Proximal Online Gradient Is Optimum for Dynamic Regret: A General Lower Bound

In online learning, the dynamic regret metric chooses the reference oracle that may change over time, while the typical (static) regret metric assumes the reference solution to be constant over the whole time horizon. The dynamic regret metric is particularly interesting for applications, such as online recommendation (since the customers’ preference always evolves over time). While the online gradient (OG) method has been shown to be optimal for the static regret metric, the optimal algorithm for the dynamic regret remains unknown. In this article, we show that proximal OG (a general version of OG) is optimum to the dynamic regret by showing that the proved lower bound matches the upper bound. It is highlighted that we provide a new and general lower bound of dynamic regret. It provides new understanding about the difficulty to follow the dynamics in the online setting.