Learning Smooth Distance Functions via Queries

TL;DR

This paper explores query-based learning of smooth distance functions, providing formal guarantees on query complexity and proposing methods that adapt to local geometry using Mahalanobis distances.

Contribution

It introduces novel algorithms for learning smooth distance functions with provable query complexity bounds under additive and multiplicative approximations.

Findings

Quadratic query complexity for additive approximation with global method.

Combined global and local approach for multiplicative approximation.

Query complexity scales with cover size and ambient space dimension.

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: $\omega$-additive approximation and $(1 + \omega)$-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.