Permutational Rademacher Complexity: a New Complexity Measure for Transductive Learning

TL;DR

This paper introduces Permutational Rademacher Complexity (PRC), a new measure tailored for transductive learning, providing tighter bounds and better understanding of model complexity in scenarios with labeled and unlabeled data.

Contribution

The paper proposes PRC as a novel complexity measure for transductive learning, establishing its properties, relationships with existing measures, and deriving new data-dependent risk bounds.

Findings

PRC offers tighter control over empirical processes in transductive learning.

PRC is more suitable than existing measures for transductive settings.

New risk bounds are derived using PRC and concentration inequalities.

Abstract

Transductive learning considers situations when a learner observes $m$ labelled training points and $u$ unlabelled test points with the final goal of giving correct answers for the test points. This paper introduces a new complexity measure for transductive learning called Permutational Rademacher Complexity (PRC) and studies its properties. A novel symmetrization inequality is proved, which shows that PRC provides a tighter control over expected suprema of empirical processes compared to what happens in the standard i.i.d. setting. A number of comparison results are also provided, which show the relation between PRC and other popular complexity measures used in statistical learning theory, including Rademacher complexity and Transductive Rademacher Complexity (TRC). We argue that PRC is a more suitable complexity measure for transductive learning. Finally, these results are combined with a standard concentration argument to provide novel data-dependent risk bounds for transductive learning.