A unified framework for Bregman proximal methods: subgradient, gradient, and accelerated gradient schemes

We provide a unified framework for analyzing the convergence of Bregman proximal first-order algorithms for convex minimization. Our framework hinges on properties of the convex conjugate and gives novel proofs of the convergence rates of the Bregman proximal subgradient, Bregman proximal gradient, and a new accelerated Bregman proximal gradient algorithm under fairly general and mild assumptions. Our accelerated Bregman proximal gradient algorithm attains the best-known accelerated rate of convergence when suitable relative smoothness and triangle scaling assumptions hold. However, the algorithm requires no prior knowledge of any related smoothness or triangle scaling constants.