Polysomnography (PSG) is the gold standard for diagnosing sleep obstructive apnea (OSA). It allows monitoring of breathing events throughout the night. The detection of these events is usually done by trained sleep experts. However, this task is tedious, highly time-consuming and subject to important inter-scorer variability. In this study, we adapted our state-of-the-art deep learning method for sleep event detection, DOSED, to the detection of sleep breathing events in PSG for the diagnosis of OSA. We used a dataset of 52 PSG recordings with apnea-hypopnea event scoring from 5 trained sleep experts. We assessed the performance of the automatic approach and compared it to the inter-scorer performance for both the diagnosis of OSA severity and, at the microscale, for the detection of single breathing events. We observed that human sleep experts reached an average accuracy of 75% while the automatic approach reached 81% for sleep apnea severity diagnosis. The F1 score for individual event detection was 0.55 for experts and 0.57 for the automatic approach, on average. These results demonstrate that the automatic approach can perform at a sleep expert level for the diagnosis of OSA.