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| Formato: | Preprint |
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2025
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| Acceso en línea: | https://arxiv.org/abs/2508.06884 |
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| _version_ | 1866911703773478912 |
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| author | Tyurin, Alexander |
| author_facet | Tyurin, Alexander |
| contents | We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $\ell$-smoothness condition $||\nabla^{2}f(x)|| \le \ell\left(||\nabla f(x)||\right),$ which generalizes the $L$-smoothness and $(L_{0},L_{1})$-smoothness. While accelerated gradient descent AGD is known to reach the optimal complexity $O(\sqrt{L} R / \sqrt{\varepsilon})$ under $L$-smoothness, where $\varepsilon$ is an error tolerance and $R$ is the distance between a starting and an optimal point, existing extensions to $\ell$-smoothness either incur extra dependence on the initial gradient, suffer exponential factors in $L_{1} R$, or require costly auxiliary sub-routines, leaving open whether an AGD-type $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ rate is possible for small-$\varepsilon$, even in the $(L_{0},L_{1})$-smoothness case.
We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ oracle complexity for small-$\varepsilon$ and virtually any $\ell$. For instance, for $(L_{0},L_{1})$-smoothness, our bound $O(\sqrt{L_0} R / \sqrt{\varepsilon})$ is provably optimal in the small-$\varepsilon$ regime and removes all non-constant multiplicative factors present in prior accelerated algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness Tyurin, Alexander Optimization and Control Machine Learning We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $\ell$-smoothness condition $||\nabla^{2}f(x)|| \le \ell\left(||\nabla f(x)||\right),$ which generalizes the $L$-smoothness and $(L_{0},L_{1})$-smoothness. While accelerated gradient descent AGD is known to reach the optimal complexity $O(\sqrt{L} R / \sqrt{\varepsilon})$ under $L$-smoothness, where $\varepsilon$ is an error tolerance and $R$ is the distance between a starting and an optimal point, existing extensions to $\ell$-smoothness either incur extra dependence on the initial gradient, suffer exponential factors in $L_{1} R$, or require costly auxiliary sub-routines, leaving open whether an AGD-type $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ rate is possible for small-$\varepsilon$, even in the $(L_{0},L_{1})$-smoothness case. We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ oracle complexity for small-$\varepsilon$ and virtually any $\ell$. For instance, for $(L_{0},L_{1})$-smoothness, our bound $O(\sqrt{L_0} R / \sqrt{\varepsilon})$ is provably optimal in the small-$\varepsilon$ regime and removes all non-constant multiplicative factors present in prior accelerated algorithms. |
| title | Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2508.06884 |