  Variational inequalities in general and saddle point problems in particular
are increasingly relevant in machine learning applications, including
adversarial learning, GANs, transport and robust optimization. With increasing
data and problem sizes necessary to train high performing models across various
applications, we need to rely on parallel and distributed computing. However,
in distributed training, communication among the compute nodes is a key
bottleneck during training, and this problem is exacerbated for high
dimensional and over-parameterized models. Due to these considerations, it is
important to equip existing methods with strategies that would allow to reduce
the volume of transmitted information during training while obtaining a model
of comparable quality. In this paper, we present the first theoretically
grounded distributed methods for solving variational inequalities and saddle
point problems using compressed communication: MASHA1 and MASHA2. Our theory
and methods allow for the use of both unbiased (such as Rand$k$; MASHA1) and
contractive (such as Top$k$; MASHA2) compressors. New algorithms support
bidirectional compressions, and also can be modified for stochastic setting
with batches and for federated learning with partial participation of clients.
We empirically validated our conclusions using two experimental setups: a
standard bilinear min-max problem, and large-scale distributed adversarial
training of transformers.
