Batch-Size Independent Regret Bounds for Combinatorial Semi-Bandits with Probabilistically Triggered Arms or Independent Arms
This provides more efficient algorithms for applications like cascading bandits and online influence maximization, though it represents an incremental improvement over existing bandit frameworks.
The paper tackles the problem of reducing batch-size dependency in combinatorial semi-bandits by introducing a novel TPVM condition that replaces the smoothness condition, resulting in regret bounds improved from O(K) to O(log K) or O(log² K) for triggered arms and completely removing K-dependency for independent arms.
In this paper, we study the combinatorial semi-bandits (CMAB) and focus on reducing the dependency of the batch-size $K$ in the regret bound, where $K$ is the total number of arms that can be pulled or triggered in each round. First, for the setting of CMAB with probabilistically triggered arms (CMAB-T), we discover a novel (directional) triggering probability and variance modulated (TPVM) condition that can replace the previously-used smoothness condition for various applications, such as cascading bandits, online network exploration and online influence maximization. Under this new condition, we propose a BCUCB-T algorithm with variance-aware confidence intervals and conduct regret analysis which reduces the $O(K)$ factor to $O(\log K)$ or $O(\log^2 K)$ in the regret bound, significantly improving the regret bounds for the above applications. Second, for the setting of non-triggering CMAB with independent arms, we propose a SESCB algorithm which leverages on the non-triggering version of the TPVM condition and completely removes the dependency on $K$ in the leading regret. As a valuable by-product, the regret analysis used in this paper can improve several existing results by a factor of $O(\log K)$. Finally, experimental evaluations show our superior performance compared with benchmark algorithms in different applications.