通过函数空间正则化，同时提升模型在下游任务的性能和分布外鲁棒性。

Robust fine-tuning aims to achieve competitive in-distribution (ID) performance while maintaining the out-of-distribution (OOD) robustness of a pre-trained model when transferring it to a downstream task. To remedy this, most robust fine-tuning methods aim to preserve the pretrained weights, features, or logits. However, we find that these methods cannot always improve OOD robustness for different model architectures. This is due to the OOD robustness requiring the model function to produce stable prediction for input information of downstream tasks, while existing methods might serve as a poor proxy for the optimization in the function space. Based on this finding, we propose a novel regularization that constrains the distance of fine-tuning and pre-trained model in the function space with the simulated OOD samples, aiming to preserve the OOD robustness of the pre-trained model. Besides, to further enhance the OOD robustness capability of the fine-tuning model, we introduce an additional consistency regularization to promote stable predictions of perturbed samples. Extensive experiments demonstrate our approach could consistently improve both downstream task ID fine-tuning performance and OOD robustness across a variety of CLIP backbones, outperforming existing regularization-based robust fine-tuning methods.