Generation through the lens of learning theory

TL;DR

This paper analyzes the concept of generation within statistical learning theory, introducing new notions of uniform and non-uniform generation, and explores their relationship with predictability and promptability.

Contribution

It formalizes generation in learning theory, introduces the Closure dimension for characterization, and compares generatability with predictability, extending results to prompted generation.

Findings

Generated classes are characterized by the Closure dimension.

Generatability and predictability are incompatible properties.

Complete characterization of prompt generatability provided.

Abstract

We study generation through the lens of statistical learning theory. First, we abstract and formalize the results of Gold [1967], Angluin [1979], Angluin [1980] and Kleinberg and Mullainathan [2024] in terms of a binary hypothesis class defined over an abstract example space. Then, we extend the notion of "generation" from Kleinberg and Mullainathan [2024] to two new settings, we call "uniform" and "non-uniform" generation, and provide a characterization of which hypothesis classes are uniformly and non-uniformly generatable. As is standard in learning theory, our characterizations are in terms of the finiteness of a new combinatorial dimension termed the Closure dimension. By doing so, we are able to compare generatability with predictability (captured via PAC and online learnability) and show that these two properties of hypothesis classes are incompatible -- there are classes that are generatable but not predictable and vice versa. Finally, we extend our results to capture prompted generation and give a complete characterization of which classes are prompt generatable, generalizing some of the work by Kleinberg and Mullainathan [2024].