Adaptivity and Universality: Problem-dependent Universal Regret for Online Convex Optimization

• URL: http://arxiv.org/abs/2511.19937v1
• Date: Tue, 25 Nov 2025 05:23:10 GMT
• Title: Adaptivity and Universality: Problem-dependent Universal Regret for Online Convex Optimization
• Authors: Peng Zhao, Yu-Hu Yan, Hang Yu, Zhi-Hua Zhou,
• Abstract summary: We introduce UniGrad, a novel approach that achieves both universality and adaptivity, with two distinct realizations: UniGrad.Correct and UniGrad.Bregman.<n>Both methods achieve universal regret guarantees that adapt to gradient variation, simultaneously attaining $mathcalO(log V_T)$ regret for strongly convex functions and $mathcalO(d log V_T)$ regret for exp-concave functions.
• Score: 64.88607416000376
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Universal online learning aims to achieve optimal regret guarantees without requiring prior knowledge of the curvature of online functions. Existing methods have established minimax-optimal regret bounds for universal online learning, where a single algorithm can simultaneously attain $\mathcal{O}(\sqrt{T})$ regret for convex functions, $\mathcal{O}(d \log T)$ for exp-concave functions, and $\mathcal{O}(\log T)$ for strongly convex functions, where $T$ is the number of rounds and $d$ is the dimension of the feasible domain. However, these methods still lack problem-dependent adaptivity. In particular, no universal method provides regret bounds that scale with the gradient variation $V_T$, a key quantity that plays a crucial role in applications such as stochastic optimization and fast-rate convergence in games. In this work, we introduce UniGrad, a novel approach that achieves both universality and adaptivity, with two distinct realizations: UniGrad.Correct and UniGrad.Bregman. Both methods achieve universal regret guarantees that adapt to gradient variation, simultaneously attaining $\mathcal{O}(\log V_T)$ regret for strongly convex functions and $\mathcal{O}(d \log V_T)$ regret for exp-concave functions. For convex functions, the regret bounds differ: UniGrad.Correct achieves an $\mathcal{O}(\sqrt{V_T \log V_T})$ bound while preserving the RVU property that is crucial for fast convergence in online games, whereas UniGrad.Bregman achieves the optimal $\mathcal{O}(\sqrt{V_T})$ regret bound through a novel design. Both methods employ a meta algorithm with $\mathcal{O}(\log T)$ base learners, which naturally requires $\mathcal{O}(\log T)$ gradient queries per round. To enhance computational efficiency, we introduce UniGrad++, which retains the regret while reducing the gradient query to just $1$ per round via surrogate optimization. We further provide various implications.

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