Federated learning allows edge devices to collaboratively train a shared model without sharing data, keeping data on the device. FedMA is a new algorithm for federated learning of modern neural networks like CNNs and LSTMs. It matches and averages hidden elements (e.g., channels in CNNs, hidden states in LSTMs) with similar feature extraction signatures to improve performance and reduce communication burden. FedMA outperforms existing federated learning methods on real-world datasets and reduces communication costs. It addresses the issue of coordinate-wise averaging in FedAvg, which can negatively affect model performance and increase communication load. FedMA uses Bayesian nonparametric methods to adapt to data heterogeneity and is more efficient than FedAvg. It is applied to CNNs and LSTMs, and it performs well on tasks like MNIST, CIFAR-10, and Shakespeare datasets. FedMA also helps mitigate data biases by learning meaningful feature relationships across clients. Experiments show that FedMA outperforms FedAvg and FedProx in terms of convergence rate and communication efficiency. It is particularly effective in heterogeneous settings and can handle large-scale data with varying distributions. FedMA is a layer-wise algorithm that first matches and averages weights of the first layer, then freezes these layers while training subsequent layers. This approach reduces communication rounds and improves performance. FedMA is also data-efficient and can handle increasing numbers of clients. It is interpretable, as it matches filters and averages them to extract meaningful features. FedMA is a promising approach for federated learning, especially in scenarios with data heterogeneity and biases.Federated learning allows edge devices to collaboratively train a shared model without sharing data, keeping data on the device. FedMA is a new algorithm for federated learning of modern neural networks like CNNs and LSTMs. It matches and averages hidden elements (e.g., channels in CNNs, hidden states in LSTMs) with similar feature extraction signatures to improve performance and reduce communication burden. FedMA outperforms existing federated learning methods on real-world datasets and reduces communication costs. It addresses the issue of coordinate-wise averaging in FedAvg, which can negatively affect model performance and increase communication load. FedMA uses Bayesian nonparametric methods to adapt to data heterogeneity and is more efficient than FedAvg. It is applied to CNNs and LSTMs, and it performs well on tasks like MNIST, CIFAR-10, and Shakespeare datasets. FedMA also helps mitigate data biases by learning meaningful feature relationships across clients. Experiments show that FedMA outperforms FedAvg and FedProx in terms of convergence rate and communication efficiency. It is particularly effective in heterogeneous settings and can handle large-scale data with varying distributions. FedMA is a layer-wise algorithm that first matches and averages weights of the first layer, then freezes these layers while training subsequent layers. This approach reduces communication rounds and improves performance. FedMA is also data-efficient and can handle increasing numbers of clients. It is interpretable, as it matches filters and averages them to extract meaningful features. FedMA is a promising approach for federated learning, especially in scenarios with data heterogeneity and biases.