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Advances in generative modeling and adversarial learning have given rise torenewed interest in smooth games. However, the absence of symmetry in thematrix of second derivatives poses challenges that are not present in theclassical minimization framework. While a rich theory of average-case analysishas been developed for minimization problems, little is known in the context ofsmooth games. In this work we take a first step towards closing this gap bydeveloping average-case optimal first-order methods for a subset of smoothgames. We make the following three main contributions. First, we show that forzero-sum bilinear games the average-case optimal method is the optimal methodfor the minimization of the Hamiltonian. Second, we provide an explicitexpression for the optimal method corresponding to normal matrices, potentiallynon-symmetric. Finally, we specialize it to matrices with eigenvalues locatedin a disk and show a provable speed-up compared to worst-case optimalalgorithms. We illustrate our findings through benchmarks with a varying degreeof mismatch with our assumptions.

Average-case Acceleration for Bilinear Games and Normal Matrices