Stochastic Hamiltonian Gradient Methods for Smooth Games

Abstract

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.

ICML 2020 - Proceedings of the 37th International Conference on Machine Learning

Cited by in corpus (11)

• Explore Aggressively, Update Conservatively: Stochastic Extragradient Methods with Variable Stepsize Scaling
• Stability and Generalization of Stochastic Gradient Methods for Minimax Problems
• Solving Structured Hierarchical Games Using Differential Backward Induction
• Hybrid Variance-Reduced SGD Algorithms For Nonconvex-Concave Minimax Problems
• Extragradient Method: $O(1/K)$ Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity
• AI-SARAH: Adaptive and Implicit Stochastic Recursive Gradient Methods
• Stochastic Extragradient: General Analysis and Improved Rates
• Stochastic Gradient Descent-Ascent and Consensus Optimization for Smooth Games: Convergence Analysis under Expected Co-coercivity
• Minimax Problems with Coupled Linear Constraints: Computational Complexity, Duality and Solution Methods
• Average-case Acceleration for Bilinear Games and Normal Matrices
• Complex Momentum for Optimization in Games