An Inductive Bias for Distances: Neural Nets that Respect the Triangle\n Inequality

Distances are pervasive in machine learning. They serve as similarity\nmeasures, loss functions, and learning targets; it is said that a good distance\nmeasure solves a task. When defining distances, the triangle inequality has\nproven to be a useful constraint, both theoretically--to prove convergence and\noptimality guarantees--and empirically--as an inductive bias. Deep metric\nlearning architectures that respect the triangle inequality rely, almost\nexclusively, on Euclidean distance in the latent space. Though effective, this\nfails to model two broad classes of subadditive distances, common in graphs and\nreinforcement learning: asymmetric metrics, and metrics that cannot be embedded\ninto Euclidean space. To address these problems, we introduce novel\narchitectures that are guaranteed to satisfy the triangle inequality. We prove\nour architectures universally approximate norm-induced metrics on\n$\\mathbb{R}^n$, and present a similar result for modified Input Convex Neural\nNetworks. We show that our architectures outperform existing metric approaches\nwhen modeling graph distances and have a better inductive bias than non-metric\napproaches when training data is limited in the multi-goal reinforcement\nlearning setting.\n

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