Gradient-Variation Regret Bounds for Unconstrained Online Learning

2026-04-13

AI Summary

This work addresses the challenge of achieving adaptive regret bounds in unconstrained online learning without prior knowledge of the comparator norm, Lipschitz constant, or smoothness parameters. To this end, the authors propose a fully parameter-free online learning algorithm featuring a closed-form update rule that constructs an adaptive regularization mechanism based on the gradient variation $V_T(u)$. This approach yields, for the first time, a dynamic regret bound proportional to $V_T(u)$ without requiring any prior information, and it naturally extends to the Strongly Adaptive (SEA) setting. For $L$-smooth convex loss functions, the resulting regret bound is $\widetilde{O}(|u|\sqrt{V_T(u)} + L|u|^2 + G^4)$, which significantly improves upon the best existing results in the SEA framework.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We develop parameter-free algorithms for unconstrained online learning with regret guarantees that scale with the gradient variation $V_T(u) = \sum_{t=2}^T \|\nabla f_t(u)-\nabla f_{t-1}(u)\|^2$. For $L$-smooth convex loss, we provide fully-adaptive algorithms achieving regret of order $\widetilde{O}(\|u\|\sqrt{V_T(u)} + L\|u\|^2+G^4)$ without requiring prior knowledge of comparator norm $\|u\|$, Lipschitz constant $G$, or smoothness $L$. The update in each round can be computed efficiently via a closed-form expression. Our results extend to dynamic regret and find immediate implications to the stochastically-extended adversarial (SEA) model, which significantly improves upon the previous best-known result [Wang et al., 2025].

Problem

• unconstrained online learning
• gradient variation
• regret bounds
• parameter-free algorithms
• dynamic regret