Abstract
We present a polynomial-time online algorithm for maximizing the conditional value at risk (CVaR) of a monotone stochastic submodular function. Given T i.i.d. samples from an underlying distribution arriving online, our algorithm produces a sequence of solutions that converges to a (1-1/e)-approximate solution with a convergence rate of O(T^{-1/4}) for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require Omega (T) space, our online algorithm only requires O(sqrt{T}) space. We extend our online algorithm to portfolio optimization for monotone submodular set functions under a matroid constraint. Experiments conducted on real-world datasets demonstrate that our algorithm can rapidly achieve CVaRs that are comparable to those obtained by existing offline algorithms.
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