We studied self-diffusion of small two-dimensional Ag islands, containing up to ten atoms, on the Ag(111) surface using self-learning kinetic Monte Carlo (SLKMC) simulations. Activation barriers are calculated using the semi-empirical embedded atom method (EAM) potential. We find that two- to seven-atom islands primarily diffuse via concerted translation processes with small contributions from multi-atom and single-atom processes, while eight- to ten-atom islands diffuse via single-atom processes, especially edge diffusion, corner rounding and kink detachment, along with a minimal contribution from concerted processes. For each island size, we give a detailed description of the important processes, and their activation barriers, responsible for its diffusion.