Randomized smoothing has shown promising certified robustness against adversaries in classification tasks. Despite such success with only zeroth-order access to base models, randomized smoothing has not been extended to a general form of regression. By defining robustness in regression tasks flexibly through probabilities, we demonstrate how to establish upper bounds on input data point perturbation (using the $\ell_2$ norm) for a user-specified probability of observing valid outputs. Furthermore, we showcase the asymptotic property of a basic averaging function in scenarios where the regression model operates without any constraint. We then derive a certified upper bound of the input perturbations when dealing with a family of regression models where the outputs are bounded. Our simulations verify the validity of the theoretical results and reveal the advantages and limitations of simple smoothing functions, i.e., averaging, in regression tasks. The code is publicly available at \url{https://github.com/arekavandi/Certified_Robust_Regression}.

通过定义回归任务中的鲁棒性，灵活地通过概率，我们展示了如何为用户指定的观察到有效输出的概率建立输入数据点扰动（使用$l_2$范数）的上界。我们还展示了在回归模型无约束操作的情况下，基本的平均函数的渐近特性。在处理输出有界的回归模型族时，我们导出了输入扰动的认证上界。我们的模拟验证了理论结果的有效性，并揭示了简单平滑函数（例如平均）在回归任务中的优势和局限性。

RS-Reg：随机平滑的概率性和稳健回归认证