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On the nature of Bregman functions

Edouard Pauwels

TL;DR

The paper characterizes when a Legendre function $h$ on a convex compact domain $C$ induces a Bregman divergence that yields convergent behavior for Fejér-type sequences. It shows that the boundary compatibility condition $D_h(y,x_k)\to 0$ (eq:convB) holds if and only if $C$ is a polytope, while strict convexity of $h$ on $C$ is equivalent to the interior condition $D_h(y,x_k)\to 0$ implying $x_k\to y$ (eq:convA). The main theorem then states that if $C$ is a polytope and $h$ is strictly convex on $C$, every weak $D$-Fejér sequence converges, connecting geometric domain structure with algorithmic convergence in Bregman-based methods such as Mirror Descent and NoLips. Extensions address unbounded domains and relaxed regularity, showing local polyhedrality suffices in broader settings, albeit with limitations for non-polyhedral domains. These results clarify when classical Fejér-analytic convergence arguments are valid and highlight the need for refined tools beyond Bregman conditions in non-polyhedral contexts.

Abstract

Let C be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.

On the nature of Bregman functions

TL;DR

The paper characterizes when a Legendre function on a convex compact domain induces a Bregman divergence that yields convergent behavior for Fejér-type sequences. It shows that the boundary compatibility condition (eq:convB) holds if and only if is a polytope, while strict convexity of on is equivalent to the interior condition implying (eq:convA). The main theorem then states that if is a polytope and is strictly convex on , every weak -Fejér sequence converges, connecting geometric domain structure with algorithmic convergence in Bregman-based methods such as Mirror Descent and NoLips. Extensions address unbounded domains and relaxed regularity, showing local polyhedrality suffices in broader settings, albeit with limitations for non-polyhedral domains. These results clarify when classical Fejér-analytic convergence arguments are valid and highlight the need for refined tools beyond Bregman conditions in non-polyhedral contexts.

Abstract

Let C be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a polytope. This provides insights on sequential convergence of many Bregman divergence based algorithm: abstract compatibility conditions between Bregman and Euclidean topology may equivalently be replaced by explicit conditions on h and C. This also emphasizes that a general convergence theory for these methods (beyond polyhedral domains) would require more refinements than Bregman's conditions.
Paper Structure (19 sections, 11 theorems, 23 equations, 1 figure)

This paper contains 19 sections, 11 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fejérian sequences:
  3. Algorithmic examples
  4. Mirror descent:
  5. Bregman gradient and NoLips algorithm:
  6. Proximal minimization with $D$-functions:
  7. Alternating projection:
  8. Sequential convergence analysis
  9. Main results
  10. Condition \ref{['eq:convB']}
  11. Intuition: incompatibility with curvature
  12. First implication
  13. Proof of the claim.
  14. Reverse implication
  15. Condition \ref{['eq:convA']}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $C \subset \mathbb{R}^n$ be convex, compact, with nonempty interior and $h$ be Legendre with domain $C$, continuous on $C$. Then:

Figures (1)

  • Figure 1: Left: Illustration of the tangential phenomenon, closedness to the boundary and to the opposite extremity of the chord. Right: in general extreme points constitute a region of high curvature and it is possible to find neighboring points in a similar configuration. This cannot happen too much under condition \ref{['eq:convB']} and in particular extreme points should not accumulate anywhere.

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Claim 1
  • Proposition 2
  • Lemma 3: Tseng and Bertsekas
  • Lemma 4
  • ...and 4 more