Learning Smooth Distance Functions via Queries
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915044281810944 |
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| author | Kumar, Akash Dasgupta, Sanjoy |
| author_facet | Kumar, Akash Dasgupta, Sanjoy |
| contents | In this work, we investigate the problem of learning distance functions within the query-based learning framework, where a learner is able to pose triplet queries of the form: ``Is $x_i$ closer to $x_j$ or $x_k$?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: $ω$-additive approximation and $(1 + ω)$-multiplicative approximation. For the additive approximation, we propose a global method whose query complexity is quadratic in the size of a finite cover of the sample space. For the (stronger) multiplicative approximation, we introduce a method that combines global and local approaches, utilizing multiple Mahalanobis distance functions to capture local geometry. This method has a query complexity that scales quadratically with both the size of the cover and the ambient space dimension of the sample space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Learning Smooth Distance Functions via Queries Kumar, Akash Dasgupta, Sanjoy Machine Learning Artificial Intelligence Information Retrieval In this work, we investigate the problem of learning distance functions within the query-based learning framework, where a learner is able to pose triplet queries of the form: ``Is $x_i$ closer to $x_j$ or $x_k$?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: $ω$-additive approximation and $(1 + ω)$-multiplicative approximation. For the additive approximation, we propose a global method whose query complexity is quadratic in the size of a finite cover of the sample space. For the (stronger) multiplicative approximation, we introduce a method that combines global and local approaches, utilizing multiple Mahalanobis distance functions to capture local geometry. This method has a query complexity that scales quadratically with both the size of the cover and the ambient space dimension of the sample space. |
| title | Learning Smooth Distance Functions via Queries |
| topic | Machine Learning Artificial Intelligence Information Retrieval |
| url | https://arxiv.org/abs/2412.01290 |