(Almost) Smooth Sailing: Towards Numerical Stability of Neural Networks Through Differentiable Regularization of the Condition Number

TL;DR

This paper proposes a differentiable regularizer based on the condition number to improve numerical stability in neural networks, enabling gradient-based optimization and demonstrating benefits in noisy classification and image denoising tasks.

Contribution

It introduces a novel, almost everywhere differentiable regularizer for the condition number, facilitating stable training of neural networks.

Findings

Improved stability in noisy classification tasks.

Enhanced denoising performance on MNIST images.

Regularizer is easy to implement and integrate.

Abstract

Maintaining numerical stability in machine learning models is crucial for their reliability and performance. One approach to maintain stability of a network layer is to integrate the condition number of the weight matrix as a regularizing term into the optimization algorithm. However, due to its discontinuous nature and lack of differentiability the condition number is not suitable for a gradient descent approach. This paper introduces a novel regularizer that is provably differentiable almost everywhere and promotes matrices with low condition numbers. In particular, we derive a formula for the gradient of this regularizer which can be easily implemented and integrated into existing optimization algorithms. We show the advantages of this approach for noisy classification and denoising of MNIST images.