Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-arity PAC learning, VC dimension and packing lemma

Artem Chernikov, Henry Towsner

TL;DR

The paper develops higher-arity VC theory via the $VC_k$-dimension for $k$-fold product spaces and generalizes Haussler's packing lemma to this setting. It then links finite $VC_k$-dimension to proper $PAC_k$-learnability under product measures, establishing an equivalence through a higher-arity packing lemma and a constructive learning function. A slice-wise packing lemma and corresponding hypergraph regularity results are presented, showing that slice-wise control leads to global regularity phenomena in hypergraphs and CSA frameworks. Collectively, these results extend classical VC theory to higher arities, clarifying the conditions under which learning and regularity hold in product spaces and connecting to several contemporary findings in the literature.

Abstract

The aim of this note is to overview some of our work in Chernikov, Towsner'20 (arXiv:2010.00726) developing higher arity VC theory (VC$_n$ dimension), including a generalization of Haussler packing lemma, and an associated tame (slice-wise) hypergraph regularity lemma; and to demonstrate that it characterizes higher arity PAC learning (PAC$_n$ learning) in $n$-fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi'15. We also point out how some of the recent results in arXiv:2402.14294, arXiv:2505.15688, arXiv:2509.20404 follow from our work in arXiv:2010.00726.

Higher-arity PAC learning, VC dimension and packing lemma

TL;DR

The paper develops higher-arity VC theory via the -dimension for -fold product spaces and generalizes Haussler's packing lemma to this setting. It then links finite -dimension to proper -learnability under product measures, establishing an equivalence through a higher-arity packing lemma and a constructive learning function. A slice-wise packing lemma and corresponding hypergraph regularity results are presented, showing that slice-wise control leads to global regularity phenomena in hypergraphs and CSA frameworks. Collectively, these results extend classical VC theory to higher arities, clarifying the conditions under which learning and regularity hold in product spaces and connecting to several contemporary findings in the literature.

Abstract

The aim of this note is to overview some of our work in Chernikov, Towsner'20 (arXiv:2010.00726) developing higher arity VC theory (VC dimension), including a generalization of Haussler packing lemma, and an associated tame (slice-wise) hypergraph regularity lemma; and to demonstrate that it characterizes higher arity PAC learning (PAC learning) in -fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi'15. We also point out how some of the recent results in arXiv:2402.14294, arXiv:2505.15688, arXiv:2509.20404 follow from our work in arXiv:2010.00726.

Paper Structure

This paper contains 6 sections, 6 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. VC$_k$ dimension
  3. Sauer-Shelah lemma for VC$_k$-dimension
  4. PAC$_n$ learning
  5. Packing lemma for families of finite VC$_k$ dimension
  6. Equivalence of finite $\operatorname{VC}_k$-dimension to $\operatorname{PAC}_k$-learning

Key Result

Corollary 1.1

The following are equivalent for a class $\mathcal{F}$ of subsets of $V_1 \times \ldots \times V_k$:

Theorems & Definitions (24)

  • Corollary 1.1: Corollary \ref{['cor: everything equiv']}
  • Definition 2.1
  • Remark 3.2
  • Remark 3.3
  • Definition 4.1
  • Definition 4.3
  • Remark 4.4
  • Definition 5.2
  • Remark 5.4
  • Remark 6.3
  • ...and 14 more