Understanding Implicit Regularization in Over-Parameterized Single Index Model

Area of Science:

• Statistics
• Machine Learning
• High-Dimensional Data Analysis

Background:

• Single index models are crucial for dimensionality reduction in high-dimensional data.
• Existing methods often rely on explicit regularization, which can be suboptimal.
• Understanding implicit regularization in over-parameterized models is an active research area.

Purpose of the Study:

• To develop novel regularization-free algorithms for high-dimensional vector and matrix single index models.
• To provide theoretical guarantees for implicit regularization in these settings.
• To analyze performance for nonlinear link functions and heavy-tailed responses.

Main Methods:

• Leveraging over-parameterization with a score function transform and robust truncation.
• Constructing an over-parameterized least-squares loss function.
• Applying regularization-free gradient descent with carefully chosen initialization and stepsize.

Main Results:

• Theoretical proof of minimax optimal statistical rates of convergence for both vector and matrix cases.
• Demonstration of implicit regularization's effectiveness.
• Experimental validation supporting theoretical findings.

Conclusions:

• Regularization-free gradient descent on over-parameterized loss functions can achieve optimal rates in high-dimensional single index models.
• The proposed methods offer a competitive alternative to traditional regularized approaches.
• Implicit regularization plays a significant role in the success of these algorithms.