We study the magnetic hysteresis in the random field Ising model in 3D. We discuss the disorder dependence of the coercive field H_c, and obtain an analytical description of the smooth part of the hysteresis below and above H_c, by identifying the disorder configurations (instantons) that are the most probable to trigger local avalanches. We estimate the critical disorder strength at which the hysteresis curve becomes continuous. From an instanton analysis at zero field we obtain a description of local two-level systems in the ferromagnetic phase. ; Comment: Phys. Rev. Lett. 96, 117202 (2006)