  In this paper, we explore two fundamental first-order algorithms in convex
optimization, namely, gradient descent (GD) and proximal gradient method
(ProxGD). Our focus is on making these algorithms entirely adaptive by
leveraging local curvature information of smooth functions. We propose adaptive
versions of GD and ProxGD that are based on observed gradient differences and,
thus, have no added computational costs. Moreover, we prove convergence of our
methods assuming only local Lipschitzness of the gradient. In addition, the
proposed versions allow for even larger stepsizes than those initially
suggested in [MM20].
