Frank Wolfe Algorithm for Nonmonotone One-Sided Smooth Function Maximization Problem

Abstract

In this paper, we study the problem of maximizing a nonmonotone one-sided-$\eta$ smooth (OSS for short) function $\psi (x)$ under a downwards-closed convex polytope constraint. The concept of OSS was first proposed by Mehrdad et al. [1, 2] to express the properties of multilinear extension of some set functions. It is a generalization of the continuous DR submodular function. The OSS property guarantees an alternative bound based on Taylor expansion. If the objective function is nonmonotone diminishing return (DR) submodular, Bian et al. [3] gave a 1/e approximation algorithm with a regret bound $O(\frac{LD^{2}}{2K})$. On general convex sets, D$\ddot{u}$rr et al. [4] gave a $\frac{1}{3\sqrt{3}}$ approximation solution with $O(\frac{LD^{2}}{(\ln K)^{2}})$ regrets. In this paper, we consider maximizing the more general OSS function, and by adjusting the iterative step of the Jump-Start Frank Wolfe algorithm, an approximation of 1/e can still be obtained in the case of a larger regret bound $O(\frac{L(\mu D)^{2}}{2K})$. (where $L,\mu , D$ are some parameters, see Table 1). The larger the parameter $\eta$ we choose, the more regrets we will receive, because of $\mu =\left( \frac{\beta }{\beta +1}\right) ^{-2\eta }$ $(\beta \in (0,1])$.