Average-case Acceleration for Bilinear Games and Normal Matrices

Carles Domingo-Enrich, Fabian Pedregosa, Damien Scieur

Keywords: Smooth games, First-order Methods, Acceleration, Bilinear games, Average-case Analysis, Orthogonal Polynomials

Abstract:

Advances in generative modeling and adversarial learning have given rise to renewed interest in smooth games. However, the absence of symmetry in the matrix of second derivatives poses challenges that are not present in the classical minimization framework. While a rich theory of average-case analysis has been developed for minimization problems, little is known in the context of smooth games. In this work we take a first step towards closing this gap by developing average-case optimal first-order methods for a subset of smooth games.
We make the following three main contributions. First, we show that for zero-sum bilinear games the average-case optimal method is the optimal method for the minimization of the Hamiltonian. Second, we provide an explicit expression for the optimal method corresponding to normal matrices, potentially non-symmetric. Finally, we specialize it to matrices with eigenvalues located in a disk and show a provable speed-up compared to worst-case optimal algorithms. We illustrate our findings through benchmarks with a varying degree of mismatch with our assumptions.

One-sentence Summary: We extend the framework of average-case optimal first-order methods to problems with non-symmetric matrices, which naturally arise in equilibrium finding for games.

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