Average-case Acceleration for Bilinear Games and Normal Matrices

TL;DR

This paper develops average-case optimal first-order methods for smooth games, especially bilinear and normal matrices, demonstrating speed-ups over worst-case algorithms and providing explicit solutions for certain matrix classes.

Contribution

It introduces the first average-case optimal methods for a subset of smooth games, including explicit solutions for normal matrices and speed-up results for matrices with eigenvalues in a disk.

Findings

Optimal method for zero-sum bilinear games matches Hamiltonian minimization

Explicit average-case optimal method derived for normal matrices

Provable speed-up over worst-case algorithms for matrices with eigenvalues in a disk

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.