Online Learning with Continuous Variations: Dynamic Regret and Reductions

Online learning is a powerful tool for analyzing iterative algorithms. However, the classic adversarial setup sometimes fails to capture certain regularity in online problems in practice. Motivated by this, we establish a new setup, called Continuous Online Learning (COL), where the gradient of online loss function changes continuously across rounds with respect to the learner's decisions. We show that COL covers and more appropriately describes many interesting applications, from general equilibrium problems (EPs) to optimization in episodic MDPs. In particular, we show monotone EPs admits a reduction to achieving sublinear static regret in COL. Using this new setup, we revisit the difficulty of sublinear dynamic regret. We prove a fundamental equivalence between achieving sublinear dynamic regret in COL and solving certain EPs. With this insight, we offer conditions for efficient algorithms that achieve sublinear dynamic regret, even when the losses are chosen adaptively without any a priori variation budget. Furthermore, we show for COL a reduction from dynamic regret to both static regret and convergence in the associated EP, allowing us to analyze the dynamic regret of many existing algorithms.