Primal-Dual Rates and Certificates
  We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group Lasso and TV-regularized problems. The theory applies to any norm-regularized generalized linear model. Our approach provides efficiently computable duality gaps which are globally defined, without modifying the original problems in the region of interest. 
Celestine D\"unner and Simone Forte and Martin Tak\'a\v{c} and Martin   Jaggi
cs.LG math.OC