Towards A Deeper Geometric, Analytic and Algorithmic Understanding of Margins

TL;DR

This paper provides a comprehensive geometric, analytic, and algorithmic analysis of the margin concept in linear feasibility problems, enhancing understanding and convergence guarantees in margin-based learning.

Contribution

It introduces new geometric characterizations of the margin, generalizes classical theorems using margins, and analyzes the Perceptron algorithm's convergence in relation to margin.

Findings

New geometric characterizations of the margin

Generalizations of Gordan's and Hoffman's theorems using margins

Convergence analysis of the Perceptron algorithm related to margin

Abstract

Given a matrix $A$, a linear feasibility problem (of which linear classification is a special case) aims to find a solution to a primal problem $w: A^Tw > \textbf{0}$ or a certificate for the dual problem which is a probability distribution $p: Ap = \textbf{0}$. Inspired by the continued importance of "large-margin classifiers" in machine learning, this paper studies a condition measure of $A$ called its \textit{margin} that determines the difficulty of both the above problems. To aid geometrical intuition, we first establish new characterizations of the margin in terms of relevant balls, cones and hulls. Our second contribution is analytical, where we present generalizations of Gordan's theorem, and variants of Hoffman's theorems, both using margins. We end by proving some new results on a classical iterative scheme, the Perceptron, whose convergence rates famously depends on the margin. Our results are relevant for a deeper understanding of margin-based learning and proving convergence rates of iterative schemes, apart from providing a unifying perspective on this vast topic.