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In this paper, we study the Empirical Risk Minimization problem in thenon-interactive local model of differential privacy. In the case of constant orlow dimensionality ($p\ll n$), we first show that if the ERM loss function is$(\infty, T)$-smooth, then we can avoid a dependence of the sample complexity,to achieve error $\alpha$, on the exponential of the dimensionality $p$ withbase $1/\alpha$ (i.e., $\alpha^{-p}$), which answers a question in [smith 2017interaction]. Our approach is based on polynomial approximation. Then, wepropose player-efficient algorithms with $1$-bit communication complexity and$O(1)$ computation cost for each player. The error bound is asymptotically thesame as the original one. Also with additional assumptions we show a serverefficient algorithm. Next we consider the high dimensional case ($n\ll p$), weshow that if the loss function is Generalized Linear function and convex, thenwe could get an error bound which is dependent on the Gaussian width of theunderlying constrained set instead of $p$, which is lower than that in [smith2017 interaction].

Empirical Risk Minimization in Non-interactive Local Differential Privacy: Efficiency and High Dimensional Case