In this work, we focus on maximizing the stochastic DS decomposition problem. If the constraint is a uniform matroid, we design an adaptive policy, namely Myopic Parameter Conditioned Greedy, and prove its theoretical guarantee \(f(\varTheta (\pi _k))-(1-c_G)g(\varTheta (\pi _k))\ge (1-e^{-1})F(\pi ^*_A, \varTheta (\pi _k)) - G(\pi ^*_A,\varTheta (\pi _k))\), where \(F(\pi ^*_A, \varTheta (\pi _k)) = \mathbb {E}_{\varTheta }[f(\varTheta (\pi ^*_A)) \vert \varTheta (\pi _k)]\). When the constraint is a general matroid constraint, we design the Parameter Measured Continuous Conditioned Greedy to return a fractional solution. To round an integer solution from the fractional solution, we adopt the lattice contention resolution and prove that there is a \((b, \frac{1-e^{-b}}{b})\) lattice CR scheme under a matroid constraint. Additionally, we adopt the pipage rounding to obtain a non-adaptive policy with the theoretical guarantee \(F(\pi )-(1-c_G)G(\pi ) \ge (1-e^{-1}) F(\pi ^*_A) - G(\pi ^*_A) - O(\epsilon )\) and utlize the \((1,1-e^{-1})\)-lattice contention resolution scheme \(\tau \) to obtain an adaptive solution \(\mathbb {E}_{\tau \sim \varLambda } [f(\tau (\varTheta (\pi )))- (1-c_G) g(\tau (\varTheta (\pi )))] \ge (1-e^{-1})^2F(\pi ^*_A,\varTheta (\pi )) - (1-e^{-1}) G(\pi ^*_A,\varTheta (\pi )) -O(\epsilon )\). Since any set function can be expressed as the DS decomposition, our framework provides a method for solving the maximization problem of set functions defined on a random variable set.

在这项工作中，我们专注于最大化随机 DS 分解问题。如果约束是均匀拟阵，我们设计一个自适应策略，即Myopic Parameter Conditioned Greedy，并证明其理论保证\(f(\varTheta (\pi _k))-(1-c_G)g(\varTheta (\pi _k))\ge (1-e^{-1})F(\pi ^*_A, \varTheta (\pi _k)) - G(\pi ^*_A,\varTheta (\pi _k))\) ，其中 \(F(\pi ^*_A, \varTheta (\pi _k)) = \mathbb {E}_{\varTheta }[f(\varTheta (\pi ^*_A)) \vert \varTheta (\ pi _k)]\)。当约束是一般拟阵约束时，我们设计参数测量连续条件贪婪来返回分数解。为了从分数解中舍入整数解，我们采用晶格竞争解决方案并证明在 a 下存在 \((b, \frac{1-e^{-b}}{b})\) 晶格 CR 方案拟阵约束。此外，我们采用 pipage 舍入来获得非自适应策略，理论保证 \(F(\pi )-(1-c_G)G(\pi ) \ge (1-e^{-1}) F( \pi ^*_A) - G(\pi ^*_A) - O(\epsilon )\) 并利用 \((1,1-e^{-1})\)-格争用解决方案 \(\ tau \) 以获得自适应解 \(\mathbb {E}_{\tau \sim \varLambda } [f(\tau (\varTheta (\pi )))- (1-c_G) g(\tau (\ varTheta (\pi )))] \ge (1-e^{-1})^2F(\pi ^*_A,\varTheta (\pi )) - (1-e^{-1}) G(\ pi ^*_A,\varTheta (\pi )) -O(\epsilon )\)。由于任何集合函数都可以表示为 DS 分解，因此我们的框架提供了一种解决随机变量集合上定义的集合函数最大化问题的方法。