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Le Cam's method, Fano's inequality, and Assouad's lemma are three widely usedtechniques to prove lower bounds for statistical estimation tasks. We proposetheir analogues under central differential privacy. Our results are simple,easy to apply and we use them to establish sample complexity bounds in severalestimation tasks. We establish the optimal sample complexity of discretedistribution estimation under total variation distance and $\ell_2$ distance.We also provide lower bounds for several other distribution classes, includingproduct distributions and Gaussian mixtures that are tight up to logarithmicfactors. The technical component of our paper relates coupling betweendistributions to the sample complexity of estimation under differentialprivacy.

Differentially Private Assouad, Fano, and Le Cam