Inexact Bregman Proximal Gradient Method and Its Inertial Variant with Absolute and Partial Relative Stopping Criteria

The Bregman proximal gradient method (BPGM), which uses the Bregman distance as a proximity measure in the iterative scheme, has recently been redeveloped for minimizing convex composite problems without the global Lipschitz gradient continuity assumption. This makes the BPGM appealing for a wide range of applications, and hence, it has received growing attention in recent years. However, most existing convergence results are obtained only under the assumption that the involved subproblems are solved exactly, which is unrealistic in many applications and limits the applicability of the BPGM. To make the BPGM implementable and practical, in this paper, we develop inexact versions of the BPGM (denoted by iBPGM) by employing either an absolute-type stopping criterion or a partial relative-type stopping criterion for solving the subproblems. The $\mathcal{O}(1/k)$ convergence rate and the convergence of the sequence are also established for our iBPGM under some conditions. Moreover, we develop an inertial variant of our iBPGM (denoted by v-iBPGM) and establish the $\mathcal{O}(1/k^{\gamma })$ convergence rate, where $\gamma \ge 1$ is a restricted relative smoothness exponent, depending on the smooth function in the objective and the kernel function. Specially, when the smooth function in the objective has a Lipschitz continuous gradient and the kernel function is strongly convex, we have γ = 2, and thus the v-iBPGM improves the convergence rate of the iBPGM from $\mathcal{O}(1/k)$ to $\mathcal{O}(1/k^2)$, in accordance with the existing results on the exact accelerated BPGM. Finally, some preliminary numerical experiments for solving the discrete quadratic regularized optimal transport problem are conducted to illustrate the convergence behaviors of our iBPGM and v-iBPGM under different inexactness settings.

Funding: The research of first author is supported in part by the National Natural Science Foundation of China under Grant 12301411, the Natural Science Foundation of Guangdong under Grant 2023A1515012026, and the Basic and Applied Basic Research Foundation of Guangzhou under Grant 2024A04J4184. The research of second author is supported in part by the Ministry of Education, Singapore, under Academic Research Fund Tier 1 Grant A00084930000.