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

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