An acceleration procedure for optimal first-order methods

We introduce an optimal first-order method that allows an easy and cheap evaluation of the local Lipschitz constant of the objective's gradient. This constant must ideally be chosen at every iteration as small as possible, while serving in an indispensable upper bound for the value of the objective function. In the previously existing variants of optimal first-order methods, this upper bound inequality was constructed from points computed during the current iteration. It was thus not possible to select the optimal value for this Lipschitz constant at the beginning of the iteration. In our variant, the upper bound inequality is constructed from points available before the current iteration, offering us the possibility to set the Lipschitz constant to its optimal value at once. This procedure, even if efficient in practice, presents a higher worse-case complexity than standard optimal first-order methods. We propose an alternative strategy that retains the practical efficiency of this procedure, while having an optimal worse-case complexity. Our generic scheme can be adapted for smoothing techniques. We perform numerical experiments on large-scale eigenvalue minimization problems, allowing us to reduce computation times by two to three orders of magnitude for the largest problems we considered over standard optimal methods.