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

arXiv: https://arxiv.org/abs/2002.05825 · HTML: https://mlsanity.com/paper/2002.05825/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2020.

## Built on by (most cited first, 24 of 24)

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- [Physics-informed Goal-Conditioned Reinforcement Learning under Hybrid Contact Dynamics](https://mlsanity.com/paper/2605.30503/) (2026)
- [GTRL: Grounding Divide-and-Conquer Value Learning with Temporal Differences](https://mlsanity.com/paper/2609.33259/) (2026)
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## Abstract (arXiv)

Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a useful constraint, both theoretically--to prove convergence and optimality guarantees--and empirically--as an inductive bias. Deep metric learning architectures that respect the triangle inequality rely, almost exclusively, on Euclidean distance in the latent space. Though effective, this fails to model two broad classes of subadditive distances, common in graphs and reinforcement learning: asymmetric metrics, and metrics that cannot be embedded into Euclidean space. To address these problems, we introduce novel architectures that are guaranteed to satisfy the triangle inequality. We prove our architectures universally approximate norm-induced metrics on ℝn, and present a similar result for modified Input Convex Neural Networks. We show that our architectures outperform existing metric approaches when modeling graph distances and have a better inductive bias than non-metric approaches when training data is limited in the multi-goal reinforcement learning setting.