广义光滑性下具有不精确Hessian的最优无参数二阶加速
Optimal Parameter-Free Second-order Acceleration with Inexact Hessians under Generalized Smoothness
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中文总结 AI 辅助
针对广义Hessian光滑性下的二阶凸优化，提出无参数自适应算法，其复杂度匹配下界，并在远离与接近解时分别实现几何与经典收敛。
中文摘要 AI 辅助
我们研究了广义Hessian光滑性下的二阶凸优化问题，其中局部Hessian Lipschitz常数可能随梯度范数线性增长。对于不精确的Hessian，我们开发了自适应算法，包括一种加速的Monteiro-Svaiter型方法，其复杂度在经典Lipschitz-Hessian设置下与具有δ-不精确Hessian的二阶方法的下界相匹配。我们的方法是无参数的，因为它们不需要输入问题特定的参数，包括Hessian不精确度水平δ和光滑性常数。相反，它们的回溯调整了一个单一的正则化参数，该参数共同捕捉了Hessian不精确性和局部Hessian光滑性。由此产生的收敛速率分为两个阶段。远离解时，依赖于梯度的光滑性项产生几何收敛。接近解时，收敛速率由普通Hessian光滑性和不精确度水平δ决定，恢复了经典的Lipschitz-Hessian保证。
英文摘要
We study second-order convex optimization under generalized Hessian smoothness, where the local Hessian Lipschitz constant may grow linearly with the gradient norm. For inexact Hessians, we develop adaptive algorithms, including an accelerated Monteiro-Svaiter-type method whose complexity matches the lower bound for second-order methods with $δ$-inexact Hessians in the classical Lipschitz-Hessian setting. Our methods are parameter-free in the sense that they require no problem-specific parameters as input, including the Hessian inexactness level $δ$ and smoothness constants. Instead, their backtracking adapts a single regularization parameter that jointly captures Hessian inexactness and local Hessian smoothness. The resulting rates split into two regimes. Far from the solution, the gradient-dependent smoothness term yields geometric convergence. Near the solution, the rate is governed by the ordinary Hessian smoothness and the inexactness level $δ$, recovering the classical Lipschitz-Hessian guarantees.
发表机构
- MBZUAI（穆罕默德·本·扎耶德人工智能大学）
- MIRAI（未来研究院）
机构由 AI 辅助整理，请以论文原文为准。
