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Minimizing a convex risk function is the main step in many basic learningalgorithms. We study protocols for convex optimization which provably leak verylittle about the individual data points that constitute the loss function.Specifically, we consider differentially private algorithms that operate in thelocal model, where each data record is stored on a separate user device andrandomization is performed locally by those devices. We give new protocols for\emph{noninteractive} LDP convex optimization---i.e., protocols that requireonly a single randomized report from each user to an untrusted aggregator.  We study our algorithms' performance with respect to expected loss---eitherover the data set at hand (empirical risk) or a larger population from whichour data set is assumed to be drawn. Our error bounds depend on the form ofindividuals' contribution to the expected loss. For the case of\emph{generalized linear losses} (such as hinge and logistic losses), we givean LDP algorithm whose sample complexity is only linear in the dimensionality$p$ and quasipolynomial in other terms (the privacy parameters $\epsilon$ and$\delta$, and the desired excess risk $\alpha$). This is the first algorithmfor nonsmooth losses with sub-exponential dependence on $p$.  For the Euclidean median problem, where the loss is given by the Euclideandistance to a given data point, we give a protocol whose sample complexitygrows quasipolynomially in $p$. This is the first protocol with sub-exponentialdependence on $p$ for a loss that is not a generalized linear loss .  Our result for the hinge loss is based on a technique, dubbed polynomial ofinner product approximation, which may be applicable to other problems. Ourresults for generalized linear losses and the Euclidean median are based on newreductions to the case of hinge loss.

Noninteractive Locally Private Learning of Linear Models via Polynomial Approximations