We derive generalizations of AdaBoost and related gradient-based coordinate descent methods that incorporate sparsity-promoting penalties for the norm of the predictor that is being learned. The end result is a family of coordinate descent algorithms that integrate forward feature induction and back-pruning through regularization and give an automatic stopping criterion for feature induction. We study penalties based on the $\ell_1$, $\ell_2$, and $\ell_\infty$ norms of the predictor and introduce mixed-norm penalties that build upon the initial penalties. The mixed-norm regularizers facilitate structural sparsity in parameter space, which is a useful property in multiclass prediction and other related tasks. We report empirical results that demonstrate the power of our approach in building accurate and structurally sparse models.
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