  Wide applications of differentiable two-player sequential games (e.g., image
generation by GANs) have raised much interest and attention of researchers to
study efficient and fast algorithms. Most of the existing algorithms are
developed based on nice properties of simultaneous games, i.e., convex-concave
payoff functions, but are not applicable in solving sequential games with
different settings. Some conventional gradient descent ascent algorithms
theoretically and numerically fail to find the local Nash equilibrium of the
simultaneous game or the local minimax (i.e., local Stackelberg equilibrium) of
the sequential game. In this paper, we propose the HessianFR, an efficient
Hessian-based Follow-the-Ridge algorithm with theoretical guarantees.
Furthermore, the convergence of the stochastic algorithm and the approximation
of Hessian inverse are exploited to improve algorithm efficiency. A series of
experiments of training generative adversarial networks (GANs) have been
conducted on both synthetic and real-world large-scale image datasets (e.g.
MNIST, CIFAR-10 and CelebA). The experimental results demonstrate that the
proposed HessianFR outperforms baselines in terms of convergence and image
generation quality.
