Efficient Distance Metric Learning by Adaptive Sampling and Mini-Batch Stochastic Gradient Descent (SGD)

TL;DR

This paper introduces adaptive sampling and mini-batch strategies within stochastic gradient descent to significantly reduce the computational cost of distance metric learning, especially the expensive PSD projections.

Contribution

It develops novel SGD-based algorithms with theoretical guarantees that effectively lower the number of PSD projections needed in DML.

Findings

Reduced number of PSD projections in DML algorithms

Theoretical guarantees for adaptive sampling and mini-batch methods

Empirical validation showing improved efficiency and comparable accuracy

Abstract

Distance metric learning (DML) is an important task that has found applications in many domains. The high computational cost of DML arises from the large number of variables to be determined and the constraint that a distance metric has to be a positive semi-definite (PSD) matrix. Although stochastic gradient descent (SGD) has been successfully applied to improve the efficiency of DML, it can still be computationally expensive because in order to ensure that the solution is a PSD matrix, it has to, at every iteration, project the updated distance metric onto the PSD cone, an expensive operation. We address this challenge by developing two strategies within SGD, i.e. mini-batch and adaptive sampling, to effectively reduce the number of updates (i.e., projections onto the PSD cone) in SGD. We also develop hybrid approaches that combine the strength of adaptive sampling with that of mini-batch online learning techniques to further improve the computational efficiency of SGD for DML. We prove the theoretical guarantees for both adaptive sampling and mini-batch based approaches for DML. We also conduct an extensive empirical study to verify the effectiveness of the proposed algorithms for DML.