We study the problems of distributed online and bandit convex optimization
against an adaptive adversary. We aim to minimize the average regret on M
machines working in parallel over T rounds with R intermittent
communications. Assuming the underlying cost functions are convex and can be
generated adaptively, our results show that collaboration is not beneficial
when the machines have access to the first-order gradient information at the
queried points. This is in contrast to the case for stochastic functions, where
each machine samples the cost functions from a fixed distribution. Furthermore,
we delve into the more challenging setting of federated online optimization
with bandit (zeroth-order) feedback, where the machines can only access values
of the cost functions at the queried points. The key finding here is
identifying the high-dimensional regime where collaboration is beneficial and
may even lead to a linear speedup in the number of machines. We further
illustrate our findings through federated adversarial linear bandits by
developing novel distributed single and two-point feedback algorithms. Our work
is the first attempt towards a systematic understanding of federated online
optimization with limited feedback, and it attains tight regret bounds in the
intermittent communication setting for both first and zeroth-order feedback.
Our results thus bridge the gap between stochastic and adaptive settings in
federated online optimization