度量熵对偶与结果不一致性的样本复杂性
Metric Entropy Duality and the Sample Complexity of Outcome Indistinguishability
摘要：
我们给出了结果不可否认性的第一个样本复杂性特征，这是Dwork，Kim，Reingold，Rothblum和Yona（STOC 2021）最近引入的机器学习的理论框架。在结果不可预测性中，学习器的目标是输出一个预测器，该预测器不能通过一类$D$预测器检查根据预测器的预测生成的结果来与目标预测器区分开。在分布特定的和可实现的设置，学习者被赋予的数据分布与预测类$P$包含的目标预测，我们表明，样本的复杂性的结果不确定性的特征在于由度量熵的$P$ w.r.t.由$D$定义的对偶Minkowski范数，等价地由$D$的度量熵w.r. t.由$P$定义的对偶Minkowski范数。这种等价性与凸几何中长期存在的度量熵对偶猜想有着有趣的联系。我们的样本复杂性表征意味着度量熵对偶的一个变体，我们表明它几乎是紧的。在分布自由设置中，我们专注于Dwork等人考虑的情况。其中$P$包含所有可能的预测因子，因此样本复杂度仅取决于$D$。在这种情况下，我们表明，样本的复杂性的结果不可否认的特点是脂肪粉碎尺寸的$D$。我们还展示了一个强大的样本复杂性之间的分离，可实现的和不可知的结果不可分割的分布自由和分布特定的设置。这与免费分发（分别）相反。分布特定的）PAC学习，其中可实现和不可知设置中的样本复杂性可以由VC维度表征（分别为度量熵）。
英文摘要：
We give the first sample complexity characterizations for outcome indistinguishability, a theoretical framework of machine learning recently introduced by Dwork, Kim, Reingold, Rothblum, and Yona (STOC 2021). In outcome indistinguishability, the goal of the learner is to output a predictor that cannot be distinguished from the target predictor by a class $D$ of distinguishers examining the outcomes generated according to the predictors' predictions. In the distribution-specific and realizable setting where the learner is given the data distribution together with a predictor class $P$ containing the target predictor, we show that the sample complexity of outcome indistinguishability is characterized by the metric entropy of $P$ w.r.t. the dual Minkowski norm defined by $D$, and equivalently by the metric entropy of $D$ w.r.t. the dual Minkowski norm defined by $P$. This equivalence makes an intriguing connection to the long-standing metric entropy duality conjecture in convex geometry. Our sample complexity characterization implies a variant of metric entropy duality, which we show is nearly tight. In the distribution-free setting, we focus on the case considered by Dwork et al. where $P$ contains all possible predictors, hence the sample complexity only depends on $D$. In this setting, we show that the sample complexity of outcome indistinguishability is characterized by the fat-shattering dimension of $D$. We also show a strong sample complexity separation between realizable and agnostic outcome indistinguishability in both the distribution-free and the distribution-specific settings. This is in contrast to distribution-free (resp. distribution-specific) PAC learning where the sample complexity in both the realizable and the agnostic settings can be characterized by the VC dimension (resp. metric entropy).
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。
