  While large language models (LLMs) are empowered with broad knowledge, their
task-specific performance is often suboptimal. It necessitates fine-tuning LLMs
with task-specific data, but such data may be inaccessible due to privacy
concerns. In this paper, we propose a novel approach to enhance LLMs with
smaller language models (SLMs) that are trained on clients using their private
task-specific data. To enable mutual enhancement between LLMs and SLMs, we
propose CrossLM, where the SLMs promote the LLM to generate task-specific
high-quality data, and both the LLM and SLMs are enhanced with the generated
data. We evaluate CrossLM using publicly accessible language models across a
range of benchmark tasks. The results demonstrate that CrossLM significantly
enhances the task-specific performance of SLMs on clients and the LLM on the
cloud server simultaneously while preserving the LLM's generalization
capability.
