The oracle Complexity of Smooth Convex Optimization in Nonstandard Settings
published: Aug. 20, 2015, recorded: July 2015, views: 1572
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First-order convex minimization algorithms are currently the methods of choice for large-scale sparse – and more generally parsimonious – regression models. We pose the question on the limits of performance of black-box oriented methods for convex minimization in non-standard settings, where the regularity of the objective is measured in a norm not necessarily induced by the feasible domain. This question is studied for `p/`q-settings, and their matrix analogues (Schatten norms), where we find surprising gaps on lower bounds compared to state of the art methods. We propose a conjecture on the optimal convergence rates for these settings, for which a positive answer would lead to significant improvements on minimization algorithms for parsimonious regression models.
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