Let @math be a symmetric probability measure of finite entropy on a group @math . We show that if @math , then the pair @math has the Liouville property (all bounded @math -harmonic functions on @math are constant). Furthermore, if @math where @math , then the entropy of the @math -fold convolution power @math satisfies @math . This improves earlier results of Gournay and of Saloff-Coste and the second author. We extend the bounds to transitive graphs and illustrate their sharpness on a family of groups.
In the work of the second author with Saloff-Coste @cite_12 , an upper bound similar to Theorem was proved. More precisely, suppose @math is a symmetric probability measure of finite second moment on @math such that @math for some @math , then by [Theorem 1.7] Saloff-Coste2014 [ H_ (n) (n ^ 1+ n )^ 1- for any >0. ] In particular, this bound implies the Liouville property of @math for @math . Our results in this paper imply a sharper upper bound @math under the same assumption, and are applicable to a larger class of decay lower bounds.
@cite_12
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Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution @math and a set of @math independent samples @math is drawn from @math . We seek an algorithm that, given @math as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution @math . How many samples @math must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples @math that are needed to answer statistical queries. The bounds improve and simplify the work of (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in , our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization