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Federated Learning (FL) is emerging as a popular, promising decentralized learning framework that enables collaborative training among clients, with no need to share private data between them or to a centralized server. However, considering many edge clients do not have sufficient computing, memory, or communication capabilities, federated learning of large models still faces significant bottlenecks. To keep such weak but crucial clients in the loop, prior works either consider a heterogeneous-client setting where clients train models with different sizes; or offload training to the server. However, the heterogeneous-client setting requires some clients to train full model, which is not aligned with the resource-constrained setting; while the latter ones break privacy promises in FL when sharing intermediate representations or labels with the server. To overcome these limitations, in this work, we formulate a realistic, but much less explored, cross-device FL setting in which no client can train a full large model nor is willing to share any intermediate information with the remote server. Under such a formulation, we develop a principal sub-model (PriSM) training methodology to collaboratively train a full large model, while assigning each client a small sub-model that is a probabilistic low-rank approximation to the full server model. When creating sub-models, PriSM first performs a principal kernel analysis in the orthogonal kernel space to obtain importance of each kernel. Then, PriSM adopts a novel importance-aware sampling process to select a subset of kernels (i.e., a kernel with high importance is assigned with a higher sampling probability). This sampling process ensures each sub-model is still a low-rank approximation to the full model, while all sub-models together achieve nearly full coverage on the principal kernels.