Efficient Empirical Risk Minimization with Smooth Loss Functions in Non-interactive Local Differential Privacy

In this paper, we study the Empirical Risk Minimization problem in the non-interactive local model of differential privacy. We first show that if the ERM loss function is (∞, T )-smooth, then we can avoid a dependence of the sample complexity, to achieve error α, on the exponential of the dimensionality p with base 1/α (i.e., α−p), which answers a question in (Smith et al., 2017). Our approach is based on Bernstein polynomial approximation. Then, we propose playerefficient algorithms with 1-bit communication complexity and O(1) computation cost for each player. The error bound is asymptotically the same as the original one. Also with additional assumptions we show a server efficient algorithm with polynomial running time. At last, we propose (efficient) non-interactive locally differential private algorithms, based on different types of polynomial approximations, for learning the set of k-way marginal queries and the set of smooth queries.