A Unified Framework for Iterate Convergence of Bregman Proximal Methods
TL;DR
A unified iterate convergence framework that applies to a broad group of kernels and composite objective functions is developed, and it is shown that the continuous-time BPM (mirror flow) converges to a stationary point for o-minimal definable objective functions, yielding the first trajectory convergence result for mirror flow.
Abstract
Iterate convergence of Bregman proximal methods (BPMs) has long remained open, especially for nonconvex objectives. Recently, \citet{chen2026skl} made progress by establishing iterate convergence for a BPM via the so-called scaled Kurdyka-\L{}ojasiewicz (SK\L{}) property, but only for the Shannon entropy kernel and linearly constrained problems. In this paper, we develop a unified iterate convergence framework that applies to a broad group of kernels and composite objective functions. Our approach extends the analytical tools in \cite{chen2026skl}, in particular the SK\L{} property, which plays a central role in ensuring convergence of the generated sequences. By introducing kernel-dependent parameterization functions, we show that the extended SK\L{} property holds for all continuous subanalytic functions, particularly when the kernel has a closed domain. We then verify that the assumptions of the framework are satisfied by standard BPMs under mild regularity conditions, thereby establishing their iterate convergence for a wide range of objective functions. Furthermore, based on the parameterization functions, we show that the continuous-time BPM (mirror flow) converges to a stationary point for o-minimal definable objective functions, yielding the first trajectory convergence result for mirror flow without imposing convexity assumptions on the objective function or isolation assumptions on stationary points. Taken together, these discrete- and continuous-time convergence results provide a unified trajectory convergence theory for BPMs.