Stochastic Continuous Submodular Maximization: Boosting via Non-oblivious Function
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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.