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On Lower Complexity Bounds for Large-Sca...
Guzman, Cristobal...

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On Lower Complexity Bounds for Large-Scale Smooth Convex Optimization by Guzman, Cristobal ( Author )
Lemonjar Open Access
Australian National University
25-08-2023
We derive lower bounds on the black-box oracle complexity of large-scale smooth convex minimization problems, with emphasis on minimizing smooth (with Holder continuous, with a given exponent and constant, gradient) convex functions over high-dimensional ||.||_p-balls, 1<=p<=\infty. Our bounds turn out to be tight (up to logarithmic in the design dimension factors), and can be viewed as a substantial extension of the existing lower complexity bounds for large-scale convex minimization covering the nonsmooth case and the 'Euclidean' smooth case (minimization of convex functions with Lipschitz continuous gradients over Euclidean balls). As a byproduct of our results, we demonstrate that the classical Conditional Gradient algorithm is near-optimal, in the sense of Information-Based Complexity Theory, when minimizing smooth convex functions over high-dimensional ||.||_\infty-balls and their matrix analogies -- spectral norm balls in the spaces of square matrices.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1307.5001
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