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

Abstract

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 $\mathrm{BDS}$ up to logarithmic factors, and asked whether every class admits sample complexity $O((\mathrm{BDS}+\log(1/\delta))/\epsilon)$. First, we show that the published lower bound $\Omega((\mathrm{BDS}+\log(1/\delta))/\epsilon)$ is incorrect as stated: we exhibit explicit classes with $\mathrm{BDS}=K-1$ 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 $\mathrm{aBDS}\le\mathrm{BDS}$, proving a constant-free three-part lower bound. On the upper-bound side we remove the ambient label count $K$ entirely, proving $O((B\log^3 B+B\log(1/\delta))/\epsilon)$ for $B=\mathrm{BDS}$, 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 $\Theta((n\min{n,\log(1/\delta)}+\log(1/\delta))/\epsilon)$, a confidence direct-sum regime where a multiplicative $\log(1/\delta)$ 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.