  We investigate multiarmed bandits with delayed feedback, where the delays
need neither be identical nor bounded. We first prove that "delayed" Exp3
achieves the $O(\sqrt{(KT + D)\ln K} )$ regret bound conjectured by
Cesa-Bianchi et al. [2019] in the case of variable, but bounded delays. Here,
$K$ is the number of actions and $D$ is the total delay over $T$ rounds. We
then introduce a new algorithm that lifts the requirement of bounded delays by
using a wrapper that skips rounds with excessively large delays. The new
algorithm maintains the same regret bound, but similar to its predecessor
requires prior knowledge of $D$ and $T$. For this algorithm we then construct a
novel doubling scheme that forgoes the prior knowledge requirement under the
assumption that the delays are available at action time (rather than at loss
observation time). This assumption is satisfied in a broad range of
applications, including interaction with servers and service providers. The
resulting oracle regret bound is of order $\min_\beta (|S_\beta|+\beta \ln K +
(KT + D_\beta)/\beta)$, where $|S_\beta|$ is the number of observations with
delay exceeding $\beta$, and $D_\beta$ is the total delay of observations with
delay below $\beta$. The bound relaxes to $O (\sqrt{(KT + D)\ln K} )$, but we
also provide examples where $D_\beta \ll D$ and the oracle bound has a
polynomially better dependence on the problem parameters.

Machine Learning;Machine Learning