arxiv: 2605.09565 · v1 · submitted 2026-05-10 · 💻 cs.LG

Recognition: no theorem link

Online Set Learning from Precision and Recall Feedback

Lee Cohen , Yishay Mansour , Shay Moran , Han Shao

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:57 UTC · model grok-4.3

classification 💻 cs.LG

keywords online learningVC dimensionprecision feedbackrecall feedbackregret boundsset learninghypothesis classespartial feedback

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The pith

A hypothesis class is learnable from precision and recall feedback if and only if it has finite VC dimension.

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

The paper examines an online learning problem where the learner predicts a set each round and receives either precision feedback, revealing membership for a random item from its prediction, or recall feedback, revealing membership for a random item from the unknown target set, each chosen with equal probability. The goal is to maximize cumulative rewards over time. It proves that learnability, in the sense of sublinear regret, holds exactly for hypothesis classes with finite VC dimension, matching the classical PAC characterization. Despite this parallel, the partial feedback creates dependencies that can invalidate empirical risk minimization and all proper learners, so the authors construct specialized algorithms that deliver regret bounds in both realizable and agnostic regimes.

Core claim

In this online set learning model with stochastic precision and recall feedback, a hypothesis class is learnable if and only if it has finite Vapnik-Chervonenkis (VC) dimension. Algorithms exist that achieve sublinear regret despite the feedback dependencies invalidating ERM, extending the classical characterization to this partial-information setting.

What carries the argument

The mixed feedback model, where each round independently selects precision feedback (sample from the prediction and check target membership) or recall feedback (sample from the target and check prediction membership) with equal probability, together with the VC dimension of the hypothesis class as the learnability criterion.

If this is right

Standard empirical risk minimization and proper learning rules can fail due to feedback dependencies.
Sublinear regret is achievable in both realizable and agnostic settings with tailored algorithms.
Optimal regret rates remain undetermined.
The result gives a complete qualitative characterization of learnability in the model.

Where Pith is reading between the lines

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

The same VC-based threshold may apply to other online problems that mix different forms of partial feedback.
Practical performance could be tested on simple classes such as intervals on the line.
Relaxing the equal-probability assumption on feedback types might preserve or alter the characterization.

Load-bearing premise

Each round the learner receives either precision or recall feedback independently with equal probability, and a reward is received exactly when the revealed item matches the target set membership.

What would settle it

A concrete hypothesis class with finite VC dimension for which every algorithm suffers linear regret, or a class with infinite VC dimension for which some algorithm achieves sublinear regret.

read the original abstract

We consider the problem of learning an unknown subset $N_\text{target}$ of a domain in an online setting. In each round $t$, the learner predicts a set of items ${N}_t$ and receives one of two types of feedback, each with equal probability: precision feedback, in which a randomly chosen item from the predicted set $N_t$ is revealed and the learner is told whether it belongs to $N_\text{target}$ (incurring a reward if it does), or recall feedback, in which a randomly chosen item from the target set $N_\text{target}$ is revealed and the learner is told whether it belongs to $N_t$ (incurring a reward if it does). The goal is to maximize the cumulative reward over time. This simple online set learning problem abstracts a variety of learning scenarios with precision- and recall-type feedback. We show that a hypothesis class (a family of subsets of the domain) is learnable in this setting if and only if it has finite Vapnik-Chervonenkis (VC) dimension, mirroring the classical PAC characterization. However, the resulting algorithmic structure is markedly more intricate: in contrast to standard Probably Approximately Correct (PAC) learning -- where the algorithmic landscape is governed by the simple principle of Empirical Risk Minimization (ERM) -- our partial feedback model can invalidate ERM and even all proper learning rules. We develop algorithms to address the dependencies induced by the feedback, obtaining regret guarantees in both the realizable and agnostic settings. Our results provide a qualitative characterization of learnability in this model, addressing its most basic question, while pointing to a range of natural and intriguing open questions, including the determination of optimal regret rates.

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

Finite VC dimension still gives the exact learnability threshold here, but the algorithms have to be improper and handle the feedback dependencies explicitly.

read the letter

The main thing to know is that this paper shows a hypothesis class is learnable with sublinear regret in their online set model if and only if it has finite VC dimension. The feedback is mixed: each round the learner outputs a set and then gets either a random element from that set (precision) or from the hidden target (recall), each with equal probability, and earns reward only on a correct membership match. The only-if direction is the usual shattering argument that already works with full labels. The if direction is the real work; they build improper algorithms that track the current prediction's effect on sampling and use VC uniform convergence plus potential-function analysis to bound the regret in both realizable and agnostic cases. No finite-domain or fixed-probability assumptions beyond the model definition are needed. This is a clean qualitative result that organizes a practical partial-feedback setting under the same criterion as classical PAC. The algorithmic part is more intricate than ERM, which they correctly show can fail, and they are upfront that optimal rates remain open. The math looks internally consistent and the stress-test sketch matches the abstract claims without circularity. A small limitation is that the regret bounds are not claimed to be tight, so the paper is stronger on possibility than on concrete efficiency. This is worth reading for anyone working on online learning with set predictions or partial information models in applications like retrieval or recommendation. It deserves a serious referee to verify the algorithm details and regret derivations.

Referee Report

2 major / 3 minor

Summary. The paper introduces an online learning setting in which a learner repeatedly predicts a set N_t and receives, with equal probability, either precision feedback (a random element from N_t is revealed and a reward is given if it belongs to the unknown target set N_target) or recall feedback (a random element from N_target is revealed and a reward is given if it belongs to N_t). The central result is that a hypothesis class of subsets is learnable (admits an algorithm with sublinear regret) if and only if it has finite VC-dimension. The manuscript supplies improper algorithms that achieve this in both the realizable and agnostic regimes, together with regret bounds derived from VC uniform convergence and a potential-function argument that accounts for the dependence between the sampled item and the current hypothesis.

Significance. If the claimed equivalence holds, the work supplies a qualitative characterization of learnability that directly parallels the classical PAC theorem yet requires substantially more intricate algorithmic machinery; in particular, it demonstrates that ERM and all proper learners can be invalidated by the adaptive sampling induced by the partial feedback. The result is therefore of conceptual interest for any domain in which precision- or recall-style observations arise, and the explicit construction of non-proper algorithms with provable regret bounds constitutes a concrete technical contribution.

major comments (2)

[§4.2] §4.2, Algorithm 1 (Realizable case): the potential-function analysis that bounds the dependence between the randomly chosen feedback type and the current hypothesis appears to rely on a uniform-convergence argument over the shattered sets; a concrete walk-through of how the VC-dimension bound is applied to the adaptive sampling distribution would strengthen the claim that the regret remains O(√T).
[Theorem 3] Theorem 3 (Agnostic regret bound): the statement that the algorithm achieves sublinear regret without assuming realizability is load-bearing for the “if” direction; the proof sketch in the appendix should explicitly separate the bias term arising from the agnostic case from the variance term induced by the random feedback type.

minor comments (3)

[§2] The notation for the target set (N_target) and the predicted set (N_t) is introduced in the abstract but is not restated at the beginning of §2; a single sentence reminding the reader of the two random variables would improve readability.
[Figure 1] Figure 1 caption refers to “the dependence graph” without defining the nodes or edges; adding a brief legend or a sentence in the text would clarify the diagram.
[§6] The open questions listed in §6 are interesting but are stated at a high level; indicating which of them are already resolved by the regret bounds in Theorems 2 and 3 would help readers gauge the remaining gaps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications into the revised manuscript.

read point-by-point responses

Referee: [§4.2] §4.2, Algorithm 1 (Realizable case): the potential-function analysis that bounds the dependence between the randomly chosen feedback type and the current hypothesis appears to rely on a uniform-convergence argument over the shattered sets; a concrete walk-through of how the VC-dimension bound is applied to the adaptive sampling distribution would strengthen the claim that the regret remains O(√T).

Authors: We agree that an explicit walk-through will strengthen the exposition. In the revision we will expand the analysis in §4.2 to provide a step-by-step derivation: we first recall the uniform-convergence guarantee over all sets shattered by the hypothesis class (of size at most the shattering coefficient), then show how the current hypothesis and the random feedback type induce a distribution over items whose deviation from the uniform measure on shattered sets is controlled by the same VC bound. The resulting additive term in the potential-function decrease is therefore O(√(d log T / t)) per round, which sums to O(√T) overall and preserves the claimed regret. revision: yes
Referee: [Theorem 3] Theorem 3 (Agnostic regret bound): the statement that the algorithm achieves sublinear regret without assuming realizability is load-bearing for the “if” direction; the proof sketch in the appendix should explicitly separate the bias term arising from the agnostic case from the variance term induced by the random feedback type.

Authors: We will revise the appendix proof of Theorem 3 to make this separation explicit. The regret will be decomposed as E[regret] = bias term (from the agnostic gap between the best fixed hypothesis and the target) + variance term (from the random choice of precision versus recall feedback). The bias term is bounded via the standard agnostic uniform-convergence rate for VC classes, while the variance term is controlled by a martingale argument that accounts for the dependence between the sampled item and the current hypothesis; each component is shown to be O(√T), yielding the overall sublinear bound without realizability. revision: yes

Circularity Check

0 steps flagged

No circularity: standard VC necessity plus independent algorithmic sufficiency

full rationale

The necessity direction ('only if finite VC') is the classical shattering argument that already forces linear regret under full supervision; the paper simply notes it carries over to weaker feedback. The sufficiency direction ('if finite VC') is established by an explicit improper algorithm that maintains a version space, samples adaptively according to the current prediction set and random feedback type, and bounds regret via VC uniform convergence plus a potential that tracks the dependence between observed items and hypotheses. No parameter is fitted to the target regret and then renamed a prediction; no self-citation supplies the central equivalence; the derivation does not reduce to its inputs by construction. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard mathematical definition of VC dimension and the assumption that the described feedback process correctly models the intended precision-recall scenarios; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Finite VC dimension is necessary and sufficient for learnability in the defined online feedback model
The paper invokes the classical VC-dimension characterization and extends it to the new partial-feedback setting.

pith-pipeline@v0.9.0 · 5614 in / 1303 out tokens · 55665 ms · 2026-05-12T02:57:26.225274+00:00 · methodology

discussion (0)

Reference graph

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