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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 prove the first non-trivial lower bounds for high-dimensional learning and estimation in both the pan-private model and the general multi-message shuffle model. Our lower bounds apply to a variety of problems—for example, we show that, private agnostic learning of parity functions over d bits requires Ω(2d/2) samples in these models, and privately selecting the most common attribute from a set of d choices requires Ω(d1/2) samples, both of which are exponential separations from the central model.

The limits of pan privacy and shuffle privacy for learning and estimation