ONLINE LEARNING IN A ONE-DIMENSIONAL PERIODIC QUADRATIC VARIATIONAL PROBLEM WITH AN ADVERSARIAL EXTERNAL FORCE

We consider a one-dimensional quadratic variational problem with periodic boundary conditions and an unknown external force. The problem is solved sequentially: at each step, a player selects a trajectory and then receives information about an external force. His goal is to minimize a quadratic functional in the sense of static or dynamic regret with respect to some comparison sequence. The solution is approximated by trigonometric polynomials. Based on the estimates of the approximation error as well as on the estimates of the Lipschitz constant, smoothness constant, and strong convexity parameter of the approximating function, we show that the regret bounding problem is reduced to a standard finite-dimensional situation. Regret bounds for several algorithms for minimizing static and dynamic regret are presented.