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Upper and lower bounds for the Bregman divergence.

Benjamin Sprung1

  • 1Institute for Numerical and Applied Mathematics, Göttingen, Germany.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary

Researchers explored upper and lower bounds for Bregman divergence in normed spaces. A simpler proof was developed for specific cases, with results applicable to broader convex functional analysis.

Area of Science:

  • Mathematical analysis
  • Convex analysis
  • Functional analysis

Background:

  • Bregman divergence is a key concept in optimization and machine learning.
  • Understanding its bounds is crucial for algorithm development.
  • Existing proofs for specific cases can be complex.

Purpose of the Study:

  • To establish and simplify proofs for upper and lower bounds of Bregman divergence.
  • To investigate these bounds for convex functionals on normed spaces.
  • To extend findings to more general functional forms.

Main Methods:

  • Analysis of Bregman divergence properties.
  • Development of a new, simplified proof technique.
  • Application of subgradient concepts in normed spaces.
Keywords:
Bregman distanceBregman divergenceTotal convexityUniform convexityUniform smoothness

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Main Results:

  • A novel and simpler proof for Bregman divergence inequalities (Xu-Roach case).
  • Established upper and lower bounds for Bregman divergence.
  • Demonstrated the transferability of results to more general convex functions.

Conclusions:

  • The study provides a more accessible proof for Bregman divergence bounds.
  • The findings enhance the theoretical understanding of convex functionals in normed spaces.
  • Results have implications for optimization and related fields.