  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 $n$ contaminated data batches under a local differential
privacy constraint. A fraction $1-\epsilon$ of the batches contain $k$ i.i.d.
samples drawn from a discrete distribution $p$ over $d$ elements. To protect
the users' privacy, each of the samples is privatized using an $\alpha$-locally
differentially private mechanism. The remaining $\epsilon n $ batches are an
adversarial contamination. The minimax rate of estimation under contamination
alone, with no privacy, is known to be $\epsilon/\sqrt{k}+\sqrt{d/kn}$, up to a
$\sqrt{\log(1/\epsilon)}$ factor. Under the privacy constraint alone, the
minimax rate of estimation is $\sqrt{d^2/\alpha^2 kn}$. We show that combining
the two constraints leads to a minimax estimation rate of
$\epsilon\sqrt{d/\alpha^2 k}+\sqrt{d^2/\alpha^2 kn}$ up to a
$\sqrt{\log(1/\epsilon)}$ factor, 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.
