  The rapid development of machine learning and deep learning has introduced
increasingly complex optimization challenges that must be addressed. Indeed,
training modern, advanced models has become difficult to implement without
leveraging multiple computing nodes in a distributed environment. Distributed
optimization is also fundamental to emerging fields such as federated learning.
Specifically, there is a need to organize the training process to minimize the
time lost due to communication. A widely used and extensively researched
technique to mitigate the communication bottleneck involves performing local
training before communication. This approach is the focus of our paper.
Concurrently, adaptive methods that incorporate scaling, notably led by Adam,
have gained significant popularity in recent years. Therefore, this paper aims
to merge the local training technique with the adaptive approach to develop
efficient distributed learning methods. We consider the classical Local SGD
method and enhance it with a scaling feature. A crucial aspect is that the
scaling is described generically, allowing us to analyze various approaches,
including Adam, RMSProp, and OASIS, in a unified manner. In addition to
theoretical analysis, we validate the performance of our methods in practice by
training a neural network.
