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标题: "Optimistic Rates for Multiclass PAC Learning"
原文链接: "https://arxiv.org/abs/2608.10869"
发布日期: "2026-08-11"
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## Computer Science > Machine Learning

## Title:Optimistic Rates for Multiclass PAC Learning

Authors:[Xiaoyu Li](https://arxiv.org/search/cs?searchtype=author&query=Li,+X), [Andi Han](https://arxiv.org/search/cs?searchtype=author&query=Han,+A), [Jiaojiao Jiang](https://arxiv.org/search/cs?searchtype=author&query=Jiang,+J), [Junbin Gao](https://arxiv.org/search/cs?searchtype=author&query=Gao,+J)

[View PDF](https://arxiv.org/pdf/2608.10869) [HTML (experimental)](https://arxiv.org/html/2608.10869v1)

> Abstract:Worst-case multiclass bounds do not become smaller when the best classifier is already nearly correct: what is missing is an optimistic rate, a guarantee whose fluctuation scales with the oracle risk itself. For a class of Natarajan dimension $d_N$ and Daniely-Shalev-Shwartz dimension $d_{DS}$, the optimal excess risk is known at the two endpoints ($d_{DS}/n$ realizable, $\sqrt{d_N/n}+d_{DS}/n$ agnostic \[HMZ24, CEH+26, Pab26\]) and open in between. We close the gap: at every fixed oracle risk $L^\star$, the optimal excess risk is $\widetilde{\Theta}(\sqrt{L^\star d_N/n}+d_{DS}/n)$, uniformly in the alphabet size, attained by a learner that knows neither $L^\star$ nor the confidence level. The upper bound composes the cover-menu-compression architecture of \[CEH+26\], at the realizable rate of \[Pab26\], with a new comparator-facing relative compression theorem: a size-$k$ compression rule that empirically dominates a comparator $h$ has population risk at most $L(h)+O(\sqrt{L(h)\Gamma}+\Gamma)$ with $\Gamma=(k\log n+\log(1/\delta))/n$, without stability; this transfers the comparison principle of the sharp binary theory \[MQZ26\] while discarding its Boolean-cube geometry, which does not lift to multiclass labels. The lower bound forces both terms using one class and one distribution at every fixed $L^\star$, by a pair-Assouad scheme calibrated to $L^\star$ and a fiber argument on the pseudo-cubes underlying the Natarajan-versus-DS separation of \[BCD+22\]. Both theorems extend to list learning: against the best $r$-tuple of hypotheses, the same architecture and the same two engines yield an optimistic rate and a lower bound of the same shape, forcing the fluctuation term that \[Pab26\] expected to be necessary against list comparators, and removing the factor $r$ from the known realizable list lower bound.

| Comments: |  |
| --- | --- |
| Subjects: | Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | [arXiv:2608.10869](https://arxiv.org/abs/2608.10869) \[cs.LG\] |
|  | (or [arXiv:2608.10869v1](https://arxiv.org/abs/2608.10869v1) \[cs.LG\] for this version) |
|  | [https://doi.org/10.48550/arXiv.2608.10869](https://doi.org/10.48550/arXiv.2608.10869) |

## Submission history

From: Xiaoyu Li \[[view email](https://arxiv.org/show-email/e7933809/2608.10869)\]  
**\[v1\]** Tue, 11 Aug 2026 12:46:03 UTC (60 KB)

[Which authors of this paper are endorsers?](https://arxiv.org/auth/show-endorsers/2608.10869) | Disable MathJax ([What is MathJax?](https://info.arxiv.org/help/mathjax.html))
