On the Learning and Learnability of Quasimetrics

Tongzhou Wang, Phillip Isola

Keywords: embedding learning, quasimetric learning, deep learning

Abstract: 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.

One-sentence Summary: We theoretically analyze various algorithms on learning quasimetrics (asymmetrical metrics), and propose an embedding-based method with strong guarantees. Experiments on graph learning and Q-learning show its effectiveness over common baselines.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2206.15478/code)