  We consider the problem of minimizing a convex function over a closed convex
set, with Projected Gradient Descent (PGD). We propose a fully parameter-free
version of AdaGrad, which is adaptive to the distance between the
initialization and the optimum, and to the sum of the square norm of the
subgradients. Our algorithm is able to handle projection steps, does not
involve restarts, reweighing along the trajectory or additional gradient
evaluations compared to the classical PGD. It also fulfills optimal rates of
convergence for cumulative regret up to logarithmic factors. We provide an
extension of our approach to stochastic optimization and conduct numerical
experiments supporting the developed theory.
