  This paper studies proximal gradient iterations for solving simple bilevel
optimization problems where both the upper and the lower level cost functions
are split as the sum of differentiable and (possibly nonsmooth) proximable
functions. We develop a novel convergence recipe for iteration varying
stepsizes that relies on Barzilai-Borwein type local estimates for the
differentiable terms. Leveraging the convergence recipe, under global Lipschitz
gradient continuity, we establish convergence for a nonadaptive stepsize
sequence, without requiring any strong convexity or linesearch. In the locally
Lipschitz differentiable setting, we develop an adaptive linesearch method that
introduces a systematic adaptive scheme enabling large and nonmonotonic
stepsize sequences while being insensitive to the choice of hyperparameters and
initialization. Numerical simulations are provided showcasing favorable
convergence speed of our methods.
