Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Simultaneous Blackwell Approachability and Applications to Multiclass Omniprediction

Lunjia Hu, Kevin Tian, Chutong Yang

TL;DR

The main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity or regret horizon for $\varepsilon$-omniprediction in a $k$-class prediction problem.

Abstract

Omniprediction is a learning problem that requires suboptimality bounds for each of a family of losses $\mathcal{L}$ against a family of comparator predictors $\mathcal{C}$. We initiate the study of omniprediction in a multiclass setting, where the comparator family $\mathcal{C}$ may be infinite. Our main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity (in statistical settings) or regret horizon (in online settings) $\approx \varepsilon^{-(k+1)}$, for $\varepsilon$-omniprediction in a $k$-class prediction problem. En route to proving this result, we design a framework of potential broader interest for solving Blackwell approachability problems where multiple sets must simultaneously be approached via coupled actions.

Simultaneous Blackwell Approachability and Applications to Multiclass Omniprediction

TL;DR

The main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity or regret horizon for -omniprediction in a -class prediction problem.

Abstract

Omniprediction is a learning problem that requires suboptimality bounds for each of a family of losses against a family of comparator predictors . We initiate the study of omniprediction in a multiclass setting, where the comparator family may be infinite. Our main result is an extension of the recent binary omniprediction algorithm of [OKK25] to the multiclass setting, with sample complexity (in statistical settings) or regret horizon (in online settings) , for -omniprediction in a -class prediction problem. En route to proving this result, we design a framework of potential broader interest for solving Blackwell approachability problems where multiple sets must simultaneously be approached via coupled actions.
Paper Structure (29 sections, 41 theorems, 132 equations)

This paper contains 29 sections, 41 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Simultaneous Blackwell approachability.
  4. Multiclass omniprediction.
  5. Other consequences.
  6. Related work
  7. Multiclass calibration.
  8. Multiclass learning.
  9. Blackwell approachability.
  10. Preliminaries
  11. Notation
  12. Omniprediction
  13. Simultaneous Blackwell Approachability
  14. Blackwell approachability
  15. Framework
  16. ...and 14 more sections

Key Result

Theorem 1

Let $\mathcal{L}$ be the family of multiclass GLM losses eq:glm_def and let $\mathcal{C}$ be the family of bounded $k \times d$ linear classifiers eq:clin_multi. Then given $T$ i.i.d. samples $(\mathbf{x}, \mathbf{y}) \sim \mathcal{D}$ for we return an $\varepsilon$-omnipredictor in time $O(dkT) + O(\frac{1}{\varepsilon})^{2k}\textup{poly}(k, \log \frac{1}{\varepsilon})$, with high probability.

Theorems & Definitions (79)

  • Theorem 1: Informal, see Theorem \ref{['thm:glm_multi']}
  • Lemma 1: Lemma 9, hu2025omnipredicting
  • Lemma 2: Theorem 4.2, Bubeck15
  • Definition 1: Omniprediction
  • Definition 2: $\mathcal{F}$-multiaccuracy
  • Definition 3: $\mathcal{W}$-calibration
  • Lemma 3
  • proof
  • Lemma 4: Theorem 1, GneitingR07
  • Proposition 1
  • ...and 69 more