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 sequenc
e 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.
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generali
zation of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. Other types of alternating e-projections algorithms are also provided by using different information geometric structures on the positive definite cone.
