# Batch-Size Independent Regret Bounds for the Combinatorial Multi-Armed Bandit Problem

## TL;DR

This paper introduces a new smoothness criterion for the combinatorial multi-armed bandit problem, enabling tighter regret bounds that are independent of batch size, especially in nonlinear reward settings like probabilistic maximum coverage.

## Contribution

The authors propose Gini-weighted smoothness, a novel criterion that improves regret bounds for CMAB algorithms by removing batch-size dependence in nonlinear reward scenarios.

## Key findings

- Achieves batch-size independent regret bounds in nonlinear CMAB problems.
- Provides dramatic improvements in upper bounds for probabilistic maximum coverage.
- Proves matching lower bounds, demonstrating tightness of the proposed algorithm.

## Abstract

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.