Supplementary Material for Priv’IT: Private and Sample Efficient Identity Testing

We will prove the theorem for the case where β = 1/3, the general case follows at the cost of a multiplicative log(1/β) in the sample complexity from a standard amplification argument. To be more precise, we can consider splitting our dataset into O(log(1/β)) sub-datasets and run the β = 1/3 test on each one independently. We return the majority result – since each test is correct with probability ≥ 2/3, correctness of the overall test follows by Chernoff bound. It remains to argue privacy – note that a neighboring dataset will only result in a single sub-dataset being changed. Since we take the majority result, conditioning on the result of the other sub-tests, the result on this sub-dataset will either be irrelvant to or equal to the overall output. In the former case, any test is private, and in the latter case, we know that the individual test is ε-differentially private. Overall privacy follows by applying the law of total probability. We require the following two claims, which give bounds on the random variables Ni and Yi. Note that, due to the fact that we draw Poisson(m) samples, each Ni ∼ Poisson(mpi) independently.