# Stochastic Hamiltonian Gradient Methods for Smooth Games

arXiv: https://arxiv.org/abs/2007.04202 · HTML: https://mlsanity.com/paper/2007.04202/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2020.

## Built on by (most cited first, 30 of 32)

- [Faster Single-loop Algorithms for Minimax Optimization without Strong Concavity](https://mlsanity.com/paper/2112.05604/) (2021)
- [Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity](https://mlsanity.com/paper/2107.00052/) (2021)
- [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (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)
- [Stochastic Extragradient: General Analysis and Improved Rates](https://mlsanity.com/paper/2111.08611/) (2021)
- [Solving stochastic weak Minty variational inequalities without increasing batch size](https://mlsanity.com/paper/2302.09029/) (2023)
- [On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averaging](https://mlsanity.com/paper/2107.00464/) (2021)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [AI-SARAH: Adaptive and Implicit Stochastic Recursive Gradient Methods](https://mlsanity.com/paper/2102.09700/) (2021)
- [Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2206.01095/) (2022)
- [Single-Call Stochastic Extragradient Methods for Structured Non-monotone Variational Inequalities: Improved Analysis under Weaker Conditions](https://mlsanity.com/paper/2302.14043/) (2023)
- [Stochastic Polyak Step-sizes and Momentum: Convergence Guarantees and Practical Performance](https://mlsanity.com/paper/2406.04142/) (2024)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
- [Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems](https://mlsanity.com/paper/2112.09579/) (2021)
- [Sharpness-Aware Minimization: General Analysis and Improved Rates](https://mlsanity.com/paper/2503.02225/) (2025)
- [Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations](https://mlsanity.com/paper/2112.13826/) (2021)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements](https://mlsanity.com/paper/2306.16502/) (2023)
- [Stochastic Extragradient with Random Reshuffling: Improved Convergence for Variational Inequalities](https://mlsanity.com/paper/2403.07148/) (2024)
- [Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization](https://mlsanity.com/paper/2206.08573/) (2022)
- [Communication-Efficient Gradient Descent-Accent Methods for Distributed Variational Inequalities: Unified Analysis and Local Updates](https://mlsanity.com/paper/2306.05100/) (2023)
- [Stochastic Smoothed Gradient Descent Ascent for Federated Minimax Optimization](https://mlsanity.com/paper/2311.00944/) (2023)
- [Extragradient Method for (L0, L1)-Lipschitz Root-finding Problems](https://mlsanity.com/paper/2510.22421/) (2025)
- [Stability and Generalization of Differentially Private Minimax Problems](https://mlsanity.com/paper/2204.04858/) (2022)
- [Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold](https://mlsanity.com/paper/2405.13392/) (2024)
- [Towards Sharper Risk Bounds for Minimax Problems](https://mlsanity.com/paper/2410.08497/) (2024)
- [Shuffling the Data, Stretching the Step-size: Sharper Bias in constant step-size SGD](https://mlsanity.com/paper/2604.10373/) (2026)
- [Frictional Q-Learning](https://mlsanity.com/paper/2509.19771/) (2025)

## Abstract (arXiv)

The success of adversarial formulations in machine learning has brought renewed motivation for smooth games. In this work, we focus on the class of stochastic Hamiltonian methods and provide the first convergence guarantees for certain classes of stochastic smooth games. We propose a novel unbiased estimator for the stochastic Hamiltonian gradient descent (SHGD) and highlight its benefits. Using tools from the optimization literature we show that SHGD converges linearly to the neighbourhood of a stationary point. To guarantee convergence to the exact solution, we analyze SHGD with a decreasing step-size and we also present the first stochastic variance reduced Hamiltonian method. Our results provide the first global non-asymptotic last-iterate convergence guarantees for the class of stochastic unconstrained bilinear games and for the more general class of stochastic games that satisfy a "sufficiently bilinear" condition, notably including some non-convex non-concave problems. We supplement our analysis with experiments on stochastic bilinear and sufficiently bilinear games, where our theory is shown to be tight, and on simple adversarial machine learning formulations.