Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness

• URL: http://arxiv.org/abs/2508.06884v1
• Date: Sat, 09 Aug 2025 08:28:06 GMT
• Title: Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness
• Authors: Alexander Tyurin,
• Abstract summary: We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $ell$-smoothness condition $||nabla2f(x)|| le ellleft(||nabla f(x)||right),$ which generalizes the $L$-smoothness and $(L_0,L_1)$-smoothness.
• Score: 57.93371273485736
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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