Batch-Size Independent Regret Bounds for Combinatorial Semi-Bandits with Probabilistically Triggered Arms or Independent Arms

TL;DR

This paper introduces new regret bounds for combinatorial semi-bandits that are independent of batch size, using novel conditions and algorithms to improve theoretical guarantees and practical performance across various applications.

Contribution

The paper proposes the TPVM condition and new algorithms BCUCB-T and SESCB that significantly reduce or eliminate the dependence on batch size in regret bounds for CMAB problems.

Findings

Regret bounds improved from O(K) to O(log K) or O(log^2 K) for CMAB with probabilistically triggered arms.

Regret for non-triggering CMAB with independent arms is made independent of K.

Experimental results demonstrate superior performance of proposed algorithms over benchmarks.

Abstract

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.