{'fraction_non_alphanumeric': 0.030277544154751892, 'fraction_numerical': 0.005046257359125316, 'mean_word_length': 5.329787234042553, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On metric entropy, Vapnik-Chervonenkis dimension, and learnability for a class of distributions

In [23], Valiant proposed a formal framework for distribution-free concept learning which has generated a great deal of interest. A fundamental result regarding this framework was proved by Blumer et al. [6] characterizing those concept classes which are learnable in terms of their Vapnik-Chervonenkis (VC) dimension. More recently, Benedek and Itai [4] studied learnability with respect to a fixed probability distribution (a variant of the original distribution-free framework) and proved an analogous result characterizing learnability in this case. They also stated a conjecture regarding learnability for a class of distributions. In this report, we first point out that the condition for learnability obtained in [4] is equivalent to the notion of finite metric entropy (which has been studied in other contexts). Some relationships, in addition to those shown in [4], between the VC dimension of a concept class and its metric entropy with respect to various distributions are then discussed. Finally, we prove some partial results regarding learnability for a class of distributions.