Abstract: The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice because of their excellent convergence and generalization qualities, but their theoretical properties are not yet well understood. We provide convergence results for step decay in the non-convex regime, ensuring that the gradient norm vanishes at an $\mathcal{O}(\ln T/\sqrt{T})$ rate. We also provide near-optimal (and sometimes provably tight) convergence guarantees for general, possibly non-smooth, convex and strongly convex problems. The practical efficiency of the step decay step-size is demonstrated in several large-scale deep neural network training tasks.