We study the ridge regression (L2 regularized least squares) problem and its dual, which is also a ridge regression problem. We observe that the optimality conditions can be formulated in several different but equivalent ways, in the form of a linear system involving a structured matrix depending on a single "stepsize" parameter which we introduce for regularization purposes. This leads to the idea of studying and comparing, in theory and practice, the performance of the fixed point method applied to these reformulations.
We compute the optimal stepsize parameters and uncover interesting connections between the complexity bounds of the variants of the fixed point scheme we consider. These connections follow from a close link between the spectral properties of the associated matrices. For instance, some reformulations involve purely imaginary eigenvalues; some involve real eigenvalues and others have all eigenvalues on the complex circle.
We show that the deterministic Quartz method - which is a special case of the randomized dual coordinate ascent method with arbitrary sampling recently developed by Qu, Richtárik and Zhang - can be cast in our framework, and achieves the best rate in theory and in numerical experiments among the fixed point methods we study.
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