The Symmetry Coefficient of Positively Homogeneous Functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910918937411584 |
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| author | Nilsson, Max Giselsson, Pontus |
| author_facet | Nilsson, Max Giselsson, Pontus |
| contents | 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, $α(h)$, of the reference function $h$. While some explicit values of $α(h)$ have been determined for specific functions $h$, a general characterization has remained elusive. This paper explores two problems: ($\textit{i}$) deriving calculus rules for the symmetry coefficient and ($\textit{ii}$) computing $α(\lVert\cdot\rVert_2^p)$ 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 $α(\lVert\cdot\rVert_2^p)$ is independent of dimension and propose an efficient algorithm for its computation. Additionally, we prove that $α(\lVert\cdot\rVert_2^p)$ 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 $p \in \{6,8,10\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Symmetry Coefficient of Positively Homogeneous Functions Nilsson, Max Giselsson, Pontus Optimization and Control 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, $α(h)$, of the reference function $h$. While some explicit values of $α(h)$ have been determined for specific functions $h$, a general characterization has remained elusive. This paper explores two problems: ($\textit{i}$) deriving calculus rules for the symmetry coefficient and ($\textit{ii}$) computing $α(\lVert\cdot\rVert_2^p)$ 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 $α(\lVert\cdot\rVert_2^p)$ is independent of dimension and propose an efficient algorithm for its computation. Additionally, we prove that $α(\lVert\cdot\rVert_2^p)$ 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 $p \in \{6,8,10\}$. |
| title | The Symmetry Coefficient of Positively Homogeneous Functions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2504.18296 |