Fast Differentiable Clipping-Aware Normalization and Rescaling

• URL: http://arxiv.org/abs/2007.07677v1
• Date: Wed, 15 Jul 2020 13:43:22 GMT
• Title: Fast Differentiable Clipping-Aware Normalization and Rescaling
• Authors: Jonas Rauber, Matthias Bethge
• Abstract summary: 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 normalized to perturbations.
• Score: 22.320256458354137
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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