  Random smoothing data augmentation is a unique form of regularization that
can prevent overfitting by introducing noise to the input data, encouraging the
model to learn more generalized features. Despite its success in various
applications, there has been a lack of systematic study on the regularization
ability of random smoothing. In this paper, we aim to bridge this gap by
presenting a framework for random smoothing regularization that can adaptively
and effectively learn a wide range of ground truth functions belonging to the
classical Sobolev spaces. Specifically, we investigate two underlying function
spaces: the Sobolev space of low intrinsic dimension, which includes the
Sobolev space in $D$-dimensional Euclidean space or low-dimensional
sub-manifolds as special cases, and the mixed smooth Sobolev space with a
tensor structure. By using random smoothing regularization as novel
convolution-based smoothing kernels, we can attain optimal convergence rates in
these cases using a kernel gradient descent algorithm, either with early
stopping or weight decay. It is noteworthy that our estimator can adapt to the
structural assumptions of the underlying data and avoid the curse of
dimensionality. This is achieved through various choices of injected noise
distributions such as Gaussian, Laplace, or general polynomial noises, allowing
for broad adaptation to the aforementioned structural assumptions of the
underlying data. The convergence rate depends only on the effective dimension,
which may be significantly smaller than the actual data dimension. We conduct
numerical experiments on simulated data to validate our theoretical results.
