On the Learning and Learnability of Quasimetrics
This addresses a foundational gap in machine learning for modeling asymmetric relationships, with potential applications in reinforcement learning and graph analysis, though it is incremental as it introduces a new method for a known bottleneck.
The paper tackled the problem of learning quasimetrics, which are asymmetric distance functions common in real-world structures like graphs and reinforcement learning, by showing that standard methods like MLPs fail to learn them consistently, and proposed Poisson Quasimetric Embedding (PQE) as a learnable alternative with strong guarantees, achieving effectiveness in experiments on random graphs, social graphs, and offline Q-learning.
Our world is full of asymmetries. Gravity and wind can make reaching a place easier than coming back. Social artifacts such as genealogy charts and citation graphs are inherently directed. In reinforcement learning and control, optimal goal-reaching strategies are rarely reversible (symmetrical). Distance functions supported on these asymmetrical structures are called quasimetrics. Despite their common appearance, little research has been done on the learning of quasimetrics. Our theoretical analysis reveals that a common class of learning algorithms, including unconstrained multilayer perceptrons (MLPs), provably fails to learn a quasimetric consistent with training data. In contrast, our proposed Poisson Quasimetric Embedding (PQE) is the first quasimetric learning formulation that both is learnable with gradient-based optimization and enjoys strong performance guarantees. Experiments on random graphs, social graphs, and offline Q-learning demonstrate its effectiveness over many common baselines.