Continuously Differentiable Exponential Linear Units

TL;DR

This paper introduces a new parametrization of Exponential Linear Units (ELUs) that is continuously differentiable for all alpha, improving their mathematical properties and ease of tuning in deep learning models.

Contribution

The authors propose a C1 continuous variant of ELUs that includes bounded derivatives, encompasses ReLU and linear functions, and is scale-similar with respect to alpha.

Findings

The new ELU parametrization is C1 continuous for all alpha.

It has bounded derivatives with respect to input.

It generalizes ReLU and linear transfer functions.

Abstract

Exponential Linear Units (ELUs) are a useful rectifier for constructing deep learning architectures, as they may speed up and otherwise improve learning by virtue of not have vanishing gradients and by having mean activations near zero. However, the ELU activation as parametrized in [1] is not continuously differentiable with respect to its input when the shape parameter alpha is not equal to 1. We present an alternative parametrization which is C1 continuous for all values of alpha, making the rectifier easier to reason about and making alpha easier to tune. This alternative parametrization has several other useful properties that the original parametrization of ELU does not: 1) its derivative with respect to x is bounded, 2) it contains both the linear transfer function and ReLU as special cases, and 3) it is scale-similar with respect to alpha.