Combinatorial Logistic Online Learning and Its Applications in Nonlinear Networked Systems

Combinatorial multi-armed bandit (CMAB) is a fundamental online learning framework that can optimize cumulative rewards in networked systems under uncertainty. Real-world applications like content delivery and channel allocation often feature binary base arm rewards and nonlinear total reward functions. This paper introduces combinatorial logistic bandits (CLogB), a contextual CMAB framework with the base arm reward modeled as a nonlinear logistic function of the context, and the feedback is governed by a general arm-triggering process. We study CLogB with smooth reward functions, covering applications such as online content delivery, online multi-LLM selection, and dynamic channel allocation. Our first algorithm, CLogUCB, uses a variance-agnostic exploration bonus and achieves a regret bound of Õ(d√κKT), where d is the feature dimension, κ reflects logistic model nonlinearity,Kis the maximum number of triggered arms, and Õ ignores logarithmic factors. This improves on prior results by Õ (√κ). We further propose VA-CLogUCB, a variance-adaptive enhancement achieving regret bounds of Õ(d√KT) under standard smoothness conditions and Õ (d√T) under stronger variance conditions, removing dependence on K. For time-invariant feature maps, we enhance computational efficiency by avoiding nonconvex optimization while maintaining Õ(d√T) regret. Experiments on synthetic and real-world datasets validate the superior performance of our algorithms, demonstrating their effectiveness and scalability for real-world networked systems.