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Many important machine learning problems can be formulated as highly constrained convex optimization problems. One important example is metric constrained problems. In this paper, we show that standard optimization techniques can not be used to solve metric constrained problem. To solve such problems, we provide a general active set framework, called Project and Forget, and several variants thereof that use Bregman projections. Project and Forget is a general purpose method that can be used to solve highly constrained convex problems with many (possibly exponentially) constraints. We provide a theoretical analysis of Project and Forget and prove that our algorithms converge to the global optimal solution and have a linear rate of convergence. We demonstrate that using our method, we can solve large problem instances of general weighted correlation clustering, metric nearness, information theoretic metric learning and quadratically regularized optimal transport; in each case, out-performing the state of the art methods with respect to CPU times and problem sizes. [ABSTRACT FROM AUTHOR]

In this paper, we study the efficiency of a Restarted SubGradient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an ∈-optimal solution with a lower complexity than the SG method. In particular, we first show that RSG can reduce the dependence of SG's iteration complexity on the distance between the initial solution and the optimal set to that between the ∈-level set and the optimal set multiplied by a logarithmic factor. Moreover, we show the advantages of RSG over SG in solving a broad family of problems that satisfy a local error bound condition, and also demonstrate its advantages for three specific families of convex optimization problems with different power constants in the local error bound condition. (a) For the problems whose epigraph is a polyhedron, RSG is shown to converge linearly. (b) For the problems with local quadratic growth property in the ∈-sublevel set, RSG has an O( 1/∈ log( 1 ∈ )) iteration complexity. (c) For the problems that admit a local Kurdyka- Lojasiewicz property with a power constant of β ∈ [0, 1), RSG has an O( 1/∈2β log( 1/∈ )) iteration complexity. The novelty of our analysis lies at exploiting the lower bound of the first-order optimality residual at the ∈-level set. It is this novelty that allows us to explore the local properties of functions (e.g., local quadratic growth property, local Kurdyka-Łojasiewicz property, more generally local error bound conditions) to develop the improved convergence of RSG. We also develop a practical variant of RSG enjoying faster convergence than the SG method, which can be run without knowing the involved parameters in the local error bound condition. We demonstrate the effectiveness of the proposed algorithms on several machine learning tasks including regression, classification and matrix completion. [ABSTRACT FROM AUTHOR]