Primal and Dual Combinatorial Dimensions

TL;DR

This paper establishes tight bounds relating primal and dual combinatorial dimensions, such as pseudo-dimension and fat-shattering, for multi-valued function classes, enhancing understanding in learning theory.

Contribution

It provides new tight bounds and generalizations linking primal and dual dimensions for multi-valued functions, extending classical results.

Findings

Bound the dual dimension in terms of the primal for multi-valued classes

Provide lower bounds matching existing upper bounds

Generalize Assouad's bound to multi-valued function classes

Abstract

We give tight bounds on the relation between the primal and dual of various combinatorial dimensions, such as the pseudo-dimension and fat-shattering dimension, for multi-valued function classes. These dimensional notions play an important role in the area of learning theory. We first review some (folklore) results that bound the dual dimension of a function class in terms of its primal, and after that give (almost) matching lower bounds. In particular, we give an appropriate generalization to multi-valued function classes of a well-known bound due to Assouad (1983), that relates the primal and dual VC-dimension of a binary function class.