Continuous Submodular Maximization: Boosting via Non-oblivious Function
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In this paper, we revisit the constrained and stochastic continuous submodular maximization in both offline and online settings. For each γ-weakly DR-submodular function f, we use the factor-revealing optimization equation to derive an optimal auxiliary function F, whose stationary points provide a (1-e^-γ)-approximation to the global maximum value (denoted as OPT) of problem max_x∈𝒞f(x). Naturally, the projected (mirror) gradient ascent relied on this non-oblivious function achieves (1-e^-γ-ϵ^2)OPT-ϵ after O(1/ϵ^2) iterations, beating the traditional (γ^2/1+γ^2)-approximation gradient ascent <cit.> for submodular maximization. Similarly, based on F, the classical Frank-Wolfe algorithm equipped with variance reduction technique <cit.> also returns a solution with objective value larger than (1-e^-γ-ϵ^2)OPT-ϵ after O(1/ϵ^3) iterations. In the online setting, we first consider the adversarial delays for stochastic gradient feedback, under which we propose a boosting online gradient algorithm with the same non-oblivious search, achieving a regret of √(D) (where D is the sum of delays of gradient feedback) against a (1-e^-γ)-approximation to the best feasible solution in hindsight. Finally, extensive numerical experiments demonstrate the efficiency of our boosting methods.
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