Online Risk-Averse Submodular Maximization

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 Ω(T) space, our online algorithm only requires O( √ T) space. We extend our on- line algorithm to portfolio optimization for mono- tone 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.