- Abstract: Training example order in SGD has long been known to affect convergence rate. Recent results show that accelerated rates are possible in a variety of cases for permutation-based sample orders, in which each example from the training set is used once before any example is reused. In this paper, we develop a broad condition on the sequence of examples used by SGD that is sufficient to prove tight convergence rates in both strongly convex and non-convex settings. We show that our approach suffices to recover, and in some cases improve upon, previous state-of-the-art analyses for four known example-selection schemes: (1) shuffle once, (2) random reshuffling, (3) random reshuffling with data echoing, and (4) Markov Chain Gradient Descent. Motivated by our theory, we propose two new example-selection approaches. First, using quasi-Monte-Carlo methods, we achieve unprecedented accelerated convergence rates for learning with data augmentation. Second, we greedily choose a fixed scan-order to minimize the metric used in our condition and show that we can obtain more accurate solutions from the same number of epochs of SGD. We conclude by empirically demonstrating the utility of our approach for both convex linear-model and deep learning tasks.