Batch-Size Independent Regret Bounds for the Combinatorial Multi-Armed Bandit Problem
This addresses the problem of loose regret bounds in combinatorial bandits for applications like influence maximization and recommendations, though it appears incremental as it refines existing approaches rather than introducing a new paradigm.
The paper tackles the combinatorial multi-armed bandit problem with nonlinear rewards by introducing a new smoothness criterion called Gini-weighted smoothness, which leads to tighter regret bounds that are independent of batch size, achieving dramatic improvements in upper bounds for problems like probabilistic maximum coverage.
We consider the combinatorial multi-armed bandit (CMAB) problem, where the reward function is nonlinear. In this setting, the agent chooses a batch of arms on each round and receives feedback from each arm of the batch. The reward that the agent aims to maximize is a function of the selected arms and their expectations. In many applications, the reward function is highly nonlinear, and the performance of existing algorithms relies on a global Lipschitz constant to encapsulate the function's nonlinearity. This may lead to loose regret bounds, since by itself, a large gradient does not necessarily cause a large regret, but only in regions where the uncertainty in the reward's parameters is high. To overcome this problem, we introduce a new smoothness criterion, which we term \emph{Gini-weighted smoothness}, that takes into account both the nonlinearity of the reward and concentration properties of the arms. We show that a linear dependence of the regret in the batch size in existing algorithms can be replaced by this smoothness parameter. This, in turn, leads to much tighter regret bounds when the smoothness parameter is batch-size independent. For example, in the probabilistic maximum coverage (PMC) problem, that has many applications, including influence maximization, diverse recommendations and more, we achieve dramatic improvements in the upper bounds. We also prove matching lower bounds for the PMC problem and show that our algorithm is tight, up to a logarithmic factor in the problem's parameters.