Conditions on Preference Relations that Guarantee the Existence of Optimal Policies

Learning from Preferential Feedback (LfPF) plays an essential role in training Large Language Models, as well as certain types of interactive learning agents. However, a substantial gap exists between the theory and application of LfPF algorithms. Current results guaranteeing the existence of optimal policies in LfPF problems assume that both the preferences and transition dynamics are determined by a Markov Decision Process. We introduce the Direct Preference Process, a new framework for analyzing LfPF problems in partially-observable, non-Markovian environments. Within this framework, we establish conditions that guarantee the existence of optimal policies by considering the ordinal structure of the preferences. Using the von Neumann-Morgenstern Expected Utility Theorem, we show that the Direct Preference Process generalizes the standard reinforcement learning problem. Our findings narrow the gap between the empirical success and theoretical understanding of LfPF algorithms and provide future practitioners with the tools necessary for a more principled design of LfPF agents.