两个维度支配不可知多类直推学习
Two Dimensions Govern Agnostic Multiclass Transductive Learning
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中文总结 AI 辅助
该研究解决了不可知多类直推学习的极小极大速率问题，证明其与不可知多类PAC学习遵循相同的双维法则，给出了最优超额误差的上界及必要条件。
中文摘要 AI 辅助
在直推分类中，对手固定一个带标签的总体，均匀隐藏一个标签，学习者可见所有剩余标签。对于二分类，不可知直推学习与PAC学习具有相同的极小极大速率。这一结论能否推广到多类学习（尤其是均匀收敛可能失效的无界标签空间情形）是一个未解决的问题。我们在对数因子范围内解决了该问题。对于每个具有DS维数$d_{DS}$和Natarajan维数$d_{\text{N}}$的多类假设类$\boldsymbol{\textit{H}}$，最优不可知直推超额误差满足$\boldsymbol{\tilde\theta}\boldsymbol{\bigg(}\frac{d_{DS}}{n}+\boldsymbol{\bigg(}\frac{d_{\text{N}}}{n}\boldsymbol{\bigg)}^{\frac{1}{2}}\boldsymbol{\bigg)}$。该结果适用于任意标签空间，且两个项均为必要项：DS伪立方体给出可实现情形下$\frac{d_{DS}}{n}$的阻碍，而带重复点与公平标签的Natarajan立方体给出不可知情形下$\boldsymbol{\bigg(}\frac{d_{\text{N}}}{n}\boldsymbol{\bigg)}^{\frac{1}{2}}$的阻碍。上界采用随机保留原则：学习者刻意忽略可见标签的恒定比例，使真实测试点在一个大的未观测块中均匀分布。我们结合可实现压缩、标签空间归约以及该有限总体划分下的菜单内不可知压缩，且新的无放回乘权引理保留了快速$\frac{d_{DS}}{n}$项。因此，不可知多类PAC学习与直推学习在对数因子范围内遵循相同的双维法则。
英文摘要
In transductive classification, an adversary fixes a labeled population, one label is hidden uniformly, and the learner sees all remaining labels. For binary classes, agnostic transductive and PAC learning have the same minimax rate. Whether this extends to multiclass learning was open, especially for unbounded label spaces where uniform convergence can fail. We resolve the question up to logarithmic factors. For every multiclass class $\mathcal H$ with DS dimension $d_{DS}$ and Natarajan dimension $d_{\mathrm N}$, the optimal agnostic transductive excess error satisfies $\widetildeΘ\left(\frac{d_{DS}}{n}+\sqrt{\frac{d_{\mathrm N}}{n}}\right).$ The result holds for arbitrary label spaces. The two terms are both necessary. A DS pseudo-cube gives the realizable $d_{DS}/n$ obstruction, while a Natarajan cube with repeated points and fair labels gives the agnostic $\sqrt{d_{\mathrm N}/n}$ obstruction. The upper bound uses a random-reservation principle. The learner deliberately ignores a constant fraction of the visible labels, which makes the true test point uniform in a large unseen block. We combine realizable compression, a label-space reduction, and inside-menu agnostic compression across this finite-population split. A new without-replacement multiplicative-weights lemma preserves the fast $d_{DS}/n$ term. Consequently, agnostic multiclass PAC and transductive learning obey the same two-dimension law up to logarithmic factors.
发表机构
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
