An Optimal Affine Invariant Smooth Minimization Algorithm.

TITLE: An Optimal Affine Invariant Smooth Minimization Algorithm

AUTHORS: Alexandre d'Aspremont, Cristóbal Guzmán, Martin Jaggi

ABSTRACT: We formulate an affine invariant implementation of the accelerated first-order algorithm in Nesterov (1983). Its complexity bound is proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We extend these results to more general problems, optimizing Holder smooth functions using p-uniformly convex prox terms, and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best Holder smoothness parameter and the best gradient Lipschitz constant. Finally, we detail matching complexity lower bounds when the feasible set is an l_p ball. In this setting, our upper bounds on iteration complexity for the algorithm in Nesterov (1983) are thus optimal in terms of target precision, smoothness and problem dimension.

STATUS: Preprint.

ArXiv PREPRINT: 1301.0465

PAPER: An Optimal Affine Invariant Smooth Minimization Algorithm in pdf