due to their unreliability. In this work, we observe that a speaker recognition network tends to model the data with reliable labels faster than those with unreliable labels. This motivates us to study a loss-gated learning (LGL) strategy, which extracts the reliable labels through the fitting ability of the neural network during training. With the proposed LGL, our speaker recognition model obtains a $46.3\%$ performance gain over the system without it. Further, the proposed self-supervised speaker recognition with LGL trained on the VoxCeleb2 dataset without any labels achieves an equal error rate of $1.66\%$ on the VoxCeleb1 original test set. Code has been made available at: https://github.com/TaoRuijie/Loss-Gated-Learning.