Zeroth-order Stochastic Approximation Algorithms for DR-submodular Optimization

In this paper, we study approximation algorithms for several classes of DR-submodular optimization problems, where DR is short for diminishing return. Following a newly introduced algorithm framework for zeroth-order stochastic approximation methods, we first propose algorithms CG-ZOSA and RG-ZOSA for smooth DR-submodular optimization based on the coordinate-wise gradient estimator and the randomized gradient estimator, respectively. Our theoretical analysis proves that CG-ZOSA can reach a solution whose expected objective value exceeds (1 e(-1) -is an element of(2))OPT -is an element of after O(-is an element of(-2) ) iterations and O(N-2/3 d is an element of(-2)) oracle calls, where d represents the problem dimension. On the other hand, RG-ZOSA improves the approximation ratio to (1-e(-1)-is an element of(2)/d) while maintaining the same overall oracle complexity. For non-smooth up-concave maximization problems, we propose a novel auxiliary function based on a smoothed objective function and introduce the NZOSA algorithm. This algorithm achieves an approximation ratio of (1-e(-1 )-is an element of ln is an element of(-1)-is an element of(2)ln is an element of(-1)) with O(d is an element of(-2)) iterations and O (N-2/3 d(3/2) is an element of(-3)) oracle calls. We also extend NZOSA to handle a class of robust DR-submodular maximization problems. To validate the effectiveness of our proposed algorithms, we conduct experiments on both synthetic and real-world problems. The results demonstrate the superior performance and efficiency of our methods in solving DR-submodular optimization problems.