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To automatically, accurately, and quickly detect local changes in time-series data continuously emitted by X-ray sources, an autoencoder-based unsupervised learning anomaly detection method is proposed. Here, we consider the X-ray burst of GRO J1744-28 as our case study. First, we tested the proposed method using simulation data and a test set based on a phenomenologically motivated light-curve fitting of different burst types. Our method exhibited superior performance, achieving F-scores of 0.969 and 0.936 for the detection of small bursts with low peak count rates such as structured bursts and microbursts, respectively. Then, based on Rossi X-ray Timing Detector observation data for GRO J1744-28 during the outburst period, we identified low-amplitude bursts using the proposed method and analyzed the burst regularity of GRO J1744-28. Our approach does not require complex modeling and has powerful feature extraction and detection capabilities, which can be used to automatically and efficiently detect changes in a data stream.