  Minimizing a convex risk function is the main step in many basic learning
algorithms. We study protocols for convex optimization which provably leak very
little about the individual data points that constitute the loss function.
Specifically, we consider differentially private algorithms that operate in the
local model, where each data record is stored on a separate user device and
randomization is performed locally by those devices. We give new protocols for
\emph{noninteractive} LDP convex optimization---i.e., protocols that require
only a single randomized report from each user to an untrusted aggregator.
  We study our algorithms' performance with respect to expected loss---either
over the data set at hand (empirical risk) or a larger population from which
our data set is assumed to be drawn. Our error bounds depend on the form of
individuals' contribution to the expected loss. For the case of
\emph{generalized linear losses} (such as hinge and logistic losses), we give
an 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 algorithm
for nonsmooth losses with sub-exponential dependence on $p$.
  For the Euclidean median problem, where the loss is given by the Euclidean
distance to a given data point, we give a protocol whose sample complexity
grows quasipolynomially in $p$. This is the first protocol with sub-exponential
dependence 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 of
inner product approximation, which may be applicable to other problems. Our
results for generalized linear losses and the Euclidean median are based on new
reductions to the case of hinge loss.
