  In many dynamic systems, such as wireless communications, 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. This model is inadequate for systems
that permit only periodic decision updates, i.e., each decision is fixed over
multiple time slots, while the environment changes between the decision epochs.
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 aggregated gradient descent to
update the decisions over time. PQGA is applicable to both constant and
time-varying update periods. Most importantly, we show that PQGA yields bounded
dynamic regret, static regret, and constraint violation. Furthermore, they are
sublinear over time if the accumulated variation of the system states and
update periods do not grow too fast. 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.?
