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Extragradient with Positive Momentum is Optimal for Games with Cross-Shaped Jacobian Spectrum

by   Junhyung Lyle Kim, et al.

The extragradient method has recently gained increasing attention, due to its convergence behavior on smooth games. In n-player differentiable games, the eigenvalues of the Jacobian of the vector field are distributed on the complex plane, exhibiting more convoluted dynamics compared to classical (i.e., single player) minimization. In this work, we take a polynomial-based analysis of the extragradient with momentum for optimizing games with cross-shaped Jacobian spectrum on the complex plane. We show two results. First, based on the hyperparameter setup, the extragradient with momentum exhibits three different modes of convergence: when the eigenvalues are distributed i) on the real line, ii) both on the real line along with complex conjugates, and iii) only as complex conjugates. Then, we focus on the case ii), i.e., when the eigenvalues of the Jacobian have cross-shaped structure, as observed in training generative adversarial networks. For this problem class, we derive the optimal hyperparameters of the momentum extragradient method, and show that it achieves an accelerated convergence rate.


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