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A Theory of Universal Agnostic Learning

Steve Hanneke, Shay Moran

TL;DR

This work delivers a complete theory of universal learning rates for agnostic binary classification, extending the realizable-case framework to general distributions. It identifies a tetrachotomy: for finite classes the optimal rate is $e^{-n}$; for infinite classes, rates are $e^{-o(n)}$ if the class does not shatter an infinite Littlestone tree, $o(n^{-1/2})$ if it shatters an infinite Littlestone tree but not an infinite VC–Littlestone (VCL) tree, and arbitrarily slow otherwise. The analysis combines Gale–Stewart game strategies (SOA and VCL-based pattern avoidance), concentration tools including a Uniform Bernstein inequality for non-identically distributed data, and transductive learning techniques for partial concept classes with finite VC dimension. Finite-class results are straightforward via empirical risk minimization, while the infinite-class cases hinge on combinatorial tree structures and careful probabilistic constructions (eluder sequences) to establish both lower and upper bounds. The paper thus map out the full landscape of universal rates, offering algorithms that achieve the identified optimal rates and laying groundwork for extending the theory to broader learning settings.

Abstract

We provide a complete theory of optimal universal rates for binary classification in the agnostic setting. This extends the realizable-case theory of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021) by removing the realizability assumption on the distribution. We identify a fundamental tetrachotomy of optimal rates: for every concept class, the optimal universal rate of convergence of the excess error rate is one of $e^{-n}$, $e^{-o(n)}$, $o(n^{-1/2})$, or arbitrarily slow. We further identify simple combinatorial structures which determine which of these categories any given concept class falls into.

A Theory of Universal Agnostic Learning

TL;DR

This work delivers a complete theory of universal learning rates for agnostic binary classification, extending the realizable-case framework to general distributions. It identifies a tetrachotomy: for finite classes the optimal rate is ; for infinite classes, rates are if the class does not shatter an infinite Littlestone tree, if it shatters an infinite Littlestone tree but not an infinite VC–Littlestone (VCL) tree, and arbitrarily slow otherwise. The analysis combines Gale–Stewart game strategies (SOA and VCL-based pattern avoidance), concentration tools including a Uniform Bernstein inequality for non-identically distributed data, and transductive learning techniques for partial concept classes with finite VC dimension. Finite-class results are straightforward via empirical risk minimization, while the infinite-class cases hinge on combinatorial tree structures and careful probabilistic constructions (eluder sequences) to establish both lower and upper bounds. The paper thus map out the full landscape of universal rates, offering algorithms that achieve the identified optimal rates and laying groundwork for extending the theory to broader learning settings.

Abstract

We provide a complete theory of optimal universal rates for binary classification in the agnostic setting. This extends the realizable-case theory of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021) by removing the realizability assumption on the distribution. We identify a fundamental tetrachotomy of optimal rates: for every concept class, the optimal universal rate of convergence of the excess error rate is one of , , , or arbitrarily slow. We further identify simple combinatorial structures which determine which of these categories any given concept class falls into.
Paper Structure (26 sections, 33 theorems, 316 equations, 2 figures)

This paper contains 26 sections, 33 theorems, 316 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background:
  3. Technical contributions and challenges:
  4. Summary of the Main Results
  5. Additional Definitions and Notation
  6. Finite Classes
  7. Supporting Lemmas
  8. Near-Exponential Rate Lower Bound
  9. Sketch:
  10. Formal Proof:
  11. Near-Exponential Rate Upper Bound
  12. The Realizable Case:
  13. The Agnostic Case:
  14. Proof of Theorem \ref{['thm:agnostic-near-exponential-upper-bound']}: $e^{-o(n)}$ Rates
  15. Slower Than Near-exponential is not Faster Than Super-Root
  16. ...and 11 more sections

Key Result

Theorem 1

For every infinite concept class $\mathcal{H}$, exactly one of the following holds.

Figures (2)

  • Figure 1: Algorithm achieving $e^{-\psi(n)}$ rate for classes $\mathcal{H}$ with no infinite Littlestone tree.
  • Figure 2: Algorithm achieving $o(n^{-1/2})$ rate for classes $\mathcal{H}$ with no infinite VCL tree.

Theorems & Definitions (38)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4: *bousquet:21
  • Definition 5: *bousquet:21
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • Lemma 9: *bousquet:21
  • Lemma 10: *bousquet:21
  • ...and 28 more