  In many dynamic systems, decisions on system operation are updated over time,
and the decision maker requires an online learning approach to optimize its
strategy in response to the changing environment. When the loss and constraint
functions are convex, this belongs to the general family of online convex
optimization (OCO). In existing OCO works, the environment is assumed to vary
in a time-slotted fashion, while the decisions are updated at each time slot.
However, many wireless communication systems permit only periodic decision
updates, i.e., each decision is fixed over multiple time slots, while the
environment changes between the decision epochs. The standard OCO model is
inadequate for these systems. Therefore, in this work, we consider periodic
decision updates for OCO. We aim to minimize the accumulation of time-varying
convex loss functions, subject to both short-term and long-term constraints.
Information about the loss functions within the current update period may be
incomplete and is revealed to the decision maker only after the decision is
made. We propose an efficient algorithm, termed Periodic Queueing and Gradient
Aggregation (PQGA), which employs novel periodic queues together with possibly
multi-step aggregated gradient descent to update the decisions over time. We
derive upper bounds on the dynamic regret, static regret, and constraint
violation of PQGA. As an example application, we study the performance of PQGA
in a large-scale multi-antenna system shared by multiple wireless service
providers. Simulation results show that PQGA converges fast and substantially
outperforms the known best alternative.
