Training foundation models on extensive datasets and then finetuning them on specific tasks has emerged as the mainstream approach in artificial intelligence. However, the model robustness, which is a critical aspect for safety, is often optimized for each specific task rather than at the pretraining stage. In this paper, we propose a method for pretraining certifiably robust models that can be readily finetuned for adaptation to a particular task. A key challenge is dealing with the compromise between semantic learning and robustness. We address this with a simple yet highly effective strategy based on significantly broadening the pretraining data distribution, which is shown to greatly benefit finetuning for downstream tasks. Through pretraining on a mixture of clean and various noisy images, we find that surprisingly strong certified accuracy can be achieved even when finetuning on only clean images. Furthermore, this strategy requires just a single model to deal with various noise levels, thus substantially reducing computational costs in relation to previous works that employ multiple models. Despite using just one model, our method can still yield results that are on par with, or even superior to, existing multi-model methods.

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