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.

翻译：我们考虑一个长期未决的问题：寻找一类概率分布中能够表征其PAC可学习性的参数。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普遍的学习可表征性概念以及多种学习任务，展示类似的结果。我们证明，不存在任何维度概念能够表征分布类学习的样本复杂度。随后，我们考虑更弱的需求：仅表征可学习性（而非定量样本复杂度函数）。我们为此类表征提出了一些自然要求，进而证明：对于分布类，不存在任何满足这些要求的可学习性表征。进一步地，我们表明该结果适用于各类其他学习问题。具体而言，我们证明：在以下任务中，均不存在任何表征可学习性的维度概念（或可学习性表征）——分布类的分类学习、受限边缘分布下的二分类学习，以及具有连续损失函数的实值函数类的可学习性。