We consider the optimization problem of cardinality constrained maximization of a monotone submodular set function $f:2^U\to\mathbb{R}_{\geq 0}$ (SM) with noisy evaluations of $f$. In particular, it is assumed that we do not have value oracle access to $f$, but instead for any $X\subseteq U$ and $u\in U$ we can take samples from a noisy distribution with expected value $f(X\cup\{u\})-f(X)$. Our goal is to develop algorithms in this setting that take as few samples as possible, and return a solution with an approximation guarantee relative to the optimal with high probability. We propose the algorithm Confident Threshold Greedy (CTG), which is based on the threshold greedy algorithm of Badanidiyuru and Vondrak [1] and samples adaptively in order to produce an approximate solution with high probability. We prove that CTG achieves an approximation ratio arbitrarily close to $1-1/e$, depending on input parameters. We provide an experimental evaluation on real instances of SM and demonstrate the sample efficiency of CTG.

翻译：我们考虑单调子模集合函数 $f:2^U\to\mathbb{R}_{\geq 0}$ (SM) 在基数约束下最大化且存在$f$噪声评估的优化问题。特别地，假设我们无法通过值预言机直接访问$f$，但可以对任意$X\subseteq U$和$u\in U$，从期望值为$f(X\cup\{u\})-f(X)$的噪声分布中采样。我们的目标是在此背景下开发算法，使其在尽可能少采样的情况下，以高概率返回相对于最优解具有近似保证的解。我们提出算法Confident Threshold Greedy (CTG)，该算法基于Badanidiyuru和Vondrak [1]的阈值贪婪算法，通过自适应采样以高概率产生近似解。我们证明CTG能够实现任意接近$1-1/e$的近似比（具体取决于输入参数）。我们在真实SM实例上进行实验评估，展示了CTG的采样效率。