  A recent line of ground-breaking results for permutation-based SGD has
corroborated a widely observed phenomenon: random permutations offer faster
convergence than with-replacement sampling. However, is random optimal? We show
that this depends heavily on what functions we are optimizing, and the
convergence gap between optimal and random permutations can vary from
exponential to nonexistent. We first show that for 1-dimensional strongly
convex functions, with smooth second derivatives, there exist permutations that
offer exponentially faster convergence compared to random. However, for general
strongly convex functions, random permutations are optimal. Finally, we show
that for quadratic, strongly-convex functions, there are easy-to-construct
permutations that lead to accelerated convergence compared to random. Our
results suggest that a general convergence characterization of optimal
permutations cannot capture the nuances of individual function classes, and can
mistakenly indicate that one cannot do much better than random.
