A Bregman Divergence View on the Difference-of-Convex Algorithm

Oisin Faust, Hamza Fawzi, James Saunderson
Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:3427-3439, 2023.

Abstract

The difference of convex (DC) algorithm is a conceptually simple method for the minimization of (non)convex functions that are expressed as the difference of two convex functions. An attractive feature of the algorithm is that it maintains a global overestimator on the function and does not require a choice of step size at each iteration. By adopting a Bregman divergence point of view, we simplify and strengthen many existing non-asymptotic convergence guarantees for the DC algorithm. We further present several sufficient conditions that ensure a linear convergence rate, namely a new DC Polyak-Lojasiewicz condition, as well as a relative strong convexity assumption. Importantly, our conditions do not require smoothness of the objective function. We illustrate our results on a family of minimization problems involving the quantum relative entropy, with applications in quantum information theory.

Cite this Paper

BibTeX
@InProceedings{pmlr-v206-faust23a, title = {A Bregman Divergence View on the Difference-of-Convex Algorithm}, author = {Faust, Oisin and Fawzi, Hamza and Saunderson, James}, booktitle = {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics}, pages = {3427--3439}, year = {2023}, editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem}, volume = {206}, series = {Proceedings of Machine Learning Research}, month = {25--27 Apr}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v206/faust23a/faust23a.pdf}, url = {https://proceedings.mlr.press/v206/faust23a.html}, abstract = {The difference of convex (DC) algorithm is a conceptually simple method for the minimization of (non)convex functions that are expressed as the difference of two convex functions. An attractive feature of the algorithm is that it maintains a global overestimator on the function and does not require a choice of step size at each iteration. By adopting a Bregman divergence point of view, we simplify and strengthen many existing non-asymptotic convergence guarantees for the DC algorithm. We further present several sufficient conditions that ensure a linear convergence rate, namely a new DC Polyak-Lojasiewicz condition, as well as a relative strong convexity assumption. Importantly, our conditions do not require smoothness of the objective function. We illustrate our results on a family of minimization problems involving the quantum relative entropy, with applications in quantum information theory.} }
Endnote
%0 Conference Paper %T A Bregman Divergence View on the Difference-of-Convex Algorithm %A Oisin Faust %A Hamza Fawzi %A James Saunderson %B Proceedings of The 26th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2023 %E Francisco Ruiz %E Jennifer Dy %E Jan-Willem van de Meent %F pmlr-v206-faust23a %I PMLR %P 3427--3439 %U https://proceedings.mlr.press/v206/faust23a.html %V 206 %X The difference of convex (DC) algorithm is a conceptually simple method for the minimization of (non)convex functions that are expressed as the difference of two convex functions. An attractive feature of the algorithm is that it maintains a global overestimator on the function and does not require a choice of step size at each iteration. By adopting a Bregman divergence point of view, we simplify and strengthen many existing non-asymptotic convergence guarantees for the DC algorithm. We further present several sufficient conditions that ensure a linear convergence rate, namely a new DC Polyak-Lojasiewicz condition, as well as a relative strong convexity assumption. Importantly, our conditions do not require smoothness of the objective function. We illustrate our results on a family of minimization problems involving the quantum relative entropy, with applications in quantum information theory.
APA
Faust, O., Fawzi, H. & Saunderson, J.. (2023). A Bregman Divergence View on the Difference-of-Convex Algorithm. Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 206:3427-3439 Available from https://proceedings.mlr.press/v206/faust23a.html.

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