The past thirty years have shown a rise in models of language acquisition in which the state of the learner is characterized as a probability distribution over a set of non-stochastic grammars. In recent years, increasingly powerful models have been constructed as earlier models have failed to generalize well to increasingly complex and realistic learning domains. I particularly note that few recent models learn with limited feedback, which measures the amount of information brought to and taken from each learning instance. In this dissertation, I adopt a geometric lens for viewing this class of learning models. Viewing previous algorithms geometrically, I diagnose their flaws and motivate a novel, natural algorithm which can overcome those flaws while operating under limited feedback, which I call the barycentric learning model. Viewing representational theories geometrically, I apply the same learning algorithm successfully to learning problems across domains in parametric, ranked-constraint, and weighted-constraint theoretical frameworks. I apply novel formal tools to analyze the algorithm's behavior, which help us understand where the algorithm demonstrates convergence and non-convergence, as well as the dynamics of learning paths within individuals, and of language change paths across generations. The success of this model demonstrates that limited feedback suffices for a larger class of learning problems than previously known, while pointing a way forward for the formal and abstract understanding of language acquisition.