arXiv 2103.003842021-03cs.LGcs.AI

自适应正则次模极大化

Adaptive Regularized Submodular Maximization

Naveen Jindal School of Management, University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）；Department of Computer Science, University of Texas at Dallas（德克萨斯大学达拉斯分校计算机科学系）

Shaojie Tang, Jing Yuan

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摘要：

在本文中，我们研究的问题之间的差异最大化的自适应子模（收入）函数和非负的模（成本）函数下的自适应设置。我们的问题的输入是一组$n$项目，其中每个项目都有一个特定的状态，从一些已知的先验分布$p$。收入函数$g$是在项目和状态上定义的，而成本函数$c$是在项目上定义的，即，每个项目都有固定的成本。每个项目的状态最初是未知的，必须选择一个项目，以观察其实现状态。策略$π$根据到目前为止所做的观察指定接下来要挑选哪个项。用$g_{avg}(π)$表示$π$的预期收入，用$c_{avg}(π)$表示$π$的预期成本。我们的目标是在$k$基数约束下确定最佳策略$π^o\in \arg\max_πg_{avg}(π)-c_{avg}(π)$.由于我们的目标函数可以取负值和正值，现有的子模最大化的结果可能不适用。为了克服这一挑战，我们与绩效获得者一起开发了一系列有效的解决方案。令$π^o$表示最优策略。对于$g$是自适应单调和自适应子模的情况，我们提出了一个有效的策略$π^l$，使得$g_{avg}(π^l) - c_{avg}(π^l) \geq (1-\frac{1}{e}-ε)g_{avg}(π^o) - c_{avg}(π^o)$只使用$O(nε^{-2}\log ε^{-1})$值进行Oracle查询.对于$g$是自适应子模的情况，我们提出了一种随机化策略$π^r$，使得$g_{avg}(π^r) - c_{avg}(π^r) \geq \frac{1}{e}g_{avg}(π^o) - c_{avg}(π^o)$ .

英文摘要：

In this paper, we study the problem of maximizing the difference between an adaptive submodular (revenue) function and an non-negative modular (cost) function under the adaptive setting. The input of our problem is a set of $n$ items, where each item has a particular state drawn from some known prior distribution $p$. The revenue function $g$ is defined over items and states, and the cost function $c$ is defined over items, i.e., each item has a fixed cost. The state of each item is unknown initially, one must select an item in order to observe its realized state. A policy $π$ specifies which item to pick next based on the observations made so far. Denote by $g_{avg}(π)$ the expected revenue of $π$ and let $c_{avg}(π)$ denote the expected cost of $π$. Our objective is to identify the best policy $π^o\in \arg\max_πg_{avg}(π)-c_{avg}(π)$ under a $k$-cardinality constraint. Since our objective function can take on both negative and positive values, the existing results of submodular maximization may not be applicable. To overcome this challenge, we develop a series of effective solutions with performance grantees. Let $π^o$ denote the optimal policy. For the case when $g$ is adaptive monotone and adaptive submodular, we develop an effective policy $π^l$ such that $g_{avg}(π^l) - c_{avg}(π^l) \geq (1-\frac{1}{e}-ε)g_{avg}(π^o) - c_{avg}(π^o)$, using only $O(nε^{-2}\log ε^{-1})$ value oracle queries. For the case when $g$ is adaptive submodular, we present a randomized policy $π^r$ such that $g_{avg}(π^r) - c_{avg}(π^r) \geq \frac{1}{e}g_{avg}(π^o) - c_{avg}(π^o)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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