  In this paper, we study the problem of maximizing continuous submodular
functions that naturally arise in many learning applications such as those
involving utility functions in active learning and sensing, matrix
approximations and network inference. Despite the apparent lack of convexity in
such functions, we prove that stochastic projected gradient methods can provide
strong approximation guarantees for maximizing continuous submodular functions
with convex constraints. More specifically, we prove that for monotone
continuous DR-submodular functions, all fixed points of projected gradient
ascent provide a factor $1/2$ approximation to the global maxima. We also study
stochastic gradient and mirror methods and show that after
$\mathcal{O}(1/\epsilon^2)$ iterations these methods reach solutions which
achieve in expectation objective values exceeding
$(\frac{\text{OPT}}{2}-\epsilon)$. An immediate application of our results is
to maximize submodular functions that are defined stochastically, i.e. the
submodular function is defined as an expectation over a family of submodular
functions with an unknown distribution. We will show how stochastic gradient
methods are naturally well-suited for this setting, leading to a factor $1/2$
approximation when the function is monotone. In particular, it allows us to
approximately maximize discrete, monotone submodular optimization problems via
projected gradient descent on a continuous relaxation, directly connecting the
discrete and continuous domains. Finally, experiments on real data demonstrate
that our projected gradient methods consistently achieve the best utility
compared to other continuous baselines while remaining competitive in terms of
computational effort.
