  Variational quantum algorithms are promising algorithms for achieving quantum
advantage on near-term devices. The quantum hardware is used to implement a
variational wave function and measure observables, whereas the classical
computer is used to store and update the variational parameters. The
optimization landscape of expressive variational ans\"atze is however dominated
by large regions in parameter space, known as barren plateaus, with vanishing
gradients which prevents efficient optimization. In this work we propose a
general algorithm to avoid barren plateaus in the initialization and throughout
the optimization. To this end we define a notion of weak barren plateaus (WBP)
based on the entropies of local reduced density matrices. The presence of WBPs
can be efficiently quantified using recently introduced shadow tomography of
the quantum state with a classical computer. We demonstrate that avoidance of
WBPs suffices to ensure sizable gradients in the initialization. In addition,
we demonstrate that decreasing the gradient step size, guided by the entropies
allows to avoid WBPs during the optimization process. This paves the way for
efficient barren plateau free optimization on near-term devices.
