While classic work in convex-concave min-max optimization relies on
average-iterate convergence results, the emergence of nonconvex applications
such as training Generative Adversarial Networks has led to renewed interest in
last-iterate convergence guarantees. Proving last-iterate convergence is
challenging because many natural algorithms, such as Simultaneous Gradient
Descent/Ascent, provably diverge or cycle even in simple convex-concave min-max
settings, and previous work on global last-iterate convergence rates has been
limited to the bilinear and convex-strongly concave settings. In this work, we
show that the Hamiltonian Gradient Descent (HGD) algorithm achieves linear
convergence in a variety of more general settings, including convex-concave
problems that satisfy a "sufficiently bilinear" condition. We also prove
similar convergence rates for the Consensus Optimization (CO) algorithm of
[MNG17] for some parameter settings of CO.

该研究论文阐述了针对非凸函数最优化问题中的后向迭代收敛的挑战性，介绍了哈密顿梯度下降算法以及协作优化算法，并证明了这些算法在某些情况下表现出线性收敛性。