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# Optimal Complexity and Certification of Bregman First-Order Methods

2 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique - ENS Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : We provide a lower bound showing that the $O(1/k)$ convergence rate of the NoLips method (a.k.a. Bregman Gradient) is optimal for the class of functions satisfying the $h$-smoothness assumption. This assumption, also known as relative smoothness, appeared in the recent developments around the Bregman Gradient method, where acceleration remained an open issue. On the way, we show how to constructively obtain the corresponding worst-case functions by extending the computer-assisted performance estimation framework of Drori and Teboulle (Mathematical Programming, 2014) to Bregman first-order methods, and to handle the classes of differentiable and strictly convex functions.
Document type :
Journal articles
Domain :

https://hal.inria.fr/hal-02384167
Contributor : Radu-Alexandru Dragomir Connect in order to contact the contributor
Submitted on : Thursday, November 28, 2019 - 11:24:44 AM
Last modification on : Thursday, November 3, 2022 - 10:57:00 AM

### Citation

Radu-Alexandru Dragomir, Adrien Taylor, Alexandre d'Aspremont, Jérôme Bolte. Optimal Complexity and Certification of Bregman First-Order Methods. Mathematical Programming, 2022, 194 (1), pp.41-83. ⟨10.1007/s10107-021-01618-1⟩. ⟨hal-02384167⟩

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