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The symmetry coefficient of positively homogeneous functions

Nilsson, Max LU orcid and Giselsson, Pontus LU orcid (2025) In Optimization Letters
Abstract

The Bregman distance is a central tool in convex optimization, particularly in first-order gradient descent and proximal-based algorithms. Such methods enable optimization of functions without Lipschitz continuous gradients by leveraging the concept of relative smoothness, with respect to a reference function h. A key factor in determining the full range of allowed step sizes in Bregman schemes is the symmetry coefficient,, of the reference function h. While some explicit values of have been determined for specific functions h, a general characterization has remained elusive. This paper explores two problems: (i) deriving calculus rules for the symmetry coefficient and (ii) computing for general p. We establish upper and lower bounds... (More)

The Bregman distance is a central tool in convex optimization, particularly in first-order gradient descent and proximal-based algorithms. Such methods enable optimization of functions without Lipschitz continuous gradients by leveraging the concept of relative smoothness, with respect to a reference function h. A key factor in determining the full range of allowed step sizes in Bregman schemes is the symmetry coefficient,, of the reference function h. While some explicit values of have been determined for specific functions h, a general characterization has remained elusive. This paper explores two problems: (i) deriving calculus rules for the symmetry coefficient and (ii) computing for general p. We establish upper and lower bounds for the symmetry coefficient of sums of positively homogeneous Legendre functions and, under certain conditions, provide exact formulas for these sums. Furthermore, we demonstrate that is independent of dimension and propose an efficient algorithm for its computation. Additionally, we prove that asymptotically equals, and is lower bounded by, the function 1/(2p), offering a simpler upper bound for step sizes in Bregman schemes. Finally, we present closed-form computations for specific cases such as.

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organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Bregman distance, First-order algorithms, Legendre function, NoLips algorithm, Symmetry coefficient
in
Optimization Letters
publisher
Springer
external identifiers
  • scopus:105021544301
ISSN
1862-4472
DOI
10.1007/s11590-025-02266-6
language
English
LU publication?
yes
id
94643949-a095-4f75-b267-4a5dcbca0dd9
date added to LUP
2026-01-08 14:27:38
date last changed
2026-01-08 14:27:57
@article{94643949-a095-4f75-b267-4a5dcbca0dd9,
  abstract     = {{<p>The Bregman distance is a central tool in convex optimization, particularly in first-order gradient descent and proximal-based algorithms. Such methods enable optimization of functions without Lipschitz continuous gradients by leveraging the concept of relative smoothness, with respect to a reference function h. A key factor in determining the full range of allowed step sizes in Bregman schemes is the symmetry coefficient,, of the reference function h. While some explicit values of have been determined for specific functions h, a general characterization has remained elusive. This paper explores two problems: (i) deriving calculus rules for the symmetry coefficient and (ii) computing for general p. We establish upper and lower bounds for the symmetry coefficient of sums of positively homogeneous Legendre functions and, under certain conditions, provide exact formulas for these sums. Furthermore, we demonstrate that is independent of dimension and propose an efficient algorithm for its computation. Additionally, we prove that asymptotically equals, and is lower bounded by, the function 1/(2p), offering a simpler upper bound for step sizes in Bregman schemes. Finally, we present closed-form computations for specific cases such as.</p>}},
  author       = {{Nilsson, Max and Giselsson, Pontus}},
  issn         = {{1862-4472}},
  keywords     = {{Bregman distance; First-order algorithms; Legendre function; NoLips algorithm; Symmetry coefficient}},
  language     = {{eng}},
  publisher    = {{Springer}},
  series       = {{Optimization Letters}},
  title        = {{The symmetry coefficient of positively homogeneous functions}},
  url          = {{http://dx.doi.org/10.1007/s11590-025-02266-6}},
  doi          = {{10.1007/s11590-025-02266-6}},
  year         = {{2025}},
}