  To address the communication burden issues associated with federated learning
(FL), decentralized federated learning (DFL) discards the central server and
establishes a decentralized communication network, where each client
communicates only with neighboring clients. However, existing DFL methods still
suffer from two major challenges: local inconsistency and local heterogeneous
overfitting, which have not been fundamentally addressed by existing DFL
methods. To tackle these issues, we propose novel DFL algorithms, DFedADMM and
its enhanced version DFedADMM-SAM, to enhance the performance of DFL. The
DFedADMM algorithm employs primal-dual optimization (ADMM) by utilizing dual
variables to control the model inconsistency raised from the decentralized
heterogeneous data distributions. The DFedADMM-SAM algorithm further improves
on DFedADMM by employing a Sharpness-Aware Minimization (SAM) optimizer, which
uses gradient perturbations to generate locally flat models and searches for
models with uniformly low loss values to mitigate local heterogeneous
overfitting. Theoretically, we derive convergence rates of $\small
\mathcal{O}\Big(\frac{1}{\sqrt{KT}}+\frac{1}{KT(1-\psi)^2}\Big)$ and $\small
\mathcal{O}\Big(\frac{1}{\sqrt{KT}}+\frac{1}{KT(1-\psi)^2}+
\frac{1}{T^{3/2}K^{1/2}}\Big)$ in the non-convex setting for DFedADMM and
DFedADMM-SAM, respectively, where $1 - \psi$ represents the spectral gap of the
gossip matrix. Empirically, extensive experiments on MNIST, CIFAR10 and
CIFAR100 datesets demonstrate that our algorithms exhibit superior performance
in terms of both generalization and convergence speed compared to existing
state-of-the-art (SOTA) optimizers in DFL.
