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Learning Smooth Distance Functions via Queries

Akash Kumar, Sanjoy Dasgupta

TL;DR

This work studies learning smooth distance functions from triplet queries, introducing two approximation regimes: an $\omega$-additive and a $(1+\omega)$-multiplicative notion of triplet-equivalence. It develops a global-cover approach for additive accuracy and a hybrid global-local strategy using local Mahalanobis surrogates to achieve multiplicative accuracy, with formal guarantees on query complexity that scale with cover sizes and ambient dimension. The framework covers non-Mahalanobis smooth distances by leveraging smoothness (via Hessians) and a finite-cover extension, and it provides concrete algorithms for learning both general distance functions and Mahalanobis distances from triplet feedback. The results highlight a principled path to interactive metric learning applicable to personalized retrieval, embedding construction, and geometry inference under limited feedback. Overall, the paper advances query-efficient methods for learning distance structures by combining global covers, local linearizations, and careful complexity analyses to handle both additive and multiplicative approximation goals in high-dimensional spaces.

Abstract

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.

Learning Smooth Distance Functions via Queries

TL;DR

This work studies learning smooth distance functions from triplet queries, introducing two approximation regimes: an -additive and a -multiplicative notion of triplet-equivalence. It develops a global-cover approach for additive accuracy and a hybrid global-local strategy using local Mahalanobis surrogates to achieve multiplicative accuracy, with formal guarantees on query complexity that scale with cover sizes and ambient dimension. The framework covers non-Mahalanobis smooth distances by leveraging smoothness (via Hessians) and a finite-cover extension, and it provides concrete algorithms for learning both general distance functions and Mahalanobis distances from triplet feedback. The results highlight a principled path to interactive metric learning applicable to personalized retrieval, embedding construction, and geometry inference under limited feedback. Overall, the paper advances query-efficient methods for learning distance structures by combining global covers, local linearizations, and careful complexity analyses to handle both additive and multiplicative approximation goals in high-dimensional spaces.

Abstract

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 closer to or ?'' We establish formal guarantees on the query complexity required to learn smooth, but otherwise general, distance functions under two notions of approximation: -additive approximation and -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.

Paper Structure

This paper contains 20 sections, 7 theorems, 16 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Informal theorem 1.
  3. Informal theorem 2.
  4. Related Work
  5. Learning a distance function
  6. Learning an embedding
  7. Preliminaries
  8. Mahalanobis distance function
  9. Comparisons of distances
  10. Query learning a distance function:
  11. Additive approximation of a smooth distance function
  12. Learning a distance function on a finite space, upto triplet equivalence
  13. A notion of smooth distance function
  14. Learning a smooth distance upto additive approximation
  15. Distance Model:
  16. ...and 5 more sections

Key Result

Theorem 5

Given any distance function $d: {\mathcal{X}} \times {\mathcal{X}} \to \mathbb{R}_{\ge 0}$ on a sample space ${\mathcal{X}}$, and a finite subset ${\mathcal{X}}_o \subset {\mathcal{X}}$, a learner can find a distance function $\hat{d}: {\mathcal{X}}_o \times {\mathcal{X}}_o \to \mathbb{R}_{\ge 0}$ t

Theorems & Definitions (15)

  • Definition 1: distance function
  • Definition 2: triplet equivalence
  • Definition 3: additive approximation
  • Definition 4: multiplicative approximation
  • Theorem 5
  • Definition 6: $(\alpha,L,\delta)$-smooth
  • Lemma 7: Taylor's theorem
  • Remark 8
  • Theorem 9
  • Corollary 10
  • ...and 5 more