Some recent results do consider residual-like elements (see discussion of related work below),butgenerallydonotapply tostandard architectures.

Nonstationarity is prevalent in sequential decision making, which poses a critical challenge to the vast majority of existing approaches developed offline.

PW A systems allow for discontinuities across the separate regions ("pieces"), and are thus a simplified

Whentheinputs associated Ex. 1), theterms

Neural Information Processing Systems http://nips.cc/

Gradient-variation online learning aims to achieve regret guarantees that scale with variations in the gradients of online functions, which is crucial for attaining fast convergence in games and robustness in stochastic o ptimization, hence receiving increased attention. Existing results often req uire the smoothness condition by imposing a fixed bound on gradient Lipschitzness, w hich may be unrealistic in practice. Recent efforts in neural network optim ization suggest a generalized smoothness condition, allowing smoothness to correlate with gradient norms. In this paper, we systematically study gradient-var iation online learning under generalized smoothness. We extend the classic optimi stic mirror descent algorithm to derive gradient-variation regret by analyzin g stability over the optimization trajectory and exploiting smoothness locally. Th en, we explore universal online learning, designing a single algorithm with the optimal gradient-va riation regrets for convex and strongly convex functions simultane ously, without requiring prior knowledge of curvature. This algorithm adopts a tw o-layer structure with a meta-algorithm running over a group of base-learners . To ensure favorable guarantees, we design a new Lipschitz-adaptive meta-a lgorithm, capable of handling potentially unbounded gradients while ensuring a second-order bound to effectively ensemble the base-learners. Finally, we provi de the applications for fast-rate convergence in games and stochastic extended adv ersarial optimization.