Fast Differentiable Clipping-Aware Normalization and Rescaling

TL;DR

This paper introduces a fast, differentiable algorithm for rescaling vectors with clipping constraints, enabling efficient training of neural networks with normalized perturbations across various frameworks.

Contribution

The authors present an analytical, differentiable method for optimal rescaling under clipping constraints, improving over slow iterative binary search approaches.

Findings

The algorithm works for any p-norm.

It is faster than iterative binary search methods.

It is compatible with multiple deep learning frameworks.

Abstract

Rescaling a vector $\vec{\delta} \in \mathbb{R}^n$ to a desired length is a common operation in many areas such as data science and machine learning. When the rescaled perturbation $\eta \vec{\delta}$ is added to a starting point $\vec{x} \in D$ (where $D$ is the data domain, e.g. $D = [0, 1]^n$), the resulting vector $\vec{v} = \vec{x} + \eta \vec{\delta}$ will in general not be in $D$. To enforce that the perturbed vector $v$ is in $D$, the values of $\vec{v}$ can be clipped to $D$. This subsequent element-wise clipping to the data domain does however reduce the effective perturbation size and thus interferes with the rescaling of $\vec{\delta}$. The optimal rescaling $\eta$ to obtain a perturbation with the desired norm after the clipping can be iteratively approximated using a binary search. However, such an iterative approach is slow and non-differentiable. Here we show that the optimal rescaling can be found analytically using a fast and differentiable algorithm. Our algorithm works for any p-norm and can be used to train neural networks on inputs with normalized perturbations. We provide native implementations for PyTorch, TensorFlow, JAX, and NumPy based on EagerPy.