# Stochastic Variance Reduction Methods for Saddle-Point Problems

arXiv: https://arxiv.org/abs/1605.06398 · HTML: https://mlsanity.com/paper/1605.06398/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2016.

## Built on by (most cited first, 30 of 60)

- [AUC Maximization in the Era of Big Data and AI: A Survey](https://mlsanity.com/paper/2203.15046/) (2022)
- [Smooth Tchebycheff Scalarization for Multi-Objective Optimization](https://mlsanity.com/paper/2402.19078/) (2024)
- [Stochastic Variance Reduction for Variational Inequality Methods](https://mlsanity.com/paper/2102.08352/) (2021)
- [The Complexity of Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2103.15888/) (2021)
- [Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity](https://mlsanity.com/paper/2107.00052/) (2021)
- [Stochastic Gradient Descent-Ascent: Unified Theory and New Efficient Methods](https://mlsanity.com/paper/2202.07262/) (2022)
- [Stability and Generalization of Stochastic Gradient Methods for Minimax Problems](https://mlsanity.com/paper/2105.03793/) (2021)
- [Lower Complexity Bounds of Finite-Sum Optimization Problems: The Results and Construction](https://mlsanity.com/paper/2103.08280/) (2021)
- [Bring Your Own Algorithm for Optimal Differentially Private Stochastic Minimax Optimization](https://mlsanity.com/paper/2206.00363/) (2022)
- [Efficient Algorithms for Federated Saddle Point Optimization](https://mlsanity.com/paper/2102.06333/) (2021)
- [Fast Distributionally Robust Learning with Variance Reduced Min-Max Optimization](https://mlsanity.com/paper/2104.13326/) (2021)
- [Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions](https://mlsanity.com/paper/2203.09436/) (2022)
- [Distributed Learning Systems with First-order Methods](https://mlsanity.com/paper/2104.05245/) (2021)
- [Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization](https://mlsanity.com/paper/2106.01761/) (2021)
- [Distributed Methods with Compressed Communication for Solving Variational Inequalities, with Theoretical Guarantees](https://mlsanity.com/paper/2110.03313/) (2021)
- [Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods](https://mlsanity.com/paper/2202.04640/) (2022)
- [Optimal Algorithms for Decentralized Stochastic Variational Inequalities](https://mlsanity.com/paper/2202.02771/) (2022)
- [First Order Methods with Markovian Noise: from Acceleration to Variational Inequalities](https://mlsanity.com/paper/2305.15938/) (2023)
- [Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization](https://mlsanity.com/paper/2201.07427/) (2022)
- [Stochastic Halpern iteration in normed spaces and applications to reinforcement learning](https://mlsanity.com/paper/2403.12338/) (2024)
- [Differentially Private Algorithms for the Stochastic Saddle Point Problem with Optimal Rates for the Strong Gap](https://mlsanity.com/paper/2302.12909/) (2023)
- [Efficient Algorithms for Empirical Group Distributionally Robust Optimization and Beyond](https://mlsanity.com/paper/2403.03562/) (2024)
- [Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusions](https://mlsanity.com/paper/2310.02987/) (2023)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization](https://mlsanity.com/paper/2205.14278/) (2022)
- [DIPPA: An improved Method for Bilinear Saddle Point Problems](https://mlsanity.com/paper/2103.08270/) (2021)
- [Adaptive, Doubly Optimal No-Regret Learning in Strongly Monotone and Exp-Concave Games with Gradient Feedback](https://mlsanity.com/paper/2310.14085/) (2023)
- [Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems](https://mlsanity.com/paper/2212.07844/) (2022)
- [Faster Stochastic Algorithms for Minimax Optimization under Polyak--Lojasiewicz Conditions](https://mlsanity.com/paper/2307.15868/) (2023)
- [Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problems](https://mlsanity.com/paper/2406.02413/) (2024)

## Abstract (arXiv)

We consider convex-concave saddle-point problems where the objective functions may be split in many components, and extend recent stochastic variance reduction methods (such as SVRG or SAGA) to provide the first large-scale linearly convergent algorithms for this class of problems which is common in machine learning. While the algorithmic extension is straightforward, it comes with challenges and opportunities: (a) the convex minimization analysis does not apply and we use the notion of monotone operators to prove convergence, showing in particular that the same algorithm applies to a larger class of problems, such as variational inequalities, (b) there are two notions of splits, in terms of functions, or in terms of partial derivatives, (c) the split does need to be done with convex-concave terms, (d) non-uniform sampling is key to an efficient algorithm, both in theory and practice, and (e) these incremental algorithms can be easily accelerated using a simple extension of the "catalyst" framework, leading to an algorithm which is always superior to accelerated batch algorithms.