Safeguarding adaptive methods: global convergence of Barzilai-Borwein and other stepsize choices

-- Leveraging on recent advancements on adaptive methods for convex minimization problems, this paper provides a linesearch-free proximal gradient framework for glob-alizing 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 Hölder 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. Convex nonsmooth optimization problems are encountered in various engineering applications such as image denoising [4], signal processing and digital communication [16], machine learning [7], and control [15], to name a few. Traditional constant stepsizes require the gradient of the function f to satisfy global Lipschitz continuity [5, Prop.