Online Optimization in Dynamic Environments: Improved Regret Rates for Strongly Convex Problems
This work addresses the problem of dynamic parameter tracking for online learning systems, offering incremental improvements in regret rates for strongly convex scenarios.
The paper tackles tracking a time-varying parameter with unknown dynamics by formalizing it as an online optimization problem, and it achieves improved regret bounds for strongly convex loss functions compared to existing convex methods.
In this paper, we address tracking of a time-varying parameter with unknown dynamics. We formalize the problem as an instance of online optimization in a dynamic setting. Using online gradient descent, we propose a method that sequentially predicts the value of the parameter and in turn suffers a loss. The objective is to minimize the accumulation of losses over the time horizon, a notion that is termed dynamic regret. While existing methods focus on convex loss functions, we consider strongly convex functions so as to provide better guarantees of performance. We derive a regret bound that captures the path-length of the time-varying parameter, defined in terms of the distance between its consecutive values. In other words, the bound represents the natural connection of tracking quality to the rate of change of the parameter. We provide numerical experiments to complement our theoretical findings.