Sampling without Replacement Leads to Faster Rates in Finite-Sum Minimax Optimization
This work addresses optimization efficiency for machine learning practitioners by improving convergence in minimax problems, though it is incremental as it builds on existing sampling strategies.
The paper demonstrates that sampling without replacement, specifically Random Reshuffling and Single Shuffling, leads to faster convergence rates than sampling with replacement in finite-sum minimax optimization, with tight rates shown for smooth strongly convex-strongly concave and nonconvex-nonconcave settings.
We analyze the convergence rates of stochastic gradient algorithms for smooth finite-sum minimax optimization and show that, for many such algorithms, sampling the data points without replacement leads to faster convergence compared to sampling with replacement. For the smooth and strongly convex-strongly concave setting, we consider gradient descent ascent and the proximal point method, and present a unified analysis of two popular without-replacement sampling strategies, namely Random Reshuffling (RR), which shuffles the data every epoch, and Single Shuffling or Shuffle Once (SO), which shuffles only at the beginning. We obtain tight convergence rates for RR and SO and demonstrate that these strategies lead to faster convergence than uniform sampling. Moving beyond convexity, we obtain similar results for smooth nonconvex-nonconcave objectives satisfying a two-sided Polyak-Łojasiewicz inequality. Finally, we demonstrate that our techniques are general enough to analyze the effect of data-ordering attacks, where an adversary manipulates the order in which data points are supplied to the optimizer. Our analysis also recovers tight rates for the incremental gradient method, where the data points are not shuffled at all.