Bregman Proximal Point Algorithm Revisited: A New Inexact Version and Its Inertial Variant

We study a general convex optimization problem, which covers various classic problems in different areas and particularly includes many optimal transport related arising recent years. To solve this we revisit the Bregman proximal point algorithm (BPPA) introduce new inexact stopping condition for solving subproblems, can circumvent underlying feasibility difficulty often appearing existing conditions when problem has complex feasible set. Our also several as special cases hence makes our BPPA (iBPPA) more flexible to fit scenarios practice. As an application standard (OT) iBPPA with entropic term bypass some numerical instability issues that usually plague popular Sinkhorn's OT community, since does not require parameter be very small obtaining accurate approximate solution. The iteration complexity of $O(1/k)$ convergence sequence are established under mild conditions. Moreover, inspired by Nesterov's acceleration technique, develop inertial variant iBPPA, denoted V-iBPPA, establish $O(1/k^{\lambda})$, where $\lambda\geq1$ is quadrangle scaling exponent kernel function. In particular, constant function strongly Lipschitz continuous gradient (hence $\lambda=2$), V-iBPPA achieves faster rate $O(1/k^2)$ just accelerated algorithms. Some preliminary experiments conducted show behaviors inexactness settings. empirically verify potential improving speed.