Random Reshuffling: Simple Analysis with Vast Improvements (2006.05988v3)

Abstract: 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.

• The paper introduces a novel theoretical framework that improves the convergence rate of Random Reshuffling from k² to k or √k.
• It demonstrates that RR outperforms SGD in strongly convex and non-convex scenarios by revealing distinctive variance properties.
• The analysis extends to Incremental Gradient methods, offering convergence bounds that align with empirical findings in large-scale optimization.

Analysis of Random Reshuffling: Simple Analysis with Vast Improvements

This paper presents an insightful analysis of the Random Reshuffling (RR) algorithm and introduces significant improvements to its theoretical understanding. RR is a method utilized to minimize finite-sum functions through iterative gradient descent steps combined with data reshuffling. It is traditionally compared against Stochastic Gradient Descent (SGD) and is acclaimed for its efficiency in both convex and non-convex optimization landscapes.

Overview of Results

The primary focus of the paper is on establishing enhanced convergence rates for RR without relying on the constraints often imposed in previous works, such as small step sizes, bounded gradients, and large epochs. By developing a novel proof technique, the authors enhance the theoretical understanding of RR, showing an improved dependence on the condition number from $k^2$ to $k$, or alternatively to $\sqrt{k}$. They further elucidate that RR exhibits a distinctive variance type, contributing to its favorable performance compared to SGD.

• Strong Convexity: The analysis confirms that for strongly convex functions, RR outperforms SGD due to better dependency on the problem's condition number and a new notion of variance particular to RR. These results are consistently near the lower bounds known for RR and its variant, the Shuffle-Once (SO) algorithm.
• Non-Strongly Convex and Non-Convex Cases: The findings also extend to non-strongly convex and non-convex objectives, where RR demonstrates superior convergence properties compared with existing literature.

The authors derive key results for the Incremental Gradient (IG) algorithm as a natural consequence of their RR analysis, despite IG not utilizing data reshuffling. These findings for IG are particularly notable as they provide convergence bounds consistent with industry observations even when applying non-random data ordering.

Implications and Theoretical Advancements

This research underscores the importance of data permutation/shuffling in optimization tasks, by providing theoretic underpinnings to a heuristic prevalent yet insufficiently understood until now. The theoretical breakthroughs achieved concerning RR’s variance explain its practical performance superiority, offering an analytical framework that supports empirical observations in machine learning.

• Practical Significance: Theoretical validation of using minimal shuffling, as embodied by SO, aligns with common practices in large-scale machine learning applications, suggesting further exploration and potential adaptation in automated optimization schedules.
• Future Directions: These insights may inspire subsequent studies to explore adaptive reshuffling strategies or modify RR and SO for specific applications like deep learning, where data order significantly impacts convergence speed and accuracy.

Conclusions

The paper contributes a substantial theoretical foundation to the understanding of data reshuffling paradigms, specifically targeting RR. It provides a powerful methodological alternative not only to SGD but also to other algorithms where data order considerations are crucial. While the practical extensions of these theoretical insights remain to be fully realized, the implications are promising for developing more robust and efficient training algorithms in the future of AI research. The presented results invite further exploration into leveraging reshuffling strategies to optimize performance across various machine learning frameworks, thereby broadening the application horizon of RR beyond its current scope.