This paper deals with bandit online learning problems involving feedback of
unknown delay that can emerge in multi-armed bandit (MAB) and bandit convex
optimization (BCO) settings. MAB and BCO require only values of the objective
function involved that become available through feedback, and are used to
estimate the gradient appearing in the corresponding iterative algorithms.
Since the challenging case of feedback with \emph{unknown} delays prevents one
from constructing the sought gradient estimates, existing MAB and BCO
algorithms become intractable. For such challenging setups, delayed
exploration, exploitation, and exponential (DEXP3) iterations, along with
delayed bandit gradient descent (DBGD) iterations are developed for MAB and
BCO, respectively. Leveraging a unified analysis framework, it is established
that the regret of DEXP3 and DBGD are ${\cal O}\big( \sqrt{K\bar{d}(T+D)}
\big)$ and ${\cal O}\big( \sqrt{K(T+D)} \big)$, respectively, where $\bar{d}$
is the maximum delay and $D$ denotes the delay accumulated over $T$ slots.
Numerical tests using both synthetic and real data validate the performance of
DEXP3 and DBGD.

该论文研究了多臂赌博问题和赌博凸优化问题中存在的未知延迟反馈问题，并开发了应对这种情况下的延迟探索、利用和指数迭代（DEXP3）和延迟赌博梯度下降（DBGD）算法。通过统一的分析框架，证明了 DEXP3 和 DBGD 算法的性能优越。