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A Characterization of Multioutput Learnability

Vinod Raman, Unique Subedi, Ambuj Tewari

TL;DR

It is shown that a multioutput function class is learnable if and only if each single-output restriction of the function class is learnable.

Abstract

We consider the problem of learning multioutput function classes in the batch and online settings. In both settings, we show that a multioutput function class is learnable if and only if each single-output restriction of the function class is learnable. This provides a complete characterization of the learnability of multilabel classification and multioutput regression in both batch and online settings. As an extension, we also consider multilabel learnability in the bandit feedback setting and show a similar characterization as in the full-feedback setting.

A Characterization of Multioutput Learnability

TL;DR

It is shown that a multioutput function class is learnable if and only if each single-output restriction of the function class is learnable.

Abstract

We consider the problem of learning multioutput function classes in the batch and online settings. In both settings, we show that a multioutput function class is learnable if and only if each single-output restriction of the function class is learnable. This provides a complete characterization of the learnability of multilabel classification and multioutput regression in both batch and online settings. As an extension, we also consider multilabel learnability in the bandit feedback setting and show a similar characterization as in the full-feedback setting.
Paper Structure (36 sections, 24 theorems, 107 equations, 9 algorithms)

This paper contains 36 sections, 24 theorems, 107 equations, 9 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Preliminaries
  4. Batch Setting
  5. Online Setting
  6. Batch Multioutput Learnability
  7. Batch Multilabel Classification
  8. Characterizing Batch Learnability for the Hamming Loss
  9. Characterizing Batch Learnability for General Losses
  10. Batch Multioutput Regression
  11. Characterizing Learnability for Decomposable Losses
  12. A More General Characterization of Learnability for Decomposable Losses
  13. Characterizing Learnability for Non-Decomposable Losses
  14. Online Multioutput Learnability
  15. Online Agnostic-to-Realizable Reduction
  16. ...and 21 more sections

Key Result

Theorem 1

(Informal) A multioutput function class $\mathcal{F} \subseteq\mathcal{Y}^{\mathcal{X}}$ is learnable if and only if each restriction $\mathcal{F}_k \subseteq\mathcal{Y}_k^{\mathcal{X}}$ is learnable.

Theorems & Definitions (36)

  • Theorem
  • Definition 1: Identity of Indiscernibles
  • Definition 2: $c$-subadditive
  • Definition 3: Agnostic Multioutput Learnability
  • Lemma 4: hopkins22a
  • Definition 5: Online Multioutput Learnability
  • Theorem
  • Theorem 6
  • Lemma 7
  • Theorem 8
  • ...and 26 more