典型稳定性
Typical Stability
摘要：
在本文中，我们介绍了一个算法的稳定性称为典型稳定性的概念。当我们的目标是发布实值查询时，（统计）在数据集上计算，这个概念不要求查询具有有限的敏感性--当用作算法稳定性的概念时，通常在差分隐私[DMNS 06，Dwork 06]下假设的条件[DFHPRR 15 a，DFHPRR 15 b，BNSSSU 16] --也不要求数据集中的样本是独立的--这是在寻求泛化错误保证时通常假设的条件。相反，典型的稳定性要求查询的输出，当在从底层分布中提取的数据集上计算时，集中在相对于该分布的期望值周围。我们讨论了典型稳定性对泛化误差的影响（即，在数据集上计算的查询值与查询相对于真实数据分布的期望值之间的差）。我们表明，典型的稳定性可以控制自适应数据分析中的泛化误差，即使当数据集中的样本不一定是独立的，当要计算的查询不一定是有界灵敏度，只要查询的结果在数据集上（即，所计算的统计量）遵循具有“轻”尾的分布。这样的查询的示例包括但不限于亚高斯和亚指数查询。我们还讨论了典型稳定性的组合保证，并证明组合定理，表征典型稳定性的参数退化下$k$倍自适应组合。我们还给出了简单的噪声添加算法，实现这一概念。这些算法类似于它们的差分私有对应物，然而，添加的噪声被不同地校准。
英文摘要：
In this paper, we introduce a notion of algorithmic stability called typical stability. When our goal is to release real-valued queries (statistics) computed over a dataset, this notion does not require the queries to be of bounded sensitivity -- a condition that is generally assumed under differential privacy [DMNS06, Dwork06] when used as a notion of algorithmic stability [DFHPRR15a, DFHPRR15b, BNSSSU16] -- nor does it require the samples in the dataset to be independent -- a condition that is usually assumed when generalization-error guarantees are sought. Instead, typical stability requires the output of the query, when computed on a dataset drawn from the underlying distribution, to be concentrated around its expected value with respect to that distribution. We discuss the implications of typical stability on the generalization error (i.e., the difference between the value of the query computed on the dataset and the expected value of the query with respect to the true data distribution). We show that typical stability can control generalization error in adaptive data analysis even when the samples in the dataset are not necessarily independent and when queries to be computed are not necessarily of bounded-sensitivity as long as the results of the queries over the dataset (i.e., the computed statistics) follow a distribution with a "light" tail. Examples of such queries include, but not limited to, subgaussian and subexponential queries. We also discuss the composition guarantees of typical stability and prove composition theorems that characterize the degradation of the parameters of typical stability under $k$-fold adaptive composition. We also give simple noise-addition algorithms that achieve this notion. These algorithms are similar to their differentially private counterparts, however, the added noise is calibrated differently.
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