We propose and analyze batch greedy heuristics for cardinality constrained maximization of non-submodular non-decreasing set functions. We consider the standard greedy paradigm, along with its distributed greedy and stochastic greedy variants. Our theoretical guarantees are characterized by the combination of submodularity and supermodularity ratios. We argue how these parameters define tight modular bounds based on incremental gains, and provide a novel reinterpretation of the classical greedy algorithm using the minorize-maximize (MM) principle. Based on that analogy, we propose a new class of methods exploiting any plausible modular bound. In the context of optimal experimental design for linear Bayesian inverse problems, we bound the submodularity and supermodularity ratios when the underlying objective is based on mutual information. We also develop novel modular bounds for the mutual information in this setting, and describe certain connections to polyhedral combinatorics. We discuss how algorithms using these modular bounds relate to established statistical notions such as leverage scores and to more recent efforts such as volume sampling. We demonstrate our theoretical findings on synthetic problems and on a real-world climate monitoring example.
翻译:我们提出并分析一系列贪婪的理论理论,以限制非子模量非递减性固定功能的最大化。我们考虑了标准的贪婪范式,以及分散的贪婪和随机贪婪变异。我们的理论保障的特点是将亚调和超模量比率结合起来。我们争论这些参数如何在增量收益的基础上界定紧凑的模块界限,并利用微小化-最大化(MMM)原则对古典贪婪算法进行新颖的重新解释。根据这一类推,我们提出了一种新的方法,利用任何可能的模块捆绑起来。在对线性巴伊西亚反面问题进行最佳实验设计的背景下,我们把亚调和超模量比率绑在一起,因为基本目标以相互信息为基础。我们还为这一环境中的相互信息开发了新的模块界限,并描述了与多元组合组合组合组合的连接。我们讨论了使用这些模块界限的算法如何与既定的统计概念有关,例如杠杆分和数量抽样等最新努力。我们展示了我们对合成问题和真实世界气候监测的理论结论。