We present a unified, black-box-style method for developing
and analyzing online convex optimization (OCO) algorithms
for full-information online learning in delayed-feedback environments.
Our new, simplified analysis enables us to substantially
improve upon previous work and to solve a number
of open problems from the literature. Specifically, we develop
and analyze asynchronous AdaGrad-style algorithms
from the Follow-the-Regularized-Leader (FTRL) and MirrorDescent
family that, unlike previous works, can handle projections
and adapt both to the gradients and the delays, without
relying on either strong convexity or smoothness of the
objective function, or data sparsity. Our unified framework
builds on a natural reduction from delayed-feedback to standard
(non-delayed) online learning. This reduction, together
with recent unification results for OCO algorithms, allows us
to analyze the regret of generic FTRL and Mirror-Descent algorithms
in the delayed-feedback setting in a unified manner
using standard proof techniques. In addition, the reduction is
exact and can be used to obtain both upper and lower bounds
on the regret in the delayed-feedback setting.