#### Authors

Shinji Ito, Daisuke Hatano, Hanna Sumita, Kei Takemura, Takuro Fukunaga, Naonori Kakimura, Ken-Ichi Kawarabayashi

#### Abstract

This paper offers a nearly optimal algorithm for online linear optimization with delayed bandit feedback. Online linear optimization with bandit feedback, or nonstochastic linear bandits, provides a generic framework for sequential decision-making problems with limited information. This framework, however, assumes that feedback can be observed just after choosing the action, and, hence, does not apply directly to many practical applications, in which the feedback can often only be obtained after a while. To cope with such situations, we consider problem settings in which the feedback can be observed $d$ rounds after the choice of an action, and propose an algorithm for which the expected regret is $\tilde{O}( \sqrt{m (m + d) T} )$, ignoring logarithmic factors in $m$ and $T$, where $m$ and $T$ denote the dimensionality of the action set and the number of rounds, respectively. This algorithm achieves nearly optimal performance, as we are able to show that arbitrary algorithms suffer the regret of $\Omega(\sqrt{m (m+d) T})$ in the worst case. To develop the algorithm, we introduce a technique we refer to as \textit{distribution truncation}, which plays an essential role in bounding the regret. We also apply our approach to cooperative bandits, as studied by Cesa-Bianchi et al. [17] and Bar-On and Mansour [12], and extend their results to the linear bandits setting.