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A One-Inclusion Graph Approach to Multi-Group Learning

Noah Bergam, Samuel Deng, Daniel Hsu

Abstract

We prove the tightest-known upper bounds on the sample complexity of multi-group learning. Our algorithm extends the one-inclusion graph prediction strategy using a generalization of bipartite $b$-matching. In the group-realizable setting, we provide a lower bound confirming that our algorithm's $\log n / n$ convergence rate is optimal in general. If one relaxes the learning objective such that the group on which we are evaluated is chosen obliviously of the sample, then our algorithm achieves the optimal $1/n$ convergence rate under group-realizability.

A One-Inclusion Graph Approach to Multi-Group Learning

Abstract

We prove the tightest-known upper bounds on the sample complexity of multi-group learning. Our algorithm extends the one-inclusion graph prediction strategy using a generalization of bipartite -matching. In the group-realizable setting, we provide a lower bound confirming that our algorithm's convergence rate is optimal in general. If one relaxes the learning objective such that the group on which we are evaluated is chosen obliviously of the sample, then our algorithm achieves the optimal convergence rate under group-realizability.
Paper Structure (24 sections, 30 theorems, 123 equations, 1 algorithm)

This paper contains 24 sections, 30 theorems, 123 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results.
  3. Related work.
  4. Setting
  5. Multi-Group Learning: Prediction Model and Transductive Model
  6. Group-realizability.
  7. Multi-group prediction error.
  8. Multi-group transductive error.
  9. The One-Inclusion Graph
  10. The Multi-group One Inclusion Graph Algorithm
  11. Multi-group Matching Problem
  12. From a Multi-group Matching to a Learner
  13. Proof of Lemma \ref{['lemma:suffLP']} through Duality
  14. Lower Bound
  15. Multi-group PAC Model
  16. ...and 9 more sections

Key Result

Lemma 1

Let $C \subseteq \lbrace0,1\rbrace^{\mathcal{X}}$ be a concept class with VC dimension $d.$ For any $n \in \mathbb{N}$, $\max_{U \in \mathcal{X}^n} \max_{W \subseteq V} \mathrm{dens}\left(G^W_C(U)\right) \leq d.$

Theorems & Definitions (55)

  • Definition 1
  • Lemma 1: *haussler1994predicting
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • ...and 45 more