Collaborative Learning with Shared Linear Representations: Statistical Rates and Optimal Algorithms

Collaborative learning enables multiple clients to learn shared feature representations across local data distributions, with the goal of improving model performance and reducing overall sample complexity. While empirical evidence shows the success of collaborative learning, a theoretical understanding of the optimal statistical rate remains lacking, even in linear settings. In this paper, we identify the optimal statistical rate when clients share a common low-dimensional linear representation. Specifically, we design a spectral estimator with local averaging that approximates the optimal solution to the least squares problem. We establish a minimax lower bound to demonstrate that our estimator achieves the optimal error rate. Notably, the optimal rate reveals two distinct phases. In typical cases, our rate matches the standard rate based on the parameter counting of the linear representation. However, a statistical penalty arises in collaborative learning when there are too many clients or when local datasets are relatively small. Furthermore, our results, unlike existing ones, show that, at a system level, collaboration always reduces overall sample complexity compared to independent client learning. In addition, at an individual level, we provide a more precise characterization of when collaboration benefits a client in transfer learning and private fine-tuning.