Permutation-Based SGD: Is Random Optimal?
This work addresses the optimization efficiency problem for machine learning practitioners, providing insights into when random permutations suffice and when better permutations can be constructed, though it is incremental in refining existing theoretical understanding.
The paper investigates whether random permutations are optimal for permutation-based SGD, finding that the convergence gap between optimal and random permutations varies from exponential to nonexistent depending on the function class, with examples showing exponentially faster convergence for specific 1-dimensional functions and accelerated convergence for quadratic functions.
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.