Nearly Minimax Optimal Submodular Maximization with Bandit Feedback
This work addresses a fundamental challenge in online optimization with bandit feedback for submodular functions, providing near-optimal regret bounds that improve upon prior results.
The paper tackles the problem of maximizing an unknown monotonic submodular set function under cardinality constraint with bandit feedback, establishing the first minimax lower bound and proposing an algorithm that matches this bound up to logarithmic factors for a restricted class.
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 $f(S_t) + η_t$ where $η_t$ 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 $\tildeΩ(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tildeΩ(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. 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.