Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.

翻译：追踪时变变分不等式的解是博弈论、优化和机器学习应用中的一个重要问题。现有研究主要关注时变博弈或时变优化问题。对于强凸优化问题或强单调博弈，这些结果在假设时变问题的变化受限（即具有次线性解路径的问题）的前提下提供了追踪保证。本工作从两个方面拓展了现有结果：首先，我们为以下两种情况提供了追踪界：（1）具有次线性解路径但不一定单调函数的变分不等式；（2）不一定具有次线性解路径长度的周期性时变变分不等式。我们的第二个主要贡献是对周期性时变变分不等式的离散动力系统的收敛行为和轨迹进行了广泛研究。我们证明这些系统可能表现出可证明的混沌行为，也可能收敛到解。最后，我们通过实验验证了理论结果。