可实现的直推分类的Minimax下界
Minimax Lower Bounds for Realizable Transductive Classification
摘要：
直推学习考虑$m$标记样本的训练集和$u$未标记样本的测试集，目标是最好地标记该特定测试集。相反，归纳学习考虑从$P(X,Y)$中提取的$m$标记样本的训练集，目标是最好地标记从$P(X)$中提取的任何未来样本。这种比较表明，转导是一种比诱导容易得多的推理类型，但事实真的如此吗？本文提供了一个否定的回答这个问题，通过证明第一个已知的极小极大下限的转导，可实现的，二进制分类。我们的下界表明，$m$应该至少是$Ω(d/ε+ \log(1/δ)/ε)$时，$ε$ -学习一个概念类$\mathcal{H}$有限VC维$d<\infty$与信心$1-δ$，所有$m \leq u$。这一结果得出了三个重要结论。首先，一般的转导和一般的归纳一样困难，因为这两个问题都有$Ω(d/m)$极大极小值。其次，使用未标记的数据无助于一般的转换，因为有监督的学习算法，如ERM和（Hanneke，2015）匹配我们的转换下限，而忽略未标记的测试集。第三，我们的转换下界意味着半监督学习的下界，这增加了关于未标记数据在机器学习中的作用的重要讨论。
英文摘要：
Transductive learning considers a training set of $m$ labeled samples and a test set of $u$ unlabeled samples, with the goal of best labeling that particular test set. Conversely, inductive learning considers a training set of $m$ labeled samples drawn iid from $P(X,Y)$, with the goal of best labeling any future samples drawn iid from $P(X)$. This comparison suggests that transduction is a much easier type of inference than induction, but is this really the case? This paper provides a negative answer to this question, by proving the first known minimax lower bounds for transductive, realizable, binary classification. Our lower bounds show that $m$ should be at least $Ω(d/ε+ \log(1/δ)/ε)$ when $ε$-learning a concept class $\mathcal{H}$ of finite VC-dimension $d<\infty$ with confidence $1-δ$, for all $m \leq u$. This result draws three important conclusions. First, general transduction is as hard as general induction, since both problems have $Ω(d/m)$ minimax values. Second, the use of unlabeled data does not help general transduction, since supervised learning algorithms such as ERM and (Hanneke, 2015) match our transductive lower bounds while ignoring the unlabeled test set. Third, our transductive lower bounds imply lower bounds for semi-supervised learning, which add to the important discussion about the role of unlabeled data in machine learning.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
