# Bandit Multiclass PAC Learning: Corrected Lower Bounds, Exact Families, and a Confidence Direct-Sum Phenomenon

arXiv: https://arxiv.org/abs/2609.29694 · HTML: https://mlsanity.com/paper/2609.29694/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 14 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [A Characterization of Multiclass Learnability](https://mlsanity.com/paper/2203.01550/) (2022)
- [Optimal Learners for Multiclass Problems](https://mlsanity.com/paper/1405.2420/) (2014)
- [A Characterization of List Learnability](https://mlsanity.com/paper/2211.04956/) (2022)
- [Optimal Identity Testing with High Probability](https://mlsanity.com/paper/1708.02728/) (2017)
- [The Real Price of Bandit Information in Multiclass Classification](https://mlsanity.com/paper/2405.10027/) (2024)
- [Fast Rates for Bandit PAC Multiclass Classification](https://mlsanity.com/paper/2406.12406/) (2024)
- [From Contextual Combinatorial Semi-Bandits to Bandit List Classification: Improved Sample Complexity with Sparse Rewards](https://mlsanity.com/paper/2502.09257/) (2025)
- [PAC Learning with Bandit Feedback: Sharp Sample Complexity in the Realizable Setting](https://mlsanity.com/paper/2605.25678/) (2026)
- [The Sample Complexity of Multiclass and Sparse Contextual Bandits](https://mlsanity.com/paper/2605.29645/) (2026)

## Abstract (arXiv)

We study realizable multiclass PAC learning with bandit feedback: the learner observes an i.i.d. instance, predicts one of K labels, and learns only whether the prediction was correct. Hanneke, Meng, Moran, and Shaeiri (arXiv:2605.25678) characterized the optimal sample complexity via the bandit DS dimension BDS up to logarithmic factors, and asked whether every class admits sample complexity O((BDS+log(1/δ))/ε). First, we show that the published lower bound Omega((BDS+log(1/δ))/ε) is incorrect as stated: we exhibit explicit classes with BDS=K-1 whose sample complexity is exponentially smaller, and locate two independent gaps in its proof. We repair the lower-bound theory around a new anchored dimension aBDS≤BDS, proving a constant-free three-part lower bound. On the upper-bound side we remove the ambient label count K entirely, proving O((Blog3 B+Blog(1/δ))/ε) for B=BDS, plus a constant-confidence bound via a new fiberization lemma; for two natural families we determine the sample complexity up to constant factors. Finally, we answer the open question in the negative under its uniform-constant reading, and show the failure is intrinsic: for an explicit affine multiplexer class we establish the full confidence profile Theta((nminn,log(1/δ)+log(1/δ))/ε), a confidence direct-sum regime where a multiplicative log(1/δ) cost is information-theoretically necessary, followed by a rank-saturation phase transition. Two classes with identical dimension profiles can have polynomially different sample complexities, so no characterization by these dimensions alone is accurate to polylogarithmic factors. We also show these results are consistent with additive-confidence list-PAC guarantees via the ListCascade bridge.