Center Smoothing for Certifiably Robust Vector-Valued Functions

Randomized smoothing has been successfully applied in high-dimensional image classification tasks to obtain models that are provably robust against input perturbations of bounded size. We extend this technique to produce certifiable robustness for vector-valued functions, i.e., bound the change in output caused by a small change in input. These functions are used in many areas of machine learning, such as image reconstruction, dimensionality reduction, super-resolution, etc., but due to the enormous dimensionality of the output space in these problems, generating meaningful robustness guarantees is difficult. We design a smoothing procedure that can leverage the local, potentially low-dimensional, behaviour of the function around an input to obtain probabilistic robustness certificates. We demonstrate the effectiveness of our method on multiple learning tasks involving vector-valued functions with a wide range of input and output dimensionalities.