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. Feedback information about the loss functions within the current update period may be delayed and incomplete. 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 for network virtualization in a large-scale multi-antenna system shared by multiple wireless service providers. Simulation results show that PQGA converges fast and substantially outperforms the current best alternative.
- Online convex optimization
- Long-term constraints
- Periodic updates
- Massive MIMO
- Wireless network virtualization