Online SuBmodular + SuPermodular (BP) Maximization with Bandit Feedback

We investigate non-modular function maximization in an online setting with $m$ users. The optimizer maintains a set $S_q$ for each user $q \in \{1, \ldots, m\}$. At round $i$, a user with unknown utility $h_q$ arrives; the optimizer selects a new item to add to $S_q$, and receives a noisy marginal gain. The goal is to minimize regret compared to an $\alpha$-approximation to the optimal full-knowledge selection (i.e., $\alpha$-regret). Prior works study this problem under a submodularity assumption for all $h_q$. However, this is not ideally amenable to applications, e.g., movie recommendations, that involve complementarity between items, where e.g., watching the first movie in a series enhances the impression of watching the sequels. Hence, we consider objectives $h_q$, called \textit{BP functions}, that decompose into the sum of monotone submodular $f_q$ and supermodular $g_q$; here, $g_q$ naturally models complementarity. Under different feedback assumptions, we develop UCB-style algorithms that use Nystrom sampling for computational efficiency. For these, we provide sublinear $\alpha$-regret guarantees for $\alpha = 1/\kappa_{f} [1 - e^{-(1 - \kappa^g) \kappa_{f}} ]$, and $\alpha = \min\{1 - \kappa_f/e, 1 - \kappa^g\}$; here, $\kappa_f, \kappa^g$ are submodular and supermodular curvatures. Furthermore, we provide similar $\alpha$-regret guarantees for functions that are almost submodular where $\alpha$ is parameterized by the submodularity ratio of the objective functions. We numerically validate our algorithms for movie recommendation on the MovieLens dataset and selection of training subsets for classification tasks.