Leveraging on recent advancements on adaptive methods for convex minimization
problems, this paper provides a linesearch-free proximal gradient framework for
globalizing the convergence of popular stepsize choices such as
Barzilai-Borwein and one-dimensional Anderson acceleration. This framework can
cope with problems in which the gradient of the differentiable function is
merely locally H\"older continuous. Our analysis not only encompasses but also
refines existing results upon which it builds. The theory is corroborated by
numerical evidence that showcases the synergetic interplay between fast
stepsize selections and adaptive methods.

通过对于凸最小化问题的自适应方法的最新进展的利用，本文提供了一种无需线搜索的近端梯度下降框架，用于全局化收敛于流行的步长选择，如 Barzilai-Borwein 和一维 Anderson 加速。该框架可以处理梯度可微函数只具有局部 Holder 连续性的问题。我们的分析不仅包含但也改进了现有结果，并以数值证据来证明快速步长选择和自适应方法之间的协同作用。