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Computable universal online learning

Dariusz Kalociński, Tomasz Steifer

TL;DR

The paper investigates whether universal online learning, guaranteed to fail only a finite number of times against an adaptive adversary, can be implemented by a computer program. It introduces computability constraints into the universal online learning framework and develops a detailed theory for $RER$ (recursively enumerable representable) classes, showing that computable realizable learnability and computable agnostic learnability diverge in general but coincide for $RER$ classes. It also provides a precise characterization of proper computable universal online learning and demonstrates separations between proper and improper computable learning within the computable regime. The results bridge abstract online learning with computability theory, revealing when inductive inference can be effectively enacted on a computer and highlighting natural limits posed by $\overline{\mathcal{H}}$ and $ ext{Π}^0_1$-class constructions.

Abstract

Understanding when learning is possible is a fundamental task in the theory of machine learning. However, many characterizations known from the literature deal with abstract learning as a mathematical object and ignore the crucial question: when can learning be implemented as a computer program? We address this question for universal online learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC'21). In this model, there is no hypothesis fixed in advance; instead, Adversary -- playing the role of Nature -- can change their mind as long as local consistency with the given class of hypotheses is maintained. We require Learner to achieve a finite number of mistakes while using a strategy that can be implemented as a computer program. We show that universal online learning does not imply computable universal online learning, even if the class of hypotheses is relatively easy from a computability-theoretic perspective. We then study the agnostic variant of computable universal online learning and provide an exact characterization of classes that are learnable in this sense. We also consider a variant of proper universal online learning and show exactly when it is possible. Together, our results give a more realistic perspective on the existing theory of online binary classification and the related problem of inductive inference.

Computable universal online learning

TL;DR

The paper investigates whether universal online learning, guaranteed to fail only a finite number of times against an adaptive adversary, can be implemented by a computer program. It introduces computability constraints into the universal online learning framework and develops a detailed theory for (recursively enumerable representable) classes, showing that computable realizable learnability and computable agnostic learnability diverge in general but coincide for classes. It also provides a precise characterization of proper computable universal online learning and demonstrates separations between proper and improper computable learning within the computable regime. The results bridge abstract online learning with computability theory, revealing when inductive inference can be effectively enacted on a computer and highlighting natural limits posed by and -class constructions.

Abstract

Understanding when learning is possible is a fundamental task in the theory of machine learning. However, many characterizations known from the literature deal with abstract learning as a mathematical object and ignore the crucial question: when can learning be implemented as a computer program? We address this question for universal online learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC'21). In this model, there is no hypothesis fixed in advance; instead, Adversary -- playing the role of Nature -- can change their mind as long as local consistency with the given class of hypotheses is maintained. We require Learner to achieve a finite number of mistakes while using a strategy that can be implemented as a computer program. We show that universal online learning does not imply computable universal online learning, even if the class of hypotheses is relatively easy from a computability-theoretic perspective. We then study the agnostic variant of computable universal online learning and provide an exact characterization of classes that are learnable in this sense. We also consider a variant of proper universal online learning and show exactly when it is possible. Together, our results give a more realistic perspective on the existing theory of online binary classification and the related problem of inductive inference.
Paper Structure (23 sections, 23 theorems, 5 equations)

This paper contains 23 sections, 23 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Setting
  4. Agnostic learning
  5. Proper versus improper learning
  6. Discussion
  7. Proofs
  8. Helpful lemmas
  9. Proof of Proposition \ref{['prop:RERuncomputable']}
  10. Proof of Theorem \ref{['thm:total equals agnostic']}
  11. Proof of Theorem \ref{['thm:separation']}
  12. Computable learner for $\mathcal{H}$.
  13. Proof of Theorem \ref{['thm:total equals partial for RER']}
  14. Proof of Theorem \ref{['thm:proper']}
  15. Proof sketch of Theorem \ref{['separation theorem proper']}
  16. ...and 8 more sections

Key Result

Proposition 1

There exists a $RER$ class that is universally online learnable but not computably universally online learnable. In fact, such a class can even be $DR$.

Theorems & Definitions (46)

  • Definition 1: universal online learning
  • Example 1
  • Example 2
  • Example 3
  • Definition 2
  • Proposition 1
  • Definition 3: agnostic universal online learnability
  • Example 4
  • Theorem 5
  • Theorem 6
  • ...and 36 more