AI 中文总结
研究在无目标函数增长、平滑度等先验知识时凸函数的一阶最小化，开发无参数束级方法，如BLW及A - BLW，能适应未知属性达最优复杂度，核心是仿射W - 证书，实验验证了方法性能。
AI 中文摘要
我们研究在没有目标函数增长、平滑度范围或相关参数的先验知识的情况下凸函数的确定性一阶最小化问题。我们开发了随时可用的无参数束级方法，这些方法能同时适应这些未知属性并达到已知的最佳神谕复杂度。对于满足二次增长的非光滑Lipschitz目标函数，所提出的束级W-证书方法（BLW）无需输入增长模量或目标精度就能实现最优复杂度。然后我们引入加速变体A-BLW。在不知道Hölder平滑度参数、二次增长模量或目标精度的情况下，A-BLW在非光滑、弱光滑和光滑情况下都能达到最优速率。两种方法的核心是仿射W-证书，它基于仿射次小函数的下降缓慢条件，在二次增长下将束模型的几何结构转化为最优间隙保证。停止时间分析进一步表明，相同的A-BLW算法无需修改，对于一般凸目标函数和满足至少二阶Hölder增长的目标函数能达到相应的已知最佳速率。数值实验说明了所提出方法的实际性能。
英文摘要
We study deterministic first-order minimization of a convex function without prior knowledge of the objective's growth, smoothness regime, or associated parameters. We develop anytime, parameter-free bundle-level methods that adapt simultaneously to these unknown properties and attain best-known oracle complexities. For nonsmooth Lipschitz objectives satisfying quadratic growth, the proposed bundle-level W-certificate method (BLW) achieves the optimal complexity without requiring the growth modulus or target accuracy as input. We then introduce an accelerated variant, A-BLW. Without knowing the Hölder smoothness parameters, the quadratic-growth modulus, or the target accuracy, A-BLW attains the optimal rates in the nonsmooth, weakly smooth, and smooth regimes. Central to both methods is an affine W-certificate, a condition based on the descent-slowness of an affine minorant that converts the geometry of a bundle model into an optimality-gap guarantee under quadratic growth. A stopping-time analysis further shows that the same A-BLW algorithm, without modification, achieves the corresponding best-known rates for general convex objectives and for objectives satisfying Hölder growth of order at least two. Numerical experiments illustrate the practical performance of the proposed methods.
Comments39 pages, 4 figures, 1 table
