The Symmetry Coefficient of Positively Homogeneous Functions

TL;DR

This work introduces and analyzes the symmetry coefficient $\alpha(h)$ of Bregman distances for positively homogeneous reference functions, focusing on the one-dimensional $|\cdot|^p$ and the multivariate $\|\cdot\|_2^p$ cases. It provides a robust 1D calculus via a bisection method to compute $\alpha(|\cdot|^p)$, proves dimension-independence for $\|\cdot\|_2^p$, and establishes asymptotic and monotonic properties showing $\alpha(|\cdot|^p)\sim(2p)^{-1}$ as $p\to\infty$ with $\alpha(h) > 1/(2p)$ for $p>2$. The paper also derives closed-form expressions for several special exponents (e.g., $p\in\{3,6,8,10\}$) and develops exact characterizations for sums of positively homogeneous Legendre functions, including a complete description of $\alpha$ for $\omega_{\beta,\gamma}$ and related sums. These results yield dimension-free step-size bounds in NoLips-type methods and deepen understanding of relative smoothness in Bregman frameworks, with practical implications for nonsmooth- and gradient-free optimization.

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, $α(h)$, 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 $p \in \{6,8,10\}$.

Figures (1)

• Figure 1: Calculation of the symmetry coefficient of ${|\cdot|^p : \mathbb{R} \to \mathbb{R}}$ for $p \in [2, 10^3]$, utilizing \ref{['alg:compute symmetry for 1d norm']}.

Theorems & Definitions (55)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof