On the nature of Bregman functions
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929739747295232 |
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| author | Pauwels, Edouard |
| author_facet | Pauwels, Edouard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_02689 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the nature of Bregman functions Pauwels, Edouard Optimization and Control 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. |
| title | On the nature of Bregman functions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2302.02689 |