Black Box Submodular Maximization: Discrete and Continuous Settings
This addresses optimization challenges in machine learning and operations research where derivative information is unavailable, though it builds incrementally on existing submodular maximization frameworks.
The paper tackles black box continuous submodular maximization without derivative access, proposing an algorithm that achieves a tight (1-1/e)OPT-ε approximation with O(d/ε³) function evaluations, and extends it to stochastic settings and discrete problems.
In this paper, we consider the problem of black box continuous submodular maximization where we only have access to the function values and no information about the derivatives is provided. For a monotone and continuous DR-submodular function, and subject to a bounded convex body constraint, we propose Black-box Continuous Greedy, a derivative-free algorithm that provably achieves the tight $[(1-1/e)OPT-ε]$ approximation guarantee with $O(d/ε^3)$ function evaluations. We then extend our result to the stochastic setting where function values are subject to stochastic zero-mean noise. It is through this stochastic generalization that we revisit the discrete submodular maximization problem and use the multi-linear extension as a bridge between discrete and continuous settings. Finally, we extensively evaluate the performance of our algorithm on continuous and discrete submodular objective functions using both synthetic and real data.