Xiaochun Niu

Zhenyu Sun, Xiaochun Niu, Ermin Wei

Generalization performance is a key metric in evaluating machine learning models when applied to real-world applications. Good generalization indicates the model can predict unseen data correctly when trained under a limited number of data. Federated learning (FL), which has emerged as a popular distributed learning framework, allows multiple devices or clients to train a shared model without violating privacy requirements. While the existing literature has studied extensively the generalization performances of centralized machine learning algorithms, similar analysis in the federated settings is either absent or with very restrictive assumptions on the loss functions. In this paper, we aim to analyze the generalization performances of federated learning by means of algorithmic stability, which measures the change of the output model of an algorithm when perturbing one data point. Three widely-used algorithms are studied, including FedAvg, SCAFFOLD, and FedProx, under convex and non-convex loss functions. Our analysis shows that the generalization performances of models trained by these three algorithms are closely related to the heterogeneity of clients' datasets as well as the convergence behaviors of the algorithms. Particularly, in the i.i.d. setting, our results recover the classical results of stochastic gradient descent (SGD).

Xiaochun Niu, Lili Su, Jiaming Xu et al.

Collaborative learning through latent shared feature representations enables heterogeneous clients to train personalized models with improved performance and reduced sample complexity. Despite empirical success and extensive study, the theoretical understanding of such methods remains incomplete, even for representations restricted to low-dimensional linear subspaces. In this work, we establish new upper and lower bounds on the statistical error in learning low-dimensional shared representations across clients. Our analysis captures both statistical heterogeneity (including covariate and concept shifts) and variation in local dataset sizes, aspects often overlooked in prior work. We further extend these results to nonlinear models including logistic regression and one-hidden-layer ReLU networks. Specifically, we design a spectral estimator that leverages independent replicas of local averages to approximate the non-convex least-squares solution and derive a nearly matching minimax lower bound. Our estimator achieves the optimal statistical rate when the shared representation is well covered across clients -- i.e., when no direction is severely underrepresented. Our results reveal two distinct phases of the optimal rate: a standard parameter-counting regime and a penalized regime when the number of clients is large or local datasets are small. These findings precisely characterize when collaboration benefits the overall system or individual clients in transfer learning and private fine-tuning.