On the Learning and Learnability of Quasimetrics

TL;DR

This paper investigates the challenges of learning quasimetrics, introduces a novel learnable embedding method called Poisson Quasimetric Embedding (PQE), and demonstrates its effectiveness through experiments on various graph types.

Contribution

The paper reveals the limitations of existing algorithms in learning quasimetrics and proposes PQE, the first method that is both learnable with gradient-based methods and has strong theoretical guarantees.

Findings

Unconstrained MLPs fail to learn consistent quasimetrics.

PQE outperforms baseline methods on diverse graph datasets.

PQE is both theoretically sound and practically effective.

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.