Stochastic Hamiltonian Gradient Methods for Smooth Games
This work addresses convergence challenges in adversarial machine learning formulations, offering incremental improvements for stochastic game optimization.
The authors tackled the problem of convergence in stochastic smooth games by proposing a novel unbiased estimator for stochastic Hamiltonian gradient descent (SHGD), showing it converges linearly to a neighborhood of a stationary point and providing the first global non-asymptotic last-iterate guarantees for classes like stochastic unconstrained bilinear games.
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.