Sequential convergence of AdaGrad algorithm for smooth convex optimization

We prove that the iterates produced by, either the scalar step size variant,\nor the coordinatewise variant of AdaGrad algorithm, are convergent sequences\nwhen applied to convex objective functions with Lipschitz gradient. The key\ninsight is to remark that such AdaGrad sequences satisfy a variable metric\nquasi-Fej\\'er monotonicity property, which allows to prove convergence.\n