# Why random reshuffling beats stochastic gradient descent

@article{Grbzbalaban2021WhyRR,
title={Why random reshuffling beats stochastic gradient descent},
author={Mert G{\"u}rb{\"u}zbalaban and Asuman E. Ozdaglar and Pablo A. Parrilo},
journal={Math. Program.},
year={2021},
volume={186},
pages={49-84}
}
• Published 29 October 2015
• Computer Science, Mathematics
• Math. Program.
We analyze the convergence rate of the random reshuffling (RR) method, which is a randomized first-order incremental algorithm for minimizing a finite sum of convex component functions. RR proceeds in cycles, picking a uniformly random order (permutation) and processing the component functions one at a time according to this order, i.e., at each cycle, each component function is sampled without replacement from the collection. Though RR has been numerically observed to outperform its with…
98 Citations

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The 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 and proves fast convergence of the Shuffle-Once algorithm, which shuffles the data only once.

### Random Shuffling Beats SGD after Finite Epochs

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It is proved that under strong convexity and second-order smoothness, the sequence generated by RandomShuffle converges to the optimal solution at the rate O(1/T^2 + n^3/ T^3), where n is the number of components in the objective, and T is the total number of iterations.

### Convergence of Random Reshuffling Under The Kurdyka-Łojasiewicz Inequality

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• 2021
Under the well-known Kurdyka-Łojasiewicz (KL) inequality, strong limit-point convergence results for RR with appropriate diminishing step sizes are established, namely, the whole sequence of iterates generated by RR is convergent and converges to a single stationary point in an almost sure sense.

### How Good is SGD with Random Shuffling?

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• 2019
This paper proves that after $k$ passes over individual functions, if the functions are re-shuffled after every pass, the best possible optimization error for SGD is at least $\Omega(1/(nk)^2+1/nk^3\right)$, which partially corresponds to recently derived upper bounds.

### Stochastic Learning Under Random Reshuffling With Constant Step-Sizes

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The analysis establishes analytically that random reshuffling outperforms uniform sampling and derives an analytical expression for the steady-state mean-square-error performance of the algorithm, which helps clarify in greater detail, the differences between sampling with and without replacement.

### Distributed Random Reshuffling over Networks

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• 2021
It is shown that D-RR inherits favorable characteristics of RR for both smooth strongly convex and smooth nonconvex objective functions, and convergence results match those of centralized RR and outperform the distributed stochastic gradient descent (DSGD) algorithm if the authors run a relatively large number of epochs.

### On the Comparison between Cyclic Sampling and Random Reshuffling

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• 2021
A norm is introduced, which is defined based on the sampling order, to measure the distance to solution and is applied on proximal Finito/MISO algorithm to identify the optimal fixed ordering, which can beat random reshuffling by a factor up to log(n)/n in terms of the best-known upper bounds.

### Proximal and Federated Random Reshuffling

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• 2022
Two new algorithms, Proximal and Federated Random Reshuffing (ProxRR and FedRR), which solve composite convex finitesum minimization problems in which the objective is the sum of a (potentially non-smooth) convex regularizer and an average of n smooth objectives are proposed.

### On the performance of random reshuffling in stochastic learning

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The analysis establishes analytically that random reshuffling outperforms independent sampling by showing that the iterate at the end of each run approaches a smaller neighborhood of size O( μ2) around the minimizer rather than O(μ).

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This paper develops a broad condition on the sequence of examples used by SGD that is sufficient to prove tight convergence rates in both strongly convex and non-convex settings, and proposes two new example-selection approaches using quasi-Monte-Carlo methods.

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