Variational inequalities are a broad and flexible class of problems that includes minimization, saddle point, fixed point problems as special cases. Therefore, variational inequalities are used in a variety of applications ranging from equilibrium search to adversarial learning. Today's realities with the increasing size of data and models demand parallel and distributed computing for real-world machine learning problems, most of which can be represented as variational inequalities. Meanwhile, most distributed approaches has a significant bottleneck - the cost of communications. The three main techniques to reduce both the total number of communication rounds and the cost of one such round are the use of similarity of local functions, compression of transmitted information and local updates. In this paper, we combine all these approaches. Such a triple synergy did not exist before for variational inequalities and saddle problems, nor even for minimization problems. The methods presented in this paper have the best theoretical guarantees of communication complexity and are significantly ahead of other methods for distributed variational inequalities. The theoretical results are confirmed by adversarial learning experiments on synthetic and real datasets.

翻译：变分不等式是一类广泛且灵活的问题，涵盖最小化、鞍点和不动点问题作为特例。因此，变分不等式被广泛应用于从均衡搜索到对抗学习等多种场景中。随着数据与模型规模的日益增长，现实中的机器学习问题（其中大部分可建模为变分不等式）迫切需要并行与分布式计算。然而，大多数分布式方法面临一个显著瓶颈——通信成本。降低总通信轮次与单轮通信成本的三项主要技术包括：利用局部函数的相似性、压缩传输信息以及执行局部更新。本文首次将所有这些方法相结合。这种三重协同此前在变分不等式与鞍问题中不存在，即使在最小化问题中也未曾实现。本文提出的方法具有最优的理论通信复杂度保证，且显著领先于其他分布式变分不等式方法。理论结果通过在合成数据集与真实数据集上的对抗学习实验得到验证。