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In this work, we develop a level-set subdifferential error bound conditionaiming towards convergence rate analysis of a variable Bregman proximalgradient (VBPG) method for a broad class of nonsmooth and nonconvexoptimization problems. It is proved that the aforementioned conditionguarantees linear convergence of VBPG, and is weaker than Kurdyka-Lojasiewiczproperty, weak metric subregularity and Bregman proximal error bound. Along theway, we are able to derive a number of verifiable conditions for level-setsubdifferential error bounds to hold, and necessary conditions and sufficientconditions for linear convergence relative to a level set for nonsmooth andnonconvex optimization problems. The newly established results not only enableus to show that any accumulation point of the sequence generated by VBPG is atleast a critical point of the limiting subdifferential or even acritical pointof the proximal subdifferential with a fixed Bregman function in eachiteration, but also provide a fresh perspective that allows us to exploreinner-connections among many known sufficient conditions for linear convergenceof various first-order methods.

Level-set Subdifferential Error Bounds and Linear Convergence of Variable Bregman Proximal Gradient Method