Understanding Implicit Regularization in Over-Parameterized Nonlinear Statistical Model

We study the implicit regularization phenomenon induced by simple optimization algorithms in over-parameterized nonlinear statistical models. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding the role of implicit regularization in the nonlinear models without excess technicality, we assume that the distribution of the covariates is known as a priori. For both the vector and matrix settings, we construct an over-parameterized least-squares loss function by employing the score function transform and a robust truncation step designed specifically for heavy-tailed data. We propose to estimate the true parameter by applying regularization-free gradient descent to the loss function. When the initialization is close to the origin and the stepsize is sufficiently small, we prove that the obtained solution achieves minimax optimal statistical rates of convergence in both the vector and matrix cases. In particular, for the vector single index model with Gaussian covariates, our proposed estimator is shown to enjoy the oracle statistical rate. Our results capture the implicit regularization phenomenon in over-parameterized nonlinear and noisy statistical models with possibly heavy-tailed data.