Stochastic $L\natural$-convex Function Minimization

We study an extension of the stochastic submodular minimization problem, namely, the stochastic L -convex minimization problem. We develop the first polynomialtime algorithms that return a near-optimal solution with high probability. We design a novel truncation operation to further reduce the computational complexity of the proposed algorithms. When applied to a stochastic submodular function, the computational complexity of the proposed algorithms is lower than that of the existing stochastic submodular minimization algorithms. In addition, we provide a strongly polynomial approximate algorithm. The algorithm execution also does not require any prior knowledge about the objective function except the L -convexity. A lower bound on the computational complexity that is required to achieve a high probability error bound is also derived. Numerical experiments are implemented to demonstrate the efficiency of our theoretical findings.