Gradient-Variation Regret Bounds for Unconstrained Online Learning

Gradient-variation online learning has attracted increasing attention for its connections to game theory and optimization. It has been studied in both non-stationary online learning, which aims to minimize dynamic regret against time-varying comparator sequences, and universal online learning, which seeks to adapt to unknown loss curvature. Prior work obtained gradient-variation bounds using different techniques tailored to each setting. We propose a simple and unified method based on surrogate losses with a quadratic penalty on decision movement, yielding gradient-variation bounds in both settings. For non-stationary online learning, the method induces an optimistic-to-gradient-variation reduction, enabling a broad class of algorithms to match the rate established by Zhao et al. (2024). For universal online learning, it yields a simple algorithm that matches the optimal universal regret bounds of Yan et al. (2024) while relaxing the extended smoothness assumption. Finally, we apply our method to online games and improve on existing guarantees.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.