重访近端束方法：Hölder 光滑性下的改进速率
Revisiting Proximal Bundle Methods: Improved Rates under H{ö}lder Smoothness
- University of California San Diego（加州大学圣迭戈分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文重新分析经典近端束方法，将空步视为独立子程序，证明其自动适应Hölder光滑性，并导出改进的复杂度界，首次在经典下降检验下给出ν∈(0,1)的保证。
AI中文摘要：
近端束方法（PBMs）是非光滑凸优化的经典算法。现有对经典PBM的分析将空步与下降检验耦合在一起。这种耦合掩盖了束更新如何逼近近端子问题。在本工作中，我们考虑复合目标函数$F=f+h$，并将每个空步循环视为一个内部束子程序，称为$\mathtt{ProxBundle}$。我们在一般束模型条件下独立于任何停止准则分析$\mathtt{ProxBundle}$，并表明它自动适应Hölder光滑性。将这一内环分析与下降步分析相结合，为经典PBM导出了更锐利的复杂度界。对于任意固定的近端参数，其总体复杂度为$\mathcal O\big(\epsilon^{-\frac{3-\nu}{1+\nu}}\big)$（当$\nu\in[0,1)$）和$\mathcal O(\epsilon^{-1})$（当$\nu=1$），其中$\nu$是Hölder光滑性指数。选择与$\epsilon$成比例的近端参数可将速率改进为$\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$（当$\nu \in [0,1)$）。这些是在经典下降检验下Hölder光滑性（$\nu\in(0,1)$）的首个保证。我们进一步引入绝对模型误差检验。由此产生的PBM变体允许简洁的不精确近端点分析，并且对于任意固定的近端参数，达到相同的复杂度$\mathcal O\big(\epsilon^{-\frac{2}{1+\nu}}\big)$。总体而言，我们的分析分离了下降步和空步的作用，并提供了对PBM在不同检验下的模块化理解。
英文摘要：
Proximal bundle methods (PBMs) are classical algorithms for nonsmooth convex optimization. Existing analyses of the classical PBM couple the null steps with the descent test. This coupling obscures how the bundle updates approximate the proximal subproblem. In this work, we consider composite objectives $F=f+h$ and view each null-step cycle as an inner bundle subroutine, called $\mathtt{ProxBundle}$. We analyze $\mathtt{ProxBundle}$ independently of any stopping criterion under general bundle model conditions and show that it automatically adapts to H{ö}lder smoothness. Combining this inner-loop analysis with the descent-step analysis yields sharper complexity bounds for the classical PBM. For any fixed proximal parameter, its overall complexity is $\mathcal O\big(ε^{-\frac{3-ν}{1+ν}}\big)$ for $ν\in[0,1)$ and $\mathcal O(ε^{-1})$ for $ν=1$, where $ν$ is the H{ö}lder smoothness exponent. Choosing the proximal parameter proportional to $ε$ improves the rate to $\mathcal O\big(ε^{-\frac{2}{1+ν}}\big)$ for $ν\in [0,1)$. These are the first guarantees under Hölder smoothness with $ν\in(0,1)$ for the classical descent test. We further introduce an absolute model-error test. The resulting PBM variant admits a clean inexact proximal-point analysis, and for any fixed proximal parameter, achieves the same complexity $\mathcal O\big(ε^{-\frac{2}{1+ν}}\big)$. Overall, our analysis separates the roles of the descent and null steps and gives a modular understanding of PBMs across different~tests.
