An Optimal Affine Invariant Smooth Minimization Algorithm

Alexandre d'Aspremont (1, 2) , Cristóbal Guzmán (3) , Martin Jaggi (4)

1 LIENS - Laboratoire d'informatique de l'école normale supérieure
2 SIERRA - Statistical Machine Learning and Parsimony
3 Georgia Institute of Technology [Atlanta]
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We formulate an affine invariant implementation of the algorithm in Nesterov (1983). We show that the complexity bound is then proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We also detail matching lower bounds when the feasible set is an ℓp ball. In this setting, our bounds on iteration complexity for the algorithm in Nesterov (1983) are thus optimal in terms of target precision, smoothness and problem dimension.

Domaines: Optimisation et contrôle [math.OC]

Alexandre d'Aspremont, Cristóbal Guzmán, Martin Jaggi. An Optimal Affine Invariant Smooth Minimization Algorithm. 2013. ⟨hal-00907547⟩