# Epsilon-saturation for stable graphs and Littlestone classes

## TL;DR

This paper introduces epsilon-saturation for stable graphs and Littlestone classes, exploring their properties, closure, and the relationship between Littlestone/stability and VC dimension, revealing surprising phenomena.

## Contribution

It defines and characterizes epsilon-saturation, showing how it preserves stability and Littlestone properties under certain conditions, and analyzes its impact on complexity measures.

## Key findings

- Epsilon-saturation forms a closure of the class under virtual elements.
- Under certain parameters, epsilon-saturation remains Littlestone or stable.
- The relationship between Littlestone/stability and VC dimension exhibits surprising phenomena.

## Abstract

Any Littlestone class, or stable graph, has finite sets which function as ``virtual elements'': these can be seen from the learning side as representing hypotheses which are expressible as weighted majority opinions of hypotheses in the class, and from the model-theoretic side as an approximate finitary version of realizing types. We introduce and study the epsilon-saturation of a Littlestone class, or stable graph, which is essentially the closure of the class under inductively adding all such virtual elements. We characterize this closure and prove that under reasonable choices of parameters, it remains Littlestone (or stable), though not always of the same Littlestone dimension. This highlights some surprising phenomena having to do with regimes of epsilon and the relation between Littlestone/stability and VC dimension.