This paper presents a state representation for reward-free Markov decision processes. The idea is to learn, in a self-supervised manner, an embedding space where distances between pairs of embedded states correspond to the minimum number of actions needed to transition between them. Unlike previous methods, our approach incorporates an asymmetric norm parametrization, enabling accurate approximations of minimum action distances in environments with inherent asymmetry. We show how this representation can be leveraged to learn goal-conditioned policies, providing a notion of similarity between states and goals and a useful heuristic distance to guide planning. To validate our approach, we conduct empirical experiments on both symmetric and asymmetric environments. Our results show that our asymmetric norm parametrization performs comparably to symmetric norms in symmetric environments and surpasses symmetric norms in asymmetric environments.

Many properties in the real world doesn't have metrics and can't be numerically observed, making them difficult to learn. To deal with this challenging problem, prior works have primarily focused on estimating those properties by using graded human scores as the target label in the training. Meanwhile, rating algorithms based on the Bradley-Terry model are extensively studied to evaluate the competitiveness of players based on their match history. In this paper, we introduce the Neural Bradley-Terry Rating (NBTR), a novel machine learning framework designed to quantify and evaluate properties of unknown items. Our method seamlessly integrates the Bradley-Terry model into the neural network structure. Moreover, we generalize this architecture further to asymmetric environments with unfairness, a condition more commonly encountered in real-world settings. Through experimental analysis, we demonstrate that NBTR successfully learns to quantify and estimate desired properties.