Efficiency Calibration of Implicit Regularization in Deep Networks via Self-paced Curriculum-Driven Singular Value Selection

The generalization of neural networks has been a major focus of research in deep learning. It is often interpreted as an implicit bias towards solutions with specific properties. Especially, in practical applications, it has been observed that linear neural networks (LNN) tend to favor low-rank solutions for matrix completion tasks. However, most existing methods rely on increasing the depth of the neural network to enhance the low rank of solutions, resulting in higher complexity. In this paper, we propose a new explicit regularization method that calibrates the implicit bias towards low-rank trends in matrix completion tasks. Our approach automatically incorporates smaller singular values into the training process using a self-paced learning strategy, gradually restoring matrix information. By jointly using both implicit and explicit regularization, we effectively capture the low-rank structure of LNN and accelerate its convergence. We also analyze how our proposed penalty term interacts with implicit regularization and provide theoretical guarantees for our new model. To evaluate the effectiveness of our method, we conduct a series of experiments on both simulated and real-world data. Our experimental results clearly demonstrate that our method has better robustness and generalization ability compared with other methods.