  We design a new algorithm for the Euclidean $k$-means problem that operates
in the local model of differential privacy. Unlike in the non-private
literature, differentially private algorithms for the $k$-means objective incur
both additive and multiplicative errors. Our algorithm significantly reduces
the additive error while keeping the multiplicative error the same as in
previous state-of-the-art results. Specifically, on a database of size $n$, our
algorithm guarantees $O(1)$ multiplicative error and $\approx n^{1/2+a}$
additive error for an arbitrarily small constant $a>0$. All previous algorithms
in the local model had additive error $\approx n^{2/3+a}$. Our techniques
extend to $k$-median clustering.
  We show that the additive error we obtain is almost optimal in terms of its
dependency on the database size $n$. Specifically, we give a simple lower bound
showing that every locally-private algorithm for the $k$-means objective must
have additive error at least $\approx\sqrt{n}$.
