Gradient-Variation Online Learning under Generalized Smoothness

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 optimization, hence receiving increased attention. Existing results often require the smoothness condition by imposing a fixed bound on gradient Lipschitzness, which may be unrealistic in practice. Recent efforts in neural network optimization suggest a generalized smoothness condition, allowing smoothness to correlate with gradient norms. In this paper, we systematically study gradient-variation online learning under generalized smoothness. We extend the classic optimistic mirror descent algorithm to derive gradient-variation regret by analyzing stability over the optimization trajectory and exploiting smoothness locally.