An improved regret analysis for UCB-N and TS-N

In the setting of stochastic online learning with undirected feedback graphs, Lykouris et al. (2020) previously analyzed the pseudo-regret of the upper confidence bound-based algorithm UCB-N and the Thompson Sampling-based algorithm TS-N. In this note, we show how to improve their pseudo-regret analysis. Our improvement involves refining a key lemma of the previous analysis, allowing a $\log (T)$ factor to be replaced by a factor ${\log }_2(\alpha )+3$ for $\alpha$ the independence number of the feedback graph.