  In this paper, we revisit Stochastic Continuous Submodular Maximization in
both offline and online settings, which can benefit wide applications in
machine learning and operations research areas. We present a boosting framework
covering gradient ascent and online gradient ascent. The fundamental ingredient
of our methods is a novel non-oblivious function $F$ derived from a
factor-revealing optimization problem, whose any stationary point provides a
$(1-e^{-\gamma})$-approximation to the global maximum of the $\gamma$-weakly
DR-submodular objective function $f\in C^{1,1}_L(\mathcal{X})$. Under the
offline scenario, we propose a boosting gradient ascent method achieving
$(1-e^{-\gamma}-\epsilon^{2})$-approximation after $O(1/\epsilon^2)$
iterations, which improves the $(\frac{\gamma^2}{1+\gamma^2})$ approximation
ratio of the classical gradient ascent algorithm. In the online setting, for
the first time we consider the adversarial delays for stochastic gradient
feedback, under which we propose a boosting online gradient algorithm with the
same non-oblivious function $F$. Meanwhile, we verify that this boosting online
algorithm achieves a regret of $O(\sqrt{D})$ against a
$(1-e^{-\gamma})$-approximation to the best feasible solution in hindsight,
where $D$ is the sum of delays of gradient feedback. To the best of our
knowledge, this is the first result to obtain $O(\sqrt{T})$ regret against a
$(1-e^{-\gamma})$-approximation with $O(1)$ gradient inquiry at each time step,
when no delay exists, i.e., $D=T$. Finally, numerical experiments demonstrate
the effectiveness of our boosting methods.
