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We deal with the problem of numerically computing the dual norm, which isimportant to study sparsity-inducing regularizations (Jenatton et al. 2011,Bachet al. 2012). The dual norms find application in optimization and statisticallearning, for example, in the design of working-set strategies, forcharacterizing dual gradient methods, for dual decompositions and in thedefinition of augmented Lagrangian functions. Nevertheless, the dual norm ofsome well-known sparsity-inducing regolarization methods are not analyticallyavailable. Examples are the overlap group $\ell_2$-norm of (Jenatton et al.2011) and the elastic net norm of Zhou and Hastie (2005). Therefore we resortto the Majorization-Minimization principle of Lange (2016) to provide anefficient algorithm that leverages a reparametrization of the dual constrainedoptimization problem as unconstrained optimization with barrier. Extensivesimulation experiments have been performed in order to verify the correctnessof operation, and evaluate the performance of the proposed method. Our resultsdemonstrate the effectiveness of the algorithm in retrieving the dual norm evenfor large dimensions.

Numerical evaluation of dual norms via the MM algorithm