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Many machine learning problems lack strong convexity properties. Fortunately,recent studies have revealed that first-order algorithms also enjoy linearconvergences under various weaker regularity conditions. While the relationshipamong different conditions for convex and smooth functions is well understood,it is not the case for the nonsmooth setting. In this paper, we go beyondconvexity and smoothness, and clarify the connections among common regularityconditions (including $\textit{strong convexity, restricted secant inequality,subdifferential error bound, Polyak-{\L}ojasiewicz inequality, and quadraticgrowth}$) in the class of weakly convex functions. In addition, we present asimple and modular proof for the linear convergence of the $\textit{proximalpoint method}$ (PPM) for convex (possibly nonsmooth) optimization using theseregularity conditions. The linear convergence also holds when the subproblemsof PPM are solved inexactly with a proper control of inexactness.

Error bounds, PL condition, and quadratic growth for weakly convex functions, and linear convergences of proximal point methods