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Effective Littlestone Dimension

Valentino Delle Rose, Alexander Kozachinskiy, Tomasz Steifer

TL;DR

Finite effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed.

Abstract

Delle Rose et al.~(COLT'23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one -- which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions.

Effective Littlestone Dimension

TL;DR

Finite effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed.

Abstract

Delle Rose et al.~(COLT'23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one -- which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions.

Paper Structure

This paper contains 6 sections, 14 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Effective threshold dimension
  4. Effective Littlestone dimension vs. computable online learning
  5. Equivalence in the bounded regime
  6. Effective Littlestone dimension and computability

Key Result

Lemma 1

Let $H$ be a hypothesis class and $L$ be an online learner for $H$ with at most $d$ mistakes, for some $d\in\mathbb{N}$. Then every function $f\in H$ coincides with $L_S$ for some sample $S$, consistent with $f$.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Proposition 3: littlestone1988learning
  • Proposition 4: littlestone1988learning
  • Proposition 5
  • proof
  • Theorem 6: hodges1997shorterAlon-et-al
  • Theorem 7
  • ...and 14 more