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Learning from Equivalence Queries, Revisited

Mark Braverman, Roi Livni, Yishay Mansour, Shay Moran, Kobbi Nissim

Abstract

Modern machine learning systems, such as generative models and recommendation systems, often evolve through a cycle of deployment, user interaction, and periodic model updates. This differs from standard supervised learning frameworks, which focus on loss or regret minimization over a fixed sequence of prediction tasks. Motivated by this setting, we revisit the classical model of learning from equivalence queries, introduced by Angluin (1988). In this model, a learner repeatedly proposes hypotheses and, when a deployed hypothesis is inadequate, receives a counterexample. Under fully adversarial counterexample generation, however, the model can be overly pessimistic. In addition, most prior work assumes a \emph{full-information} setting, where the learner also observes the correct label of the counterexample, an assumption that is not always natural. We address these issues by restricting the environment to a broad class of less adversarial counterexample generators, which we call \emph{symmetric}. Informally, such generators choose counterexamples based only on the symmetric difference between the hypothesis and the target. This class captures natural mechanisms such as random counterexamples (Angluin and Dohrn, 2017; Bhatia, 2021; Chase, Freitag, and Reyzin, 2024), as well as generators that return the simplest counterexample according to a prescribed complexity measure. Within this framework, we study learning from equivalence queries under both full-information and bandit feedback. We obtain tight bounds on the number of learning rounds in both settings and highlight directions for future work. Our analysis combines a game-theoretic view of symmetric adversaries with adaptive weighting methods and minimax arguments.

Learning from Equivalence Queries, Revisited

Abstract

Modern machine learning systems, such as generative models and recommendation systems, often evolve through a cycle of deployment, user interaction, and periodic model updates. This differs from standard supervised learning frameworks, which focus on loss or regret minimization over a fixed sequence of prediction tasks. Motivated by this setting, we revisit the classical model of learning from equivalence queries, introduced by Angluin (1988). In this model, a learner repeatedly proposes hypotheses and, when a deployed hypothesis is inadequate, receives a counterexample. Under fully adversarial counterexample generation, however, the model can be overly pessimistic. In addition, most prior work assumes a \emph{full-information} setting, where the learner also observes the correct label of the counterexample, an assumption that is not always natural. We address these issues by restricting the environment to a broad class of less adversarial counterexample generators, which we call \emph{symmetric}. Informally, such generators choose counterexamples based only on the symmetric difference between the hypothesis and the target. This class captures natural mechanisms such as random counterexamples (Angluin and Dohrn, 2017; Bhatia, 2021; Chase, Freitag, and Reyzin, 2024), as well as generators that return the simplest counterexample according to a prescribed complexity measure. Within this framework, we study learning from equivalence queries under both full-information and bandit feedback. We obtain tight bounds on the number of learning rounds in both settings and highlight directions for future work. Our analysis combines a game-theoretic view of symmetric adversaries with adaptive weighting methods and minimax arguments.

Paper Structure

This paper contains 47 sections, 4 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bandit Feedback.
  3. Overview of our contributions.
  4. Organization of the paper.
  5. Main Results
  6. Symmetric Counterexample Generators
  7. Example 1 (Random counterexample generators angluin_dohrn_2017).
  8. Example 2 (Decision-tree--induced counterexample generators).
  9. Statement of Results
  10. Infinite classes.
  11. Future Research.
  12. Proof Overview
  13. Upper Bounds
  14. Full Information
  15. A zero-sum game.
  16. ...and 32 more sections

Key Result

theorem 1

Let $\mathcal{H}$ be a finite hypothesis class with $\mathrm{Ldim}(\mathcal{H}) = d$, and let $\mathcal{A}$ be a symmetric adversary. Then, there exists a learning rule such that for every target concept $c \in \mathcal{H}$, the interaction between the learner and $\mathcal{A}$ terminates after at m

Figures (1)

  • Figure 1: A schematic example of a (ternary) decision tree when $\mathcal{Y}=\{0,1,2\}$. Internal nodes are labeled by instances, and edges are labeled by outcomes in $\mathcal{Y}$. Leaves correspond to prediction paths and carry no instance labels.

Theorems & Definitions (11)

  • definition 1: Symmetric counterexample generators
  • definition 2: Order-induced counterexample generators
  • definition 3: Littlestone Dimension
  • theorem 1: Full-information
  • theorem 2: Bandit feedback
  • Theorem : Full-information (restatement of Theorem \ref{['thm:full']})
  • lemma 1
  • proof
  • Claim 1
  • Claim 2
  • ...and 1 more