# Relatively Smart II: Tractable or Semi-Supervised Instance-Optimal Learning

arXiv: https://arxiv.org/abs/2609.10886 · HTML: https://mlsanity.com/paper/2609.10886/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 17 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [The Optimal Sample Complexity of PAC Learning](https://mlsanity.com/paper/1507.00473/) (2015)
- [Bagging is an Optimal PAC Learner](https://mlsanity.com/paper/2212.02264/) (2022)
- [Testing distributional assumptions of learning algorithms](https://mlsanity.com/paper/2204.07196/) (2022)
- [A Moment-Matching Approach to Testable Learning and a New Characterization of Rademacher Complexity](https://mlsanity.com/paper/2211.13312/) (2022)
- [Testable Learning with Distribution Shift](https://mlsanity.com/paper/2311.15142/) (2023)
- [Efficient Testable Learning of Halfspaces with Adversarial Label Noise](https://mlsanity.com/paper/2303.05485/) (2023)
- [An Efficient Tester-Learner for Halfspaces](https://mlsanity.com/paper/2302.14853/) (2023)
- [Tester-Learners for Halfspaces: Universal Algorithms](https://mlsanity.com/paper/2305.11765/) (2023)
- [Majority-of-Three: The Simplest Optimal Learner?](https://mlsanity.com/paper/2403.08831/) (2024)
- [When can unlabeled data improve the learning rate?](https://mlsanity.com/paper/1905.11866/) (2019)
- [Learning with Explanation Constraints](https://mlsanity.com/paper/2303.14496/) (2023)

## Abstract (arXiv)

We continue the study of relatively smart learning, introduced by Dughmi and Pour (2026), which asks a supervised learner to compete, marginal by marginal, with every distribution-fixed error guarantee soundly certifiable from unlabeled data. They showed that the One-Inclusion Graph (OIG) learner is relatively smart with a quadratic sample-complexity blowup, and that no relatively smart learner can do better, leaving open whether ERM or another natural or tractable learner achieves comparable guarantees. They also left open whether the blowup can be restricted to unlabeled data. Our firs results shows that ERM---and in fact any proper consistent learner---is relatively smart for binary classification in the distribution-free setting. We show that a small certifiable error with m samples implies a similarly small error on the uniform distribution over a random sample of size O(m2), yielding a cover of size at most 2m+1 on that sample. This suffices to control the error of proper consistent learners with O(m2) samples. We then show that semi-supervised relatively smart learning is information-theoretically possible with a quadratic blowup only in unlabeled sample complexity and no blowup in labeled sample complexity. The learner uses a natural generalization of OIG to a leave-most-out transductive problem, where labels of part of a finite pool are revealed and the remaining labels are predicted. Finally, this label efficiency comes at a cost in simplicity and tractability. If the hypothesis class is accessed only through an agnostic ERM oracle, any semi-supervised relatively smart learner with substantially sub-quadratic labeled-sample blowup requires super-polynomially many oracle calls. This holds even when the marginal is given explicitly, and thus also yields an intractability result for distribution-fixed learning that may be of independent interest.