Random Reshuffling Dominates SGD in Convex Optimization: A Decades-Old Mystery Resolved

Mathematics > Optimization and Control

arXiv:2606.32005v1 (math) [Submitted on 30 Jun 2026]

Title:Random Reshuffling Dominates Stochastic Gradient Descent

Authors:Zijian Liu View a PDF of the paper titled Random Reshuffling Dominates Stochastic Gradient Descent, by Zijian Liu View PDF HTML (experimental)

Abstract:Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$). A particularly popular strategy in $\textsf{Shuffling SGD}$ is Random Reshuffling ($\textsf{RR}$), which has achieved great empirical success across numerous experiments. Despite its strong performance, $\textsf{RR}$ has long been considered a heuristic due to a lack of theoretical support. Over the last decade, people have finally established provable convergence rates for $\textsf{RR}$, thus justifying its observed superiority. However, for smooth convex optimization, two clouds over the convergence theory of $\textsf{RR}$ remain to this day. More precisely, according to the current theory, $\textsf{Shuffling SGD}$ under $\textsf{RR}$ converges only when the stepsize is smaller than a threshold proportional to $1/n$, where $n$ is the number of summands in the objective (or the number of data points). Consequently, the optimally tuned theoretical rate of $\textsf{Shuffling SGD}$ under $\textsf{RR}$ is strictly worse than that of $\textsf{SGD}$ when the number of epochs is smaller than another threshold proportional to $n$. These two restrictions heavily limit the applicability of existing theories and leave a critical mismatch with practice. In this work, for the first time, we prove that $\textsf{RR}$ dominates $\textsf{SGD}$ in smooth convex optimization under any reasonable stepsize after any finite number of epochs, thereby addressing a longstanding open question.

Comments:	COLT 2026
Subjects:	Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
Cite as:	arXiv:2606.32005 [math.OC]
	(or arXiv:2606.32005v1 [math.OC] for this version)
	https://doi.org/10.48550/arXiv.2606.32005 Focus to learn more arXiv-issued DOI via DataCite

Submission history

From: Zijian Liu [view email]
[v1] Tue, 30 Jun 2026 17:38:22 UTC (37 KB)
Full-text links:

Access Paper:

View a PDF of the paper titled Random Reshuffling Dominates Stochastic Gradient Descent, by Zijian Liu
• View PDF
• HTML (experimental)
• TeX Source

view license

Current browse context:

math.OC < prev   |   next >
new | recent | 2026-06 Change to browse by: cs
cs.LG
math
stat
stat.ML

References & Citations

• NASA ADS
• Google Scholar
• Semantic Scholar

export BibTeX citation Loading...

BibTeX formatted citation

× Data provided by:

Bookmark

Bibliographic Tools

Bibliographic and Citation Tools

Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media

Code, Data and Media Associated with this Article

alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos

Demos

Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers

Recommenders and Search Tools

Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?)

• Author
• Venue
• Institution
• Topic

About arXivLabs

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)