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<p>The subject of this paper is regularity-preserving aggregation of <i>regular</i> norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to “type 2” spaces [9, Chapter 9] playing an important role in (1) high-dimensional convex geometry and probability in Banach spaces, see [8, 9, 12, 13, 15], and in (2) design of proximal first order algorithms for large-scale convex optimization with dimension-independent, or nearly so, complexity. Regularity, with moderate parameters, of a norm makes applicable, in a dimension-independent fashion, various geometric, probabilistic, and optimization-related results, thus motivating the subject of this paper—aggregations of regular norms resulting in controlled (and moderate) inflation of regularity parameters.</p>

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Aggregating regular norms

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Anatoli Juditsky,
  • Arkadi Nemirovski

摘要

The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to “type 2” spaces [9, Chapter 9] playing an important role in (1) high-dimensional convex geometry and probability in Banach spaces, see [8, 9, 12, 13, 15], and in (2) design of proximal first order algorithms for large-scale convex optimization with dimension-independent, or nearly so, complexity. Regularity, with moderate parameters, of a norm makes applicable, in a dimension-independent fashion, various geometric, probabilistic, and optimization-related results, thus motivating the subject of this paper—aggregations of regular norms resulting in controlled (and moderate) inflation of regularity parameters.