There has been a recent wave of interest in intermediate trust models for differential privacy that eliminate the need for a fully trusted central data collector, but overcome the limitations of local differential privacy. This interest has led to the introduction of the shuffle model (Cheu et al., EUROCRYPT 2019; Erlingsson et al., SODA 2019) and revisiting the pan-private model (Dwork et al., ITCS 2010). The message of this line of work is that, for a variety of low-dimensional problems---such as counts, means, and histograms---these intermediate models offer nearly as much power as central differential privacy. However, there has been considerably less success using these models for high-dimensional learning and estimation problems. In this work, we show that, for a variety of high-dimensional learning and estimation problems, both the shuffle model and the pan-private model inherently incur an exponential price in sample complexity relative to the central model. For example, we show that, private agnostic learning of parity functions over $d$ bits requires $\Omega(2^{d/2})$ samples in these models, and privately selecting the most common attribute from a set of $d$ choices requires $\Omega(d^{1/2})$ samples, both of which are exponential separations from the central model. Our work gives the first non-trivial lower bounds for these problems for both the pan-private model and the general multi-message shuffle model.

中文翻译：

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泛隐私和随机隐私对学习和估计的限制

最近对用于差异隐私的中间信任模型产生了兴趣，该模型消除了对完全可信的中央数据收集器的需求，但克服了本地差异隐私的局限性。这种兴趣导致引入了 shuffle 模型（Cheu 等人，EUROCRYPT 2019；Erlingsson 等人，SODA 2019）并重新审视了泛私有模型（Dwork 等人，ITCS 2010）。这一系列工作的信息是，对于各种低维问题——例如计数、均值和直方图——这些中间模型提供的能力几乎与中心差分隐私一样多。然而，将这些模型用于高维学习和估计问题的成功率要低得多。在这项工作中，我们表明，对于各种高维学习和估计问题，相对于中心模型，shuffle 模型和 pan-private 模型本质上都会导致样本复杂性的指数价格。例如，我们表明，在这些模型中，对 $d$ 位的奇偶函数的私有不可知学习需要 $\Omega(2^{d/2})$ 样本，并从一组 $d 中私下选择最常见的属性$choices 需要 $\Omega(d^{1/2})$ 样本，两者都是与中心模型的指数分离。我们的工作为泛私有模型和一般多消息洗牌模型提供了这些问题的第一个非平凡下界。在这些模型中，对 $d$ 位上的奇偶函数的私有不可知学习需要 $\Omega(2^{d/2})$ 样本，并且从一组 $d$ 选择中私下选择最常见的属性需要 $\Omega( d^{1/2})$ 样本，两者都是与中心模型的指数分离。我们的工作为泛私有模型和一般多消息洗牌模型提供了这些问题的第一个非平凡下界。在这些模型中，对 $d$ 位上的奇偶函数的私有不可知学习需要 $\Omega(2^{d/2})$ 样本，并且从一组 $d$ 选择中私下选择最常见的属性需要 $\Omega( d^{1/2})$ 样本，两者都是与中心模型的指数分离。我们的工作为泛私有模型和一般多消息洗牌模型提供了这些问题的第一个非平凡下界。