# Learning Theory of the SVRG: Generalization and Convergence Analysis

arXiv: https://arxiv.org/abs/2605.28513 · HTML: https://mlsanity.com/paper/2605.28513/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 49 references that are in the map.
- First posted on arXiv 27 May 2026.

## Builds on (most cited first, 30 of 31)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses](https://mlsanity.com/paper/2006.06914/) (2020)
- [Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1603.06160/) (2016)
- [Katyusha: The First Direct Acceleration of Stochastic Gradient Methods](https://mlsanity.com/paper/1603.05953/) (2016)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [Stability and Generalization of Learning Algorithms that Converge to Global Optima](https://mlsanity.com/paper/1710.08402/) (2017)
- [Data-Dependent Stability of Stochastic Gradient Descent](https://mlsanity.com/paper/1703.01678/) (2017)
- [Generalization Bounds of SGLD for Non-convex Learning: Two Theoretical Viewpoints](https://mlsanity.com/paper/1707.05947/) (2017)
- [Variance-Reduced Methods for Machine Learning](https://mlsanity.com/paper/2010.00892/) (2020)
- [A Proximal Stochastic Gradient Method with Progressive Variance Reduction](https://mlsanity.com/paper/1403.4699/) (2014)
- [A Unified Convergence Analysis for Shuffling-Type Gradient Methods](https://mlsanity.com/paper/2002.08246/) (2020)
- [Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods](https://mlsanity.com/paper/1712.09677/) (2017)
- [Sharp Analysis for Nonconvex SGD Escaping from Saddle Points](https://mlsanity.com/paper/1902.00247/) (2019)
- [Algorithmic stability and hypothesis complexity](https://mlsanity.com/paper/1702.08712/) (2017)
- [On the Optimization and Generalization of Multi-head Attention](https://mlsanity.com/paper/2310.12680/) (2023)
- [Three-Way Trade-Off in Multi-Objective Learning: Optimization, Generalization and Conflict-Avoidance](https://mlsanity.com/paper/2305.20057/) (2023)
- [Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning](https://mlsanity.com/paper/1402.4419/) (2014)
- [Stability vs Implicit Bias of Gradient Methods on Separable Data and Beyond](https://mlsanity.com/paper/2202.13441/) (2022)
- [Understanding Generalization of Federated Learning via Stability: Heterogeneity Matters](https://mlsanity.com/paper/2306.03824/) (2023)
- [Stability & Generalisation of Gradient Descent for Shallow Neural Networks without the Neural Tangent Kernel](https://mlsanity.com/paper/2107.12723/) (2021)
- [Generalization and Stability of Interpolating Neural Networks with Minimal Width](https://mlsanity.com/paper/2302.09235/) (2023)
- [Understanding How Consistency Works in Federated Learning via Stage-wise Relaxed Initialization](https://mlsanity.com/paper/2306.05706/) (2023)
- [Convergence of Stochastic Proximal Gradient Algorithm](https://mlsanity.com/paper/1403.5074/) (2014)
- [Minimax Estimation for Personalized Federated Learning: An Alternative between FedAvg and Local Training?](https://mlsanity.com/paper/2103.01901/) (2021)
- [Stochastic Dual Coordinate Ascent with Adaptive Probabilities](https://mlsanity.com/paper/1502.08053/) (2015)
- [Sharper Analysis for Minibatch Stochastic Proximal Point Methods: Stability, Smoothness, and Deviation](https://mlsanity.com/paper/2301.03125/) (2023)

## Abstract (arXiv)

Variance reduction (VR) methods employ stochastic gradients with decreasing variance, and they have been widely applied to solve large-scale optimization problems in machine learning because of their efficiency. Existing theoretical studies of VR methods are mainly focused on the convergence analysis, leaving the generalization behavior largely unexplored. In this paper, we bridge this gap by developing the first non-vacuous generalization analysis of the representative VR method: Stochastic Variance Reduced Gradient (SVRG), through the lens of algorithmic stability. In particular, we establish sharp stability bounds of the SVRG in both convex and strongly convex settings by exploiting its algorithmic structure. The obtained bounds are data-dependent, because the training errors are incorporated along the trajectory. Our analysis clarifies the interplay between optimization and generalization, leading to optimal excess population risk bounds in both settings. Our approach differs substantially from existing analyses of stochastic algorithms in the sense that we decompose the SVRG update as an SGD-like step plus a zero-mean correction term and then introduce novel Lyapunov functions to absorb the additional gradient terms induced by the reference points. Our analytical framework can be generalized to other VR methods, and we demonstrate the generalization by the well-known Stochastic Average Gradient Accelerated (SAGA) method.