  The theory of discrete-time online learning has been successfully applied in
many problems that involve sequential decision-making under uncertainty.
However, in many applications including contractual hiring in online
freelancing platforms and server allocation in cloud computing systems, the
outcome of each action is observed only after a random and action-dependent
time. Furthermore, as a consequence of certain ethical and economic concerns,
the controller may impose deadlines on the completion of each task, and require
fairness across different groups in the allocation of total time budget $B$. In
order to address these applications, we consider continuous-time online
learning problem with fairness considerations, and present a novel framework
based on continuous-time utility maximization. We show that this formulation
recovers reward-maximizing, max-min fair and proportionally fair allocation
rules across different groups as special cases. We characterize the optimal
offline policy, which allocates the total time between different actions in an
optimally fair way (as defined by the utility function), and impose deadlines
to maximize time-efficiency. In the absence of any statistical knowledge, we
propose a novel online learning algorithm based on dual ascent optimization for
time averages, and prove that it achieves $\tilde{O}(B^{-1/2})$ regret bound.
