While the localization of radiological sources has traditionally been handled with statistical algorithms, such a task can be augmented with advanced machine learning methodologies. The combination of deep and reinforcement learning has provided learning-based navigation to autonomous, single-detector, mobile systems. However, these approaches lacked the capacity to terminate a surveying/search task without outside influence of an operator or perfect knowledge of source location (defeating the purpose of such a system). Two stopping criteria are investigated in this work for a machine learning navigated system: one based upon Bayesian and maximum likelihood estimation (MLE) strategies commonly used in source localization, and a second providing the navigational machine learning network with a "stop search" action. A convolutional neural network was trained via reinforcement learning in a 10 m × 10 m simulated environment to navigate a randomly placed detector-agent to a randomly placed source of varied strength (stopping with perfect knowledge during training). The network agent could move in one of four directions (up, down, left, right) after taking a 1 s count measurement at the current location. During testing, the stopping criteria for this navigational algorithm was based upon a Bayesian likelihood estimation technique of source presence, updating this likelihood after each step, and terminating once the confidence of the source being in a single location exceeded 0.9. A second network was trained and tested with similar architecture as the previous but which contained a fifth action: for self-stopping. The accuracy and speed of localization with set detector and source initializations were compared over 50 trials of MLE-Bayesian approach and 1000 trials of the CNN with self-stopping. The statistical stopping condition yielded a median localization error of ~1.41 m and median localization speed of 12 steps. The machine learning stopping condition yielded a median localization error of 0 m and median localization speed of 17 steps. This work demonstrated two stopping criteria available to a machine learning guided, source localization system.