Understanding Convergence and Generalization in Federated Learning through Feature Learning Theory

Federated Learning (FL) has attracted significant attention as an efficient privacypreserving approach to distributed learning across multiple clients. Despite extensive empirical research and practical applications, a systematic way to theoretically understand the convergence and generalization properties in FL remains limited. This work aims to establish a unified theoretical foundation for understanding FL through feature learning theory. We focus on a scenario where each client employs a two-layer convolutional neural network (CNN) for local training on their own data. Many existing works analyze the convergence of Federated Averaging (FedAvg) under lazy training with linearizing assumptions in weight space. In contrast, our approach tracks the trajectory of signal learning and noise memorization in FL, eliminating the need for these assumptions. We further show that FedAvg can achieve near-zero test error by effectively increasing signal-tonoise ratio (SNR) in feature learning, while local training without communication achieves a large constant test error. This finding highlights the benefits of communication for generalization in FL. Moreover, our theoretical results suggest that a weighted FedAvg method, based on the similarity of input features across clients, can effectively tackle data heterogeneity issues in FL. Experimental results on both synthetic and real-world datasets verify our theoretical conclusions and emphasize the effectiveness of the weighted FedAvg approach. * Corresponding author 2 )), with σ p being the strength of noise. Note that C c=1 µ (c) µ (c) ⊤ /∥µ (c) ∥ 2 2 is introduced to ensure noise vector orthogonal to signal