We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + \eta_t$ where $\eta_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tilde{\Omega}(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tilde{\Omega}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

翻译：我们考虑在随机Bandit反馈下，最大化一个未知的单调、子模集函数 $f: 2^{[n]} \rightarrow [0,1]$，并带有基数约束。在每个时间步 $t=1,\dots,T$，学习者选择一个集合 $S_t \subset [n]$，满足 $|S_t| \leq k$，并获得奖励 $f(S_t) + \eta_t$，其中 $\eta_t$ 是均值为零的亚高斯噪声。目标是最小化学习者相对于通过鲁棒贪婪最大化 $f$ 获得的、满足 $|S_*| = k$ 的最大值 $f(S_*)$ 的近似值的遗憾。迄今为止，文献中最佳的遗憾界为 $k n^{1/3} T^{2/3}$。通过简单地将每个集合视为一个独立臂，可以推断出使用标准多臂Bandit算法也能实现 $\sqrt{ {n \choose k} T }$ 的遗憾。在本工作中，我们首次为该设定建立了极小极大下界，其尺度为 $\tilde{\Omega}(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$。对于一个稍受限制的算法类，我们证明了一个更强的遗憾下界 $\tilde{\Omega}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$。此外，我们提出了一种算法 Sub-UCB，其遗憾为 $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$，能够在忽略对数因子的情况下匹配该受限算法类的遗憾下界。