Online DR-Submodular Maximization with Stochastic Cumulative Constraints
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In this paper, we consider online continuous DR-submodular maximization with linear stochastic long-term constraints. Compared to the prior work on online submodular maximization, our setting introduces the extra complication of stochastic linear constraint functions that are i.i.d. generated at each round. To be precise, at step t∈{1,...,T}, a DR-submodular utility function f_t(·) and a constraint vector p_t, i.i.d. generated from an unknown distribution with mean p, are revealed after committing to an action x_t and we aim to maximize the overall utility while the expected cumulative resource consumption ∑_t=1^T 〈 p,x_t〉 is below a fixed budget B_T. Stochastic long-term constraints arise naturally in applications where there is a limited budget or resource available and resource consumption at each step is governed by stochastically time-varying environments. We propose the Online Lagrangian Frank-Wolfe (OLFW) algorithm to solve this class of online problems. We analyze the performance of the OLFW algorithm and we obtain sub-linear regret bounds as well as sub-linear cumulative constraint violation bounds, both in expectation and with high probability.
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