An Inductive Bias for Distances: Neural Nets that Respect the Triangle\n Inequality
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Bibliographic record
Abstract
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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it