  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.
