Optimal Affine Invariant Smooth Minimization Algorithms

Abstract

We formulate an affine-invariant implementation of the accelerated first-order algorithm in [Y. Nesterov, Dokl. Math., 27 (1983), pp. 372–376]. 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 Hölder 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 Hölder smoothness parameter and the best gradient Lipschitz constant. Finally, we detail matching complexity lower bounds when the feasible set is an $ell_p$ ball. In this setting, our upper bounds on iteration complexity for the algorithm in [Y. Nesterov, Dokl. Math., 27 (1983), pp. 372–376] are thus optimal in terms of target precision, smoothness, and problem dimension.