Beyond Exponential Graph: Communication-Efficient Topologies for Decentralized Learning via Finite-time Convergence

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-$\left(k+1\right)$ Graph. Unlike the existing topologies, the Base-$\left(k+1\right)$ 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-$\left(k+1\right)$ 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-$\left(k+1\right)$ Graph enables various decentralized learning methods to achieve higher accuracy with better communication efficiency than the existing topologies. Our code is available at https://github.com/yukiTakezawa/BaseGraph.