The rate of convergence of Nesterov's accelerated forward-backward method is actually faster than 1/k2

The it forward-backward algorithm is a powerful tool for solving optimization problems with a it additively separable and it smooth + it nonsmooth structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov has been proved useful to improve the theoretical rate of convergence for the function values from the standard mathcal O(k-1) down to mathcal O(k-2). In this short paper, we prove that the rate of convergence of a slight variant of Nesterov's accelerated forward-backward method, which produces it convergent sequences, is actually o(k-2), rather than mathcal O(k-2). Our arguments rely on the connection between this algorithm and a second-order differential inclusion with vanishing damping.