  We consider the problem of minimizing a high-dimensional objective function,
which may include a regularization term, using (possibly noisy) evaluations of
the function. Such optimization is also called derivative-free, zeroth-order,
or black-box optimization. We propose a new $\textbf{Z}$eroth-$\textbf{O}$rder
$\textbf{R}$egularized $\textbf{O}$ptimization method, dubbed ZORO. When the
underlying gradient is approximately sparse at an iterate, ZORO needs very few
objective function evaluations to obtain a new iterate that decreases the
objective function. We achieve this with an adaptive, randomized gradient
estimator, followed by an inexact proximal-gradient scheme. Under a novel
approximately sparse gradient assumption and various different convex settings,
we show the (theoretical and empirical) convergence rate of ZORO is only
logarithmically dependent on the problem dimension. Numerical experiments show
that ZORO outperforms the existing methods with similar assumptions, on both
synthetic and real datasets.
