# Local Regularization Does Not Characterize Multiclass PAC Learnability

arXiv: https://arxiv.org/abs/2607.23449 · HTML: https://mlsanity.com/paper/2607.23449/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 5 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [A Characterization of Multiclass Learnability](https://mlsanity.com/paper/2203.01550/) (2022)
- [Optimal Learners for Multiclass Problems](https://mlsanity.com/paper/1405.2420/) (2014)
- [Regularization and Optimal Multiclass Learning](https://mlsanity.com/paper/2309.13692/) (2023)
- [Local Regularizers Are Not Transductive Learners](https://mlsanity.com/paper/2502.07187/) (2025)

## Built on by (most cited first, 2 of 2)

- [Two Dimensions Govern Agnostic Multiclass Transductive Learning](https://mlsanity.com/paper/2608.25326/) (2026)
- [Algorithmic Principles For Multiclass Learning Are Hard To Come By: Limits of Regularization and Proper Learning](https://mlsanity.com/paper/2608.26516/) (2026)

## Abstract (arXiv)

Local regularization assigns each hypothesis a test-point-dependent score and predicts with a minimum-score hypothesis consistent with the sample. Asilis et al. asked whether this principle characterizes multiclass PAC learnability. We give a negative answer. There is a countable class of Daniely--Shalev-Shwartz dimension at most two with realizable PAC sample complexity [ O!left(frac1varepsilonlogfrac1δright), ] that no local regularizer learns. Hypotheses are edges of complete graphs and instances are tournaments. At a test tournament, the scores fix an edge ranking while the training sample independently removes competitors. Cyclic triangles force enough inversions that surviving competitors produce constant population error at arbitrarily large sample sizes.