Small Gradient Norm Regret for Online Convex Optimization
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917258846011392 |
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| author | Gao, Wenzhi He, Chang Udell, Madeleine |
| author_facet | Gao, Wenzhi He, Chang Udell, Madeleine |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_13519 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Small Gradient Norm Regret for Online Convex Optimization Gao, Wenzhi He, Chang Udell, Madeleine Machine Learning Optimization and Control 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. |
| title | Small Gradient Norm Regret for Online Convex Optimization |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2601.13519 |