THE RATE OF CONVERGENCE OF NESTEROV'S ACCELERATED FORWARD-BACKWARD METHOD IS ACTUALLY FASTER THAN 1/k2

The forward-backward algorithm is a powerful tool for solving optimization problems with an additively separable and smooth plus nonsmooth structure. In the convex setting, a simple but ingenious acceleration scheme developed by Nesterov improves the theoretical rate of convergence for the function values from the standard O(k(-1)) down to 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 convergent sequences, is actually o(k(-2)), rather than O(k(-2)). Our arguments rely on the connection between this algorithm and a second-order differential inclusion with vanishing damping.