Robust estimation of discrete distributions under local differential privacy

Flore Sentenac, Julien Chhor

Abstract: Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from contaminated data batches under a local differential privacy constraint. A fraction of the batches contain iid samples drawn from a discrete distribution over elements. To protect the users’ privacy, each of the samples is privatized using an -locally differentially private mechanism. The remaining batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be . Under the privacy constraint alone, the minimax rate of estimation is . We show, up to a factor, that combining the two constraints leads to a minimax estimation rate of , larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.