Generalizing and Improving Jacobian and Hessian Regularization

TL;DR

This paper introduces a generalized framework for Jacobian and Hessian regularization, enabling flexible target matrices and efficient spectral norm minimization, leading to improved adversarial robustness and computational efficiency.

Contribution

It extends Jacobian and Hessian regularization to arbitrary target matrices and introduces Lanczos-based spectral norm minimization for scalable, stable regularization.

Findings

Effective regularization of large Jacobian and Hessian matrices.

Improved adversarial robustness in image classifiers.

Superior performance of the proposed methods over prior approaches.

Abstract

Jacobian and Hessian regularization aim to reduce the magnitude of the first and second-order partial derivatives with respect to neural network inputs, and they are predominantly used to ensure the adversarial robustness of image classifiers. In this work, we generalize previous efforts by extending the target matrix from zero to any matrix that admits efficient matrix-vector products. The proposed paradigm allows us to construct novel regularization terms that enforce symmetry or diagonality on square Jacobian and Hessian matrices. On the other hand, the major challenge for Jacobian and Hessian regularization has been high computational complexity. We introduce Lanczos-based spectral norm minimization to tackle this difficulty. This technique uses a parallelized implementation of the Lanczos algorithm and is capable of effective and stable regularization of large Jacobian and Hessian matrices. Theoretical justifications and empirical evidence are provided for the proposed paradigm and technique. We carry out exploratory experiments to validate the effectiveness of our novel regularization terms. We also conduct comparative experiments to evaluate Lanczos-based spectral norm minimization against prior methods. Results show that the proposed methodologies are advantageous for a wide range of tasks.