  Decentralized learning has recently been attracting increasing attention for
its applications in parallel computation and privacy preservation. Many recent
studies stated that the underlying network topology with a faster consensus
rate (a.k.a. spectral gap) leads to a better convergence rate and accuracy for
decentralized learning. However, a topology with a fast consensus rate, e.g.,
the exponential graph, generally has a large maximum degree, which incurs
significant communication costs. Thus, seeking topologies with both a fast
consensus rate and small maximum degree is important. In this study, we propose
a novel topology combining both a fast consensus rate and small maximum degree
called the Base-$(k + 1)$ Graph. Unlike the existing topologies, the Base-$(k +
1)$ Graph enables all nodes to reach the exact consensus after a finite number
of iterations for any number of nodes and maximum degree k. Thanks to this
favorable property, the Base-$(k + 1)$ Graph endows Decentralized SGD (DSGD)
with both a faster convergence rate and more communication efficiency than the
exponential graph. We conducted experiments with various topologies,
demonstrating that the Base-$(k + 1)$ Graph enables various decentralized
learning methods to achieve higher accuracy with better communication
efficiency than the existing topologies.
