Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems

Niao He (1) , Anatoli B. Juditsky (2) , Arkadii S. Nemirovskii (3)

1 Georgia Institute of Technology [Atlanta]
2 SAM - Statistique Apprentissage Machine
3 ISyE - School of Industrial and Systems Engineering [Georgia Tech]

In the paper, we develop a composite version of Mirror Prox algorithm for solving convexconcave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called “composite minimization” (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions “as if” there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) O(1/ε) efficiency estimate of its prototype. We demonstrate that the proposed approach can be successfully applied to Lasso-type problems with several penalizing terms (e.g. acting together l₁ and nuclear norm regularization) and to problems of semi-separable structures considered in the alternating directions methods, implying in both cases methods with the O(1/ε) complexity bounds.

Mots clés: numerical algorithms for variational problems; composite optimization; minimization problems with multi-term penalty; proximal methods

Domaines: Optimisation et contrôle [math.OC]

Niao He, Anatoli B. Juditsky, Arkadii S. Nemirovskii. Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems. Computational Optimization and Applications, 2015, 61 (2), pp.275-319. ⟨10.1007/s10589-014-9723-3⟩. ⟨hal-01335905⟩