Machine learning is a powerful tool that can be applied to pattern search and mathematical optimization for making predictions on new data with unknown labels. In the field of medical imaging, one challenge with applying machine learning techniques is the limited size and relative expense of obtaining labeled data. For example, in glenoid labral tears, current imaging diagnosis is best achieved by imaging through Magnetic Resonance (MR) Arthrography, a method of injecting contrast-enhancing material into the joint that can potentially cause discomfort to the patient, and adds expense compared to a standard MRI. This work proposes limiting the use of MR arthrography through a medical diagnostic approach, based on convolutional neural networks and transfer learning from a separate medical imaging dataset to improve efficiency and effectiveness. Results indicate an effective method applied to a small dataset of unenhanced shoulder MRI in order to diagnose labral tear severity while potentially significantly reducing cost and reducing unnecessary invasive imaging techniques. The proposed method ultimately can reduce physician workload while ensuring that the least number of patients as possible need to be subjected to an additional invasive contrast-enhanced imaging procedure.