Online risk-averse submodular maximization

Abstract

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 (

$$1-1/e$$

1 - 1 / e

)-approximate solution with a convergence rate of

$$O(T^{-1/4})$$

O (

T

- 1 / 4

)

for monotone continuous DR-submodular functions. Compared with previous offline algorithms, which require

$$\Omega (T)$$

Ω ( T )

space, our online algorithm only requires

$$O(\sqrt{T})$$

O (

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.