Random Reshuffling: Simple Analysis with Vast Improvements

By Konstantin Mishchenko, Ahmed Khaled, Peter Richtárik. 2020

164 citations recorded by the source corpus.

Random Reshuffling (RR) is an algorithm for minimizing finite-sum functions that utilizes iterative gradient descent steps in conjunction with data reshuffling. Often contrasted with its sibling Stochastic Gradient Descent (SGD), RR is usually faster in practice and enjoys significant popularity in convex and non-convex optimization. The convergence rate of RR has attracted substantial attention recently and, for strongly convex and smooth functions, it was shown to converge faster than SGD if 1) the stepsize is small, 2) the gradients are bounded, and 3) the number of epochs is large. We remove these 3 assumptions, improve the dependence on the condition number from $\kappa^2$ to $\kappa$ (resp.\ from $\kappa$ to $\sqrt{\kappa}$) and, in addition, show that RR has a different type of variance. We argue through theory and experiments that the new variance type gives an additional justification of the superior performance of RR. To go beyond strong convexity, we present several results for non-strongly convex and non-convex objectives. We show that in all cases, our theory improves upon existing literature. Finally, we prove fast convergence of the Shuffle-Once (SO) algorithm, which shuffles the data only once, at the beginning of the optimization process. Our theory for strongly-convex objectives tightly matches the known lower bounds for both RR and SO and substantiates the common practical heuristic of shuffling once or only a few times. As a byproduct of our analysis, we also get new results for the Incremental Gradient algorithm (IG), which does not shuffle the data at all.

Subfields: Artificial Intelligence, Computational Mechanics, Computational Theory and Mathematics, Optimization & Training, Statistical Learning Theory

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Citations in the corpus

Characterizing & Finding Good Data Orderings for Fast Convergence of Sequential Gradient Methods (2022, methodology): “The bound for IG is only investigated in the worst-case order selection, leading to an inferior bound O( 1 T 2/3 ) [17].”
SGD with shuffling: optimal rates without component convexity and large epoch requirements (2020, background): “[10]‡ O ( e− K 4κ + log (2)(nK) nK2 ) K ≥ 1”
A Unified Convergence Analysis for Shuffling-Type Gradient Methods (2020, methodology): “Here, we use diminishing learning rates in Theorem 2 instead of constant ones as in (HaoChen and Sra, 2019; Mishchenko et al., 2020; Nagaraj et al., 2019).”