A Robust Framework for Analyzing Gradient-Based Dynamics in Bilinear Games
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In this work, we establish a frequency-domain framework for analyzing gradient-based algorithms in linear minimax optimization problems; specifically, our approach is based on the Z-transform, a powerful tool applied in Control Theory and Signal Processing in order to characterize linear discrete-time systems. We employ our framework to obtain the first tight analysis of stability of Optimistic Gradient Descent/Ascent (OGDA), a natural variant of Gradient Descent/Ascent that was shown to exhibit last-iterate convergence in bilinear games by Daskalakis et al. <cit.>. Importantly, our analysis is considerably simpler and more concise than the existing ones. Moreover, building on the intuition of OGDA, we consider a general family of gradient-based algorithms that augment the memory of the optimization through multiple historical steps. We reduce the convergence – to a saddle-point – of the dynamics in bilinear games to the stability of a polynomial, for which efficient algorithmic schemes are well-established. As an immediate corollary, we obtain a broad class of algorithms – that contains OGDA as a special case – with a last-iterate convergence guarantee to the space of Nash equilibria of the game.
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