Permutation-Based SGD: Is Random Optimal?

TL;DR

This paper investigates the effectiveness of permutation-based stochastic gradient descent, revealing that the optimality of random permutations varies significantly depending on the function class, with some permutations offering exponential acceleration.

Contribution

The paper demonstrates that the optimality of random permutations in SGD depends on the function class, providing examples where specific permutations outperform random choices.

Findings

For 1D strongly convex functions, certain permutations yield exponentially faster convergence.

For general strongly convex functions, random permutations are proven to be optimal.

Constructed permutations can accelerate convergence for quadratic, strongly convex functions.

Abstract

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.