On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithms

Building upon recent works on linesearch-free adaptive proximal gradientmethods, this paper proposes adaPG$^{q,r}$, a framework that unifies andextends existing results by providing larger stepsize policies and improvedlower bounds. Different choices of the parameters $q$ and $r$ are discussed andthe efficacy of the resulting methods is demonstrated through numericalsimulations. In an attempt to better understand the underlying theory, itsconvergence is established in a more general setting that allows fortime-varying parameters. Finally, an adaptive alternating minimizationalgorithm is presented by exploring the dual setting. This algorithm not onlyincorporates additional adaptivity, but also expands its applicability beyondstandard strongly convex settings.