Submodular optimization generalizes many classic problems in combinatorial optimization and has recently found a wide range of applications in machine learning (e.g., feature engineering and active learning). For many large-scale optimization problems, we are often concerned with the adaptivity complexity of an algorithm, which quantifies the number of sequential rounds where polynomially-many independent function evaluations can be executed in parallel. While low adaptivity is ideal, it is not sufficient for a distributed algorithm to be efficient, since in many practical applications of submodular optimization the number of function evaluations becomes prohibitively expensive. Motivated by these applications, we study the adaptivity and query complexity of adaptive submodular optimization. Our main result is a distributed algorithm for maximizing a monotone submodular function with cardinality constraint $k$ that achieves a $(1-1/e-\varepsilon)$-approximation in expectation. This algorithm runs in $O(\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation oracle in expectation. The approximation guarantee and query complexity are optimal, and the adaptivity is nearly optimal. Moreover, the number of queries is substantially less than in previous works. Last, we extend our results to the submodular cover problem to demonstrate the generality of our algorithm and techniques.

翻译：摘要：子模优化推广了组合优化中的许多经典问题，近期在机器学习领域（如特征工程和主动学习）中发现了广泛应用。对于许多大规模优化问题，我们通常关注算法的自适应性复杂度，该指标量化了在并行计算中可同时执行多项式数量独立函数评估的连续轮数。虽然低自适应性是理想特性，但这并不足以保证分布式算法的高效性，因为在许多子模优化的实际应用中，函数评估次数会变得极其昂贵。受这些应用的启发，我们研究了自适应子模优化的自适应性与查询复杂度。我们的主要成果是一种分布式算法，用于最大化具有基数约束$k$的单调子模函数，该算法在期望意义下实现了$(1-1/e-\varepsilon)$近似比。该算法在$O(\log(n))$自适应轮次内运行，并在期望意义下仅需$O(n)$次函数评估预言机调用。其近似保证和查询复杂度达到最优，自适应性近乎最优。此外，查询次数显著少于先前研究。最后，我们将成果推广至子模覆盖问题，以展示算法与技术的普适性。