On growth error bound conditions with an application to heavy ball method

Abstract: In this paper, we investigate the growth error bound condition. By using the proximal point algorithm, we first provide a more accessible and elementary proof of the fact that Kurdyka-{\L}ojasiewicz conditions imply growth error bound conditions for convex functions which has been established before via a subgradient flow. We then extend the result for nonconvex functions. Furthermore we show that every definable function in an o-minimal structure must satisfy a growth error bound condition. Finally, as an application, we consider the heavy ball method for solving convex optimization problems and propose an adaptive strategy for selecting the momentum coefficient. Under growth error bound conditions, we derive convergence rates of the proposed method. A numerical experiment is conducted to demonstrate its acceleration effect over the gradient method.