Learning in Non-Stationary Wireless Control Systems via Newton's Method

This paper considers wireless control systems over an unknown time-varying non-stationary channel. The goal is to maximize control performance of a set of independent control systems by allocating transmitting power within a fixed budget. Since the channel's time-varying distribution is unknown, samples of the channel are taken at every epoch. By reverting the resulting stochastic optimization problem in its Lagrange dual domain, we demonstrate that it takes the equivalent form of minimizing a certain empirical risk measure, a well-studied problem in machine learning. Newton's method is used to quickly learn approximately optimal power allocation policies over the sampled dual function as the channel evolves over time over windows of epochs. The quadratic convergence rate of Newton is used to establish, under certain conditions on the sampling size and rate of channel variation, an instantaneous learning and tracking of these optimal policies. Numerical simulations demonstrate the effectiveness of the learning algorithm on a low-dimensional wireless control problem.