arxiv: 2605.13384 · v1 · submitted 2026-05-13 · 💻 cs.LG

Recognition: unknown

Teaching and Learning under Deductive Errors

Jan Arne Telle , Brigt H{\aa}vardstun , Jose Hernandez-Orallo

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:24 UTC · model grok-4.3

classification 💻 cs.LG

keywords machine teachingPAC learningdeductive errorsteaching setscomputational complexitylarge language modelsfew-shot learning

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The pith

Teachers can find small example sets that still guide learners making deductive errors to approximately correct hypotheses with high probability.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a machine teaching framework that relaxes the usual assumption of error-free deduction by the learner. It revises the PAC model to incorporate a known error rate, proving that a suitably chosen teaching set suffices for the learner to reach an approximately correct hypothesis with high probability. This matters for real learners such as humans and large language models that fail consistency checks or behave stochastically. The authors also map out the computational complexity of finding optimal teaching sets and supply parameterized algorithms together with matching lower bounds.

Core claim

In an overhauled PAC teaching setting that accounts for a fixed deductive error level, there exist teaching sets of bounded size such that, with high probability, any learner whose internal deductions are wrong with at most that error rate will output a hypothesis that is approximately correct with respect to the target concept.

What carries the argument

The PAC teaching set under deductive errors: a finite collection of labeled examples chosen by the teacher so that the learner's possibly erroneous deductions still produce an approximately correct hypothesis with high probability.

If this is right

Six decision problems about finding minimum-size or minimum-cost PAC teaching sets admit XP algorithms parameterized by teaching-set size.
The algorithms have tight runtime bounds under the Exponential Time Hypothesis.
The same framework covers both teacher-driven machine teaching and learner-driven scenarios.
Selected teaching and learning protocols reproduce observed behavior in LLM few-shot teaching sessions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The model suggests that estimating a learner's error rate in advance could let teachers or prompt engineers use fewer examples while retaining reliability.
It raises the question of how to adapt teaching sets when error rates differ across hypotheses rather than staying fixed.
The complexity results indicate that exhaustive search over teaching-set size remains feasible only for small sets, pointing toward heuristic or approximate methods for larger problems.
The experimental protocols could be reused to benchmark new LLM teaching strategies against the deductive-error baseline.

Load-bearing premise

The learner's deductive error rate can be estimated in advance and remains consistent enough for the PAC guarantee to apply across different hypotheses.

What would settle it

An experiment in which learners with a fixed, known deductive error rate fail to output approximately correct hypotheses even after exposure to the size-optimal teaching sets computed by the model would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.13384 by Brigt H{\aa}vardstun, Jan Arne Telle, Jose Hernandez-Orallo.

**Figure 2.** Figure 2: Each data point is a teaching set, in (a) of size one: a single positive [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

Most models of machine teaching and learning assume the learner makes no errors in its internal deductive inference. However, humans and large language models in few-shot learning regimes are two important examples of learners where this does not hold. They fail on some consistency checks, and they can fail stochastically. In this paper we introduce a teaching and learning framework that takes these deductive errors into account. We specifically study the case of machine teaching, as different characterizations of the teacher can account for both machine teaching and learning. In an overhauled Probably Approximately Correct (PAC) setting, we study theoretically that, for some estimated error level, the teacher must find a PAC teaching set that with high probability will lead the learner to guess a hypothesis that is approximately correct. We study the computational complexity of six different problems related to computing optimal PAC teaching sets. We give XP algorithms parametrized by size of teaching set, with tight runtime bounds under standard complexity assumptions like ETH. These results are complemented with a small experimental study of which teaching and learning protocols can best represent the observed behavior in some LLM teaching sessions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends PAC teaching to handle fixed-rate deductive errors with new complexity results, but the fixed error assumption is a real limitation.

read the letter

The one thing your colleague needs to know is that this paper introduces a version of PAC machine teaching that accounts for learners making deductive errors, and it provides computational complexity results for finding good teaching sets in that setting. They do well by formalizing how a teacher can still achieve approximate correctness with high probability when the learner has a known error rate in its deductions. This is relevant for modeling LLMs and humans. The complexity analysis covers six problems, with XP algorithms parametrized by the size of the teaching set and lower bounds based on ETH. Those results appear carefully derived and add to the literature on teaching complexity. Including a small experimental study with LLMs is a reasonable step to connect the theory to practice. On the soft spots, the framework depends on having a fixed, hypothesis-independent error estimate upfront. The stress test note is right to flag this: if the error rate changes with the hypothesis, the uniform convergence needed for the PAC bound breaks down. The paper does not seem to provide ways to handle varying errors or to estimate the rate reliably from data. The experiments are described as small, which limits how much they can show about real-world applicability. This paper is for researchers in computational learning theory who work on machine teaching. Someone interested in the algorithmic side of teaching with imperfect learners will find the complexity section useful. It should go through peer review because the new framework and the algorithmic contributions are developed enough to merit detailed referee comments on the proofs and the modeling assumptions.

Referee Report

2 major / 2 minor

Summary. The paper introduces a framework for machine teaching and learning that accounts for deductive errors in learners (e.g., humans or LLMs), which may fail consistency checks stochastically. In an extended PAC setting, it claims that for a given estimated error level, there exists a PAC teaching set such that the learner outputs an approximately correct hypothesis with high probability. It analyzes the computational complexity of six problems for computing optimal such teaching sets, providing XP algorithms parametrized by teaching-set size together with tight ETH-based lower bounds, and reports a small experimental study comparing teaching protocols on LLM sessions.

Significance. If the central PAC existence claim and complexity characterizations hold, the work supplies a theoretical basis for teaching under imperfect deduction and gives precise XP/ETH results that delineate tractability. The experimental component, though limited in scale, provides initial empirical grounding for LLM-relevant protocols. The explicit parametrization and matching lower bounds constitute a technical strength.

major comments (2)

[theoretical PAC framework] Overhauled PAC setting (abstract and theoretical section): the existence of a PAC teaching set is asserted for a fixed, a-priori estimated error level that is treated as hypothesis-independent. No derivation is supplied showing that uniform convergence still holds when the error probability may vary with the chosen hypothesis, which is required for the claimed guarantee to apply uniformly over the hypothesis class.
[computational complexity analysis] Complexity results for the six problems: the XP algorithms are stated to be parametrized by teaching-set size with ETH-tight bounds, yet the manuscript provides neither explicit runtime expressions nor pseudocode for the algorithms, preventing verification that the parametrization is correctly applied and that the bounds are indeed tight.

minor comments (2)

[abstract] The abstract refers to 'six different problems' without enumerating them; an explicit list would improve readability.
[experimental study] The experimental study is described as 'small' but supplies no details on the number of LLM sessions, the precise protocols tested, or statistical significance of the observed differences.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the two major comments point by point below and will revise the manuscript to incorporate the requested clarifications and details.

read point-by-point responses

Referee: [theoretical PAC framework] Overhauled PAC setting (abstract and theoretical section): the existence of a PAC teaching set is asserted for a fixed, a-priori estimated error level that is treated as hypothesis-independent. No derivation is supplied showing that uniform convergence still holds when the error probability may vary with the chosen hypothesis, which is required for the claimed guarantee to apply uniformly over the hypothesis class.

Authors: We appreciate the referee highlighting this point. In the framework we introduce, the deductive error probability is modeled as a fixed, a priori estimate ε that is independent of the hypothesis; it represents a uniform bound on the learner's stochastic failure rate across consistency checks. Under this modeling assumption the standard uniform-convergence argument of PAC learning applies directly, because the deviation probability is bounded uniformly rather than hypothesis-dependently. Nevertheless, we agree that an explicit short derivation would improve clarity. In the revised version we will insert a brief proof sketch in the theoretical section that shows how the PAC guarantee (existence of a teaching set yielding an approximately correct hypothesis with high probability) follows from the fixed-ε assumption and the usual ε-δ uniform convergence over the hypothesis class. revision: yes
Referee: [computational complexity analysis] Complexity results for the six problems: the XP algorithms are stated to be parametrized by teaching-set size with ETH-tight bounds, yet the manuscript provides neither explicit runtime expressions nor pseudocode for the algorithms, preventing verification that the parametrization is correctly applied and that the bounds are indeed tight.

Authors: We agree that the lack of explicit runtimes and pseudocode makes independent verification difficult. In the revised manuscript we will supply the precise runtime expressions for each of the six problems (e.g., O(f(k)·n^c) where k is the teaching-set size and c is a small constant) and include pseudocode for the main XP algorithms (dynamic-programming enumeration over subsets of size at most k) in a new appendix. These additions will also make the matching ETH lower bounds easier to inspect. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PAC extension with fixed error-rate parameter

full rationale

The paper defines a teaching framework by augmenting classical PAC learning with a fixed, a priori estimated deductive error probability that is treated as an exogenous model parameter. All central results—the existence of PAC teaching sets that succeed with high probability, and the XP algorithms for computing optimal teaching sets parametrized by set size—are derived directly from this augmented model using standard uniform-convergence arguments. No key quantity is obtained by fitting to the paper’s own outputs, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-defined ansatz. The framework is therefore self-contained against external PAC benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim depends on an estimated error level treated as a parameter in the overhauled PAC setting and on the domain assumption that learner errors are stochastic but bounded.

free parameters (1)

estimated error level
Used to determine the size and properties of the PAC teaching set that guarantees approximate correctness with high probability.

axioms (1)

domain assumption Learner deductive errors occur at a fixed estimated rate and can be modeled within the PAC framework
Invoked to overhaul the standard PAC setting for teaching sets.

pith-pipeline@v0.9.0 · 5491 in / 1109 out tokens · 25179 ms · 2026-05-14T20:24:35.372634+00:00 · methodology

discussion (0)

Reference graph

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