  Projection-free online learning, which eschews the projection operation via
less expensive computations such as linear optimization (LO), has received much
interest recently due to its efficiency in handling high-dimensional problems
with complex constraints. However, previous studies assume that any queried
gradient is revealed immediately, which may not hold in practice and limits
their applications. To address this limitation, we generalize the online
Frank-Wolfe (OFW) algorithm and the online smooth projection-free (OSPF)
algorithm, which are state-of-the-art LO-based projection-free online
algorithms for non-smooth and smooth functions respectively, into a delayed
setting where queried gradients can be delayed by arbitrary rounds.
Specifically, the main idea of our generalized OFW is to perform an update
similar to the original OFW after receiving any delayed gradient, and play the
latest decision for each round. Moreover, the essential change on OSPF is to
replace the sum of queried gradients, which is originally utilized in each
update, with the sum of available gradients. Despite their simplicities, our
novel analysis shows that under a relatively large amount of delay, the
generalized OFW and OSPF enjoy the same regret bound as OFW and OSPF in the
non-delayed setting, respectively.
