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.
@cite_27: We propose a new primal-dual algorithmic framework for a prototypical constrained convex optimization template. The algorithmic instances of our framework are universal since they can automatically adapt to the unknown Holder continuity degree and constant within the dual formulation. They are also guaranteed to have optimal convergence rates in the objective residual and the feasibility gap for each Holder smoothness degree. In contrast to existing primal-dual algorithms, our framework avoids the proximity operator of the objective function. We instead leverage computationally cheaper, Fenchel-type operators, which are the main workhorses of the generalized conditional gradient (GCG)-type methods. In contrast to the GCG-type methods, our framework does not require the objective function to be differentiable, and can also process additional general linear inclusion constraints, while guarantees the convergence rate on the primal problem.
@cite_10: Online learning is a well established learning paradigm which has both theoretical and practical appeals. The goal of online learning is to make a sequence of accurate predictions given knowledge of the correct answer to previous prediction tasks and possibly additional available information. Online learning has been studied in several research fields including game theory, information theory, and machine learning. It also became of great interest to practitioners due the recent emergence of large scale applications such as online advertisement placement and online web ranking. In this survey we provide a modern overview of online learning. Our goal is to give the reader a sense of some of the interesting ideas and in particular to underscore the centrality of convexity in deriving efficient online learning algorithms. We do not mean to be comprehensive but rather to give a high-level, rigorous yet easy to follow, survey.
Distributed algorithms. For @math -problems exceeding the memory capacity of a single computer, a communication-efficient distributed scheme leveraging this Lipschitzing trick is presented in @cite_27 @cite_10 .