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• W1810986334 abstract "For the general problem of minimizing a convex function over a compact convex domain, we will investigate a simple iterative approximation algorithm based on the method by Frank & Wolfe 1956, that does not need projection steps in order to stay inside the optimization domain. Instead of a projection step, the linearized problem defined by a current subgradient is solved, which gives a step direction that will naturally stay in the domain. Our framework generalizes the sparse greedy algorithm of Frank & Wolfe and its primal-dual analysis by Clarkson 2010 (and the low-rank SDP approach by Hazan 2008) to arbitrary convex domains. We give a convergence proof guaranteeing {epsilon}-small duality gap after O(1/{epsilon}) iterations. The method allows us to understand the sparsity of approximate solutions for any l1-regularized convex optimization problem (and for optimization over the simplex), expressed as a function of the approximation quality. We obtain matching upper and lower bounds of {Theta}(1/{epsilon}) for the sparsity for l1-problems. The same bounds apply to low-rank semidefinite optimization with bounded trace, showing that rank O(1/{epsilon}) is best possible here as well. As another application, we obtain sparse matrices of O(1/{epsilon}) non-zero entries as {epsilon}-approximate solutions when optimizing any convex function over a class of diagonally dominant symmetric matrices. We show that our proposed first-order method also applies to nuclear norm and max-norm matrix optimization problems. For nuclear norm regularized optimization, such as matrix completion and low-rank recovery, we demonstrate the practical efficiency and scalability of our algorithm for large matrix problems, as e.g. the Netflix dataset. For general convex optimization over bounded matrix max-norm, our algorithm is the first with a convergence guarantee, to the best of our knowledge." @default.
• W1810986334 created "2016-06-24" @default.
• W1810986334 creator A5073756389 @default.
• W1810986334 date "2011-08-04" @default.
• W1810986334 modified "2023-09-27" @default.
• W1810986334 title "Convex Optimization without Projection Steps" @default.
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• W1810986334 hasPublicationYear "2011" @default.

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