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Aggregating regular norms

Anatoli Juditsky, Arkadi Nemirovski

Abstract

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 important role in 1) high-dimensional convex geometry and probability in Banach spaces [0.9.12.13.15], 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, numerous geometric, probabilistic, and optimization-related results, which motivates our interest in aggregating regular norms with controlled (and moderate) inflation of regularity parameters.

Aggregating regular norms

Abstract

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 important role in 1) high-dimensional convex geometry and probability in Banach spaces [0.9.12.13.15], 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, numerous geometric, probabilistic, and optimization-related results, which motivates our interest in aggregating regular norms with controlled (and moderate) inflation of regularity parameters.
Paper Structure (22 sections, 10 theorems, 114 equations)

This paper contains 22 sections, 10 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Problem statement and main results
  3. Regular norms
  4. Main results
  5. Remark:
  6. Illustration: regularity of ellitopic and spectratopic norms
  7. Ellitopes and spectratopes
  8. Ellitopes.
  9. Spectratopes.
  10. Calculus of ellitopes/spectratopes.
  11. Ellitopic and spectratopic norms.
  12. The result
  13. Appendix
  14. Proof of Theorem \ref{['the1Gen']}
  15. Preliminary step
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $\theta(\cdot):{\mathbf{R}}^K_+\to{\mathbf{R}}$ be a convex continuous homogeneous, of degree 1, function which is monotone on ${\mathbf{R}}^K_+$ and positive outside of the origin. Let $\|\cdot\|_i$ be $(\varkappa,\varsigma)$-regular norms on ${\mathbf{R}}^{n_i}$, $1\leq i\leq K$. Then the aggr (clearly, this indeed is a norm on ${\mathbf{R}}^{n_1+...+n_K}$) is $(c_1[\ln(K+1)+\varkappa],c_2\v

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.1
  • Theorem 3.1
  • Lemma A.1
  • Corollary A.1
  • Lemma A.2
  • Lemma A.3
  • Lemma A.4
  • Lemma A.5