Large web-scale datasets have driven the rapid advancement of pre-trained language models (PLMs), but unauthorized data usage has raised serious copyright concerns. Existing dataset ownership verification (DOV) methods typically assume that watermarks remain stable during inference; however, this assumption often fails under natural noise and adversary-crafted perturbations. We propose the first certified dataset ownership verification method for PLMs under a gray-box setting (i.e., the defender can only query the suspicious model but is aware of its input representation module), based on dual-space smoothing (i.e., DSSmoothing). To address the challenges of text discreteness and semantic sensitivity, DSSmoothing introduces continuous perturbations in the embedding space to capture semantic robustness and applies controlled token reordering in the permutation space to capture sequential robustness. DSSmoothing consists of two stages: in the first stage, triggers are collaboratively embedded in both spaces to generate norm-constrained and robust watermarked datasets; in the second stage, randomized smoothing is applied in both spaces during verification to compute the watermark robustness (WR) of suspicious models and statistically compare it with the principal probability (PP) values of a set of benign models. Theoretically, DSSmoothing provides provable robustness guarantees for dataset ownership verification by ensuring that WR consistently exceeds PP under bounded dual-space perturbations. Extensive experiments on multiple representative web datasets demonstrate that DSSmoothing achieves stable and reliable verification performance and exhibits robustness against potential adaptive attacks. Our code is available at https://github.com/NcepuQiaoTing/DSSmoothing.

Deep neural networks (DNNs) are vulnerable to backdoor attacks, where an attacker manipulates a small portion of the training data to implant hidden backdoors into the model. The compromised model behaves normally on clean samples but misclassifies backdoored samples into the attacker-specified target class, posing a significant threat to real-world DNN applications. Currently, several empirical defense methods have been proposed to mitigate backdoor attacks, but they are often bypassed by more advanced backdoor techniques. In contrast, certified defenses based on randomized smoothing have shown promise by adding random noise to training and testing samples to counteract backdoor attacks. In this paper, we reveal that existing randomized smoothing defenses implicitly assume that all samples are equidistant from the decision boundary. However, it may not hold in practice, leading to suboptimal certification performance. To address this issue, we propose a certified backdoor defense method with sample-specific smoothing noises, termed Cert-SSBD. Cert-SSBD first employs stochastic gradient ascent to optimize the noise magnitude for each sample, ensuring a sample-specific noise level that is then applied to multiple poisoned training sets to retrain several smoothed models. After that, Cert-SSBD aggregates the predictions of multiple smoothed models to generate the final robust prediction. In particular, in this case, existing certification methods become inapplicable since the optimized noise varies across different samples. To conquer this challenge, we introduce a storage-update-based certification method, which dynamically adjusts each sample’s certification region to improve certification performance. We conduct extensive experiments on multiple benchmark datasets, demonstrating the effectiveness of our proposed method. Our code is available at https://github.com/NcepuQiaoTing/Cert-SSBD.