The Symmetry Coefficient of Positively Homogeneous Functions

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, $\alpha(h)$, of the reference function $h$. While some explicit values of $\alpha(h)$ 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 $\alpha(\lVert\cdot\rVert_2^p)$ 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 $\alpha(\lVert\cdot\rVert_2^p)$ is independent of dimension and propose an efficient algorithm for its computation. Additionally, we prove that $\alpha(\lVert\cdot\rVert_2^p)$ 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 $p \in \{6,8,10\}$.