Generalization Bounds for Metric and Similarity Learning

Abstract

Recently, metric learning and similarity learning have attracted a large amount of interest. Many models and optimization algorithms have been proposed. However, there is relatively little work on the generalization analysis of such methods. In this paper, we derive novel generalization bounds of metric and similarity learning. In particular, we first show that the generalization analysis reduces to the estimation of the Rademacher average over “sums-of-i.i.d.” sample-blocks related to the specific matrix norm. Then, we derive generalization bounds for metric/similarity learning with different matrix-norm regularizers by estimating their specific Rademacher complexities. Our analysis indicates that sparse metric/similarity learning with L1\documentclass[12pt]minimal \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}-69pt \begin{document}$L^1$\end{document}-norm regularization could lead to significantly better bounds than those with Frobenius-norm regularization. Our novel generalization analysis develops and refines the techniques of U-statistics and Rademacher complexity analysis.