Understanding Implicit Regularization in Over-Parameterized Single Index Model

TL;DR

This paper investigates how over-parameterized gradient descent algorithms implicitly regularize high-dimensional single index models, achieving optimal statistical rates without explicit regularization, especially for heavy-tailed data.

Contribution

It introduces a regularization-free approach using over-parameterization and gradient descent for vector and matrix single index models with theoretical guarantees.

Findings

Achieves minimax optimal convergence rates.

Outperforms classical regularized methods empirically.

Handles heavy-tailed data effectively.

Abstract

In this paper, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. 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 of the role played by implicit regularization without excess technicality, we assume that the distribution of the covariates is known 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 addition, our experimental results support our theoretical findings and also demonstrate that our methods empirically outperform classical methods with explicit regularization in terms of both $\ell_2$-statistical rate and variable selection consistency.