  We consider distributed online learning for joint regret with communication
constraints. In this setting, there are multiple agents that are connected in a
graph. Each round, an adversary first activates one of the agents to issue a
prediction and provides a corresponding gradient, and then the agents are
allowed to send a $b$-bit message to their neighbors in the graph. All agents
cooperate to control the joint regret, which is the sum of the losses of the
activated agents minus the losses evaluated at the best fixed common comparator
parameters $u$. We observe that it is suboptimal for agents to wait for
gradients that take too long to arrive. Instead, the graph should be
partitioned into local clusters that communicate among themselves. Our main
result is a new method that can adapt to the optimal graph partition for the
adversarial activations and gradients, where the graph partition is selected
from a set of candidate partitions. A crucial building block along the way is a
new algorithm for online convex optimization with delayed gradient information
that is comparator-adaptive, meaning that its joint regret scales with the norm
of the comparator $||u||$. We further provide near-optimal gradient compression
schemes depending on the ratio of $b$ and the dimension times the diameter of
the graph.
