刻画分布学习的不可能性——一个长期存在问题的简单解决方案
Impossibility of Characterizing Distribution Learning -- a simple solution to a long-standing problem
AI总结：
本文证明不存在能刻画分布类 PAC 可学习性或样本复杂度的参数/维度，并将该不可能性推广到分类、受限边际二分类及连续损失实值函数学习等任务。
摘要：
我们考虑寻找一个概率分布类的参数以刻画其 PAC 可学习性这一长期存在的问题。我们给出了一个相当令人惊讶的答案——不存在这样的参数。我们的技术使我们能够针对若干刻画可学习性的一般概念以及若干学习任务证明类似结果。我们表明，不存在能够刻画分布类学习样本复杂度的维度概念。随后，我们考虑仅刻画可学习性（而非定量的样本复杂度函数）这一更弱的要求。我们为此类刻画提出了一些自然要求，并进一步证明，对于分布类，不存在满足这些要求的可学习性刻画。此外，我们表明我们的结果适用于各种其他学习问题。特别地，我们证明，对于以下任何任务，都不存在刻画其样本复杂度的维度概念（或可学习性刻画）：分布类的分类学习、关于受限边际分布集合的二分类学习，以及具有连续损失的实值函数类的可学习性。
英文摘要：
We consider the long-standing question of finding a parameter of a class of probability distributions that characterizes its PAC learnability. We provide a rather surprising answer - no such parameter exists. Our techniques allow us to show similar results for several general notions of characterizing learnability and for several learning tasks. We show that there is no notion of dimension that characterizes the sample complexity of learning distribution classes. We then consider the weaker requirement of only characterizing learnability (rather than the quantitative sample complexity function). We propose some natural requirements for such a characterization and go on to show that there exists no characterization of learnability that satisfies these requirements for classes of distributions. Furthermore, we show that our results hold for various other learning problems. In particular, we show that there is no notion of dimension characterizing (or characterization of learnability) for any of the tasks: classification learning for distribution classes, learning of binary classifications w.r.t. a restricted set of marginal distributions, and learnability of classes of real-valued functions with continuous losses.
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。
